This is the crux of the problem. A parsec is a way of measuring the distance of objects FROM you, not distances you travel.

The idea of using the Parsec as some standard unit of distance is insanely stupid, and nobody who actually understands what a parsec is would ever think of it being used by a galactic empire or even for interstellar navigation as some standard unit of distance, because that isn’t what it is.
Only fan boys who can’t let the fact that their favourite film said something hilariously stupid would think that such a mind-bogglingly dumb idea would make any sense whatsoever.

If you have a galactic empire, or are navigating through interstellar space, the only standard unit of distance you’re going to use is the light year, because it is a STANDARD unit of distance that everyone can measure, and not a RATIO dependent on the distance the observer moves which will be different for everyone.

And who they hell is going to come up with a standard unit of distance that is 3.26 light years anyway? That’s just a silly idea. It’s so close to a light year that you’d just talk in light years. That’s like deciding to come up with an extra standard unit of distance in the metric system that equals 2.7 kilometers. Who the hell does that?
You’d have thought the fact that a parsec equated to such an odd numerical value would have made people think that maybe it’s not what they think it is….

It’s just a convenient measure of distance we can treat as a unit because the way we use it relates solely to the distances to stars when observed from our Earth, where we all live and which moves a pretty set amount throughout the year. Take away all those factors, and the parsec ceases to have any coherent meaning as a unit of distance.

There is a final problem with the whole apologetic around the “parsec” line, and it has to do with hyperspace.
As the argument has been put in one blog, “Traveling at hyperspace is much more complicated than just pressing a button and going directly from point A to point B. A ship’s computer has to be programmed with a route to avoid the known obstacles along that route.”

This completely misunderstands what hyperspace is. You don’t travel at hyperspace, you travel through hyperspace.

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Wormholes – those curious portals to the other side of the galaxy or universe. More formerly known as an Einstein-Rosen bridge, they theoretically join two regions of space allowing one to traverse great distances instead of moving along the fabric of space.

The illustrations below shows this very well. Traveling from A to B in the conventional sense will get you there in time…

Traveling from planet A to planet B via flat space.

But, fold space by using a wormhole and you now have a much shorter path to cross to go from A to B.

The new wallpaper shows the mouth of an active wormhole from the point of view of a gas giant and its habitable moon in a nearby solar system.

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In any n-dimensional space, a hyperspace is any n+1 (or more) dimensional space in which the original n-dimensional space is embedded. Think of a 2 dimensional space, like the surface of a piece of paper, and hyperspace is the 3 dimensional space it exists in.

Now think of 2 points on that piece of paper. You want to plot a course between them, but you’ve put some objects between those 2 points. If you want to travel between them through that 2 dimensional space, you have to avoid the obstacles.

But if you travel through the hyperspace of 3 dimensional space, you can plot a course in which you don’t have to think about avoiding those obstacles at all – so the idea of having to plot a course to avoid objects that exist only in a dimensional space that you aren’t going to be traveling through is completely nonsensical.

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Qu'est qu'un trou de ver ?

Un trou de ver est une une étoile qui s'est affaissée sur elle-même ce qui a donné une singularité. Plus simplement c'est un tunnel où l'on peut voyager plus vite que la lumière reliant un point A, situé a proximité de la terre, à un point B, à 10 000 années lumière de la terre (c'est un exemple ). Pour y aller avec un vaisseau sans trou de ver allant a 80 000 km/h (c'est une image) il vous faudra environ.... d'après mes calculs... plusieurs millions d'années pour y aller. Mais grâce aux trous de ver ce temps peut être énormément réduit.

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You can expand this idea to think about a 4 or more dimensional space in which our 3 dimensional space exists, and the same rules apply when you travel through that 4 or more dimensional hyperspace – no need to avoid obstacles that only exist in the 3 dimensional space.

So no, traveling through hyperspace literally IS as simple as “just pressing a button and going directly from point A to point B” – that’s the whole freakin’ point of hyperspace!
To claim it’s not that simple is to prove that you really have no idea what hyperspace is.

The thing is that, in Star Wars, hyperspace doesn’t mean hyperspace – it just means going really fast, which isn’t what hyperspace means at all.
(There’s also a topological definition of hyperspace, but that’s still nothing like the hyperspace of Star Wars.)

Seriously, Star Wars fans, I’d have stuck to accepting that Lucas made a dumb mistake thinking that a Parsec was a unit of time if I were you, because this new stupid idea where you try to appear clever just proves how ignorant you are of what a Parsec is, and even what geometry is.

And in fact, if one of the fans had just decided to say that a “parsec” is a unit of time in the Star Wars galaxy, rather than engage in this incredible feat of mental gymnastics that fundamentally misunderstands what a parsec is in astronomy and astrophysics, I’d have been absolutely fine.
Really, I would have been completely OK with that explanation.

After all, they’re in a galaxy far, far away, right? They can have whatever units of time they want and call them whatever they want – so they can happily have a unit of time that just happens to be called a “parsec”. There’s absolutely nothing wrong with that.

Sure, we’d still all know that George Lucas originally wrote the line because he didn’t understand what a parsec is, but it would be perfectly acceptable as an answer. Not just because it’s OK for them to have whatever units of time they want, but because we know that Star Wars isn’t sci-fi – it’s fantasy.
Similarly with hyperspace. Stop pretending it relates in any way to the scientific concept of hyperspace, and just accept that this is a fantasy series where General Relativity and the rules of space-time geometry don’t exist, and which isn’t using the scientific terms in any way close to what they actually mean.
The problem only comes when you try to pretend you’re clever, and make it very obvious that you haven’t got a clue.

Stop pretending Star Wars is sci-fi, because when you do, you make up ridiculous apologetic arguments for it’s flaws that really just completely misunderstand and misrepresent science – and that’s why people have to write blog posts explaining what things like parsecs actually are, after you butcher their meaning to pretend your favourite fantasy has some scientific relevance.

Here is the basic gist of hyperspace:

Say you want to get from point A to point B, like going from Earth to Dakara. Kinda like if you were an ant trying to get across a piece of paper.


  


If you need to travel in real space which would take hundreds of thousands of years to get to point B. That is totally out of the picture for us, so they came up with the subspace theory.
To get from point A to Point B we rip a hole in our space time and drop into subspace and move in that then rip another hole and drop back in our dimension. We move in subspace much like we would move in real space except we can travel at many times the speed of light. That is like switching from the electromagnetic light spectrum to the microwave spectrum and back again.

And for that ant he would just have to bend the piece of paper and then step from one edge of the paper to the other edge.


   


However this is based on what we know right now out side of Stargate.

Shortcut through the universe

The concept of hyperspace ( multi dimensions ) can be used to travel a long distances in the universe by curving the 4 dimensional space-time. The space-time separating the two points in universe is curved in a higher dimension so that the 2 points come close to one another. Thus making the travel in our universe less time consuming

This concept can be better understood by considering a flatlander ( 2 dimensional being ) living on a rectangular paper. This flatlander can move only along the 2 dimensional paper ( front, back, left, right ) and cannot visualize the 3rd dimension ( up, down ). He has to travel from one corner of the paper to another. He has to travel along the diagonal for this. But if he can find some technology to curve the paper ( in 3rd dimension- upwards ) and bring the 2 corners close to one another, then he can walk off from one corner to another directly and avoid traveling along the diagonal. Thus his journey becomes shorter.

This concept when extended to our 4 dimensional universe requires the space-time to be bent in a higher dimension. The energy required to do this is enormous. We have not yet developed the technology required. This curving of space-time is similar to the opening of worm holes and like teleporting seen in the movies. But in reality we are far away from achieving this.

A geodesic is a path minimising line connecting two points on a sphere. Everything falling under gravity is following one of these lines in curved space-time.

Wormholes, proposed by Einstein and Rosen in 1935 as the Einstein-Rosen Bridge, and supported by the mathematics of physics and cosmology, is a hypothetical feature of gravity that could collapse spacetime in such a way as to connect two disparate regions of space—in effect, circumventing time.

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Curved Space

In order to understand this idea of curved space in two dimensions you really have to appreciate the limited point of view of the character who lives in such a space. Suppose we imagine a bug with no eyes who lives on a plane, as show in Figure 6-1. He can move only on the plane, and he has no way of knowing that there is any way to discover any “outside world.” (He hasn’t got your imagination.) We are, of course, going to argue by analogy. We live in a three-dimensional world, and we don’t have any imagination about going off our three-dimensional world in a new direction; so we have to think the thing out by analogy. It is as though we were bugs living on a plane, and there was a space in another direction. That’s way we will first work with the bug, remembering that he must live on his surface and can’t get out.

As another example of a bug living in two dimensions, let’s imagine one who lives on a sphere. We imagine that he can walk around on the surface of the sphere, as in Figure 6-2, but that he can’t look “up,” or “down,” or “out.”

Now we want to consider still a third kind of creature. He is also a bug like the others, and also lives on a plane, as our first bug did, but this time the plane is peculiar. The temperature is different at different places. Also, the bug and any rules he uses are all made of the same material which expands when it is heated. Whenever he puts a ruler somewhere to measure something the ruler expands immediately to the proper length for the temperature at that place. Whenever he puts any object–himself, a ruler, a triangle, or anything–the thing stretches itself because of the thermal expansion.

In general relativity, bodies always follow geodesics in four-dimensional space-time. In the absence of matter, these geodesics in four-dimensional space-time correspond to straight lines in threedimensional space. In the presence of matter, four-dimensional space-time is distorted, causing the paths of bodies in three-dimensional space to curve in a manner that in the old Newtonian theory was explained by the effects of gravitational attraction. This is rather like watching an airplane flying over hilly ground. The plane might be moving in a straight line through three-dimensional space, but remove the third dimension-height-and you find that its shadow follows a curved path on the hilly two-dimensional ground. Or imagine a spaceship flying in a straight line through space, passing directly over the North Pole. Project its path down onto the two-dimensional surface of the earth and you find that it follows a semicircle, tracing a line of longitude over the northern hemisphere. Though the phenomenon is harder to picture, the mass of the sun curves space-time in such a way that although the earth follows a straight path in four-dimensional space-time, it appears to us to move along a nearly circular orbit in three-dimensional space.

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Änderung der Bewegungsrichtung:

Jedes Quantum bewegt sich ausschließlich geradlinig durch den Raum. Dabei stellt sich nun die Frage, wie kommt es bei einer Bewegung zur Änderung einer Richtung?

Zur Änderung der Richtung der Bewegung kommt es, wenn die Geometrie der Umgebung unterschiedlich stark gekrümmt ist.

Ich spreche auch von der Topografie des Raums, weil dort Täler zu sehen sind, die überbrückt werden könnten, wie es z.B. die rote Line anzeigt, was aber auch ‚früheres ankommen’ bedeutet.

Ja, es wird leichter verstanden, als detailliert beschrieben. Beachten Sie dabei aber, dass immer der vorderste Teil des Quantums oder genauer gesagt, die erste Position, die Richtung, das ‚Zuerst’ bestimmt.

Wenn sich nicht jeder Teil eines Quantums über einem Tal befindet, dann können sich manche Teile auch auf unterschiedlichem Niveau befinden und von daher werden manche Punkte früher erreicht. Und weil damit nun das ‚Zuerst’ beeinflusst wurde, wird es zu einer Änderung der Richtung kommen. Der Prozess der Krümmung wird in jeden Fall anders verlaufen als es auf einem ungekrümmten Untergrund der Fall sein würde.

Die kürzeren Strecken verändern den Zeitpunkt der ersten Ursache, das ‚Zuerst’. Und genau das führt letztendlich zur Änderung der Richtung der Bewegung. Aber  das, was aus bonitistischer Sicht heraus nur weniger Zeit bedeutet, wurde bisher (2016) noch als Einsteinsche Zeitdehnung interpretiert.

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Before we get into the issue of travel time we need to point out the science fiction idea of wormholes is rather different to general relativity. There are several kinds of wormhole metrics known in GR, but they just connect two asymptotically flat regions of spacetime. If we use a rubber sheet model it would look something like this:



The wormhole metric tells us the spacetime geometry around the wormhole, but it does not tell us how, if at all, the regions A and B are connected. The usual science fiction idea of a wormhole is something like this:



The green bit of the geometry is described by GR, but the red bit is entirely fanciful. There is nothing in GR to tell us how the two regions A and B might curve round and meet as shown in the second diagram. This is entirely the domain of science fiction.

Your question asks us to compare the time taken by travelling through the wormhole to the time taken for the long way round. The problem is there is no long way round in any of the wormholes known to us, so your question cannot be answered.

However we can calculate the time taken for an observer to pass through the wormhole from region A to region B, so we can decide whether wormholes are at least in principle a viable way to travel. But there is one more complication because there are two different definitions of time involved. Suppose you and I start well away from the wormhole, and we both have clocks. You head off through the wormhole while I sit back and wait to see what happens.

I can time your progress with my clock, and this time is called my coordinate time. You can time your progress with your clock and the time you record is called the proper time. The complication is that in general our two times won't agree and in fact they can be wildly different. For example suppose you were jumping into a black hole rather than a wormhole. Your clock would record a finite (and short!) time for you to fall through the horizon and to a messy but quick death at the singularity. However as measured by my clock you would take an infinite time just to reach the event horizon and you would never pass through it. The disagreement between our clocks couldn't be more extreme.

This happens with wormholes as well. Some types of wormholes have a horizon like a black hole so while you pass through the wormhole in a finite proper time I would never see you reach its entrance let alone pass through it. That makes it a pretty useless travel device since you could never go back. Although it's not strictly speaking a wormhole the Reissner-Nordström charged black hole behaves in this way.

But other types of wormhole are much more benign. For example consider the Morris-Thorne wormhole as described in How do spatial curvature and temporal curvature differ?. Not only does this have no horizon, but it's actually pretty boring. If you head towards the wormhole with some initial velocity v

then your velocity remains constant and both of us would observe you to pass through the wormhole much as if you were simply travelling in flat spacetime. The time you take to get through depends on the scale of the wormhole, which depends on how much exotic matter we assembled to build it. In principle this time can be made arbitrarily short.

So while we can't answer the question you asked, we can say that for the right sort of wormhole the travel time can be short on everyday timescales. So this wormhole is a perfectly good way for travelling around the universe.

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The Physics of Interactionism

Ulrich Mohrhoff

Physics has been invoked both to refute and to support psycho-physical interactionism, the view that mind and matter are two mutually irreducible, interacting domains. Thus it has been held against interactionism that it implies violations of the laws of physics, notably the law of energy conservation. I examine the meaning of conservation laws in physics and show that in fact no valid argument against the interactionist theory can be drawn from them. In defence of interactionism it has been argued that mind can act on matter through an apparent loophole in physical determinism, without violating physical laws. I show that this argument is equally fallacious. This leads to the conclusion that the indeterminism of quantum mechanics cannot be the physical correlate of free will; if there is a causally efficacious non-material mind, then the behaviour of matter cannot be fully governed by physical laws. I show that the best (if not the only) way of formulating departures from the ‘normal’, physically determined behaviour of matter is in terms of modifications of the electromagnetic interactions between particles. I also show that mental states and events are non-spatial, and that departures from the ‘normal’ behaviour of matter, when caused by mental events, are not amenable to mathematical description.

I: Introduction

There is another hard problem, in addition to the problem of how anything material can have the subjective, first-person phenomenology of consciousness (Chalmers, 1995). It is the problem of how anything material can have freedom. By ‘freedom’ I mean a person’s ability to behave in a purposive, non-random fashion that is not determined by neurophysiological structure and physical law. I do not mean the absence of other determining factors, as this would render freedom synonymous with randomness.¹

I decide to raise my right arm and up it goes. This decision — a mental event — appears to me to be both the cause of the ensuing physical event and a causal primary. (A causal primary is an event the occurrence of which is not necessitated by antecedent causes.) I can think of various reasons for raising my arm (I may want to catch a ball), and these may involve antecedent causes (e.g., the ball was thrown in my direction); but if there is anything that made it inevitable that I should raise my arm, I know nothing of it.

To be sure, ignorance of an antecedent cause does not prove its nonexistence. But what does? We can aspire to establish that events of type C are regularly followed by events of type E. If we succeed, we are free (are we?) to imagine a hidden string between individual events such that each event of type C is the cause of an event of type E. Failure to establish the existence of a type of event C the instances of which are regularly followed by events of type E, on the other hand, is not a proof that events of type E lack antecedent causes. It doesn’t prove anything beyond our ignorance of the antecedent causes of events of type E. Proving an event a causal primary is an impossible task. But from this it does not follow that there are no such events. What does follow is that empirical science cannot aspire to know that an event is a causal primary. That is why scientists may ignore causal primaries. But this is no reason for philosophers to dismiss their possible existence.

The absence of causal primaries is often called ‘the causal closure of the physical world’, where ‘physical’ means ‘non-mental’ rather than ‘governed by the laws of physics’. This causal closure is a trivial consequence of the lack of scientific interest which results from being unable to identify causal primaries. What is causally closed is the scientifically known world, not the world as such. Yet there are many philosophers who look upon causal closure as an ontological truth and go on to invoke it as an argument against interactionism, the doctrine that mind and matter are two mutually irreducible, interacting spheres. Interactionists as I understand them are motivated primarily by a desire to make room for free will, the denial of which is both counter-intuitive and at odds with notions (moral and otherwise) that are central to the fabric of our active lives. While repudiating the causal closure of the physical world, interactionists nevertheless shrink from contesting the validity of the laws of physics, not realizing that this is contingent on the presumption of causal closure.

That is how we come to witness futile fights between philosophers of mind who reject interactionism on the ground that it is incompatible with the laws of physics and, in particular, the law of the conservation of energy, and interactionists who meekly defend their position, claiming that, by exploiting the loophole of quantum-mechanical indeterminism, non-material mind is capable of influencing matter without violating conservation laws. Both the charge and the defence are misconceived. The law of energy conservation is either true by virtue of the meaning of ‘energy’, and therefore is not threatened by interactionism, or it is contingent upon the causal closure of the physical world, and therefore is no threat to interactionism. The loophole hypothesis, on the other hand, violates basic physical laws other than the law of energy conservation.

This should not come as a surprise. To be causally efficacious, mental events that are causal primaries must make a difference to the behaviour of matter and thus to the behaviour of its constituent particles. The effects of such events on the behaviour of particles have to be expressed in the language of physics, for this is the only language suitable for describing the behaviour of particles. But the laws of physics presuppose causal closure and describe the behaviour of matter in the absence of causal primaries. Hence it follows that the behaviour of matter in the presence of a causally efficacious non-material mind cannot be fully governed by those laws.

The hard problem of consciousness and the hard problem of freedom appear at first sight to be logically independent. To embrace the irreducibility of consciousness, one need not deny the causal closure of the physical world, and one need not attribute to consciousness a causal role, as has been stressed by Chalmers (1997). On the other hand, it is possible to have physical events interspersed with non-physical causal primaries that lack subjective properties. Take Eccles’ (1994) theory in which ‘psychons’ in the mind affect physical processes in the brain. As Chalmers (1997) has pointed out, the question of whether psychons have any experiential qualities is irrelevant to the causal story.

But this apparent independence of causality and subjectivity is called into question every time someone utters the word ‘consciousness’. To see this, suppose that consciousness is irrelevant to the causal story. Then it is explanatorily irrelevant to our claims about consciousness: the physical act of making a judgement about experience is not sensitive to the experience itself (Kirk, 1996). In other words, there are two mutually irrelevant kinds of experience, the experience^E that we actually have and the experience^L about which we make statements. Zombie philosophers make judgements about experience^L, but it would be self-contradictory for them to conceive of the distinction between experience^E and experience^L. These conclusions seem to constitute a reductio ad absurdum of the supposition that consciousness is irrelevant to the causal story. Hollywood, it seems, has got it right: zombies are shuffling affectless brutes, not smart philosophers of mind (DeLancey, 1996). Taking the hard problem of consciousness seriously thus appears to make it necessary to take the hard problem of freedom as seriously.

And so we have more than sufficient reason to address the latter problem as vigorously as the former has been addressed in recent issues of this journal and elsewhere. In the present article I will apply myself to the preliminary task of ‘deconstructing’ physics-based arguments purporting to prove the nonexistence of freedom. Section II reviews the argument from energy conservation — the claim that it is inconsistent with the interactionist doctrine — and the counter-argument that purports to show that interactionism and free will are consistent with the unbroken reign of physical law.

Section III refutes the arguments against interactionism that invoke conservation laws. It begins with an examination of what physicists mean by ‘energy’ and ‘momentum’. The respective conservation laws are shown to be consequences of these meanings. They are necessarily true whenever ‘energy’ and ‘momentum’ are well-defined concepts. For these concepts to be well defined, it is however not necessary that the quantities they denote are conserved everywhere and under all circumstances. If they fail to be conserved, it can be for either of two reasons. It may be that energy and momentum are indeed meaningless; the curved space–time of Einstein’s general theory of relativity provides an instructive example of this possibility. Or it may be that they are conserved somewhere but not everywhere. Then they are meaningful even where they are not conserved, as for example where matter is causally open to a nonmaterial mind.

Section IV refutes the argument purporting to show that quantum mechanics offers a way of reconciling interactionism with the unbroken reign of physical law. According to this argument, an intention to act can be causally efficacious by merely modifying the probabilities associated with individual quantum events. I show that, on the contrary, an intention to act cannot be causally efficacious without modifying the statistics of ensembles of such events. And this is the same as saying that it cannot be causally efficacious without modifying some physical laws.

Section V shows how the departures from the laws of physics due to non-material mind can (and must) be formulated in the language of physics. The appropriate mathematical entity is the electromagnetic four-vector potential (or, simply, the electromagnetic field). As a summary representation of possible effects on moving particles that makes no reference whatever to causes, the electromagnetic field necessarily represents the effects of both material and non-material causes. Section VI shows that a causally efficacious non-material mind is not something that exists in space, and that its action on matter is not amenable to mathematical description.

A more technical discussion of the physics is available (Mohrhoff, 1997).

II: Energy Conservation and the Interactionist Hypothesis

Attempts to address the mind–body problem along interactionist lines have traditionally been faulted for taking liberties with physical conservation laws, notably the principle of the conservation of energy (also known as the first law of thermodynamics). M. Bunge (1980, p. 17) and D.C. Dennett (1991, p. 35) speak for the prosecution.

If immaterial mind could move matter, then it would create energy; and if matter were to act on immaterial mind, then energy would disappear. In either case energy... would fail to be conserved. And so physics, chemistry, biology, and economics would collapse.

Let us concentrate on the returned signals, the directives from mind to brain. These, ex hypothesi, are not physical; they are not light waves or sound waves or cosmic rays or streams of subatomic particles. No physical energy or mass is associated with them. How, then, do they get to make a difference to what happens in the brain cells they must affect, if the mind is to have any influence on the body? A fundamental principle of physics is that any change in the trajectory of any physical entity is an acceleration requiring the expenditure of energy, and where is this energy to come from? It is this principle of the conservation of energy that accounts for the physical impossibility of ‘perpetual motion machines’, and the same principle is apparently violated by dualism. This confrontation between quite standard physics and dualism has been endlessly discussed since Descartes’ own day, and is widely regarded as the inescapable and fatal flaw of dualism.

Dualists have taken these strictures to heart. Even Karl Popper, by proclaiming himself not to be ‘in the least impressed by the danger of falling foul of the first law of thermodynamics’ (Popper and Eccles, 1983, p. 564), implicitly acknowledges the danger. From the early days of quantum mechanics, the strategy of the defence has consisted in claiming that quantum-mechanical indeterminism allows non-material mental events to act on matter (specifically the brain) without violating conservation laws. Eddington (1935) was probably the first to speculate publicly that the mind may influence the body by affecting quantum events within the brain through a causal influence on the probability of their occurrence.

More recently H. Margenau (1984) has suggested that the mind may be ‘regarded as a field in the accepted physical sense of the term’, yet not be ‘required to contain energy in order to account for all known phenomena in which mind interacts with brain’ (p. 97): ‘In very complicated physical systems such as the brain, the neurons and the sense organs, whose constituents are small enough to be governed by probabilistic quantum laws, the physical organ is always poised for a multitude of possible changes, each with a definite probability’ (p. 96).

Standard axiomatizations of quantum mechanics recognize two kinds of change: the probabilistic collapse of a quantum-mechanical superposition which occurs during a measurement, and the deterministic evolution of the quantum state which takes place between measurements (von Neumann, 1955). Margenau proposes that the causal efficacy of mind rests on the following sequence of steps: (i) The relevant physical system develops, in accordance with the deterministic evolution of states, into a superposition of alternative states, each associated with a probability. (ii) Mind alters the physically determined probabilities, possibly by superimposing its own probability field on the physically determined probability field. (iii) The resulting superposition collapses to one of its elements in accordance with the probabilistic change of states. In this way, Margenau argues, mind can act on the brain without disturbing the balance of energy. D. Hodgson (1996) likewise invokes the mind’s ability to load the quantum dice.

Seizing on Margenau’s proposal, J.C. Eccles, in collaboration with F. Beck (Beck and Eccles, 1992; Eccles, 1994), has put forward one of the most elaborate and specific hypotheses of mind–brain interaction to date. It capitalizes on the basic unitary activity of the cerebral cortex, exocytosis. Exocytosis is the emission of chemical transmitters into the synaptic cleft by a vesicle of the presynaptic vesicular grid, a paracrystalline structure situated inside the terminal expansion (bouton) of a nerve fibre. It is an all-or-nothing event, which has been found to occur with a probability of about one fourth to one third when a bouton is activated by a nerve impulse. Eccles and Beck assume this probability to be of quantum-mechanical origin. They cite increasing evidence for a trigger mechanism that may involve quantum transitions between metastable molecular states, and propose a model for the trigger mechanism based on the tunnelling of a quasi-particle through a potential barrier.² According to their model, during a period of the order of femtoseconds the quasi-particle is distributed over both sides of the barrier. One side corresponds to the activated state of the trigger, the other side to the non-activated state. At the end of this period exocytosis has been triggered with the aforesaid probability. Eccles and Beck propose that mental intentions act through a quantum probability field altering the probability of exocytosis during this brief period.

While the postsynaptic effect due to the change in probability of exocytosis by a single vesicle is many orders of magnitude too small for modifying the patterns of neuronal activity even in small areas of the brain, there are many thousands of vesicles per bouton and many thousands of similar boutons on a pyramidal cell (the principal type of neuron of the cerebral cortex), and there are about 200 neurons in the region of a dendron, the basic anatomical unit of the cerebral cortex (Eccles, 1994, p. 98). The hypothesis of mind–brain interaction according to Eccles and Beck is that mental intention becomes neurally effective by momentarily increasing the probabilities for exocytosis in the hundreds of thousands of boutons in a whole dendron.

In summary it can be stated that it is sufficient for the dualist-interactionist hypothesis to be able to account for the ability of a non-material mental event to effect a changed probability of the vesicular emission from a single bouton on a cortical pyramidal cell. If that can occur for one, it could occur for a multitude of the boutons on that neuron, and all else follows in accord with the neuroscience of motor control (Eccles, 1994, p. 78).

It is reassuring that all of the richness and enjoyment of our experiences can now be accepted without any qualms of conscience that we may be infringing conservation laws! (Eccles, 1994, p. 170).

III: Conservation of Energy and Momentum: A Closer Look

Originally, momentum was defined as ‘mass-times-velocity’. It soon became apparent that (within Newtonian physics) this was a conserved quantity. Then the special theory of relativity superseded Newtonian physics, and mass-times-velocity was no longer conserved. By this time, however, the property of being conserved was accorded much greater importance than the original definition in terms of mass and velocity. Momentum accordingly was redefined so as to match its original definition in the low-speed limit, where the two theories make identical predictions, as well as to retain its status of a conserved quantity.

But a redefinition that consists in the substitution of one theory-dependent definiens for another, can only be a halfway stop. It must be possible to define the definiendum at a more basic level, independently of the specific principles of either theory and hence in a way that is valid for both. It indeed soon transpired that the different mathematical embodiments of momentum in the respective theories of Newton and Einstein were specific instances of a quantity that could be invariantly defined for a large class of theories. In 1918 E. Noether discovered a deep connection between symmetries³ and conservation laws. This exists in all theories that can be derived from a mathematical expression known to physicists as the Lagrangian. In all such theories (and these include not just all experimentally well-confirmed theories to date but all theories esteemed worthy of consideration by contemporary physicists), a continuous symmetry⁴ implies the existence of a locally conserved quantity.⁵ And one of these locally conserved quantities implied by the continuous symmetries of the Lagrangian is called ‘momentum’. Thereafter it was possible to claim that this has always been the true definition, even when the concept was insufficiently differentiated from its then sole instantiation, mass-times-velocity.

The same holds true of energy. Both energy and momentum are defined as conserved quantities. They are conserved by definition. Either they make sense and are conserved, or they don’t make sense. They don’t make sense whenever the mathematically described world (or, equivalently, the Lagrangian) does not possess the symmetries that imply their respective conservation laws; in other words, whenever the corresponding symmetry transformations, applied to a mathematical description of a physical situation, yields not just a different description but a different physical situation.

The symmetry that gives meaning to ‘momentum’ is known as the homogeneity of space; it consists in the mechanical equivalence of all locations in space, or in the fact that every closed mechanical system behaves in the same way anywhere. The symmetry that gives meaning to ‘energy’ is known as the homogeneity of time; this consists in the mechanical equivalence of all moments of time, or in the fact that every such system behaves in the same way anytime. Translate the coordinate origin in space and/or time, and what you get is a different description of the same physical situation. This has the nature of a postulate: differences in the outcomes of identical experiments performed at distinct locations and/or times are to be ascribed to the different physical conditions (known or unknown) prevailing at these distinct locations and/or times, not to these locations and/or times per se. An instance of the synthetic a priori judgement that everything that happens has a cause, this postulate has more to do with what we (investigating humans) make of our experiences than with any particular experience of ours. If we did not assume the existence of a cause, we would not look for one; and if we did not assume the existence of physical causes to explain the spatial or temporal inhomogeneities we observe, we would not look for such causes but rest content with attributing those inhomogeneities to space or time per se.

And so it would seem that the homogeneity of space and the homogeneity of time are a priori certain; that momentum and energy are therefore always well defined; and that they are always conserved. However, there are riders to this series of conclusions. Whatever is a priori certain is so only with regard to our mental constructs. Whether or not these can be thought of as descriptions of objective reality is another matter. Also, before anything can be derived from the said homogeneities, they must be given formal expression within the framework of a physical theory. And there is no a priori guarantee that this is possible. In fact, there are reasons to surmize the opposite, as will become apparent in what follows.

There is nothing controversial about the way in which space and time are rendered manifestly homogeneous (that is, the way in which their homogeneities find mathematical expression in a physical theory). Either one introduces a privileged class of coordinate systems (called ‘inertial systems’) or one lets a mathematical entity known as the metric tensor (or simply, the metric) do the privileging (by taking a particularly simple form in the privileged systems). However, what is capable of manifesting homogeneity also lends itself to the manifestation of inhomogeneities. The metric needed to manifest the flatness⁶ of space or space–time could instead serve to manifest the curvature of a Riemannian space or space–time. This is the same as saying that the metric texture of space or space–time offers a handle for the formulation of an interaction law. Matter could act on matter via the intermediate representation of the metric in much the same way as electric charges act on electric charges via the intermediate representation of the electromagnetic field. The curvature at any space– time point p, determining partly if not fully the motion of matter at p, could depend on the distribution and motion of matter elsewhere and at earlier times. It could thus represent a causal influence on the motion of matter at p due to the earlier distribution and motion of matter elsewhere.

This a priori possibility is an actual feature of the objective world. The interaction in question is gravity; the theory just outlined is the general theory of relativity. Now, gravity appears to be quite indispensable to the creation of what Squires (1981) has called an ‘interesting world’. Without gravity there would exist no stars, no planets, nor (for all we can imagine) any sites hospitable to something as interesting as life. In view of this it might be asserted that curvature is implied by our own existence, or that since we are here, space–time cannot be flat.

At any rate, the metric connection lends itself to the manifestation either of spatio–temporal homogeneity or of gravity. As far as the description of objective reality is concerned, the choice is not ours but Nature’s. And Nature has opted for gravity. The metric which could have offered a handle for the incorporation, in our mental picture of reality, of a homogeneous space and a homogeneous time, is already used up. From this and what has been said earlier one might draw the conclusion that in situations in which gravity plays a significant role, energy and momentum are undefined. But such a conclusion would ignore that even curved space–time is locally flat,⁷ and that, as a consequence, the energy and the momentum of all non-gravitational fields are locally (as opposed to globally) conserved. This is sufficient for them to be well-defined. What is ill-defined in any generic space–time is the gravitational energy/momentum, and hence the total energy/momentum. The energy/momentum associated with a curved region of space–time is, strictly speaking, definable only in model space–times that are flat ‘at the edges’.⁸

At certain junctures in the history of physics the law of energy conservation has been called in question. Bohr at one time felt that he had to renounce it, and not a few particle physicists despaired of it before the neutrino was proposed and, in due course, discovered. It should not be supposed that these physicists were unaware of the deep connection between the conservation laws for energy and momentum and the homogeneity of time and of space. Rather they were driven to consider the possibility that these homogeneities were not, after all, respected by Nature. Bohr thought that the problems facing atomic theory were ‘of such a nature that they hardly allow us to hope that we shall be able, within the world of the atom, to carry through a description in space and time that corresponds to our ordinary sensory perceptions’ (in Honner, 1982). If the feasibility of such a description cannot be taken for granted, the homogeneity of space and of time cannot be taken for granted either.

More recently, in connection with the so-called measurement problem in quantum mechanics, the stochastic generation (and hence non-conservation) of energy has emerged as a theoretical possibility (Ghirardi et al., 1986; Pearle, 1989). This amounts to introducing stochastic inhomogeneities in the ‘flow’ of time, and to redefining energy as the quantity whose conservation would be implied if those inhomogeneities were absent. If such a definition is adopted, the view that the conservation of energy is part of the meaning of ‘energy’, can no longer be entertained.

The situation, then, is this: If the energy conservation law is part of the meaning of ‘energy’, the interactionist hypothesis cannot imply a violation of this law. And if physicists can invoke inhomogeneities in the ‘flow’ of time and define energy in such a way that it is conserved only when and where those inhomogeneities are absent, interactionists can do the same. The causal efficacy of non-material mind could be based on its generating similar (but not stochastic) inhomogeneities. As long as there exists an experimental realm in which mind-generated inhomogeneities are absent or negligible (and from a physicist’s point of view, given present experimental limitations, they may well be negligible everywhere), energy remains well-defined even where matter is causally open to non-material mind. If no such realm existed, attributing energy to matter would be gratuitous, since in this case any mathematical expression would do. None could be tested, because the proof that one has the right expression lies in the experimental corroboration of its conservation. But if the formula for the energy associated with matter is testable somewhere, nothing prevents one from using the same formula everywhere, including where matter is open to the action of non-material mind and energy is not necessarily conserved.

IV: Interactionism Violates Physical Laws

While the argument from energy conservation does not succeed, the notion that mental events can influence physical events through the loophole of quantum-mechanical indeterminism, without in any manner whatsoever infringing on the deterministic regime of physical laws, is chimerical, as is shown presently.

Consider a causally efficacious mental event (say, the intention to flex the right index finger). If this occurs in the mind associated with any healthy body, the intended action takes place. If the same intention occurs in the minds associated with an ensemble of healthy bodies, all of those bodies flex their right index fingers as a result. There is no randomness in the causal concatenation between intention and intended action. Throughout the ensemble, the same mental event brings about the same physical event.

Consequently, if the causal efficacy of a mental intention is postulated to involve modifications of quantum-mechanical probabilities associated with ‘collapsible’ wave functions, these modifications are statistically significant. In the simplest case in which the modifications amount to the selection of one out of two possible outcomes in a single collapse, the same outcome is selected every time the intention occurs. In the Eccles–Beck model, in which the intended action is the effect of many weak modifications, accumulated over a large number of collapses, the fact that the same action is produced every time entails that the individual modifications likewise exhibit statistically significant trends.

A clear distinction must be maintained between sets of active sites in the same brain and the statistical ensembles of active sites relevant to the present discussion. The latter involve different brains or, more precisely, different instances of identical brains. Consider an ensemble of such brains. Then consider an ensemble of vesicles such that each vesicle is from a different brain and all vesicles occupy identical positions in their respective brains. There are as many such vesicle ensembles as there are vesicles in each brain. Let us compare two cases. In the first case all brains are influenced by a certain mental intention; in the second case none of the brains is influenced by it, other things being equal. What needs to be compared in particular is the behaviour of each vesicle ensemble in the two cases. If none of the vesicle ensembles shows

any difference in the percentage of ‘firing’ vesicles, the intention cannot be causally efficacious. If it is causally efficacious, the intended effect takes place whenever the intention is present in the minds associated with those brains, and only then. In this case there must be some vesicle ensembles for which the percentages of ‘firing’ vesicles differ in the two cases.

In a word, if single-case probabilities get modified, there are statistical ensembles whose behaviours get modified. What gets modified is not merely individual quantum events but the statistics of entire ensembles of such events. And these statistics, unlike the individual events, are fully determined by physical laws. Changing them means changing the physical laws.⁹ Altering the single-case probabilities associated with individual measurement-like events without changing the laws of physics is possible only if the relative frequencies associated with every ensemble of identical such events remain unaltered. But this is possible only if the individual modifications of probability are themselves probabilistic. Suppose that some of the single-case probabilities are increased and some are decreased such that the overall probability remains unchanged. Then the laws of physics remain unchanged, but there can be no talk about causation, mental or otherwise. Whatever ‘causes’ such statistically insignificant modifications of probability cannot be causally efficacious. To be causally efficacious, an event must make a difference every time it occurs. It must make a difference to the behaviour of some ensemble, that is, it must be statistically significant. The basic tenet of the interactionist position — causal openness of the material to the non-material mental — thus entails a violation (that is, an occurrence of modifications) of physical laws.¹⁰ Probability distributions, determined jointly by initial conditions and some quantum-mechanical equation of motion such as the Schrödinger equation, are altered. One might leave it at that. But one might also wonder if any such alteration could not be formulated just as well in terms of the well-known physical quantities that determine probability distributions during the deterministic phase of their evolution. This is the case, as I proceed to show.

V: Interactionism without Quantum Collapses

As an illustration of how the altered probability distributions entailed by the interactionist hypothesis could arise within the formalism that physicists use to calculate probability distributions, rather than as ad hoc modifications of the results of the calculations, we will now consider an open one-particle system. A system consisting of just one particle obviously cannot accommodate the creation or annihilation of particle pairs, but it seems reasonable to assume that minds do not cause either type of event. (The energy needed for pair creation is available in cosmic rays and high-energy physics laboratories, not in brains. The antiparticles needed for annihilation events are not normally present in brains.) We further assume that mental events do not induce particles to change type. This is tantamount to ruling out the so-called strong and weak forces as vehicles of mental causation, for it is these that cause type conversions. (The weak force can for instance convert electrons into neutrinos.)

The strong and weak forces are unlikely vehicles of mental causation because both of them are short-range forces. The strong force is confined to the interior of certain subatomic particles, the mesons and the baryons. A residue of this force, the so-called nuclear force, is confined (in brains if not in neutron stars) to the interior of the atomic nucleus, as is the weak force. None of these forces is effective at the scale of chemical processes; none therefore is relevant to the chemistry of the brain. The goings-on inside atomic nuclei have no influence on when neurons fire, or how likely they are to fire, which is how the causal efficacy of the mind must make itself felt.

And since the most general formulation of effects on the motion of a spinless particle already includes the possible effects on a particle with spin, we can confine our discussion to that type of particle which is represented by a single wave function (rather than one of those multicomponent wave functions known as spinors). Such a particle is known as a scalar particle.

The entire physics of a quantum-mechanical system is formally contained in a mathematical expression known as the probability amplitude. This amplitude allows physicists to calculate (at least in principle) the likelihood with which the system transits from any initial state to any final state in any given interval of time. The entire physics of a scalar particle is in fact known if one knows the amplitude associated with how likely the particle is to travel from point x to point y in any given time span.

It is a remarkable fact about quantum mechanics that this amplitude (let’s represent it by the symbol <y|x>) can be calculated by ‘summing over’ (that is, adding up contributions from) all space–time curves that connect x at the starting time with y at the time of the particle’s arrival — as if the particle went from x to y by travelling along every possible path (Feynman and Hibbs, 1965). Each curve simply contributes a complex number of unit magnitude. Such a number is fully specified by what is called its phase. The phase of a curve is the sum of the phases associated with its segments, and this fact makes it possible to think of the phase of a curve as its length. For an uncharged particle this mechanical length of a curve in space–time is simply proportional to the geometric length of the same curve, and the proportionality factor is simply the particle’s mass.¹¹

Clearly, the only way of influencing the motion of a scalar particle (charged or uncharged) is to modify the mechanical lengths of curves in space–time.¹² This can be done in one of two ways: in the manner of gravity, by changing the geometric lengths of curves and thereby warping space–time itself, or by changing the mechanical lengths without changing the geometric lengths.¹³ When it is weak

Figure 1. The upper diagram (A) shows a few of the curves contributing to <y|x>, the amplitude associated with the probability that a particle initially located at x is later found at y. By Euclidean standards, the shortest curve is the straight line c0. The possible effects on the motion of a particle are mathematically represented by a non-Euclidean way of measuring lengths. In terms of mechanical lengths, the shortest curve connecting x and y may be c1,or it may be c2,or it may be c1 for particles of a certain type and c2 for particles of a different type.

Diagram B: Because gravity affects all particles alike, its effect on the mechanical lengths of curves can be thought of as a warping of space-time itself. The surface with the dip represents space-time. The extra dimension into which it is warped is not physical; its sole purpose is to make it possible to visualise the warping of space–time. The dip could be due to a massive object at its centre. Because of the dip, c0 is no longer the shortest curve connecting x and y. A classical particle travelling from x to y will take the shortest curve on either side of the dip, and this makes it seem as if a force, gravity, were pulling the particle towards the centre of the dip as it travels around it.

enough to permit a human brain to function normally, gravity plays no significant role in a region of space the size of a brain, which is why we only need to consider the latter option.¹⁴

As an illustration of the kind of effect caused by changes in the mechanical lengths of space–time curves, imagine a plane (see fig. 1A). In it imagine two points x and y and a bundle of curves beginning at x and ending at y. One of these curves (call it c1) will have a shorter mechanical length than every other curve. By no means does this have to be the straight line c0. Next suppose that the mechanical lengths of all curves are increased in such a way that those of curves entirely to the left of c1 increase more than those of curves entirely to the right of c1. As a result, the mechanically shortest curve will no longer be c1 but a different curve c2 to the right of c1. One of the effects of altering the mechanical lengths of space–time curves is thus equivalent to bending the curve of minimum mechanical length between any two space–time points. (Usually there is just one such curve, but see fig. 1B for a situation in which the shortest curve connecting x and y is not unique.)

In the so-called classical limit, in which quantum mechanics degenerates into classical mechanics, the only contributions to <y|x> that ‘survive’ come from the curve (or curves) of minimum mechanical length. (More precisely, from curves that are shorter than their nearest neighbours.) This explains why a classical particle travels from x to y (in the specified time span) along the mechanically shortest curve (or one of the mechanically shortest curves) between x and y. What gets bent are the space–time trajectories of classical particles. But bending the space–time trajectory of a classical particle is the same as accelerating the particle, and this is the reason why in classical physics one talks about acceleration-causing forces instead of modifications of mechanical lengths.

How does one mathematically represent modifications of the mechanical lengths of curves that leave the geometric lengths unchanged? The answer is straightforward: by means of some field.¹⁵ This field (let’s call it A) associates with every infinitesimal curve segment (depending on both the location and the direction of the segment) the extra bit of mechanical length that the segment has for a charged particle. (For an uncharged particle, recall, the mechanical length of the segment is simply its geometric length times the particle’s mass. Uncharged particles do not ‘experience’ the non-gravitational modifications of mechanical lengths.)

The field A is known to physicists as the electromagnetic vector (or four-vector) potential. It contains exactly the same information as the electric and magnetic fields together.¹⁶ The electric field is what bends the projections of classical trajectories on space–time planes that include a time axis (that is, it accelerates charges in a fixed direction), while the magnetic field is what bends the projections of classical trajectories on spatial planes (that is, it accelerates charges in directions perpendicular to their directions of motion).

The vector potential (equivalent to the electromagnetic field) is thus the summary representation of all possible non-gravitational effects on the motion of a scalar particle, including all effects caused by mental events. Physicists habitually associate the vector potential not only with the way in which it influences the motion of charged particles but also with a particular way (given by Maxwell’s laws) in which it is generated by the motion and distribution of charges. They don’t question (and as physicists, concerned solely with the behaviour of inanimate matter, need not question) the assumption that this is also the only way of generating it. But, in fact, anything —beit physical, mental or whatever — that has a (non-gravitational) effect on the motion of a particle, necessarily contributes to the electromagnetic vector potential.¹⁷ If a mental event is to influence the behaviour of the quasi-particle in Eccles’ model of a trigger mechanism for exocytosis, it must modify the barrier — a potential barrier — penetrated by the quasi-particle.

When the electromagnetic field was introduced by Maxwell, it was thought of as the property of a mechanical substrate pervading space. When Einstein discarded this substrate, the erstwhile property became a physical entity in its own right. The symbol took on a life of its own; the mathematical description took the place of the thing described. Today many physicists believe that reality is mathematical. While the present investigation ought not to be biased in favour of any such metaphysical claim, it is safe to say that the empirical reality investigated by science is, first of all, a complex of mental constructs. (I am not saying that it is ‘nothing but’ mental constructs.) What these constructs have in common, and what distinguishes them from mere fantasies, is that they are objectifiable, that is, they are capable of being thought of as features of an objective world. The vector potential is such a construct (after quantization, at any rate), and from the role it plays in our account of particle motion it is clear that it cannot be partial to any particular type of causal agent. It serves to represent the effects of mental causes just as well as those of physical causes.

Now that we know that the second manner of modifying the mechanical lengths of space–time curves is, in actual fact, the way of the electromagnetic force, we have another reason for dismissing gravity (the first manner) as irrelevant to mental causation. Considering that exocytosis is controlled by the influx of Ca²⁺ ions into a synaptic vesicle (Eccles, 1994, pp. 149–53), mental causation is likely to be effected through a modification of the physically determined forces exerted on ions (that is, on charges), particularly those involved in the propagation of nerve impulses. But the electromagnetic interaction between, say, two protons is about 10³⁶ times stronger than their gravitational interaction. Hence if the mentally generated modification of the force exerted on a charged particle were of gravitational nature, the mental self would have to generate an implausibly strong gravitational field (about that many times stronger than the physically generated one), while it would only need to generate an electromagnetic field that is weak in comparison with the physically generated one.

Yet another reason why the electromagnetic interaction is the more likely vehicle of mental causation is the selectivity of the electromagnetic force. While this acts on charges only, gravity affects everything. If one wants to make an ion move through a neutral medium, one had better not also accelerate the medium, as this would simply cause a congestion; if one tries to move both the ion and the medium, nothing will move.

However, all said, nothing fundamental stands in the way of the notion that the mind contributes to any or all of the four fundamental forces, inasmuch as the weak and strong forces no less than the metric tensor and the electromagnetic vector potential are simply ways of formulating possible effects on the behaviour of particles, whether their origin be physical, mental or whatnot. For reasons indicated above I believe however that the electromagnetic field is the single most effective vehicle of mental causation, and that therefore the other possibilities are not worth considering.

If non-physical causes do indeed contribute to the vector potential, the well-known dynamical laws of the vector potential (that is, Maxwell’s laws or their quantum-mechanical counterparts) are violated, in the sense that they describe some but not all contributions to the vector potential. It is worth emphasising that there are neither theoretical nor experimental reasons to rule out such a violation. While empirical evidence of non-physical contributions to the vector potential may as yet be lacking, absence of evidence is not the same as evidence of absence. Evidence of absence is not available because systems in which such contributions might occur are notoriously complex, difficult to analyse, and no less difficult to experiment with. It could be argued, moreover, that if the non-physical contributions to A amounted to a substantial modification of the physically determined component of A, mind would be able to actuate matter through a less complex physiology. While the complexity of the body is no argument against interactionism, it certainly suggests that a non-material mind cannot cause more than minute modifications of the physically determined component of A.

As for theoretical derivations of the dynamical law for A, they tell us no more than what was initially assumed. Because A can be considered as a quantum-mechanical system in its own right, its dynamics is known if one knows how to calculate the amplitude for the transition from any initial field configuration to any final field configuration in any given time span. As there are contributions to the amplitude <y|x> from all curves connecting x and y, so there are contributions to this transition amplitude from all ‘histories’ of the field A (that is, from all curves in the infinite-dimensional space of field configurations). And as before, each contribution only depends on the mechanical length of the corresponding history/curve.

A crucial difference however arises when it comes to finding the correct mathematical expression for the mechanical lengths of field histories. The formula for the mechanical lengths of space–time curves ‘experienced’ by a scalar particle contains the representation of all possible effects on the motion of a scalar particle. We can be sure that none have been left out. On the other hand, we can be sure that we have the right formula for the mechanical lengths of field histories only if all sources contributing to the field are represented in it, and only if the effects represented by the field are linked to their causes according to universal mathematical laws. In order to be able to derive Maxwell’s equations (along with their quantum-mechanical counterparts) we must therefore assume (i) that the motion of a particle cannot be affected by anything except the motion and distribution of particles, and (ii) that the action of particles on particles is amenable to mathematical description. Hence the argument that mind cannot affect the behaviour of charged particles because this is governed by Maxwell’s laws, obviously begs the question.

VI: Mind, Space and Mathematical Description

The aim of this section is to show (i) that mind is non-spatial and (ii) that the action of a non-material mind on matter is not amenable to mathematical description. The latter conclusion in fact is a consequence of the former.

On the interactionist view, mind is non-spatial and, as causal agent, independent. (‘Independent’ here means that its acts of will are not fully determined by physiological microstructure and physical law. ‘Non-spatial’ means that the mental cause of an effect on the motion of particles in the brain does not consist in the spatial distribution and/or the state of motion of objects in space.) From this it follows that the condition that the effects represented by the electromagnetic field must be linked to their causes via universal mathematical laws, cannot be satisfied for the direct effects of volitions. For one thing, if this condition is to be satisfied, the causes must be, at the very least, amenable to mathematical description. Since this is essentially synonymous with spatio–temporal description, they must have positions in space. For another thing, if the link between a causal primary in the mind and its physical effect were amenable to mathematical description, one could write down a Lagrangian for the mind as causal agent. But if that were possible, this causal agent would be just another kind of matter subject to just another kind of physical law — something whose existence neither dualists nor materialists are likely to endorse. For the dualists, it would be too materialistic; for the materialists, too dualistic.

It is in fact unnecessary to assume the non-spatiality of the mental, as is shown presently. If the self were an object in space, it would have to make sense to talk about the position of the self relative to other objects in space. Let us see why it does make sense to talk about the relative positions of particles. Like the non-material self, a fundamental particle can’t be seen. Its position relative to other material objects can nevertheless be inferred from its observable effects, for instance from a trail of droplets in a cloud chamber. But this inference is possible only because there exists a physical law that relates the position of the particle to the positions of its effects. Applying our knowledge of this law to observational data (the positions of the droplets), we can infer the (approximate) position of the particle. And how did we come to know this law? It is an extrapolation from regularities observed in the relative positions of larger charged objects that can be seen.

By the same token, attributing to the self a position appears to make sense only if there exists a law relating the position of the self to the positions of observable effects caused by the self. If we knew such a law, we could infer the self’s position from its effects. But how could we discover such a law? By observing regularities in the positions, relative to observable material objects, of larger selves that can be seen? There may be psychophysical laws (Chalmers, 1995) relating mental states to physical configurations in the brain, but so far nobody has suggested that these laws involve the positions of mental states. I suppose that this is because there simply is no way of making sense of the position of a mental state. Only the physical effects that the self, ex hypothesi, is capable of producing, are localizable in space.

Not all theorists of consciousness would agree. M. Lockwood (1989, p. 101), for one, takes special relativity to imply that mental states must be in space given that they are in time. This conclusion, however, appears to rest on a too naive identification of two distinct concepts of time. What ‘time’ means in the context of psychological experience is not the same as what it means in the context of special relativity. Without an in-depth study of their relation (not offered by Lockwood), only the physical effects of mental states can be said to necessarily exist in space–time. See Clarke (1995) for a refutation of Lockwood’s arguments in support of the spatio– temporal localization of mental events.¹⁸

A priori, the modifications of the electromagnetic field ‘experienced’ by certain constituents of the body could be effected in two ways: the non-material self could contribute to the electromagnetic field as a separate source, or it could modify the way in which the field is built up by material sources. However, to act as a separate source, the self would have to exist in space, and this notion has just been rejected. Hence it follows that material particles are the only sources of the electromagnetic field, and that the non-material self can only influence the summary effect — represented by the electromagnetic field — of the action of particles on particles.

The causal efficacy of the self thus rests on the causal efficacy of the particles, or on the ability of the particles to modify their individual contributions to the electromagnetic field. The causal behaviour of particles (meaning, the way particles influence each other’s motion, as distinct from the way particles move) accordingly comes in two modes: a physical mode which obeys the laws of physics, and a nonphysical mode through which modifications of the physical mode are effected. But this means that the only causal agents in existence are the fundamental particles, and that the non-material self cannot be as non-material as dualists would have it. Interactionism thus cannot be the last word. The implications of this, as well as the possible relationship between the self and the body’s constituent particles, will be explored in another article (Mohrhoff, submitted).

VII: Summary and Outlook

The following results have been obtained:

1. (The conservation of energy and momentum is a consequence of the homogeneity of time and of space. This is warranted for systems that are causally closed. As to material systems that are open to causal influences from non-material mind, either energy/momentum is/are ill-defined or there is no reason why it/they should be conserved.
2. Assuming that part but not all of matter is causally open to non-material mind, it makes sense to attribute (non-conserved) energy and momentum even to physical systems that interact with non-material mind.
3. The causal efficacy of non-material mind implies departures from the statistical laws of quantum physics. These departures are capable of being formulated in terms of modifications, by the conscious self, of the electromagnetic interactions between particles; and they are more consistently formulated in this manner.
4. Because the electromagnetic field is a summary representation of effects on the motion of particles, the effects caused by mental events are necessarily among the effects represented by it. It is not that one cannot formulate the effects of the self in terms of a separate probability field, to use Margenau’s (1984) term. The point is that this field would be indistinguishable from a contribution to the electromagnetic field, which makes it obvious that departures from the laws of physics are involved. Thinking of the effects of the self as contributions to the electromagnetic field is preferable for two reasons. First, it eschews the contentious notion that measurement-like events take placein the unobserved brain. Second, it leads to a more unified treatment of causality. There is no reason whatever for having probabilities determined twice over, once during their deterministic evolution by the physically determined vector potential, and once at the end through a superimposed probability field generated by the self.
5. Quantum-mechanical indeterminism cannot be the physical correlate of free will. Free will implies departures from the laws of physics.
6. Mind is non-spatial. There is no point in attributing positions to mental states and events.
7. The departures from the physical laws caused by non-physical mental events are not amenable to mathematical description. It is worth emphasizing that they are not therefore random. They could be necessitated by something of a primarily qualitative nature, something that manifests itself in quantitative, spatio–temporal terms but is not reducible to these terms.

Although there are no compelling theoretical or experimental reasons why mental events should not be capable of causing departures from physical laws, it may remain difficult for interactionists and proponents of free will, at least for some time to come, to disabuse the contemporary physicist, biologist, or philosopher of science of the doctrine of physicalism, which has been a reigning orthodoxy for well over a century. So much was this doctrine taken for granted, that until recently it was considered as almost indecorous to waste much thought over the dismissal of its antithesis. Thus, after stating that ‘very few people any longer suppose that living things violate any laws of physics (as some thinkers supposed as late as the nineteenth century)’,¹⁹ Hilary Putnam (1992, p. 83) makes known why this should be so: ‘Physics can, in principle, predict the probability with which a human body will follow any given trajectory.’ Are we to suppose that the mountaineer who fell to his death would have been able to choose a less ruinous trajectory if only Eccles’ hypothesis of mind–brain interaction had been true?

What interactionists and proponents of free will claim, in effect, is that the nonmaterial self becomes materially effective by modifying the electromagnetic interactions between constituents of the body. Not only is this consistent with the assumption that the trajectory of the body’s centre of mass is fully determined by physical laws, but also it agrees with our sense of free will which interactionists wish to take seriously. I decide to raise my hand and it goes up; but nothing in my experience leads me to expect that I could alter my trajectory once I have jumped off a cliff.²⁰

Yet there is cause for optimism. If the hard problem of consciousness is taken as seriously as it now is, the hard problem of freedom is bound to follow suit. Many researchers in cognitive studies now admit the irreducibility of consciousness. And most of the philosophers who speculate about the shape of a fundamental theory of consciousness invoke some form of panpsychism.²¹ Yet, with few exceptions, these philosophers still find it necessary to reduce conscious events to ‘causal danglers’: they affirm that pain is not reducible to its physical correlate yet deny that it causes us to pull our hands out of fires. Such a position is inherently unstable, as Lowe (1995) has pointed out. It is under intense pressure either to lapse back into materialism (which restores the causal efficacy of conscious feelings by identifying them with their physical correlates) or to take the further step of admitting the causal efficacy of consciousness. The present article has shown that, from the point of view of physics, nothing stands in the way of taking this long overdue step.

Acknowledgement

I wish to thank Jean Burns for many helpful suggestions.

Notes

[1] I have no quarrel with compatibilism, the view that free will is compatible with determinism. My freedom may well consist in being governed by what I intrinsically am (what the Indian contemporaries of Plato and Aristotle would have called my ‘self-nature’ or ‘self-law’, svabhava, svadharma) rather than by universal laws or a combination of universal laws and randomness.

References

Beck, F. (1994), ‘Quantum mechanics and consciousness’, Journal of Consciousness Studies, 1 (2), pp. 253–5.

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What is ‘time’ – exploring the fourth dimension.

To begin, ‘time’ and the fourth dimension are two completely separate ideas, unlike common belief. Before I begin discussing the fourth dimension, I want to clear up what ‘time’ actually is. Time is a subjective measurement created by humans to record, keep track of and plan events occurring in the universe. If another intelligent race exists elsewhere in the universe, chances are they have their own measurement of ‘time’, but their ‘second’ would be completely different to ours. Think of how both the imperial and metric system of measuring distance are both valid ways of doing so, but have very different intervals. But if these measuring systems did not exist, that doesn’t mean that distance would suddenly cease to exist. It is exactly the same with time in relation to the fourth dimension. The fourth dimension is ‘duration’ – i.e. the opportunity for the unfolding of events to occur, and time is simply our way of measuring it..

Almost everybody on the planet has come to believe that we exist in a three-dimensional universe, as that is what we perceive. However, this is not the case – we exist within the extended idea of the fourth dimension. I’m going to break it down and create a step-by-step guide to ‘reaching’ the fourth dimension.

Remember: this description is just a way to help with visualising the dimensions. First, we start at zero; the ‘zeroth’ (or zero) dimension, which is a point of indeterminate size. Now, take two zero-dimensional points and connect them with a line – we have now reached the 1st dimension, length. Now imagine a series of one-dimensional lines bunched together side by side, Planck lengths apart. We are left with a two-dimensional plane. Do the same again but with these two-dimensional planes and we arrive at the third dimension.

Okay so we’ve reached the third dimension. Surely we don’t need to go any higher – we exist in a three-dimensional world, don’t we? Short answer: sort of. As I said earlier, the fourth dimension is the unfolding of events, or ‘duration’. Without this duration, the universe as we know it would not exist. Of course, in the absence of duration, the Big Bang would not have happened as it would not have had the opportunity to. If we, hypothetically, take the universe as it exists now and view it from the third dimension, we would in fact not be able to see anything. Light needs an opportunity, this fourth-dimensional duration, to travel. If it cannot travel, it remains completely static, meaning there would be no way for our eyes to perceive anything. Similarly, all universal bodies, from subatomic particles to galaxies, would still exist but would also, of course, be static; there would be no movement, interaction or development whatsoever. How could there be without duration?

Because the third and fourth dimensions are so connected (space-time), it is imperative to explore how the two interact. Physicist Julian Barber, on the third dimension, suggested: “In my view of the universe, it’s just like a huge collection of snapshots which are immensely, richly structured. They’re not in any communication with each other, they’re worlds unto themselves… In some very deep sense, the universe, a quantum universe, is static. Nothing changes.”

From this analogy, imagine that these ‘snapshots’ are arranged like frames (Planck frames) in a flip-book, all one unit of Planck time away from the next. The fourth dimension is like the hand flipping through the flip-book. It is the tool to move through these frames to create the apparently continuous ‘motion’ of duration – what we perceive as the progression of ‘time’.

Conclusion (TL:DR): The fourth dimension is ‘duration’, i.e. the opportunity for events to unfold. Time is completely unrelated to the fourth dimension – it is simply our way of measuring it, similarly to how miles measure distance.

How many dimensions is your world?

There are many things we take for granted. Take, for example, the world around us. How many dimensions is it? You might say, easy – that’s three, as in 3D. In a sense (literally) that’s right. We pass every hour of our lives in a 3D space – at least the one represented by our mind’s eye. We may move from here to there or perhaps we just sit still, letting time go by. Without hardly realising it,  another dimension has entered the picture – time itself. At the very least, we see that our lives take place in a world made up of 3D spatial dimensions plus 1D time. 4D. We know instinctively that if we stand still on the same spot for long enough, we will age. But what about if somebody were to tell you that our world is 5D or 11D? And, that the number of dimensions don’t have to stop there – it is quite possibly ∞D?

Imagining a World with 4 Spatial Dimensions

IMAGINING A 4TH DIRECTION:

Right, now what? Well, now I ask you to stop and try to imagine a 4th direction, honestly have a go at it for a few seconds… can’t do it? We have no more directions in the 3rd dimension in which we can expand our hypercube (obviously), but it can be done in the 4th dimension of spacetime; we call this a tesseract.

There are several ways of trying to take a perspective on viewing this expansion, but that would be very difficult to explain in this article. So instead, I will take a practical approach (that said, it is interesting to ponder this 4th expansion, and there are some good motion graphics on the web if you give it a search–like here).

spacial dimensions via instructables


Now then, picture yourself looking down on a piece of paper, and this piece of paper is home to world existing only in 2d spacetime. Even though the 2nd dimension exists within the 3rd dimension, these beings would have no idea you exist because they will have no comprehension of another direction of space (they couldn’t look up to see you because up is not a concept they can fathom). It is easy for us to say ‘height’ because we experience height – remember how difficult it was for you to think of a 4th direction?

If you poked your finger through their world, it would just appear as a flat disk with no height when viewed from the side. Also, if they were on the outside of a square with a dot inside, you could quite easily reach inside that square and pull out the dot. The 2-D beings would have no idea how this was possible because to them, you just somehow passed the only dimensional boundaries they know of. You would be able to do this because the 2nd dimension has a cross-section through our space.

I know that this may be a bit confusing, but it is useful to imagine how a being that existed in the 4th dimension would appear to us. We couldn’t say quite for sure how they would look, but I was once told they would look like ‘a bunch of skin blobs’. The 4-D creature would be able to see everything in our 3 dimensional space because they would exist outside our boundaries – as well as seeing inside any object.

This would be gruesome, but they would have the power to see inside your body and remove any one of your organs without ever having to penetrate your skin. Just as you could remove the dot by pulling it into your dimension, they could do the same to you (possibly some kind of revenge in a 2-D/4-D alliance?). Similarly, these creatures may well exist in our own universe and can live here without detection, but again, we would never be able to see them, just as 2-D beings could never see us.

Dimensions (physics): What is 1D, 2D, 3D, and 4D?

What is 1d, 2d, 3d, and 4d? How is it easily understood by a beginner?

The first three dimensions are relatively straightforward in our ordinary experience of geometry.

Length/Distance: One dimensional geometry would be confined to points on a line. Inches, meters, distinctions of nearer or farther, etc are measures of distance or length.
Width/Area: Two dimensional geometries are expressed as flat planes which have length and width but no depth. A shadow is an example of a two dimensional appearance. 2d shapes are typically measured in square units, such as cm2

or others like acres.

Depth/Volume: Three dimensional geometries add the dimension of depth or height so that they describe objects with volume. Volume should not be confused with weight as two objects can be the same volume but one can be much heavier than the other. A gallon of mercury is much heavier than a gallon of milk. 3d measures include cubic units cm3, pints, quarts, tablespoons, and liters.

Like many people, I grew up thinking of time as 'the fourth dimension'. This is a notion that can be said to go back before Einstein, as physicists had already been representing time in formulas as a one dimensional variable, t.

The fourth dimension is the position in time occupied by a three-dimensional object.

The time-line, like the number-line, is an axis of successive points, but unlike a ruler the dimension which a clock measures does not persist in our immediate experience. Instead, we think of time flows or time lines which move in a linear sequence that than cannot be seen all at once like spatial locations can be. We infer the passage of time by keeping records and devices such as timers, clocks and calendars which use units such as years, hours, minutes, and seconds.

4. Spacetime/Gravity: After Einstein, the concept of time became inextricably linked with space, resulting in spacetime. Spacetime is a logical consequence of the constancy of light speed for all observers, and gravity can be conceived of as distortions in that continuum of universal public measurement. The Earth's gravity for example, is not a force pulling objects through empty space, but a feature of spatial and temporal relations themselves: a consequence of how motion and the speed of light are defined and translated in different frames of reference.


The laws of physics and the speed of light must be the same for all uniformly moving observers, regardless of their state of relative motion. For this to be true, space and time can no longer be independent. Rather, they are "converted" into each other in such a way as to keep the speed of light constant for all observers. (This is why moving objects appear to shrink, as suspected by FitzGerald and Lorentz, and why moving observers may measure time differently, as speculated by Poincaré.) Space and time are relative (i.e., they depend on the motion of the observer who measures them) — and light is more fundamental than either. This is the basis of Einstein's theory of special relativity ("special" refers to the restriction to uniform motion).

Einstein did not quite finish the job, however. Contrary to popular belief, he did not draw the conclusion that space and time could be seen as components of a single four-dimensional spacetime fabric. That insight came from Hermann Minkowski (1864-1909), who announced it in a 1908 colloquium with the dramatic words: "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality...

Einstein initially dismissed Minkowski's four-dimensional interpretation of his theory as "superfluous learnedness" (Abraham Pais, Subtle is the Lord..., 1982). To his credit, however, he changed his mind quickly. The language of spacetime (known technically as tensor mathematics) proved to be essential in deriving his theory of general relativity.

The physics of spacetime however, can be considered separately from a model of time, which is not so much a fourth dimension as it would be an n +1 dimension, where n is any dimension of space. A cartoon might be two dimensional, but it has a beginning, middle, and end, so that rather than a fourth dimension, cartoon time is only a third, or 2+1 dimension. It may be more correct ultimately to think of time or experience as the primordial proto-dimension through which not only all other dimensions but dimensionality itself is hosted.
 

The concepts of 1d, 2d, 3d, and 4d then are really mathematical abstractions used in modeling any phenomenon which has multiple senses of order. When applied to physics, they can be seen as three spatially enumerable vectors and one space-time relativistic vector. With string theory, there could be several more compactified physical dimensions which are so small that we cannot detect them. There is the concept of a tesseract or hypercube which bears the same relation to a cube that a cube does to a square. An actual tesseract would not be possible to construct with our 3d bodies, but we can build a 3d representation of it, or draw a 3d representation of that.

"The Fourth Dimension can refer to time as another dimension, along with length, width, and depth. This idea of time as a fourth dimension is usually attributed to the "Theory of Special Relativity" proposed in 1905 by the German physicist Albert Einstein (1879-1955). However, the idea that time is a dimension goes back to the nineteenth century, as we see in the novel The Time Machine (1895) by British author H.G. Wells (1866-1946), wherein a scientist invents a machine that lets him travel to different eras, including the future. The Cubists may not have known about Einstein's theory, but were aware of the popular idea of time travel. They also understood Non-Euclidean geometry, which the artists Albert Gleizes and Jean Metzinger discussed in their book Cubism (1912). There they mention the German mathematician Georg Riemann (1826-1866) who developed the hypercube.

   
Simultaneity in Cubism was one way to illustrate the artists' understanding of the Fourth Dimension. In this sense, the Fourth Dimension concerns how two kinds of perception work together as we interact with objects or people in space. That is, to know things in real time, we must bring our memories from past time into the present. For example, when we sit down, we don't look at the chair as we lower ourselves on to it. We assume the chair will still be there when our bottoms hit the seat.

Another definition for "the Fourth Dimension" is the very act of perceiving (consciousness) or feeling (sensation). Artists and writers often think of the fourth dimension as the life of the mind. - Art History Definition: The Fourth Dimension

I have thought of the feeling or consciousness as a fifth dimension which encapsulates the other four. In physics, this dimension is collapsed to a single point as the 'observer' or 'reference frame'. In my view (Multisense Realism), this fifth dimension actually transcends dimensionality itself. MSR proposes that subjectivity and objectivity are ranges along a deeper continuum of sense and sense-making. Even the notion of dimension itself is only a sense-making framework which is transcended by direct sensation and experience. We can describe things like flavors and colors to each other but they cannot be represented quantitatively.

If there are dimensions to human privacy, they are not as clear cut as the first four, but could roughly be considered 5. Sensitivity (pain, pleasure over time), 6. Emotion (feelings about sensitivity), 7. Thought (Ideas which detach from direct experience), 8. Value (Thoughts, feelings, and sensations which change behavior). These would be added to the three ordinary dimensions used to measure public bodies, but those dimensions (length, width, depth) are only surface dimensions through which private experiences are made public. Nothing 'lives' in public bodies, it is more like a theater for the mechanics of persistence and interaction between many layers of experience on many levels of relative privacy.
 
 

An Interesting Dimension 0D 1D 2D 3D 4D IntroductionPosition VectorScalar Product Next

The First Dimension: Length

Straight Line

The first dimension is length, or x-axis—a straight line, with no other characteristics.

The Second Dimension: Height

Square

Height, or y-axis, can be added to the length to produce a two-dimensional object, such as a triangle or square.
The Third Dimension: Depth

Cube

Depth, or z-axis, can be added to the previous two dimensions to produce objects that have volume, like a cube, pyramid, or sphere. This is the end of the dimensions that are directly physically perceptible by human beings. All dimensions beyond the third are theoretical.
The Fourth Dimension: Time

Cube Time

The fourth dimension is the position in time occupied by a three-dimensional object.

Geometrical Dimensions Point - Line - Square - Cube - Hypercube -... 0D 1D 2D 3D 4D 5D EXTRUSION

Constructing a 4D Corner: creates a 3D corner creates a 4D corner ? 2D 3D 4D 3D Forcing closure:

If a line is 1D, a plane 2D, and the space 3D, how can I imagine the 4th dimension

The Point
It all starts with the point. We say that the point is in 0D (no dimension).

The Line
To make a line starting from that point we can go in any directions around it. We than stretch that point into that direction and draw a line between the two points. We can then say that we had total freedom of movement to create that line. We would also be drawn to conclude that this freedom of movement was in a 3D volume. (Spherical coordinate system.)

The Square
Making the square is a similar process but this time we have stay perpendicular to the previous line. We then stretch the line anywhere around the original line and draw two extra lines liking the copied points. We had some freedom of movement to create that square and conclude that this freedom of movement was in a 2D plane. (Circular coordinate system)

The Cube
Repeating the procedure again will generate a cube. This time our movement have to stay perpendicular to the squares plane. We then stretch the square anywhere along a perpendicular line from the plane and draw four lines liking the copied points. Again, we had some freedom of movement to create that square. We would conclude that this freedom of movement was on a 1D line. (Linear coordinate system)

What is a dimension, and how many are there?

As you've probably noticed, we live in a world defined by three spatial dimensions and one dimension of time. In other words, it only takes three numbers to pinpoint your physical location at any given moment. On Earth, these coordinates break down to longitude, latitude and altitude representing the dimensions of length, width and height (or depth). Slap a time stamp on those coordinates, and you're pinpointed in time as well.

To strip that down even more, a one-dimensional world would be like a single bead on a measured thread. You can slide the bead forward and you can slide the bead backward, but you only need one number to figure out its exact location on the string: length. Where's the bead? It's at the 6-inch (15-centimeter) mark.


Now let's upgrade to a two-dimensional world. This is essentially a flat map, like the playing field in games such as Battleship or chess. You just need length and width to determine location. In Battleship, all you have to do is say "E5," and you know the location is a convergence of the horizontal "E" line and the vertical "5" line.

Now let's add one more dimension. Our world factors height (depth) into the equation .While locating a submarine's exact location in Battleship only requires two numbers, a real-life submarine would demand a third coordinate of depth. Sure, it might be charging along on the surface, but it might also be hiding 800 feet (244 meters) beneath the waves. Which will it be?

Could there be a fourth spatial dimension? Well, that's a tricky question because we currently can't perceive or measure anything beyond the dimensions of length, width and height. Just as three numbers are required to pinpoint a location in a three-dimensional world, a four-dimensional world would require four.

At this very moment, you're likely positioned at a particular longitude, latitude and altitude. Walk a little to your left, and you'll alter your longitude or latitude or both. Stand on a chair in the exact same spot, and you'll alter your altitude. Here's where it gets hard: Can you move from your current location without altering your longitude, latitude or altitude? You can't, because there's not a fourth spatial dimension for us to move through.

But the fact that we can't move through a fourth spatial dimension or perceive one doesn't necessarily rule out its existence. In 1919, mathematician Theodor Kaluza theorized that a fourth spatial dimension might link general relativity and electromagnetic theory [source: Groleau]. But where would it go? Theoretical physicist Oskar Klein later revised the theory, proposing that the fourth dimension was merely curled up, while the other three spatial dimensions are extended. In other words, the fourth dimension is there, only it's rolled up and unseen, a little like a fully retracted tape measure. Furthermore, it would mean that every point in our three-dimensional world would have an additional fourth spatial dimension rolled away inside it.

String theorists, however, need a slightly more complicated vision to empower their superstring theories about the cosmos. In fact, it's quite easy to assume they're showing off a bit in proposing 10 or 11 dimensions including time.

Wait, don't let that blow your mind just yet. One way of envisioning this is to imagine that each point of our 3-D world contains not a retracted tape measure, but a curled-up, six-dimensional geometric shape. One such example is a Calabi-Yau shape, which looks a bit like a cross between a mollusk, an M.C. Escher drawing and a "Star Trek" holiday ornament [source: Bryant].

Think of it this way: A concrete wall looks solid and firm from a distance. Move in closer, however, and you'll see the dimples and holes that mark its surface. Move in even closer, and you'd see that it's made up of molecules and atoms. Or consider a cable: From a distance it appears to be a single, thick strand. Get right next to it, and you'll find that it's woven from countless strands. There's always greater complexity than meets the eye, and this hidden complexity may well conceal all those tiny, rolled-up dimensions.

Yet, we can only remain certain of our three spatial dimensions and one of time. If other dimensions await us, they're beyond our limited perception -- for now.

Explore the links on the next page to learn even more about the universe.

Relativity, space, time and gravity

Throughout the development of mechanics and electromagnetism the role of space and time had been clear and simple. Space and time were simply the arena within which the drama of physics was played out. Speaking metaphorically, the principal 'actors' were matter and ether/fields; space and time provided the setting but didn't get involved in the action. All that changed with the advent of the theory of relativity.

The theory was developed in two parts. The first part is called the special theory of relativity, or, occasionally, the restricted theory, and was introduced in 1905. The second part is called the general theory, and dates from about 1916. Both parts were devised by the same man, Albert Einstein.

The origins of the special theory of relativity can be traced back a long way. In 1632, Galileo wrote:

'Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies and other small flying animals. Have a large bowl of water with some fish in it: hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drop falls into the vessel beneath; and, in throwing something to your friend, you need throw no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all these things carefully (though there is no doubt that when the ship is standing still everything must happen in this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still."

Galileo Galilei (1632), Dialogue Concerning the Two Chief Systems of the World.

In other words, any phenomenon you care to study occurs in just the same way in a steadily moving ship as in a stationary ship. The underlying physical laws and fundamental constants must therefore be exactly the same for all uniformly moving (or stationary) observers. This fact, which dozing train passengers may accept with gratitude, is the central idea of the theory of special relativity. Indeed, it is called the principle of relativity. This leaves one obvious question: how did Einstein gain both fame and notoriety for promoting an idea that was nearly three hundred years old?

The answer is that a lot of physics had been discovered between the time of Galileo and that of Einstein. Most notably Maxwell's theory of electromagnetism had achieved the feat of predicting the speed of light using fundamental constants of electromagnetism, constants that could be measured using simple laboratory equipment such as batteries, coils and meters. Now, if the principle of relativity were extended to cover Maxwell's theory, the fundamental constants of electromagnetism would be the same for all uniformly moving observers and a very strange conclusion would follow: all uniformly moving observers would measure the same speed of light. Someone running towards a torch would measure the same speed of light as someone running away from the torch. Who would give credence to such a possibility?

Einstein had the courage, self-confidence and determination to reassert the principle of relativity and accept the consequences. He realised that, if the speed of light were to remain the same for all uniformly moving observers, space and time would have to have unexpected properties, leading to a number of startling conclusions, including the following:

Moving clocks run slow. If I move steadily past you, you will find that my wristwatch is ticking slower than yours. Our biological clocks are also ticking, and you will also find that I am ageing less rapidly than you.

Moving rods contract. If an observer on a platform measures the length of a passing railway carriage, he or she will measure a shorter length than that measured by a passenger who is sitting inside the carriage.

Simultaneity is relative. Suppose you find two bells in different church towers striking at exactly the same time (i.e. simultaneously). If I move steadily past you, I will find that they strike at different times (i.e. not simultaneously). It is even possible for you to find that some event A happens before some other event B and for me to find that they occur in the opposite order.

The speed of light in a vacuum is a fundamental speed limit. It is impossible to accelerate any material object up to this speed.

If these consequences seem absurd, please suspend your disbelief. It took the genius of Einstein to realise that there was nothing illogical or contradictory in these statements, but that they describe the world as it is. Admittedly we don't notice these effects in everyday life but that is because we move slowly: relativistic effects only become significant at speeds comparable with the speed of light (2.998 × 108 metres per second). But not everything moves slowly. The electrons in the tube of a TV set are one example, found in most homes, where relativistic effects are significant.

One of the first people to embrace Einstein's ideas was his former teacher, Hermann Minkowski (1864-1909). He realised that although different observers experience the same events, they will describe them differently because they disagree about the nature of space and the nature of time. On the other hand, space and time taken together form a more robust entity:

'Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.'

Hermann Minkowski, Space and Time in A. Einstein et al. (1952), The Principle of Relativity, New York, Dover Publications.

The union of space and time of which Minkowski spoke is now generally referred to as space-time. It represents a kind of melding together of space and time, and since space is three-dimensional, and time is one-dimensional, space-time is four-dimensional. Any particular observer, such as you or I, will divide space-time into space and time, but the way in which that division is made may differ from one observer to another and will crucially depend on the relative motion of the observers.

A very rough attempt at representing diagrammatically this change of attitude towards space and time is shown in Figure 1. Before Einstein introduced special relativity, the phrase 'the whole of space at a particular time' was thought to have exactly the same meaning for all observers. After Einstein's work it was felt that each observer would understand what the phrase meant, but that different observers would disagree about what constituted the whole of space at a particular time. All observers would agree on what constituted space-time, but the way in which it was sliced up into space and time would differ from one observer to another, depending on their relative motion. No observer had the true view; they were all equally valid even though they might be different.

Figure:1

Figure 1  (a) The pre-Einsteinian view of space and time. Not only are space and time separate and distinct, they are also absolute. All observers agree on what constitutes space and what constitutes time, and they also agree about what it means to speak of 'the whole of space at a particular time'. (b)The post-Einsteinian view in which space and time are seen as aspects of a unified space-time. Different observers in uniform, relative motion will each slice space time into space and time, but they will do so in different ways. Each observer knows what it means to speak of 'the whole of space at a particular time', but different observers no longer necessarily agree about what constitutes space and what constitutes time.

In retrospect, special relativity can be seen as part of a gradual process in which the laws of physics attained universal significance. The earliest attempts to understand the physical world placed Man and the Earth firmly at the centre of creation. Certain laws applied on Earth, but different laws applied in the heavens. Copernicus overturned this Earth-centred view and Newton proposed laws that claimed to apply at all places, and at all times. Special relativity continues this process by insisting that physical laws should not depend on the observer's state of motion - at least so long as that motion is uniform. It is therefore not surprising that Einstein was led to ask if physical laws could be expressed in the same way for all observers, even those who were moving non-uniformly. This was the aim of his general theory of relativity.

Einstein realised that many of the effects of non-uniform motion are similar to the effects of gravity. (Perhaps you have experienced the sensation of feeling heavier in a lift that is accelerating upwards.) With unerring instinct he treated this as a vital clue: any theory of general relativity would also have to be a theory of gravity. After more than ten years of struggle, the new theory was ready. According to general relativity, a large concentration of mass, such as the Earth, significantly distorts space-time in its vicinity. Bodies moving through a region of distorted space-time move differently from the way they would have moved in an undistorted space-time.

For example, meteors coming close to the Earth are attracted to it and deviate from uniform, steady motion in a straight line. Newton would have had no hesitation in saying that these deviations are due to gravitational forces. In Einstein's view, however, there is no force. The meteors move in the simplest way imaginable, but through a distorted space-time, and it is this distortion, generated by the presence of the Earth, that provides the attraction. This is the essence of general relativity, though the mathematics required to spell it out properly is quite formidable, even for a physicist.

The central ideas of general relativity have been neatly summarised by the American physicist John Archibald Wheeler. In a now famous phrase Wheeler said:

'Matter tells space how to curve.

Space tells matter how to move.'

Purists might quibble over whether Wheeler should have said 'space-time' rather than 'space', but as a two-line summary of general relativity this is hard to beat (see Figure 2). If you tried to summarise Newtonian gravitation in the same way all you could say is: 'Matter tells matter how to move'; the contrast is clear.
 

Figure 2 A highly schematic diagram showing space-time curvature near the Sun and indicating the way in which this can lead to the bending of starlight as it grazes the edge of the Sun. (The bending has been hugely exaggerated for the sake of clarity.) The observation of this effect in 1919, during a total eclipse of the Sun, did much to make Einstein an international celebrity.

General relativity is a field theory of gravity. At its heart are a set of equations called the Einstein field equations. To this extent general relativity is similar to Maxwell's field theory of electromagnetism. But general relativity is a very unusual field theory. Whereas electric and magnetic fields exist in space and time, the gravitational field essentially is space and time. Einstein was well aware of the contrast between gravity and electromagnetism, and spent a good deal of the later part of his life trying to formulate a unified field theory in which gravity and electromagnetism would be combined into a single 'geometric' field theory. In this quest he was ultimately unsuccessful, but general relativity remains a monumental achievement.

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These 4 properties are:

• Amplitude: this is the size of the displacement for the disturbance. The units for amplitude depend on the type of wave. For a string, the units would be meters.
• Wave speed: if you were to watch one displacement, it would be moving. The wave speed is the speed (that seems redundant). The unit for wave speed is meters per second.
• Wavelength: this is the distance from one disturbance to the next measured in meters
• Frequency: if you were to count how many waves passed a stationary point in each second, that would be the frequency (in cycles per second or Hertz).

Tough question (for me), as the STC concept is not "four-dimensional" in any sense other than that it uses some strange "time" coordinate to show the relative "spacetime" location of an event (coordinates x, y, z, time). It is more a mathematical construct than a "model" of reality. Popular press loves to use phrases like "time, the fourth dimension," which is misleading at best.

Einstein envisioned his Universe as a "solid block" (see Block Universe)) where nothing moved (!), that everything that can happen already has happened (!), and continues to happen (!), past as well as future events (!). Time, to Einstein, was a virtual "slice" of this block. Here's an illustration from a SCAM article (Paul Davies, Time — That Mysterious Flow), visualizing the concept:

A very small "block" cut out of the Universe, where the "rod" is Earth, and the "stretched slinky" is the moon. Frozen solid for eternity. Our perception of "flow of time" is said to stem from the idea that the "present time-slice" is always on the move, with a speed that depends on how fast the observer on that slice is moving (see any contradiction here?).

To quote Paul Davies (from above linked article): "Nothing in known physics corresponds to the passage of time. Indeed, physicists insist that time doesn’t flow at all; it merely is. Some philosophers argue that the very notion of the passage of time is nonsensical and that talk of the river or flux of time is founded on a misconception."

Einstein wrote (in a letter to a friend): "People like us who believe in physics know that the distinction between the past, the present and the future is only a stubbornly persistent illusion."

Personally, I find Einstein's philosophy of time (called Eternalism) somewhat hard to swallow, not only because it implies a completely deterministic Universe (that "free will" is non-existent), but mainly because the whole idea is preposterous (sorry, Albert!).

Will the frozen block universe model be replaced by the model of the fourth expanding dimension which exalts photons, quantum mechanics, and time?

The block universe contains no free will nor arrow of time, nor does it provide a model for the quantum nonlocality and the probabilistic behavior of a photon.

The model of a fourth dimension expanding at c (Einstein’s dx4/dt=ic) gives us quantum nonlocality and probability alongside relativity.

IS THE BLOCK UNIVERSE DESCRIPTION OF TIME CORRECT?

"People like us who believe in physics know that the distinction between
the past, the present and the future is only a stubbornly persistent illusion"
                                                                                 Albert Einstein

Every past or possible future event also has a place like feeling to it. Time-scape feels like it is a place where it may be possible to go.

This dimension like view of time has spawned numerous science fiction stories and movies on time travel.

This view of time suggests that dinosaurs are still alive and roaming the earth in some other time dimensions; it also suggest that there are multiplecopies of us and the whole universe smeared across multiple dimensions of time.

In Special Relativity (SR) the block universe view of time arises from an interpretation of the Lorentz transformation equation known as the Rietdijk–
Putnam argument (or the Andromeda paradox.) By this innovation of SR just walking on the earth toward or away from the Andromeda galaxy which is
2.5 million light years away we can shift our line of simultaneity so that our time can be in sync with either past or future of beings living in Andromeda.
This interpretation of SR suggests that past and future exists as a part of the block universe. Lorentz transformation is interesting but has not been
proven experimentally and this interpretation of SR cannot be verified. All of other SR’s predictions of slowing of time, length contraction and gain in
mass with motion can be derived without Lorentz transformation and are experimentally verifiable.

In the block universe time is laid out as a time-scape similar to landscape and it is obvious that there cannot be a free will. This has led to some
innovations or variations in the theme of the block universe in which the future is changeable. If time-scape is already laid out then what causes our
conscious experience to move through this time-scape and why we cannot willfully move our consciousness anywhere anytime?

The time of the block universe leads to some interesting conclusions. The universe in its time dimensions should have numerous future civilizations
millions or billions of years more technologically advanced then us. At least some of these civilizations should be capable of travelling through the
block universe and we should have seen some evidence for that, unless there is some law of the universe which prohibits time travel. Block universe
also leads to the possibility of time travel paradoxes like the grandfather paradox in which a person travels to the past and kills his grandfather thereby
changing the future so that the time traveler would not exist and thus not travel to the past to kill his grandfather.

SLOWING OF TIME IN THE BLOCK UNIVERSE

Next time you look at a tall building or a mountain try to visualize that time is running more slowly even slightly so near the bottom of these structures
then at the top. The stability of these structures depends on the fact that space-time is continuous; being slow in time does not lead to lagging behind
and disappearing into the past. Imagine if the bottom of a mountain or a building vanished into the past.

Theory of relativity predicts slowing of time with motion and gravity. These predictions have been confirmed in particle accelerators as well as gravity
experiments. If there is a block universe why particles and masses with slower time do not disappear into the past? In gravitational fields space is
clearly continuous between areas of slower and faster time, going against the concept of a persistent past dimension.

Black holes with their intense gravity that bring time to a screeching halt do not disappear from our present into the past. We need to have clarity in
our minds as to what the slowing of time means in a block universe. Does passage of time mean our consciousness is moving across time
dimensions? Slowing of time without sliding into the past or the future suggests that time is a process and not a dimension. This may be a
significant point against the block universe view of time when taken together with other aspects of time described above.

Now you have your finished product. Looks pretty cool, doesn't it? But how exactly does this two-dimensional image represent a four-dimensional figure?

It goes like this: take a pencil and paper and draw a single point on the paper. The point is a zero-dimensional entity, meaning it has no physical definition - no length, width, or height.

Now make a second point and connect it with the first point with a straight line. The line is a one-dimensional object; it only has one physical characteristic, which is length.

Draw a second line (one that's as far away from the first as its own length) at a right angle to the first and connect their vertices (corners or ends), and you get a two-dimensional square with length and width.

If you could repeat this process by drawing the same square perpendicular to the first square and connecting their vertices with each other, you would get a cube with length, width and height - the three dimensions of our physical world. Of course, it is impossible to draw a 3-D object on a 2-D paper, so we'll settle for an imperfect "projection" of a cube, drawn by placing the second square at a 45-degree angle to the first and connecting them with lines the same length as their sides.

The tesseract you just drew is, essentially, a continuation of this process. In a manner of speaking, it is an image of what would happen if you were to draw a second cube perpendicular to the first and connect their vertices. This is shown in the second picture; the first cube is blue, the second is red, the purple is where they overlap.

Like I said before, our physical world has three dimensions. These three are perpendicular (at right angles) to each other. The mysterious fourth dimension would be perpendicular to all three of these dimensions at once! But don't even try to imagine it; because we live in a 3-D world, it would be impossible for us to imagine such a direction, since technically it can't actually exist. You might as well try to make a square imagine a cube!

Here's an alternative way of thinking about it. If you were to take the six 2-D squares from the third image and fold them in a 3-D way, you would make a cube. In the same way, if you were to take the eight 3-D cubes in the fourth image and fold them in a 4-D way, you'd make a tesseract.

Since we can't imagine how a tesseract would actually look in all of its 4-D glory, all we can do is create a 2-D projection of it, the way we did for the cube. Some information is lost in the transition, but it's better than nothing.

A dimension can be either spatial (relating to space) or temporal (relating to time). Spatially, we’re 3D creatures, but temporally, we’re 1D. So, we’re in fact already 4D creatures.

Considering your reference to Flatland, I believe you mean a creature that is spatially one dimensional higher. Then, spatially, they’re 4D creatures. But just for simplicity’s sake, let’s consider that any dimension I mention below is spatial.

Now that you’ve watched the video, you have an idea of how a (spatially) 3D creature perceives a (spatially) 2D creature, and vice versa. You just raise everything one dimensional higher. For a 4D creature, all of us would appear differently from how we appear to each other. To its eyes, all parts of our bodies would be visible: like how we look at a 2D object, the 4D creature can view us, plus our insides, in all angles at the same time. We would appear in a rather different shape for that creature. And, just like how we can easily stick our finger inside a 2D figure, the creature can also fiddle with every part of our body without breaking us apart; it can directly interact with our bones without cutting through our skin. So, concealed properties such as bank vaults and prisons are basically defenseless in terms of physical guard against such a creature. A 4D creature can easily steal from a bank without breaking into it, and no 3D confinement will work against it. In fact, that creature will be literally invisible to us, just like how a 2D creature can’t see us (because we’re above it, and in Flatland, there is no direction such as up or down… not that a 2D creature can perceive such a direction anyway).

And, like how a 2D creature sees us, we will be able to see only a part of that higher dimensional creature’s body. In fact, that part of its body that we can see will be 3D for us, just like how only a 2D fraction of us would be visible to a 2D being.

Additional insight: Because shadows thrown by a 3D object appears to be 2D to us, it’s just logical to say that shadows of a 4D object should appear to be 3D. Here’s a good example of the shadow of a 4D hypercube, called a tesseract:

Zamanda sonsuz geçmişe doğru yol almak sanki içe doğru sonsuzca küçülmek gibi düşünülebilir.İç içe geçmiş anlar/zaman boyutları!  Sonsuz geleceğe doğru açılmakta sanki dışa doğru sonsuzca büyümek, genişlemek gibi düşünülebilir... içe ve dışa doğru iç içe büyüyüp küçülen bir  mekan tabirini dördüncü boyutu tarif için kullanabiliriz.

The image shows what the shadow of a tesseract would appear as in a 3D space. This 3D shadow consists of two nested cubes, with all their angles connected by lines. In the 4D space, all the lines of the tesseract would have equal length and all angles right angles. Of course, as 3D beings, we can’t perceive such a thing.

In other words, we might actually just be shadows of some 4D creatures. Or, shadows of some shadows of some 5D creatures. Maybe those 5D creatures are also shadows of some 6D creatures. And maybe those 6D creatures, too, are… hmm, we shouldn’t be overthinking.

(In string theory, higher dimensions are something much different from what we’re talking about here, so you might want to check them out).

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Posted by Rob Bryanton - 2011  :

The most important thing to remember as we're talking about these ten dimensions is that they're spatial dimensions. Some people get confused by that word, "spatial" because they think it's intended to only apply to the third dimension: the length, width and depth of the "space" we see around us. In fact, some physicists do prefer to use the term "space-like" when talking about the extra dimensions. No question, when you take a three-dimensional space and add an additional "right angle" to it, you are entering a realm which is difficult for us to picture, and it certainly behaves in ways that are beyond the limitations of 3D space: that's what Imagining the Tenth Dimension is all about.


Likewise, some people have difficulty with discussing the first and second dimension. "How can something with no depth even exist?", they ask. It is a bit of a mind-bender! We start with a point that has no size and no dimension. We make a second point some place else, and the line that passes through those points is a representation of the first dimension. If you can imagine a third point that isn't on the line you've just created, you have a way of thinking about the second dimension: a line passing through this new point and the old line defines a plane. But if the lines you're thinking about are like pencil lines, which already have a length, width and depth, then you're really not visualizing the right thing. Do you know what I mean by that?

If I say "imagine you're on a boat in the middle of the ocean", and you say "I don't own a boat and I hate the water", what does that prove? The concept of boats and oceans doesn't change whether you're willing to imagine them or not. Likewise, if I say "imagine something that has length and width and no depth", and you say "I refuse to imagine that because something that has no depth can't exist", where have we gotten? Nowhere. But does refusing to discuss an idea mean the idea doesn't exist? Of course not! We can have a perfectly good discussion about dragons, which don't appear to exist in our world, because dragons are an idea which we are capable of describing and thinking about.

Same goes for the second dimension. It's not part of our 3D world, it's something separate, but it's still something we can think about and talk about. In "What Would a Flatlander Really See?", we looked at the imaginary 2D creatures invented by Edwin Abbott for a book he published way back in 1884: "Flatland: A Romance of Many Dimensions".  Could Flatlanders really exist? Perhaps no more than dragons. But we can have a perfectly good discussion about the idea of Flatlanders, and what it would be like to live in a world that has length and width, but no depth. What would it be like to look around you within that world, where all you can make out is lines all within the same plane? That is a mind-bending exercise, good food for thought regardless of whether it would really be possible for some kind or awareness to exist within such a ridiculously limited frame of reference or not.

Over the next nineentries, we're going to look at each of the dimensions from the second all the way up to the tenth, and see what kind of a mental castle we can build for ourselves, one brick at a time. Along the way, I want you to keep reminding yourself about something called the "point-line-plane postulate", which uses the same kind of logic as the "line/branch/fold" of the Imagining the Tenth Dimension project. This postulate is the accepted method used to imagine any number of spatial dimensions, using the same repeating pattern:

0 - a Point: Whatever spatial dimension you're currently thinking about, imagine a geometric point within that dimension. Remember, when we say this point has "no size", what we really mean it that the point's size is indeterminate. "Indeterminate" means that any and all sizes you care to imagine, from the infinitely large to the infinitesimally small, are true for that point. Let's say that this is a point within "dimension x".

1 - a Line: From the current dimension you're examining, find another point not within that dimension. An easy way to do that is to imagine the first point at its largest possible size within the constraints of its own dimension, and then ask where a different point would be that isn't encompassed by that first point in its infinitely large state. Once you've found a second point, draw a line through both points, and now call what you're looking at "dimension x+1".

2 - a Plane: Again, think of both of those points encompassing their largest possible version within the constraints of the current dimension, and find a point that isn't part of what those two points are encompassing. Now you're thinking about "dimension x+2".

This logic can start from any spatial dimension, and it can be repeated infinitely: that is, once you've imagined "dimension x+2" you can rename it "dimension x" and repeat the pattern as many times as you want. Here's an important thing to remember: if we're not assigning any meaning to the dimensions we're visualizing, there's no reason to stop at ten. However, with this project, by the time we've arrived at the tenth dimension, we do find a way to say that we have arrived at the most all-encompassing version of the information that becomes reality, or the underlying symmetry state from which our universe or any other patterns emerge through the breaking of that symmetry.

And I do hope you'll enjoy the journey as we work our way through this logical presentation, one step after another.

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Defining the 4th Dimension and Beyond

2012/12/16

 
Actually, the real title of this was going to be “Defining the 4th Dimension and Beyond, Using Spacial Geometry.” While that sounds incredibly wimpy to the mathematicians out there, it would surely scare away most average readers.

But, this was envisioned by a child.

Even though it makes use of Pascal’s triangle, tesseracts, and other mathematical concepts, it mostly uses reasoning and basic knowledge, as it came out of a child’s mind. Be assured, it is accessible to everyone.

Be assured also, it has a valid scientific model of the fourth dimension, along with several cosmological implications that are derived from this model.

Background
When I was in junior high school (dinosaur era), I was reading science fiction a lot—something that my teachers didn’t feel was all that useful. And that sci-fi stuff got me thinking—something else my teachers didn’t feel was all that useful either. LOL You would think they’d have appreciated that I was doing some real pondering about the universe, but they wanted me to be doing more useful thinking… like memorizing dates in history or practicing looking up log tables.

Much as they tried to entice me with their subordinate clauses and subjunctive verbs, my mind kept drifting back to the concepts of time and space, which fascinated me. I was always really good about being able to wrap my brain around concepts and visualize them, so I started trying to imagine what a four-dimensional world would look like…and how “time” factored in (as that was what scientists were theorizing were the properties of the fourth dimension).

Well, I did it. And when I was done, I had an actual model of the fourth dimension—along with practical guidelines for defining other higher dimensions as well.

I now know that one element of my overall concept was already developed many years before I did mine—it was obscure mathematics, so no one knew much about it at the time. However, my model went considerably beyond that.

Most importantly, my particular model may yet help science by providing a base algorithm. I don’t believe there is anything that has been able to clearly define the fourth dimension, or reliably predict higher dimensions…or been presented in this way. My model offers the ability to truly visualize a spatial fourth dimension from several perspectives. Additionally, it does point toward deriving possible insights into cosmological mysteries.

To make sure you understand the concepts in the specific way that I developed and used them, I will explain how I got there, so you won’t be giving me the same funny looks my teachers did. Be assured, it’s still amazingly simple.

First, you will need to know what a “dimension” is.

The Definition of a “Dimension”
NOTE: If you already would know a parallel line and a flatlander if you saw one, you can skip to Part 3.

We live in and experience a world we see as “three-dimensional.” Traditionally, that has been defined very much the same way you think of when you define a box—length, width, height (three dimensions). Although, the way my model works out, that isn’t the ideal way to view it when defining multiple dimensions, but we’ll get to that later. Just keep in mind that in our three-dimensional world, the “volume” we live in has length-width-height, and we call that: space.

Space, itself, is made up of both one-dimensional and two-dimensional elements.

A one-dimensional object is a line. Just a line. If we use the box above as an example, either what we called the “length” or the “width” edges of it could be considered a one-dimensional object. A one-dimensional object is any single line—having no physical depth, width, height, or anything else associated with it—only length. It is just a very, very thin line (mathematical lines are infinitely thin).

A two-dimensional object only has length and width, and is just the area on a single “plane” (like a tabletop). For instance, each side (face) of that box is a plane—a two-dimensional-like surface that defines/outlines the space of that box. If the sides of that box had no mass, they would be truly two-dimensional planes.

The size of a plane that would define OUR whole space would be a flat plane that would extend outward, forever straight, all the way to the end of our universe…a super-sized side of a box. Of course, scientists really do have a sense of humor, so they have now confused the issue by informing us that space may actually be CURVED. We won’t go into that one at this point since it doesn’t affect our current discussion, but I know there are some of you out there that love those kind of paradoxes.

Now, when trying to think in universe-sized “spaces,” it gets to be a bit mind-boggling. In order to define something beyond our own “space,” we have to define what space is, so we know where it ends—or at least, where it is NOT. So, how big is space—do we know of something that is outside of our space?

Well, what we commonly call “outer space”, by cosmological definition, is still really OUR own, single, 3-D space. Within the context of mathematical definitions of lines, planes, and spaces, they are considered infinite. A pure “line” would be assumed to extend to infinity.

The length-width-height where we exist is a single “space”—until we get to a wall of some kind. Therefore, our solar system, our own Milky Way galaxy, and all the other galaxies, all the dust and matter…all the way as far as the eye can see… In fact, the whole universe is the 3-D space in which we exist—it’s OUR SPACE. Space can be pretty big. LOL

If it can take up the whole universe, then you’re probably wondering how there can be any place that isn’t our space. Where would the fourth dimension be hiding? Where do we have to go to get outside our “space”?

Current cosmology has evidence that our universe may be finite—meaning there is a limit…a boundary…a definable edge of some kind to our universe. The way they explain it, there was a “big bang” that exploded, which started our universe (another point we will discuss later). The matter/energy from that big bang *IS* our universe—it defines our universe. And ever since then, the matter/energy has been expanding outward from the center of that big bang…which means our universe is expanding…which in turn, means our “space” is expanding. This general concept, as surreal as it seems, is probably accurate. And except for the little detail that our space, therefore, isn’t precisely definable (as it is constantly growing), there could very well be an “out there” that isn’t part of our own universe. That means our three-dimensional space would exist within something that goes beyond three dimensions.

What would the fourth dimension be? It would logically have to be something that would include our space—in the same way that our space includes any lines and planes of the first and second dimensions.

The accepted theory within the scientific community is that “time” is (or is part of) the fourth dimension. So, going back to that box we started with… It exists in that space. But if we were to take a picture of the box there now, and then someone were to remove the box, and then we took a picture of that same space after it was gone, then we would see how that space would have been altered by time. That is the general principle of how the fourth dimension factors into our universe, although there are several variations.

Some current theories embrace parallel universes, as well as alternate universes. But would that imply there are other big bangs and “universes”? Possibly. It could also mean a single event that spawned multiple timeframes. One concept of a “parallel universe” assumes a starting point (e.g. the big bang), but then at every juncture where there is a possibility to change the course of an event via “choice,” two paths/threads are started—one where the path was unaltered, and another where a choice was made, which altered it in some way. For every choice/change, there would be a different timeline created for that alternate universe—spawning an infinite number of universes.

Hopefully, I have all of you following along now, as those definitions are crucial to understanding where this is going.

PART 2

Visualizing Dimensions 1, 2, and 3
It would help a bit if we could try to visualize what it might be like to exist in another dimension. Seeing things from other perspectives is hard enough in a familiar world, but we’re trying to visit worlds where we couldn’t even exist in our physical state. This is a good brain exercise—as even the futuristic scientists from our favorite sci-fi stories would have trouble replicating a two-dimensional being.

Here is a common example that allows us to visualize fairly easily how a two-dimensional world would see a three-dimensional object. This visualization was made famous in a novella called Flatland,
by Edwin A. Abbot…

Existence in the second dimension would be like living on a flat plane, where there would be no up-down at all—only sideways. Imagine standing on a very, very flat piece of land. The initial trouble with imagining this, is that we couldn’t actually “stand” in a two-dimensional world—we’d all be pancakes. A true two-dimensional state would be so “squished down” to be truly, ultimately flat, such that it’s hard to explain how anything could really even be there. But for the sake of this exercise, we have presence and vision on that two-dimensional plane.

Now, if a 3-D box were to “visit” our plane, what would we see? Let’s think about this. The box is comprised of six sides—each one a plane. Depending on the box’s position, a couple of those planes might be exactly parallel to our own plane, while the others would be generally perpendicular to our plane. That means it would appear totally different to us—based on the orientation of the box when we have our close encounter.

We wouldn’t be able to see up or down, so the only parts we would see are at our level, where the box intersects our plane.

Imagine that the box is set down into our 2-D world, so the bottom side of the box (a plane) lined up perfectly with our plane—like when someone puts a box down onto a table. Then, it would exist in our 2-D world as a whole plane. It would create a tile-like square where it sat—and only that square—displacing everything that existed there at the place it intersected.

If a 3-D alien came along and plopped down a box around you, all you would see of that box would be the lines where your plane intersected the sides of the box. It would form an outline of a square around you (a two-dimensional wall), and you’d be trapped. Eek!

On the other hand, if you were outside the area where the box “touched down,” you would only see the intersection of the box plane with your 2-D world. That would be a line.

That is how the box alien would look for those of us viewing it in two-dimensions. If we “two-dimensionals” were then asked what a box (aka cube) looked like, we would be convinced it looked like just a line or a line-drawn square.

As you can see… it’s all in your point of view (or as Einstein might have said, “It’s all relative”).

Of course, living in a one-dimensional world would be even worse. Travel would only be possible along a single, infinitely straight line. It would be like living in a pipe—where you can’t even turn around. Infinitely boring

Now that you have had some practice “seeing” things from one of the dimensions as you exist in another, you can now view some of my different perspectives for visualizing the 4th dimension.

PART 3

Hypothesis and Basis for My Model
Standard Geometry tells us about the properties/rules of points, lines planes and spaces…

• A point has no form, as it is infinitely small.
• A line has only length, is comprised of at least 2 points, and is infinitely long.
• A plane has two dimensions (length and width, x,y only), a flat surface area with no thickness that extends infinitely, and its area is comprised of points and lines.
• A space has three dimensions—length, width, and height (x,y,z), and its area is comprised of points, lines, and planes.

We also know that…

• A minimum of 2 points are needed for a line to exist.
• A minimum of 2 lines (parallel or intersecting), or 1 line+1 point, are needed for a plane to exist.
• A minimum of 2 planes (intersecting or parallel) are needed for a space to exist.

So, if we had a minimum of 2 spaces, what would now exist? (And following that logic, I wondered what a “parallel” or “intersecting” space would look like.)

This is how I tried to visualize the 4th dimension. Since the dimensions already were defined by numbers (and to make it easier on myself so I wouldn’t have to re-program my brain when thinking this through), I used the existing numbers. However, a point didn’t have any number, but since it was infinitesimally small, and since a zero would work for the sequence, I assigned the “dimension” 0 to a point.

I did feel good when I later found out that the science field had also decided to adopt that convention. And, it makes this part of the explanation easier.

Using the dimensional numbers to represent the elements that define each dimension, we have…

I kept trying to visualize what “Time” would look like to us if we weren’t trapped in a three-dimensional viewpoint. I also kept trying to visualize what we would look like to someone in a “time” dimension. But that was too hard without any hints. I needed some hints.

I got a little farther along when I tried to visualize going from one space to another. In order to do that, you would have to go through *something.* Time? Space-time?

To help visualize that part, I went back and thought about pretending I was that two-dimensional (flat) being, living on a single plane. That plane would, then, be my entire universe—any other plane would be a different universe. In order to get from one plane to another, the “something” that I would have to traverse would be a space. That is because, to get off that plane, the only direction I could go is “up” or “down”—aka 3-D. Trouble is, I would not have any knowledge about “up” or “down” because my perspective is limited within my plane-world. Likewise, in my 3-D world, it’s hard to imagine a fourth dimensional direction I would need to traverse to get to a “different” space.

I considered how I would get between any of the other dimensions…

Going between two Points — you would have to travel along a Line.
Going between parallel Lines — you would have to travel on a Plane.
Going between parallel Planes — you would have to travel through Space.
Going between [parallel?] Spaces — you would have to travel across ????

In the process of that, it occurred to me that a space could also be defined as a whole bunch of planes stacked on top of one-another. Now, I was getting somewhere…i was starting to think “outside the box.” LOL

So, what is it I would need to traverse to go between spaces? It would be something that contained a bunch of spaces? That *something* would, of course, be the 4th dimension, but how could I define it? Like the planes all stacked up to create a space…what if I were to stack a whole bunch of spaces—what would it create? What would it look like?

That brought back a very scary image in my mind. When I was a kid, my grandmother used to drag me to the bank before going shopping, where I would have to wait in sheer boredom while she visited with the teller. The bank had one of those heavy green glass tabletops that was just about kid-height. I remember putting my eyes right up to the edge and looking in. What I saw was both the most fascinating and the scariest image I have ever seen. Inside a huge, dark abyss, were an infinite number of reflections of the glass plane and the hardware holding the glass. It was bottomless and topless, with the eerie planes stretching out forever. I imagined our spaces stacked like that, and tried to visualize exactly what a whole bunch of spaces might form. A table sitting in a 4-D room? A honeycombed hive of 4-dimensional bees?

For one thing, I just couldn’t figure out how to get a relation to time out of that. I kept visualizing multiple universes all clumped together, but there wasn’t a unique time element that was obvious. Sure, you would take time to traverse them, but it would take time to traverse along our two-dimensional plane. Time appears to be an inherent property of all existence, but it could just be the highest level of the first four dimensions. The scientists were fairly sure time was it, but it still didn’t feel to me as though time was the main structure for a 4th dimension.

What if there was a space-time dimension that sort of bridged the third dimension with a fifth (time) dimension? Would that work? Maybe.

PART 4

The Minimal/Triangular Space Model
I thought it might be easier if I could pare down space to a minimal size. If I could better define what a “space” actually is. So, what is a space?
(I know…”it’s something to a void”…*ew, bad joke*)

Seriously, space is a three-dimensional area. How do you define three dimensions? One way, of course, is with x, y, and z coordinates But I needed something much more basic—something that would be the minimal, base structure of space, and preferably of each dimension.

I thought about the least number of elements needed to create each dimension.

What would be the least possible number of points needed to define each dimensional object…

Aha! I was getting closer to something concrete. I could interpolate that the fourth dimension would be defined with a minimum of 5 points.

Now, I only had to figure out where that fifth point needed to go. Sure, it wasn’t going to be simple, but I only had to worry about placing one point, rather than wrapping my mind around a whole stack of spaces.

Where was that point supposed to go? I kept looking at my box and my stack of spaces. It wasn’t obvious, but I was going to figure it out…it was only one stupid point…how hard could that be?

When the answer didn’t come instantly, I dabbled a bit with looking at it from other similar perspectives. For instance, what would a least number of points *look* like? Maybe that would help me visualize it better.

I sketched a very simple representation for myself that basically looked like this…

What suddenly hit me was that I was drawing TRIANGLES. And since I was reducing this down to least elements, the concept of an equilateral triangle dawned on me. What if this new dimension was easiest to define if EVERYTHING was equal? In other words, what if, just like in an equilateral triangle, the spacing of the points were equal, the length of the lines were equal, the size of the planes were equal, etc.?

This was so simple now, it *had* to be solvable. I was sure I had something. I kept looking at the three-dimensional equilateral triangle. NOW, where do I put that fifth point?

I remember staring at it numerous times during school the next day. Where do I put a point that is equally distant from all other points in an equilateral tetrahedron? If I put it outside the tetrahedron, some of the lines and planes would be too long. If I put it on the surface of the it, some of the lines and planes would be too short. If I put it inside the tetrahedron (like in the lower right drawing in the diagram above), ALL of the lines and planes would be too short. There didn’t seem to be any place left—but I was sure there had to be.

But then something clicked. The only place that was *equally* too short for all points and planes was the dead-center. What if that center could somehow burrow in deeper than just our space—poke through to…yup…the FOURTH DIMENSION?

Aha!!! Okay, now I could plop that point in there—a virtual equilateral tetrahedron in the fourth dimension—then go back and fill in the rest of my model.

What would be the least number of elements for any of these definitions?

Least number for each one…
 

I really had something going. I could easily interpolate that the “magic” number of spaces for this next dimension would be “5.” I even played a little by expanding my sequence in order to see if going much farther out would give me any additional hints or help at all. Just for fun, I also played with going to “negative” dimensions–wondering if that might apply on a sub-atomic scale.

Okay, so we have a minimum number of spaces (5) needed to create a 4th dimension, and my model with the inward-point that poked out into another universe fit that number series. Wahoo! I would try to draw the fourth dimension.

What would it actually look like? I was getting an image of the extended space bulging back out on the “other” side…that is…way inside. Anti-matter and parallel worlds were popular sci-fi concepts then, and I had a mental picture of an “anti-space” projection on the other side of that pinhole. I drew it (it reminded me of a crystalline structure).

Given that I now had a three-dimensional representation of it, I tried to rationalize what the fourth dimension would really be that would encompass our real-world universe. It somehow involved poking a hole through our existing universe.

A new concept, called a “black hole,” was making news at the time. I realized that the black holes, which crushed everything down to a very small point—crushing so hard, it could be poking a hole right through to another “space” in another universe, which would mean that both spaces together would comprise a fourth dimensional existence. That was it—that was my real-world visualization. The fourth dimension would, therefore, include whatever was on the other side of that black hole.

So, was time the fourth dimension? Umm, that didn’t seem quite right, based on what I was visualizing. I thought it was more likely a time-space, as the “event horizon” would imply that. It could also be something more exotic. But this was close enough. I was so excited, I had to share it with an adult. I had a good enough model with enough supporting evidence, that I felt confident to tell someone “in authority.”

PART 5

White Holes—Tying It All Together With Some Tantalizing Theories
I was not in the class of the best math teacher at my school, but I needed the best. So, I went to see her and tell her what I had found and presented my “time-space” rationale and drawings. However, without really investigating what I had, she told me I was wrong, and reiterated the wisdom of that era. I realized I had hit a brick wall, but I also lost confidence in pursuing it.

I was totally bummed. Based on how it all fit together and seemed so intuitive, I was still pretty sure I was right. And yet, I didn’t know who else to go to, and it didn’t seem worth trying to convince anyone else anyway.

Fast-forward to around 1980. I was watching a “Cosmos” episode, and Carl Sagan pulls out a tesseract. I had never heard of that before. Whoa!!!! That’s it! I was right! That’s my model! I was elated…also very upset that I hadn’t been taken seriously.

But…but…that’s a CUBE. It should be a TRIANGLE! If the model is to work, it should be represented with triangles, shouldn’t it? But back then, there wasn’t any World Wide Web, so there was no place to look it up and see what was known without spending a lot of time in a research library. Not something I could justify spending time doing if it wasn’t going to add much to people’s knowledge. The scientists seemed to have been well onto it, so even if they didn’t have the rest, it wouldn’t be too much longer before they got there. Once again, I let it go—but this time, it was with the realization that I really had been on the right track.

Shortly after that, I met my partner, whose background is astronomy and physics. We discussed it a couple of times, which helped to fill in some gaps for me about the known universe. During one of those discussions, some things hit me…

* Are photons a real-world example of dimension 0?
* And What if our “big bang” in this space/universe of ours, is actually a black hole that imploded in another universe and is now leaking in from another space/universe—a WHITE hole here?!
* What if all of the black holes we see here have little spaces poking out into other universes—becoming ‘white holes’ and spawning new spaces? (Note that my 4-D triangle model creates spaces in the shape of an inverted triangle within the 4th dimension—a bubble universe).
* And could there be any other spaces ‘intruding’ on the edge of our space?

Although well-versed in astronomy, my partner isn’t a cosmologist, and we didn’t have easy access to any. So, once again, back onto the proverbial shelf this project went.

Fast-forward to around 1990. My partner was doing some work at NASA, and when talking with some researchers about the Hubble Telescope and black holes, I was now confident enough to present the model to them. They didn’t think anything much was being done with dimensions and tesseracts at the time, and seemed enthusiastic about my concepts, but I don’t think they ever followed through with it, as they never got back to me either way. Yet it was worth a shot.

On to the year 2000. I met someone who is incredibly strong in both science and engineering. I showed my model to him. He basically dismissed it because he had seen it before, “Oh, that’s just Pascal’s triangle.” Not knowing how Pascal’s triangle could have related to the fourth dimension, I was confused that he didn’t wonder about that himself and look more carefully. But, now I was encouraged that my model was based on something that the mathematical community accepted as a legitimate foundation, and had a historical background as well. At that time, there wasn’t much online about Pascal’s triangle, but between the library and the online info, I read up on Pascal and the “Khayyam triangle” (the early version of it). There wasn’t much about how it related to the fourth dimension.

40+ Years — I Hope Someone Will Find This Valuable
Unfortunately, that was an intense time in my life, and my personal situation had to take precedence. Therefore, spending time searching down someone who might possibly be interested in my little model was not a rational use of my resources. I kept telling myself that “one of these days,” even without the help of an expert, I will get some of this info up on the Web so it could fill in the gaps for those who could make use of it. I had a bunch of other vital info to get up on the Web “one of these days” too, so, I was adamant it would eventually happen.

Now it’s 2010. Still didn’t find anyone to assist me privately, but I’m ready to put my findings up. I wrote out my little “dissertation.” While doing so, I thought of another analogy to that two-dimensional world. If a three-dimensional cube would be perceived as a square that is confining the area within a two-dimensional world, maybe what we see as the edge of our universe is similar. The boundary of our universe could be our 3-D view of the fourth dimension. And I could see where that could relate to time, as time governs the rate of our universe’s expansion.

I checked online to see what was current about the tesseract, and found there is a bunch more work on it, but not a lot about equilateral triangles. I then looked up the Khayyam triangle. I see there is now a Pascal’s Tetrahedron and a lot of other related work. Hmm, I FINALLY was able to get this all written out for public disclosure, and a bunch of my “revelations” are already in articles on Wiki. *sigh*

Did I do this in vain? I don’t think so. I looked carefully, and so far, no one has connected all the dots (pun intended) that bring all these elements together in one place. And even though there is plenty of math that goes far beyond what I could ever contribute, I offer the basic theory, and a visualization of the concepts in a way that hasn’t been done before. Most importantly, this provides a foundation that contains not only the definitions and rules for working with dimensions—but the ability to predict structure and the way to visualize it—on which others can now build. With the help of this model, what we observe about our universe (on both macro and micro levels) may be more easily understood.

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Q: How can we have any idea what a 4D hypercube or any n-D object “looks like”? What is the process of developing a picture of a higher dimensional object?

Posted on May 31, 2011 by The Physicist

A picture of a 3D object is a “projection” of that object onto a 2D page.  Projection to an artist means taking a picture or drawing a picture.  To a mathematician it means keeping some dimensions and “pancaking” others.

So when you take a picture the “up/down” and “left/right” dimensions are retained, but the “forward/back” dimension is flattened.  Mathematicians, being clever, have formalized this into a form that is independent of dimension.  That is, you can take an object in any number of dimensions and “project out” any number of dimensions, until it’s something we can picture (3 or fewer dimensions).
 

Top: An object in 3 dimensions. To see it, cross your eyes by looking “through” the screen until the two images line up. Middle: By “projecting out” the z axis (toward/away) the object is collapsed into two dimensions. This is what cameras do. Bottom: By projecting out the y axis (up/down) the object is collapsed again into 1 dimension. This is akin to what a 2D camera would see, photographing from below.

We’re used to a 3D-to-2D projection (it’s what our eyeballs do).  A 4D-to-2D projection, like in the picture above, would involve 2 “camera/eyeball like” projections, so it’s not as simple as “seeing” a 4D object.

As for knowing what a 4D, 5D, … shape is, we just describe its properties mathematically, and solve.  It’s necessary to use math to describe things that can’t be otherwise pictured or understood directly.  If we had to completely understand modern physics to use it, we’d be up shit creek.  However, by describing things mathematically, and then following the calculations to their conclusions, we can get a lot farther than our puny minds might otherwise allow.

Lines, squares, cubes, hyper-cubes, hyper-hyper-cubes, etc. all follow from each other pretty naturally.  The 4D picture (being 4D) should be difficult to understand.

For example, to describe a hypercube you start with a line(all shapes are lines in 1D).

To go to 2D, you’d slide the line in a new direction (the 2nd dimension) and pick up all the points the line covers.  Now you’ve got a square.

To go to 3D, you’d slide the square in a new direction (the 3rd dimension) and pick up all the points the square covers.  Cube!

To go to 4D, same thing: slide the cube in the new (4th) direction.  The only difference between this and all the previous times is that we can no longer picture the process.  However, mathematically speaking, it’s nothing special.

Answer gravy: This isn’t more of an answer, it’s just an example of how, starting from a pattern in lower dimensions, you can talk about the properties of something in higher dimensions.  In this case, the number of lines, faces, etc. that a hyper-cube will have in more than 3 dimensions.

Define as an N dimensional “surface”.  So, is a point, is a line, is a square, is a cube, and so on.

Now define as the number of N-dimensional surfaces in a D-dimensional cube.

For example, by looking at the square (picture above) you’ll notice that , , and .  That is, a square (2D cube) has four corners, four edges, and one square.

The “slide, connect, and fill in” technique can be though of like this: when you slide a point it creates a line, when you slide a line it creates a square, when you slide a square it creates a cube, etc.  Also, you find that you’ll have two copies of the original shape (picture above).

So, if you want to figure out how many “square pieces” you have in a D-dimensional cube you’d take the number of squares in a D-1 dimensional cube, double it (2 copies), and then add the number of lines in a D-1 dimensional cube (from sliding).

.  Starting with a 0 dimensional cube (a point) you can safely define .

It’s neither obvious nor interesting how, but with a little mathing you’ll find that , where “!” means factorial.  So, without ever having seen a hypercube, you can confidently talk about its properties!  For example; a hypercube has 8 cubic “faces”, 24 square faces, 32 edges, and 16 corners.

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What is 1D, 2D, 3D and 4D ?


What is 1D ?

The full form of 1D is one-dimensional. The one-dimensional is referred to a line. Usually, the one-dimensional have only x-axis that can measured with many factors like Inches, meters, Centi-meter etc.

What is 2D ?

The full form of 2D is two-dimensional. The two-dimension is referred to several lines that exists on the plane. You can also called it as bi-dimensional space. 2d shapes have x-axes and y-axes that measured in square units, such as cm2 or others like acres. 2D can only be viewed from one angle.

What is 3D ?

The full form of 3D is three-dimensional. The three-dimensions is referred to a visual object that has the appearance of depth and field. 3d shapes have x-axes, y-axes and z-axes.
Remember, human eye can see only two-dimensions. Don’t get confused about 3d that why human can see 3d. Actually, it is two-dimensions, because of colors and angle of viewing point when looking 3D.

What is 4D ?

The full form of 4D is four-dimensional. In four-dimensions the four-dimension is time and and other three are x-axes, y-axes, and z-axes. The universe is the best example of 4D.

The four dimensional cube

In math, dealing with more than three dimensions is relatively simple. A point in two dimensions is simply (x, y), two coordinates, one for left-right and one for up-down. For three dimensions, we add a coordinate (x, y, z), and the z coordinate can be thought of as forward-backward, a direction perpendicular to both our first two coordinates. We can't see any direction that is perpendicular to all three of our physical dimensions, but that doesn't stop mathematicians from blithely adding another coordinate (x, y, z, t). In Einstein's spacetime, t would be time and the two directions would be past and future. But there are other concepts of a fourth dimension as well, and although we have a hard time seeing a fourth perpendicular, the math of it is fairly straightforward.

Here is the method for building a four dimensional cube, known as a hypercube or a tesseract. They are fun to draw and kind of pretty if done carefully.

Zero Dimensional VCube (dot)

We start in zero dimensions. Zero dimensions is a single dot, which should have no height or width. In all the constructions of this nature, the zero dimensional thing is a dot.

One Dimensional Cube (Line Segment)


To make a one dimensional cube, we take two zero dimensional cubes, two dots, and connect them with a line segment.


Two Dimensional Cube (Square)

We follow the same pattern to make the two dimensional cube, which is better known as a square. We take two one dimensional cubes, and connect points to the corresponding points on the other one dimensional cube. When we went from zero to one, we didn't have to worry about "corresponding", since there was only one point on each.


Three Dimensional Cube (Cube, Duh!)

And now we have the three dimensional cube, the thing we just call a cube. (In the 2-D world of The Simpsons, it is named a frinkahedron, named after the scientist Professor Frink, since it is a bizarre and imaginary thing only understood by poindexters.) Two 2-D cubes are connected to each other point by point.

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What is a hypercube?

We now look at a different sequence of objects. Again the figures we get will be skeletal, in the sense that the construction will give the vertices and edges of each figure.

In 0 dimensions we have a point. (That is still all there is.)

In 1 dimension, we take the point and move it unit distance, drawing in the trace path of the vertex, to obtain a line segment.

In 2 dimensions, we take the line segment and move it a unit distance perpendicular to itself, drawing in the trace paths of the vertices, to obtain a square.

In 3 dimensions, we take the square and move it a unit distance perpendicular to itself, drawing in the trace paths of the vertices, to obtain a cube.

The general figure defined in this way is called a hypercube. If you want to specify the dimension d, you can speak of a d - hypercube. For example a cube is a 3-hypercube.



• Now, what statement would define the figure in 4 dimensions? (You may have problems understanding the statement!)

The first four members of the sequence of figures we obtain will look like this (we have added some shading for the faces):

How would we picture the 4-dimensional hypercube? Again there are several options: we could construct a 3-dimensional model, or draw a 2-dimensional picture.

First of all, look at the picture of the cube above (right). We can ‘see’ that this represents a cube because we know what a 3-dimensional cube looks like. But really, it is just a 2-dimensional picture. We can think of it as two congruent and similarly placed squares with corresponding vertices joined together.

• 1. This might suggest what a 3-dimensional picture of a 4-dimensional hypercube will look like. Can you describe it?

2. What would a 2-dimensional picture of a 4-dimensioal hypercube look like? (How many vertices? How are they joined?) Try drawing a couple of pictures.

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Cubes in 4 dimensions.

Mathematicians often work with the cube, when describing various physical situations, and indeed SOMA was first discovered at a mathematical lecture of space divided in cubes.

One other interesting aspects of the cube is that it is one of the objects of which we can glimpse the fourth dimension.

We will probably never know if the fourth dimension is anything else than the brainchild of mathematicians, but investigating the properties of a fourth dimension makes it a lot easyer for the scientists to describe the way objects interact here in our 3-D world.

When scientists describe the world and the universe in which we live, they often have to take the fourth 'Space-dimension' into account. No one can see this dimension, but just as you can fold a flat two-dimensional piece of paper into a three-dimensional cube, then the mathematicians can compute from our 3-D world, into the 4-D so called 'hyperworld'

The mathematical concept of dimensions is actually quite simple. A dot has no dimension because you cannot move anywhere on it. A straight line has the dimension 1. because you can move in one direction.- (length wise)

Extending the line at a rightangled direction gives us the sheet, like a piece of paper, and here we have 2 dimensions.- (length and width)
Accordingly, we can extend the sheet in a direction perpendicular to the sheet surface in order to get the cube, now with 3 dimensions.- (length, width and height)

Now, if we take this a step further. Extending the cube in a direction perpendicular to ALL the existing axes then we enter the hyper space having 4 dimensions.

Let us see how dimensions govern the evolution of a 3-D cube, how we may view the shadow of a 4 dimensional cube, and how a 4 dimensional cube will look when it is "unfolded" to our 3-D world.

When I was in middle school, I read the wonderful little book Flatland by Edwin A. Abbott. The story is narrated by a square in a two-dimensional world who encounters a sphere from the third dimension. At first, the square cannot conceive of a world with higher dimensions, until the sphere uses simple mathematical reasoning to help the square imagine a cube. For the sake of clarity, I will retell the story from the point of view of someone trying to explain what a four-dimensional cube looks like to us.

If we say that a cube is a three-dimensional square, then we might also imagine that a one-dimensional square is a line segment and a zero-dimensional square is a point. Now, we can count how many points are required to draw each of these figures:
The pattern is simple: 1, 2, 4, 8, … Therefore, we can guess that if there was a four-dimensional cube, it should be made up of 16 points. Similarly, we can count the number of boundary regions that enclose each of these figures:

Here, the pattern is 0, 2, 4, 6, … So we should guess that if there was a four-dimensional cube, it should have 8 cubes that enclose its boundary. This four-dimensional cube is also called a hypercube, and it has many different visual representations:
What I found fascinating is that no human being has ever seen a hypercube. Yet the beauty of mathematics is that it allows us to describe worlds outside of our experience.

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Introduction to Geometry:
Points, Lines, Planes and Dimensions
 

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The Dimensions of Spacetime

There is a problem with the spacetime diagram: it only has one explicit spatial coordinate x. The way the light cone is drawn suggests, properly, that there is a second spatial coordinate, say y, that points out of the plane of the figure. But what about the third spatial coordinate? It has to be perpendicular to the ct axis and the x axis and the y axis. There is no simple way to draw such a circumstance.

The following figures indicates one way to approach a representation of such a four-dimensional object.

We begin with a zero-dimensional object, a point.
We move the point one unit to the right to generate a one-dimensional line.
Moving the line one unit perpendicular to itself generates a two-dimensional square.
We move the square one unit perpendicular to itself, and we represent the three dimensional cube as shown.
Finally, if the moving of the square down and to the left was used to get from a square to a cube, then we represent moving the cube perpendicular to itself as moving it down and to the right. The result is called a tesseract.

In about 1884 Edwin Abbott wrote a lovely little book called Flatland: a Romance of Many Dimensions; the book has been reprinted many times and is readily available. In it he imagines a world with only two spatial dimensions. One of Flatland's inhabitants, named A. Square, became aware of the existence of a third spatial dimension through an interaction with a higher dimensional being, a Sphere. He attempts to explain this third dimension to the other inhabitants of Flatland, which of course promptly got him put in jail. The difficulties A. Square had in visualising the third spatial dimension is analogous to the difficulties we have in visualising a four-dimensional spacetime.

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Hunting Tesseracti
By Wyrd Smythe

If you’re anything like me, you’ve probably spent a fair amount of time wondering what is the deal with tesseracts? Just exactly what the heck is a “four-dimension cube” anyway? No doubt you’ve stared curiously at one of those 2D images (like the one here) that fakes a 3D image of an attempt to render a 4D tesseract.
Recently I spent a bunch of wetware CPU cycles, and made lots of diagrams, trying to wrap my mind around the idea of a tesseract. I think I made some progress. It was an interesting diversion, and at least I think I understand that image now!

FWIW, here’s a post about what I came up with…

w = k = t (time)

No promises that this will be coherent, useful, or even interesting, but it is long. For myself, I find writing (or talking) about a topic helps clarify it, so this is mostly an exercise for the writer.
The inspiration for this came from a Greg Egan book (Diaspora) that mentions tesseracts (you run into them in science fiction sometimes; one of my childhood SF short story collections had a story featuring a tesseract house).

More to the point, Egan mentions that a tesseract is composed of 8 cubes, 24 squares, 32 lines, and 16 points.
That got me wondering what the count table looked like for all those regular square shapes. (In this case, “square” has more the “right-angle” meaning than the four-sided shape, although that shape is one of the shapes involved.)

The table below lists the square shape objects along the left and their component parts across the top. Each row indicates how many instances of the component shape are in a given object:

Square Shapes

(The numbers in square brackets are factors of the bold numbers.)

I looked at that table for a while trying to figure out a formula describing the mathematical progressions.

Points were easy. They just double each row. But lines? What formula gives you 1, 4, 12, 32? Squares are even worse: 1, 6, 24? I’m not an expert mathematician, so I never came up with a simple formula that explains the column sequences.

The color coding shows a last attempt. I noticed the diagonal of identities (light blue). Obviously it takes one square to make a square. When I factored the numbers as shown I found another diagonal of identities (light green). That also seemed to give each column a base number (2, 4, 6, 8). Made me think I was on to something!

But the progression in the next diagonal (light yellow) is 2, 3, 4, which is nice and regular, but how did we jump from 1, 1, 1, 1 to that? The next diagonal (light red) was worse: 4, 8. Regular sequences, sure, but not well-related.

I gave up, because it was clear the sequences were due to geometry and increasing dimensions, so maybe there wasn’t a simple formula describing the sequences. Turns out there is, but I didn’t find it until much later.

§

Next, I considered how to get from a point to a line, from a line to a square, from a square to a cube, and from a cube to a tesseract (and on to higher-dimensional objects!).

So start with a point. It has no dimensions and, thus, no coordinates. The idea of an axis or center point has no meaning.

Sweep a point through a new dimension (x) to make a line.

To create a 1D line, “sweep” (that is, move) the 0D point through a new dimension (call it x).

Sweep it a specific distance, call it L (for length).

This sweep also generates a second point in addition to the original starting point.

So, as the table shows, a line is: 1 line and 2 points.

This idea of sweeping a shape through a new dimension is the basis of creating all these “square” higher-dimensional shapes.

Sweeping a zero-dimensional point results in a one-dimensional line.

Sweep a line through a new dimension (y) to make a square.

To create a 2D square, sweep the 1D line — along with its 2 points — a distance of L through a new dimension (call it y).

The new position of the line gives us a new line.

Sweeping the original line’s 2 points through y generates 2 new lines (green) and two new points (bottom).

(This is the same as before; sweeping a point through x made a line. Here the sweep is through y.)

So, as the table shows, a square is: 1 square, 4 lines, and 4 points.

Important: Sweeping a shape includes all of its component parts. Sweeping each part produces new higher-dimension component shapes in the final shape.

All sweep shapes have components of each lower-dimensional shape. A line has points, and a square has both lines and points.

A cube, therefore, will have squares, lines, and points.

Sweep a square through a new dimension (z) to make a cube.

To create a 3D cube, sweep the 2D square — and component parts — a distance of L through a new dimension (call this one z).

The new position of the square gives us a new square.

Sweeping the original square’s 4 lines through z creates 4 new squares (on top, bottom, and both sides).

Sweeping the square’s 4 points creates 4 new lines (in blue) as well as four new points.

So, as the table shows, a cube is: 1 cube, 6 squares, 12 lines, and 8 points. (As expected, the cube contains all the lower-dimensional shapes.)

The process continues to higher-dimensional shapes, but the dimensions become imaginary since there are only three axes of freedom in three dimensions. That’s what three-dimensional means!

So the diagrams are going to get challenging; some imagination is required to see exactly what they try to depict. For example:

This diagram tries to show moving a cube through w. The diagonal (purple) represents w, but this doesn’t occur in 3D space. To make the diagram more clear, many component parts are not rendered. (See version below.) The 4D tesseract is only vaguely implied here.

To make a 4D tesseract, sweep the 3D cube — and parts — through a fourth new dimension (call it w, for weird).

The new position of the cube gives us a new cube.

Sweeping the cube’s 6 squares through w gives 6 new cubes. These cubes are weird! One of their three dimensions is in w!

Sweeping the original cube’s 12 lines creates 12 new squares (which are the side faces of the weird cubes), and sweeping the 8 points creates 8 new lines (plus 8 new points).

So, as the table shows, a tesseract is: 1 tesseract, 8 cubes, 24 squares, 32 lines, and 16 points.

This version shows all 8 lines (purple) created by sweeping the points of the blue cube through the w dimension. It also shades in the 12 “squares” created by sweeping the blue lines. Remember that the blue and red cubes are actually in the same 3D location! Only w separates them.

Understanding a tesseract requires some imagination. The diagrams above try to illustrate the sweep through the purple dimension w, but what really happens?

One observation is that, with regard to the sweep object, both the old and new share the same dimensional coordinates. The only difference is that they differ in the new coordinate — which is fixed to a single, different, value in each.

For example, a line has a set of x-coordinates. When swept through y to make a square, the new line has the same x-coordinates. But the original line has one y-coordinate (for all its points) and the new line has another.

Likewise the cube-creating square has a set of xy-coordinates. Both the original square and the ending square share all of them, but each has a different, fixed, z coordinate.

The traditional way to depict a tesseract. (But, again, the red and blue cubes actually share the same 3D space!)

Therefore, in a tesseract, the old and new cubes share xyz-coordinates. They differ only in their w coordinate.

That means, from a 3D perspective, the sweep through w doesn’t move the cube!

The traditional diagram of a tesseract shows a smaller cube inside a larger cube. The large cube is the original cube, the smaller inner cube is the new cube.

(Or vice versa; works either way.)

The name, tesseract, which comes from Greek and means “four rays,” comes from the fact that each point has four lines connecting to it. (Each point of a cube has three; each point of a square has two; each point of a line has one.)

These more traditional diagrams are similar to the larger diagrams above, except the red cube is shown inside the blue cube. Both diagrams “lie” about the red cube!

This rendering highlights (in purple) one of the “cubes” created by sweeping the outer cube face to the inner one.

The reality is that both the outer and inner cube are the same size and share the same xyz-coordinates!

In fact, just as all the squares of a cube are the same shape and size, all the cubes of a tesseract are the same shape and size.

This applies to the six new “weird” cubes created by sweeping the six faces (squares) of the original cube through w.

These six cubes connect a face of the outer cube to the matching face of the inner cube. The traditional diagram shows these as truncated pyramid shapes connecting outer and inner faces, but they are actually square cubes (with the same size as the original cube)!

They are created by sweeping a 2D square to make a 3D cube. The difference is that one of the new cube’s dimensions is in w!

That means, from a 3D perspective, those six cubes have no thickness in one dimension!

This version suggests expanding the inner cube to the same size as the outer one. A purple sweep cube is shown being flattened at bottom.

Start with that traditional diagram and expand the inner cube to make it the same size as the outer cube.

In the process, the six cubes formed by sweeping the original cube’s squares decrease until they are completely flat in one of the 3D dimensions.

The top and bottom cubes become flat in the up-down (z) dimension. The front and back cubes become flat in y, and the left and right become flat in x.

But all six are full-sized cubes with length in w accounting for the missing x, y, or z, dimension.

It makes it very interesting to speculate what might happen if a tesseract actually existed as a house-sized (hollow!) object.

If there were portals (4D doors!) between the 8 cubes, what would happen upon stepping from either the inner or outer cubes (which exist in the x, y, z dimensions we know) to one of the “flat” sweep cubes?

Would the portal have to convert the missing 3D dimension to the w dimension? (Perhaps that’s a natural translation of 4D doors?)

Or would it be flat in some weird way (like Abbott’s Flatland)?

From the 3D perspective, occupants of the sweep cubes would certainly look flat. It’s anyone’s guess what it would feel like to the occupants!

Even weirder, proceeding from the inner cube, straight through a sweep cube, to the outer cube (or vice versa) returns to the same 3D space. Remember that both the inner and outer cubes occupy the same 3D points! All points inside the tesseract occupy the same 3D space as the cube!

Another odd thing to ponder is what doors to outside space would be like for the eight tesseract cubes. It’s especially strange with regard to the six “flat” cubes!

Points on opposite sides diagonally bound a cube.

Another observation about these square shapes is that whatever their dimensional, two points of that dimension, on diagonally opposite corners, bound the shape:

• two x points bound a 1D line
• two xy points bound a 2D square
• two xyz points bound a 3D cube
• two xyzw points bound a 4D tesseract

The first point is found on the original shape, while the second point is found on that shape in its new position. Further, the second point crosses all available diagonals of that shape.

For example, in a cube, if the first point is on a corner of the original square, the second point is on the diagonal point of that square in its new position.

The bounding points for a tesseract would look similar to the diagram shown here for a cube, except that the lower right point would actually be on the inner cube. The black diagonal line crosses w as well as x, y, and z.

Mathematically, these points represent the minimum and maximum spacial extent of the shape (exactly what we mean by bounding). The interesting thing is that (with square shapes) it takes only two points, regardless of the number of dimensions!

§

It’s possible now to construct a formula for the number of component shapes in an object.

The process of sweeping means this count of components is affected by more than just the previous amount.

For example, the number of cubes in an object depends on the previous number, plus any new cubes resulting from moving an existing cube, plus any new cubes resulting from sweeping squares.

The count of squares, likewise, depends on the previous number, plus new ones from moving those, plus new squares from sweeping lines.

Effectively, as with points, each component type doubles its members in the new position of the main shape. Each component type also increases by the next lower component (in dimension) creating new instances of the higher one through sweeping (lines create squares, etc).

The result is this:

2Sn + Sn-1

Where S refers to the current count of a given component shape. The subscripts (n) index the shapes as shown in the Square Shapes table. The exception is with points, where n=0. In this case the formula is just 2S₀.

Which shows why the number of points just grows by a nice factor of two, while the others have more complex progressions!

For example, given a line (1 line, 2 points), creating a new square involves:

• 1 square
• 2(1 line) + (2 points) = 2 + 2 = 4 lines
• 2(2 points) = 4 points

As the table says. For a cube (given a square):

• 1 cube
• 2(1 square) + (4 lines) = 2 + 4 = 6 squares
• 2(4 lines) + (4 points) = 8 + 4 = 12 lines
• 2(4 points) = 8 points

As the table says. The tesseract is left as an exercise for the reader!

For extra credit, extend the shape-creating process into the fifth dimension (call it u for unusually weird). Apply all aspects of the this discussion to the new shape.

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A Tesseract is by definition, a 4-Dimensional cube. Basically, it is an extension of the 3D cube to one more perpendicular axis, and hence, we need 4 points to determine the position of its vertices.

Consider 4 points taken in the 1-Dimensional space. We bend these points taking it one dimension higher, creating a square.


With the square, a 2-Dimensional object, we create a higher dimensional object by bending 6 of these in space: A cube.

Now, if we proceed further, we can create a higher dimensional object by bending 3D cubes in space. When this is done on 3D cubes, we get a Hypercube, which you refer to as a Tesseract.


Clearly, as we climb into higher dimensions, the vertices of the objects require more points to be defined in. Sadly, humans cannot visualize this, because we were built to "see" only in 3 dimensions. You can visualize one as seen in this GIF (it is in double rotation), or the video linked below:

Hyperspace

N-Space:

No discussion on hyperspace can be take place without first discussing the nature of the normal universe or N-Space.  For centuries it was accepted that the universe was made up of three dimensions, length, width, and depth, later many agreed that a fourth dimension of time also existed, though the veracity of its existence as dimension has always been debated.  The emergence of superstring theory led theoretical physicists to consider the existence of up to seventeen dimensions.  While the exact number of dimensions in normal space has never been concretely determined of agreed upon, it is universally accepted that there are at least three at a minimum with a maximum finite number N.  Therefore normal space is referred to as N-space by the majority of peoples and the scientific community.

N-space is governed by the laws of quantum, inertial, and relativistic physics.  These three forms of physics have determined the design of nearly every space craft and system ever built but have also created an upper limit on a ship’ performance, in particular their top speed.  Relativity set an upper limit on how fast any object with mass could travel, the speed of light.  It also states that the closer an object with mass comes to the speed of light the more energy it requires, with an upper limit of infinite energy required to reach light speed.  Relativity and experimentation also showed that there was no way for any object to exceed the speed of light unless it always existed above light speed, like tachyons.


N Universes:

Prior to the big bang and the formation of the known universe(s) the universe existed as a perfect multidimensional singularity (the exact number of dimensions has never been agreed upon or determined but current theory places an upper limit of 100.)  The pre-expansion singularity universe is often referred to as the S-dot or S-Space.  The existence of the S-dot and the number of dimensions that made it up places the upper limit on the number of universes created after the Big Bang, and the number of dimensions that each is composed of.  It is therefore theoretically possible that another universe of N-dimensions exists, but whether or not it is governed by the same physical laws and the exact nature of its make up have yet to be determined.

What is known is that at least two universes were created with the Big Bang, our universe of N-space, and the minimum of N+1 dimensional universe of hyperspace or simply N+ Space.  While the two known universes of N and N+ Space are separate they are still shown to be effected by one another.  The extent to which the two universes effect one another is not entirely known however it is known that gravity from both universes effects the other. 

The effect of hyperspacial gravitation on N-space has never been fully determined, but is apparent as it contributes to the expansion of N-Space.  The existence of some form of mass in hyperspace helps account for the mass missing from N-space not accounted for by Dark Matter.  In hyperspace, the effect of N-space gravitation is more apparent as hyperspace voids form in the volumes of space in which large enough gravitational masses are present.  This does not appear to be a hard and fast rule however as will be discussed later.

The exact nature of hyperspace and the physics that govern it have never been realized, though numerous theories abound.  What is known is that the speed of light in hyperspace appears to be infinite and can be reached at relatively minimal power in comparison to the infinite energy required in N-space.  Hyperspace is also known to have drifts and currents that flow throughout it; the composition of the hyperspace ether is unknown, as is the source of the drift flow.  It is also apparent that the distance between two points in hyperspace is not the same as in N-space, though there is no accurate way in which to measure this since hyperspace is almost completely featureless.  The best way to demonstrate this is to use the classic 2-D to 3-D paper model.

Before that however it is necessary to show why hyperspace is the only practical manner of Faster Than Light travel available by disproving the fallacy of transwarp drive.  To illustrate this take a piece of paper and mark a point on either end of the sheet, the shortest distance between these two points is a straight line the length of the paper.  To illustrate space warping roll the paper back upon itself placing the two points much closer together then they were before.  The new shortest distance will still be a straight line, but by compressing space in this way the distance in between is much shorter then the unwarped two-dimensional distance.

This might seem to be a simple process, but the reality of warping space time is not so simple and while it might be easy to fold a piece of paper in half, the energy required to warp space time is far more intensive.  This amount of energy must of course also be maintained throughout the use of the warp drive, and be provided by the craft using the warp drive.  Experimental measurement of space-time warping around planets and stars has found that the amount of warping is minimal despite the presence of huge amounts of gravitational and nuclear forces.  It is therefore inconceivable that any spacecraft could generate enough energy to warp space-time to the degree necessary to make long large FTL space travel possible.

With those insurmountable energy requirements in mind it then comes down to the realization that in order to travel faster then light one must leave N-Space altogether.  In order to do that however one must cross the dimensional barrier that separates the two universes, the problem there becomes how to do so without causing permanent damage to the fabric of space-time.  Just as in space warping though no ship would be able to generate enough energy to break the space-time barrier that separates the two universes, so they make no attempt to. 

In this case take the paper model, and crush it up into a tight ball, this is how N-space appears to hyperspace.  Now anything outside of the paper is hyperspace with the two points representing two tears into and out of hyperspace.  Even though the paper is crunched up to the hyperspace observer the paper space observer still has to take the long straight line distance between the two points but the hyperspace traveler has numerous routes available.  In the case where the two points are touching the distance between them in hyperspace is zero, or infinite requiring the navigator to go Around the Universe and Back Again (AUBA).  The paper is not static however it is constantly  shifting to the hyperspace observer, changing the hyperspatial positions of the tears.

The nature of the existence of the tears means that no energy needs to be expended on the part of the traveler to open the tears.  The energy needed to open the tears was already expended long before during their initial formation during the universes expansion.  There is also no need to close the tear, nor is their any risk of the tear closing on its own, the law of entropy prevents a tear from closing without massive amounts of energy pouring into it.  In the paper example, energy was expended to draw the dot, and energy would need to be expended again to erase them, therefore, so long as no more energy is added or removed the tears will remain open.

Structures and Formation:

The creation of tear requires a massive amount of energy an amount of energy that cannot be generated by artificial means.  During the formation and expansion of the universe billions of stars were, and still are, being formed and destroyed.  As these massive celestial bodies raced through the universe they would come close to one another and as they raced past at high speeds and rotational velocities their gravitational, electromagnetic, strong and weak nuclear forces tore at each other.  In some cases the two stars would start to orbit one another, but in most the momentum of the spinning stars was too great to overcome and the stars raced past each other.  That expended energy, while unable to draw the stars into each other, was not wasted however and ripped at space-time itself, ripping open holes or tears in the fabric of space time.  These tears became the bridges between N-space and hyperspace that are essential to FTL travel.  During the formation and continued expansion of the universe uncountable tears were, and continued to be formed throughout the universe.

Tears are not the only spatial anomaly formed when stars pass by one another however.  The directions the stars travel in relation to one another as well as their spin cause another phenomenon to form instead, bubbles.  Bubbles are different from tears at a very fundamental level, instead of creating a bridge between the two universes they are a pocket of hyperspace.  These anomalies form in cases where the passing stars do not have enough energy to tear space-time but instead fold it over onto itself creating a pocket of hyperspace within N-space.  The very nature of hyperspace as an N+ dimensional universe means that while to an N-space observer they might appear separate from hyperspace they are in fact very much a part of the larger whole.  If one were able to enter a bubble without bursting it they would be immediately connected to the rest of hyperspace.  In cosmic terms bubbles are short lived, existing for a far shorter period of time then tears, from only a few micropulses to a period of millions of annura as opposed to tears which for all intents and purposes might exist until the end of the universe.

Bubbles are a far more common occurrence then tears, but for the longest time were not recognized as being a form of hyperspace.  The amount of energy required to open a tear is so great that it is believed that only the interaction of massive stellar bodies can ever form them, though there are experiments to try and form artificial tears.  For bubbles this not so, the energy required to generate even a small bubble low enough that it can be generated by artificial means.  Small bubbles can even be formed inside the strong gravitational field of a planetary body, the presence of the gravitational field however destabilizes these bubbles causing them to rapidly collapse.  Naturally occurring bubbles inside of planetary gravity field are often formed during electrical storms and for centuries were misunderstood, and accounted for some cases of ball lightening, ELFs, and Blue Sprites (phenomenon that occur above cloud during lightening storms).


The tears and bubbles did not stay static space however and as the universe expanded they drifted along with it.  Carried along by the gravitational fields and solar winds of their companion stars the tears and bubbles drifted throughout the universe.  As they drifted through the universe matter of sufficient relative velocity enter the open tears.  As N-space matter slipped into the hyperspatial ether, it interacted destructively, resulting in the release of tremendous amounts of energy that began to close them.  Though the tears drifted from their original positions in the universe, they did not tend to drift far from their companion stars and took up orbits around them.  This proximity to the local stars however caused a great many tears to close during the great universal expansion as matter in the local star system fell into them.  Most of those that survived did so by drifting into volumes of space where gravity was either extremely weak or non-existent.

Null zones exist in two forms:  The first is out beyond the strong gravitational pull of a solar system and its companion satellites, or in some cases, in deep space far between planets.  The others are the true gravitational null zones that were created as planets formed around their birthing stars.  These null zones exist where the gravitational pull of celestial bodies come together and cancel each other out.  These maintain stable orbits around their local stars and or planets.  The lack of gravity in these areas make them ideal for tears and has prolonged their existence as it keeps matter from drifting through them.

Wormholes are another special case of hyperspace, and exist when two tears in hyperspace are joined together in hyperspace with no measurable separation between them.  What this means in a practical sense, is that any ship entering a wormhole can travel through it, and in effect hyperspace, with no form of protection since it never actually enters hyperspace.  Since hyperspace, like N-Space, is in constant flux the tears can and do eventually separate, resulting in open of two possible outcomes:  First and more commonly, they revert back to normal tears forever drifting through hyperspace.  The second, rarer, option allows for tears that separate over a long period of time, in universal terms, to create a tunnel of N-Space through hyperspace.  A touching tear wormhole is extremely stable and will exist as long, if not longer then a normal tear would, assuming the two tears remain in contact.  Tunneling wormholes weaken over time due the constant interaction between the N-space matter in the tunnel sheath and the hyperspatial ether.  This results in the destabilization of the wormhole which results in not only the collapse of the wormhole but the possible closure of the two tears as well.


Mass:
The number of dimensions inherent in a universe are what determine that universes physical laws.  Everything that exists in the universe, , from the most basic of elemental particles to the largest and most complex star, is an N-dimensional object of mass, where N is a finite number.  There are however exceptions to this rule, N- dimensional “objects”.  These objects exist everywhere, have no mass and are produced by any object with mass that interacts with light, shadows.  Shadows are regarded as the absence of light, and like light have no mass, and it is because of this fact that they are able to exist in N-Space.

Mass is therefore the key to how any matter, no matter how many dimensions more or less then N interacts with N-dimensional mater.  It is this understanding of the mass effect that is critical to travel in hyperspace; an N+ dimensional universe.  When matter of different dimensions comes into each other’s field of influence their own personal gravity will repel each other.  This repulsive force inevitably reduces the energy level of the matter.  When that energy level falls below the level at which the matter can continue to repel one another, the higher dimensional matter has the potential to absorb the lower dimensional.  In effect, particles of mass from N-Space repel and or absorb matter from any universe of N-minus dimensions.  The same can be said of hyperspace, which will absorb or repel any matter that interacts with its own matter of fewer than N+ dimensions.


Protection and Propulsion:

The value of hyperspace to beings that live in N-space should be obvious at this point, rapid, if not instantaneous, travel between star systems light-years apart.  As discussed in the previous section however any N-dimensional matter that enters hyperspace is at first repelled and eventually absorbed by the hyperspatial ether, destroying it.  Therefore, a means of protecting a starship that enters hyperspace had to be devised that got around the key of mass.  Only massless particles, i.e. light and radiation, prove immune to destruction in hyperspace.  Therefore by sheathing a craft in massless particles a ship should prove able to traverse hyperspace.


There are multiple methods by which to protect a ship from hyperspace all of them rely on sheathing the ship in massless particles, and the most readily available massless particles are in the form of EM Radiation.  The earliest hyperspace explorers protected their ships by covering them with massive light emitting panels, but these light panels had to be built and integrated in such a way that they did not create interference patterns which would create “holes” in the light barrier.  These holes would allow the hyperspace ether to penetrate the light shield and the consequences were often disastrous as the ether would engulf and consume the ship.  Once nano-sheet became available in large enough quantities it became possible to use IR radiation as a shield by shunting waste heat into the skin of the ship so that it emitted massive amounts of IR radiation from all across the hull.  In this way interference zones were not a problem but this method was impractical for covert and combat ships as it created an immediate target for enemy sensors. 

The advent of EMT (Electro-Magnetic Torus) fields convinced many that an effective hyperspace shield had been developed.  The opposite proved true and any ship attempting to enter hyperspace using an EMT field was destroyed due to the very nature of the field creating periodic gaps around the ship. 

The true boon to hyperspace shielding came in the form of the Gravitational Deflector Field (GDF).  It had long been known that gravitational waves could be used to repel the hyperspace ether but no one had been able to use them to protect a ship because of the massive power requirements and the interference zones created by the plate type GDFs used aboard capital ships.  Experimentation revealed that GDF did not have to be high powered in order to protect a ship from hyperspace, but all the emitters had to be attuned to avoid the interference zones that spelt disaster to earlier light based shields.  This attenuation ended up requiring massive amounts of power and in some cases additional integrated shield generators.


Propulsion in hyperspace now becomes a concern as any drive system must not interfere with the shield and must be made to be effective in N+ dimensional space.  As the laws of inertial physics seem to apply within hyperspace, a standard N-Space reaction drive would seem ideal, so long as it does not interfere with the protecting field. 

As discussed, when N-space matter first comes into the sphere of influence of matter within hyperspace the two repel each other.  The repulsion process drains all of the N-space matter’s energy to the point where it can be captured and absorbed by the N+ dimensional matter.  The amount of energy the matter initially possessed when the absorption began process dictates how much energy it will discharge in the absorption process, from a benign emission to a massive release.  It is this repulsive force that keeps the majority of N-Space matter that happens upon a tear from ever even entering hyperspace.  This same process results in any matter ejected from a ship to be forced back towards their emission source.  This same repulsive force provides the thrust needed to maneuver about in hyperspace.


Therefore a dedicated hyperdrive need not be necessary, so long as the ship’s N-space drive does not interfere with the hyperspace shield.  Reactionless drives also appear to function in hyperspace.  To what degree is up for debate, as few races use reactionless drives.  They appear slower in hyperspace, though true measurement of speed in currently impossible.


Perception/Navigation:

The question now becomes what does hyperspace look like and how does one navigate through it.  The answer to the first question is simple, hyperspace is invisible to an N-Space observer but N-space is still visible through the tears.  So what an observer sees is the tears, an uncountable number of tears and nothing else.  Once in hyperspace every tear in the universe becomes visible.   The light of nearly every star in the universe fills hyperspace with almost blinding light. 

The reason for this is simple.  The physiology of N-Space beings prevents them from being able to anything from a higher dimension.  While hyperspace may be filled with uncounted marvels to gaze upon, they are invisible and thus hyperspace itself looks like nothing but a great absence of color.

Some objects within hyperspace are visible to an N-dimensional observer however and the nature of these objects convinces many scientists that hyperspace is only an N+1 dimensional universe.  These objects appear N-dimensional but have no detectable mass, they are the shadows of matter that exists within N+1 hyperspace.  Just as two-dimensional shadows in N-space are not necessarily accurate representations of an N-dimensional objects appearance, the N-dimensional mass shadows are not usable to represent the N+1 dimensional matter that produces them. 


Mass shadows pose a serious hazard to navigation as has been proven by ships traveling through hyperspace crashing into something of great mass in hyperspace that destroys the ship.  The crews realized that something was present because of the mass shadow, but with no idea of the light source they cannot determine the actual location of object so great care is always taken around mass shadows, big and small.


In principal navigation through hyperspace ought to be simple enough.  Point one’s ship towards the tear one wishes to exit and fly towards it.  The reality however is not so simple as perception problems soon arrive within the eyes and brains of N-space beings traveling through N+ hyperspace.  There is no accurate way to measure distances in hyperspace, attempts to do so never yield the same results and the perspective faults generated in the brain create curious visual anomalies for the traveler. 


The presence of mass shadows also effects the perception of the viewer.  The masses of these objects often enough bend even N-dimensional light to a degree that will distort the apparent position of an object.  A viewer can see the tear they wish to journey through straight ahead of their ship, but as they near, they might discover that the need to take a more roundabout route.  This can be for a number of reasons, but is most often due to the presence of a hyperspatial mass.  Other anomalies make it appear like the tear a ship is searching for is directly in front of the ship, when the reality is that is located behind another tear, gravitational lensing distorting its apparent position.

The constant state of flux induced on hyperspace by its erratic ether makes transiting through hyperspace even more difficult as it causes tears to drift.  During one trip a tear can be immediately adjacent to the target tear.  Drift could send it to an entirely different N+ relative position later. 


The solution to this is the navigation buoys that straddle the tears transmitting their location in N-Space back into hyperspace for the traveling ship to discover and home in on.  The buoys transmit coded information into hyperspace using specific radio frequencies as well as light signals to identify their positions in N-space and hyperspace.  Scanners also transmit data about surrounding tears to the receiving ship in order to better aide in navigation. 

Specially keyed and protected computers aboard hyperspace capable ships maintain massive data libraries on these buoys and decode their unique signatures in order to provide the crew with the navigational information for each buoy.  The frequencies on which a buoy operates are tightly controlled and monitored.  This is to prevent interference with or use by as yet undiscovered races and governments whose own buoys work upon similar principals. 
 

 

Time Distortion:

The nature of hyperspace as N+ space makes the measurement of anything difficult and the measurement of time is no exception.  The exact way in which time flux occurs in hyperspace has never been determined.  It appears however that passing through particularly strong ether streams can result in even greater time fluctuations.  Experimental evidence has shown that ships traveling through hyperspace experience a reverse of relativistic time dilation.  They will appear to be gone in hyperspace for only a matter of pulses when to the hyperspace observer they were in hyperspace for several hects.  The reverse is also true but this phenomenon is rarely seen in hyperspace and is usually only seen in N-space when encountering bubbles.

On several occasions ships have run into bubbles and disappeared, caught half in and half out of hyperspace, but did not burst the bubble.  The bubble will eventually bounce the intruding craft back into N-space.  While the crew may have only experienced a few centi-pulses, if they perceive any time at all, the ship may have disappear for annura to an N-space observer.  These cases of lost time and long term disappearance of individuals and ships are well documented and are seen not only in deep space but inside planetary gravitational fields.  Also unlike hyperspace time distortion effects, it is possible to determine the amount of time a ship will disappear inside a bubble as well as the amount of time the crew will experience.  These figures can be determined based on the ship’s mass, entry velocity upon impact, strength of the local gravitational fields, and size of the bubble.
 

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What effect does time dilation have on something trying to cross a black hole's event horizon? Would that material cross, or would time essentially "freeze"?

An object crossing a black hole’s event horizon, the point of no return, will simply pass through from its own perspective, unaffected by time dilation. However, its appearance to outside observers is strongly affected by the black hole’s presence. Light signals sent from the object at even time intervals (from the object’s perspective) will be received further and further apart in time as the object approaches the event horizon. The strong gravitational field near the event horizon curves space, increasing the distance light must travel to reach the observer. The curvature and distance to the observer — and hence the signal’s travel time — approach infinity at the event horizon, so an outside observer will never see an object actually fall into a black hole. The object instead will appear to freeze at the event horizon.

Gravitational Time Dilation

Speed of Light and Other Things Or How To Locate A GPS Receiver

All GPS satellites are broadcasting a pseudo random string of  0s and 1s which is known only to the receiver. By figuring the delay between broadcasted and the received signal, the distance from the satellite can be calculated by multiplying it by the speed of light. Now this is usually done for 4 satellites and the position of the GPS receiver can be figured out in 3 dimensions (latitude, longitude,altitude).

Sounds Simple. Wait Time Is Relative !

GPS systems are meant to be accurate within 5 to 8 meters. If you consider the speed of light (300,000,000 m/s) that means that translates to a time accuracy of something like 20 to 30 ns. A nanosecond is a billionth of a second and this matters when it comes to GPS.

GPS satellites contain super accurate clocks onboard called Atomic Clocks which use decay time of energy levels of individual atoms to be accurate within 1 ns. So everything seems good then.

Einstein’s Relativity. Time Is Not Same Anywhere.

Remember Interstellar, where the astronauts being close to massive black hole slowed time so much that a minute there translated to several years on Earth. Einstein’s relativity shows up when it comes to GPS and without correcting for it we would never be able to accuracy we have today.

Special Relativity

Special Relativity tells us that ‘the faster we go, time slows down.”
The satellites are in orbital velocities of  14,000 km/hr. Although this is nowhere near the speed of light this makes the clock on the satellite tick 7 microseconds slower than a clock on earth per day.

So what does this have to do with general relativity? One of the predictions of general relativity is that massive objects (like the Earth) warp space and time. The warpage of time means that clocks down here on the surface of the Earth (deep down in the gravitational well), tick slower than clocks carried on satellites high above the Earth.

General relativity tells us time moves more slowly deep down in the gravitational well. If you are going to navigate using clock signals from satellites (GPS) you have to account for this!

General Relativity

Put simply, General Relativity states that ‘closer you are to a massive object, time slows down’.
The GPS receiver is much closer to a massive object, Earth than the satellite. Now this makes the Earth Clock tick 45 microseconds slower per day than the space clock.

Now putting it all together, the total delay is 45-7=38 microseconds.

The Space Clock is always 38 microseconds ahead of the Earth clock per day.

In conclusion, if these effects are not taken into account, GPS would stop being accurate after 2 minutes of use. A receiver gets all these information from all the 4 satellites and works out the delays, relative times and velocities to give the final location.

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Time warping principle

LET'S DO THE TIME WARP Relativity causes clocks in motion to tick slower than stationary clocks (top); clocks that are nearer to a massive object such as Earth also run slower (bottom).

Using superprecise atomic clocks, scientists have witnessed time dilation — the bizarre speeding up or slowing down of time described by Einstein’s theories of relativity. The experiments are presented in the Sept.

Exploring the peculiar effects of Einstein’s relativity is no longer rocket science. Tabletop experiments at a lab in Colorado have illustrated the odd behavior of time, a strangeness typically probed with space travel and jet planes.

Relativity causes clocks in motion to tick slower than stationary clocks (top); clocks that are nearer to a massive object such as Earth also run slower (bottom).]

Time-Warping Occurs in Daily Life

Now advances in laser technology and the field of quantum information science have allowed researchers to demonstrate Einstein’s theories at much more ordinary scales.

The researchers used two optical atomic clocks sitting atop steel tables in neighboring labs at the National Institute of Standards and Technology in Boulder, Colorado. Each clock has an electrically charged aluminum atom, or ion, that vibrates between two energy levels more than a million billion times per second. A 75-meter-long optical cable connects the clocks, which allows the team to compare the instruments’ timekeeping.

In the first experiment, physicist James Chin-wen Chou and his colleagues at NIST used a hydraulic jack to raise one of the tables 33 centimeters, or about a foot. Sure enough, the lower clock ran slower than the elevated one — at the rate of a 90-billionth of a second in 79 years. In a second experiment the team applied an electric field to one clock, sending the aluminum ion moving back and forth. As predicted, the moving clock ran slower than the clock that was at rest.

“It’s pretty breathtaking precision,” says physicist Daniel Kleppner of MIT. Of course scientists are well aware of these relativistic effects, he notes. The clocks on GPS devices are also affected by relativity, and appropriate adjustments are made to keep them working properly.

The experiments have more implications for precision instrumentation than they do for relativity, notes Chou. But they are a nice reminder that relativity is always at hand. “People tend to just ignore relativistic effects, but relativistic effects are everywhere,” he says. “Every day, people are moving; they are doing things like climbing stairs. It’s interesting to think about — are frequent flyers getting younger [because they move so much] or aging faster [because they spend so much time in the air]?”

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Gravity is the curvature of spacetime.

This is the heart of general relativity. So how does it work? General relativity is summarized mathematically by 10 coupled, non-linear, partial differential equations known as the Einstein Field Equations, succinctly written as

Fortunately for us, this mathematics can be captured in a simple, two-line mantra to guide intuition:

Space tells matter how to move.

Matter tells space how to curve.

Espaciotiempo de la Relatividad General

Hasta aquí el espaciotiempo es el espaciotiempo, el contenedor de la física. No participa ni interactúa con otros sistemas físicos. Con la Relatividad General la cosa cambia. La Relatividad General nos dice que la geometría del espaciotiempo puede cambiar debido a la presencia de energía y flujos de energía.

Eso se consigue tratando el elemento esencial de la geometría del espaciotiempo como un elemento físico que tiene dinámica y que interactúa con otros campos como electrones, campos electromagnéticos, fluídos, ondas, etc. ¿Qué objeto condensa la información sobre la geometría de un espacio? Ese objeto es la métrica. La métrica la cosa matemática que nos dice como se miden distancias, intervalos de tiempo, ángulos, áreas, volúmenes y todas las magnitudes geométricas. Si representamos la métrica por , podemos verlo así:

metric Lo que nos dice la Relatividad General es que la métrica puede cambiar de una zona a otra del espaciotiempo por la presencia de masas y energías. Por lo tanto, el espaciotiempo se curvará, se podrá retorcer, se podrá estirar, se podrá comprimir, etc.

Si estamos en un espacio plano un cuerpo que no esté sometido a fuerzas seguirá una línea recta.



Pero si nuestro espacio es curvo, porque la métrica cambia de punto a punto debido a la presencia de energías y flujos de energías, un cuerpo que se mueva libremente por el espaciotiempo sin estar sometido a fuerzas no podrá seguir líneas rectas. Dichas líneas no existe, lo que seguirá será curvas denominadas geodésicas que son las líneas “más rectas” de una geometría en un espacio curvo.



Otro hecho interesante es que en un espacio curvo, alrededor de un entorno pequeño de cada punto podemos describir un espacio plano. Es lo que pasa cuando estamos en una pradera y la vemos plana.

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In geometric gravity — general relativity — you can imagine spacetime like a large, deformable sheet. A particle can move anywhere on that sheet, so long as it stays on the sheet.  In places where the sheet is flat (“flat space”) the particle moves in an absolutely straight line.

But what happens if a particle encounters a large depression on the sheet? The only rule is the particle has to stay in contact with the sheet. It continues to travel in the straightest line it can, but if its path dips down into the depression, the direction the particle is travelling is slightly altered, such that when it emerges on the far side, it is travelling in a new direction that is not parallel to its original course!  Space tells matter how to move, with its shape.

Far from sources of gravity (edges of the sheet) spacetime is flat, and objects travel on straight lines. Small masses warp spacetime into a gravitational well (left dimple), while larger masses make larger gravitational wells (right dimple). If a particle comes close to a gravitational well, the curvature of spacetime bends its pathway. If a particle gets trapped in a gravitational well, the curvature of spacetime forces it to travel on a closed pathway — an orbit.

  

 

 

How do you curve spacetime?  With matter. The large, deformable sheet of spacetime is dimpled wherever there is a large concentration of mass; the larger the mass, the larger the dimple. Matter tells space how to curve, with its mass.  The larger the dimple, the larger deflection a particle passing nearby will feel. This at last, is the long awaited connection to the way we think about Newtonian gravity — the source of gravity is always matter, as we expected.

So we have done away with the concept of a “gravitational force field” and replaced it with the idea of “motion on a curved spacetime.” An astute reader will ask a pertinent question: if general relativity is really the way gravity works, why didn’t we discover it first? Where did Newtonian gravity come from?

Both Newtonian gravity and general relativity make exactly the same predictions when gravity is weak and speeds are slow.  In fact, mathematically, general relativity looks just like Newtonian gravity at slow speeds and in weak gravity. These are precisely the conditions we encounter in the solar system, which is why Newtonian gravity was discovered first, instead of general relativity!

Space-time geometry

General relativity (see this old post for a very brief introduction) is a geometric theory of gravitation. From what we know, and even if many unanswered questions remain (the most fundamental being how to reconcile general relativity with quantum physics), it is the simplest theory consistent with experimental data.

At the heart of general relativity lies the notion of space-time. It is a mathematical model that combines space and time into an interwoven continuum. Technically speaking, Einstein’s theory describes space-time as a  a (pseudo-riemannian) manifold.

Einstein’s field equation links the density and flux of matter-energy (stress-energy tensor) and the curvature of space-time:

where is Einstein’s curvature tensor and is the cosmological constant (which was at first omitted by Einstein in what he later called the “greatest mistake of his life”).

In absence of matter-energy, space-time is described by a flat pseudo-euclidean manyfold. More precisely, it is describe by Minkowski’s space-time from special relativity).

In such space-time, the trajectory of a free (i.e. non-accelerating) particle is a straight line : the shortest path between two points.

In presence of a source of gravity, space-time is described by a curved pseudo-riemanian manifold, which solves Einstein’s field equation:

In curved spaces, geodesics are the generalization of the notion of straight line in flat spaces. Timelike geodesics in general relativity describe the motion of a inertial particles. In the above illustration, the rockets’ paths are “straight lines” in a curved space-time.

Geometry of Einstein’s field equation

There are many solutions to Einstein’s field equation, corresponding to different sets of conditions (assumptions on the distribution of matter-energy for example). Each solution describes a particular geometry of space-time. Their dynamics is the cornerstone of relativistic cosmology.

Einstein’s field equation can be written in terms of tensors, or developed as a system of 10 nonlinear partial differential equations in 4 independent variables. Such a system is very difficult solve. Finding exact solution usually requires specific physical conditions, sometimes with simplifying assumptions.

There are two ways to search for these solutions:

• fixing the form of the stress–energy tensor and studying the solutions (i.e. space-time geometries)
• fixing some geometrical properties of a given space-time and finding a matter source that could provide these properties

In the first case, one usually assumes physical conditions, either observed or simplified ones.

In the second case, there is a wide liberty. For example, one can assume that the universe is homogeneous, isotropic, and accelerating and try to realize what matter can support such a structure (dark energy).

Let’s point out a few Einstein’s field equation exact solutions:

• Minkowski solution –  which describes an empty space-time with no cosmological constant
• Schwarzschild solution –which describes the geometry of space-time around a spherical mass
• Kerr solution – which describes the geometry of space-time around a rotating object
• Reissner–Nordström solution – which describes the geometry around a charged spherical mass
• Kerr–Newman solution – which describes the geometry around a charged, rotating object
• Friedmann–Lemaître–Robertson–Walker solution (or FLRW) – which describes the universe an homogeneous, isotropic expanding or contracting fluid. In cosmology, these fluid solutions are often used ascosmological models. Indeed, this model is sometimes called the Standard Model of cosmology
• De Sitter solution – which describes as spatially flat universe and neglects ordinary matter. In this solution, the universe is dominated by the cosmological constant. It is thought to correspond to dark energy in our universe or the inflaton field in the early universe (seen the “Inflation” paragraph later on)
• Anti-De Sitter solution – which describes a universe with a negative (attractive) cosmological constant, corresponding to a negative energy density and positive pressure of the vacuum. Anti-De Sitter (AdS) is best known in for its role in string and quantum gravity theories, namely through the AdS/CFT correspondence and the holographic principle (we’ll get on this subject in a later post of this series on “hidden realities”)

In fact imagination (also called mathematics) is only limited by physics: one has to ask oneself whether such solution (universe) corresponds to actually possible physical conditions or not.

Solutions can indeed exhibit causally “suspect” features such as closed timelike curves or universes with points of separation (“trouser-worlds”, which are usually ruled out).

Gödel universe is one of them. It describes a universe with a privileged direction (roughly speaking, a universe with a rotating axis). Among many strange properties, Gödel’s universe exhibits (a lot of) closed timelike curves, which would allow some forms of time travel :

In this universe,  there is actually no physical way to define whether a given event happened “earlier” or “later” than another event. Einstein himself (who knew Gödel very well) wrote: “Such cosmological solutions [ … ] have been found by Mr. Gödel. It will be interesting to weigh whether these are not to be excluded on physical grounds.”

Thus, some physicists add “good-sense” conditions. For example, we should mention Igor Novikov‘s consistency principle. This principle aims to solve time-travel paradoxes. It assumes either that there is only one timeline, or that any alternative timelines (such as those postulated by the many-worlds interpretation of quantum mechanics) are not accessible (event with null probability).

One could also assume that solutions should be a Lorentzian manifolds, i.e. smooth manifolds.
But it turns out that solutions which are not everywhere smooth can also be very fruitful. Which leads to our next paragraph on singularities.

Singularities

A singularity is a location where the curvature of space-time becomes infinite (for all coordinate systems). In this place, space-time even stops being a manifold. But infinity is one of the things physicists dislike most: they usually don’t exist in the observable world. Most of the time, this is a sign for a missing piece in the theory. Indeed, singularities occur in extreme conditions, where quantum effects dominate. This is somehow general relativity crying for help and stopping being relevant. It’s a call for quantum gravity, a theory we barely have clues about.

This doesn’t stop physicist to tackle this problem. Either in making semi-classical calculations, building (hopefully consistent) theories from the ground up … or observing Mother Nature under extreme conditions and looking for new physics.

Let’s get back first to general relativity and see how it behaves. Singularities can be found in the Schwarzschild metric, the Reissner–Nordström metric, the Kerr metric, the Kerr–Newman metric, …
The first step towards a mathematical characterization under which circumstances general relativity breaks down was achieved in the Penrose-Hawking singularity theorems. In this set of theorems, Roger Penrose and Steven Hawking  proved that, under very general conditions, singularities in any general relativity-like theories are ineluctable.
Penrose theorem mostly deals with black holes, whereas Hawking’s deals with the universe as a whole (Big Bang) and works backwards in time (Big Crunch).
 Of course, keep in mind that these theorems arose from general relativity alone and these results probably break down when quantum physics is somehow added. Hawking actually revised his own position later on, stating “that there in fact no singularity at the beginning of the universe“.

Singular objects and singular events

As we have seen, general relativity cannot be used solely to show a singularity. Because of this, I think we’d rather speak of “singular objects” and “singular events” – uncanny stellar objects and events – than “singularities”. These singular objects would be:

• Black holes
• White holes
• Wormholes

And singular events would be:

• Big bang
• Big rip
• Big freeze
• Big crunch
• Big bounce

We will deal with singular events later on, at the “Relativistic cosmology” paragraph.

Let’s start with huge beasts: black holes. A black hole is usually defined as regions of space-time exhibiting extreme gravitational effects, caused by the collapse of a huge stellar object:

Stars are formed from the collapse of interstellar matter. The compression caused by this collapse raises the temperature until nuclear reactions ignite. The collapse comes then to a halt : the outward thermal pressure balances the gravitational forces and the star reaches a dynamic equilibrium. When all its energy sources reach exhaustion, the equilibrium is broken. The star – depending of its initial mass – will either expand or collapse (again), reaching new states of its evolution. These “stellar remnants” could then be:

• White dwarfs
• Neutron stars
• Red giants
• Red dwarfs
• Brown dwarfs
• Black dwarfs
• Supernovae
   
• … and, of course, black holes

Black holes

The Schwarzschild radius is the radius of a sphere such that, if all the mass of an object were to be compressed within it, the escape velocity from the surface of the sphere would equal the speed of light. Once a stellar remnant collapses below this radius, light cannot escape and the object is no longer directly visible, thereby forming a black hole. This radius defines a perimeter called the “event horizon” (more on this later):

The no-hair theorem states that a black hole has only three independent physical properties: mass, charge, and angular momentum. These properties are very special since they are visible from outside a black hole. A charged black hole would then repels or attracts other charges just like any other charged object.

The simplest static black holes (also called Schwarzschild black holes) have mass but neither electric charge nor angular momentum. According to the no-hair theorem, this means that there is no observable difference between the gravitational field of such a black hole and that of any other spherical object of the same mass. The popular notion of a black hole “sucking in everything” in its surroundings is therefore only correct near the Schwarzchild radius. Far away, the external gravitational field is identical to that of any other body of the same mass (called quiet region). In between lies the ergosphere,

where it would theoretically be possible to harness energy and mass from a (rotating) black hole.

Just like space-time around a static black hole is described by a Schwarzchild metric

• charged black holes are described by the Reissner–Nordström metric
• rotating black holes are described by the Kerr metric
• black holes with both charge and angular momentum are described by Kerr–Newman metric

Now, let’s add a little quantum physics to this classical point of view. Even if nobody really knows how gravity can be incorporated into quantum mechanics, in the quiet zone, gravitational effects can be weak enough for calculations to be reliably performed in the framework of quantum field theory in curved spacetime.

Hawking showed that quantum effects – and contrary to what classical general relativity predicts – can allow black holes to emit radiations. For this, one has to think first about the quantum vacuum.

The quantum vacuum is the quantum state with the lowest possible energy. Contrary to one could expect, it all  but a simple empty space. At the heart of quantum mechanics lies the Werner Heisenberg‘s uncertainty principle. It is the formalization of the fact it is not possible to measure simultaneously the value of pairs of quantum variables with certainty. The standard deviations of such conjugated variables  are connected through Heisenberg’s relations:

Applied to Energy and time, it gives :

This means that during a very short period of times, there will be enough energy to create a particle-antiparticle pair that would annihilate shortly after. The quantum vacuum can then be pictured as being filled with virtual particles popping into and out of existence :

Following Yakov Zeldovich and Alexei Starobinsky, Steven Hawking and Jacob Bekenstein imagined a particle-antiparticle (virtual) pair to appear close to the event horizon of a black hole. One of the virtual pair “falls into” the black hole while the other escapes:

In order to preserve total energy, this causes the black hole to lose mass. For  an outside observer, it would appear that the black hole has just emitted a particle : a real photon, produced via the annihilation of the remaining pair with another virtual (anti-)particle outside of the black hole. This process (which is in fact much more complex) iscalled Hawking radiation.

Through this process, a black hole would lose mass and eventually evaporate. The time for a black hole of mass to dissipate is:

This a slow process though, and black holes are huge. For a black hole of one solar mass, it would take more than current age of the universe…

This radiation is somehow paradoxical, because, as the no-hair state, a black hole is solely characterized by three external parameters (mass, electric charge and angular momentum). It means than form the outside, any information entering inside the black hole is kept stored inside. But as the black hole eventually evaporates, this information is ultimately lost. This information paradox ignited a 40 years long and heated debate (and a bet lost by Hawking in 2004) between renowned physicists Steven Hawking, John Preskill, Kip Thorne, Gerard ‘t Hooft and Leonard Susskind.

This battle between these friends and colleagues leaded to the holographic principle (and a later post will be dedicated to it), where the three dimensions of space could be reconstructed from a two-dimensional world without gravity – much like a hologram.
A few days ago, Hawking made a communication on that particular subject at the KTH Royal Institute of Technology.
His idea is that information never actually makes it inside the black hole :“I propose that the information is stored not in the interior of the black hole as one might expect, but on its boundary, the event horizon […]”.  His suggestion is that the information about particles passing through is translated into an hologram that would sit on the event horizon.

Using the holographic principal, one can describe the evaporation of the black hole in the two-dimensional world without gravity, for which the usual rules of quantum mechanics apply. This process is deterministic, with small imperfections in the radiation encoding the history of the black hole. Holography tells us that information is not lost in black holes, a priori solving the information paradox.

A paper will be published next month, fully describing the findings.

Although I’m not a great fan of string theories, one shall note that there are alternative descriptions of black holes which a priori solve the information paradox, like superstring fuzzballs – where singularity at the heart of a black hole is replaced by theorizing that the entire region within the black hole’s event horizon is actually a ball of strings.
As weird as they might be, black holes are not purely theoretical singular objects. By nature, black holes do not directly emit any signals other than the Hawking radiation, which is weak and difficult to measure. Nevertheless, indirect observation is possible because the interact of such massive objects with its environment (accretion of matter, gravitational lenses, …)

Although we do not fully understand black hole physics (absence or presence of singularities, Hawking radiation, evaporation, information paradox, …), there is a full list of candidates.
 

White holes

White holes possible existence was put forward by Igor Novikov as part of a solution to the Einstein field equations. A white whole is, roughly speaking, the opposite of a black hole. According to Sean Carroll : “A black hole is a place where you can go in but you can never escape; a white hole is a place where you can leave but you can never go back. Otherwise, [both share] exactly the same mathematics, exactly the same geometry.”

White hole existence outside equations is nevertheless highly speculative, though some think the big bang is somehow a while hole.

Some also have proposed that when a black hole forms, a big bang may occur at the core, which would create a new baby universe that would expands outside of the parent universe.

There is no known observation corroborating the existence of white holes and white hole theories are shaky. All this has to be taken with a huge grain of salt.

Wormholes

Lorentzian wormholes (Einstein-Rosen bridges) are other typical singular objects. They which describe “shortcuts” connecting two separate points in space-time (even many light-years apart), different universes, or even different points in time. In a 2-dimensional surface, a wormhole would appear as a hole in that surface, lead into a world-tube, then re-emerge at another location on the 2-dimensional surface with a similar hole. An actual wormhole would be analogous to this, but with the 3 spatial dimensions. The entry and exit points would be visualized as spheres in 3-dimensional space. The possibility of traversable wormholes in general relativity was first demonstrated by Kip Thorne.

Wormholes lead to many paradoxes and introduce non-linearities at the quantum level. David Deutsch nevertheless showed that time paradoxes can be solved within the many-world interpretation of quantum mechanics, where a particle returning from the future would not return to its universe of origination but to a another world. String theorist Joseph Polchinski discovered an “Everett phone” (a theoretical universe-to-universe communicator) in Steven Weinberg’s formulation of nonlinear quantum mechanics. Such a possibility (and a scientifically correct visualization of spherical 3-dimensional wormhole entrance) is depicted in Christopher Nolan‘s remarkable film Interstellar. But from what we know, there is no observation corroborating the existence of wormholes, which remains theoretical.

Horizons

Horizons are boundaries in space-time beyond which events cannot affect an outside observer (and conversely, the observer cannot affect these events).

We have already seen how Einstein’s special relativity defines a causal structure. If at a given event E (that will be called “present” for the observer) a flash of light is emitted, it will expend at the speed of light in the form of a growing sphere. Since nothing can go faster than the speed of light, this expanding sphere is a boundary beyond which nothing affects or can be affected by E.

If one tries to visualize this sphere of light in a 3-space where two horizontal axes are chosen to be spatial dimensions and the vertical axis is chosen to be time, the expending light-sphere will be represented by an expending circle, which, as time go, will span a light-cone centered on the event E :

Relatively to the event E, the light cone will classifies event into distinct areas:

• The green light cone defines the future: events affected by information emitted at E
• The red light cone defines the past: events than can affect the present
• All other events are in the absolute elsewhere: events that cannot effect or be affected by E

The above classification holds true in any frame of reference. An event judged to be in the light cone by one observer, will also be judged to be in the same light cone by all other observers, no matter their frame of reference.

Now let’s introduce gravity.

Light Deflection and Space-time Curvature

This is a common approach to try to gain some visualization of the curvature of space-time by a gravitational mass. It starts by depicting space as a two-dimensional elastic sheet. If a massive ball is placed on this sheet, it will produce an indentation or curvature. If a smaller ball is rolled by the larger one, its path will be deflected by the indentation of the larger ball. While not adequate to depict the curvature of 4-dimensional space-time, it at least is a start. The bending of light above is greatly exaggerated.
 

Einstein's calculations in his newly developed general relativity indicated that the light from a star which just grazed the sun should be deflected by 1.75 seconds of arc. It was measured by Eddington during the total eclipse of 1919 and has been reaffirmed during most of those which have occurred since.

This bending of light can produce a gravitational lensing effect if a distant galaxy or quasar is closely aligned with a massive galaxy closer to us. If one galaxy is directly behind another, the result can be a circle of light called an Einstein ring.

The Curvature of Space

Square, living in two-dimensional Flatland, was astounded to learn there was a three-dimensional universe that included his world within it. Humans, since Einstein, have questioned whether our three-dimensional world exists inside a multi-dimensional universe. Some theories indicate that we do, while others go beyond that and suggest that we are part of a multiverse (i.e. that there are many parallel universes in which we exist). It is a strange and complex picture!

Over 100 years ago, Einstein (1879 – 1955) began to question many things, including the laws of motion, the concept of time and the nature of gravity. His work on gravity challenged the notions of Newton (1643 – 1727) that gravity was a force of attraction that acted instantly at a distance. Instead, he speculated that the fabric of space could be distorted by the presence of bodies – in much the same way that the surface of a trampoline is distorted by a human who jumps onto it. More massive objects, such as stars, create deep ‘gravity wells’ which affect other bodies that come near.  A ball will tend to roll in a straight line on a flat surface (as in diagram (a)), but its path will be deflected, or bent, as it approaches the region near such a well (diagram (b)). It might even become trapped, and roll around in circles or ellipses around the sides of such wells (like orbiting planets). Einstein predicted that light itself would be bent by massive objects near its path. This prediction was verified many years later by observations of the positions of stars near the edge of the sun taken during a solar eclipse, compared to observations of those same stars at a time when the sun was not near their line of sight with the earth.

Space is not just “empty” – but has characteristics that enable it to be distorted, bent, stretched and rippled – like the surface of a quiet pond of water. Different locations in space would obey different laws of geometry, depending on the curvature of space in those locations.

Our universe is a strange and wondrous place. Three-dimensional human beings are trying to decipher the mysteries of our universe using all of the tools available to us. We are like Square, living in Flatland, trying to imagine a universe beyond what our eyes perceive. The tools for our imaginations are Logic, Physics and, of course, Mathematics.

What is time? If you're a practising physicist, it's a quantity in your equations, t. This is the variable that you use for one of the four dimensions of the manifold of space-time, the term coined by mathematician Hermann Minkowski after Albert Einstein's theories of relativity began to show that time and space are fungible. And yet we can move freely back and forth in space but not in time. Why?

In Time Travel, science writer James Gleick reviews the science of time by focusing (mostly) on the science fiction of time travel. He starts from, and often returns to, H. G. Wells's The Time Machine, which predates Einstein's 1905 special theory of relativity by a decade. It's a pleasurable romp over Wells's fourth dimension and polished Victorian machinery; 'golden age' science-fiction authors such as Isaac Asimov, who provided the templates for modern treatments of time travel; and the Doctor Who franchise (A. Jaffe Nature 502, 620–622; 2013). Gleick also explores more highbrow offerings from writers such as David Foster Wallace and Jorge Luis Borges (who envisaged time as a “Garden of Forking Paths”), and filmmaker Chris Marker, whose 1962 sci-fi short La Jetée inspired 1995 time-travel noir 12 Monkeys.

Gleick doesn't exactly wear his knowledge lightly, but he does cram a lot in, especially in discussions of the physics. Einstein's 1915 general theory of relativity seems to allow for “closed timelike curves”, paths that start at one place and time, and end at exactly the same place and time. Unfortunately, actually creating space-time with such a curve — that is, a time machine — may be impossible, an idea formulated in Stephen Hawking's “chronology protection conjecture”. In this, the Universe conspires to make time machines impossible to build: they require physically impossible states of matter, or their creation may also generate a black hole around the machine, making it impossible to access.

Mark Garlick/SPL

A 'wormhole' — a favourite time-travel device.

But even the normal perceived flow of time in one direction is mysterious. Most of the microscopic equations of physics have a fundamental symmetry: they can't tell whether time is moving forwards or backwards (mathematically, they look identical if we replace t with −t). But this is not how we experience time. We move inexorably from past to future; we remember the past and have no direct knowledge of the future. One exception to time-reversal symmetry is thermodynamics, whose second law says that entropy always increases with time. Astronomer Arthur Eddington opined that this alone is responsible for the 'arrow of time'. The problem is that the second law is not really about physics, but probability — and hence knowledge. We know less about the details of a high-entropy system than a low-entropy one, so it's harder to extract useful work.

The symmetry of time is also broken in quantum mechanics, which describes a physical system by its wavefunction, but gives us probabilities, not definite results. When we make a quantum measurement, we sometimes say that the wavefunction collapses, a process that has only one direction. But this is about knowledge, too, in contemporary ways of understanding quantum mechanics such as the many-worlds interpretation — the idea that every possible outcome exists out there in the multiverse. When we make a measurement, we gain information about the system.

Gleick spends some pages on the 'problem of now', the question of how the equations of physics seem to give us a Universe in which time isn't just one of four space-time dimensions. Instead, it is special: why do we always live at a specific moment, only remembering the past and waiting for the future? The issue nags at many physicists, including me. Sometimes, I'm convinced that 'now' is a non-problem. Once quantum mechanics and thermodynamics have given time a direction, 'now' isn't physics, but a combination of time's arrow with psychology and physiology. The past is what is encoded in our memories. To a rock, an electron or a galaxy, there is no now. But occasionally I wonder whether this is sufficient.

Physicist Richard Muller also seems exercised by this conundrum. His Now attempts to lay out a solution. He starts with a pop-science introduction to the required physics: the broad theories of relativity and quantum mechanics, and the specific roles of cosmology and particle physics in our Universe, such as those of the Higgs boson and its mass-giving field. His introductions to modern physics are probably too technical for most lay readers, despite relegating most of the harder maths to a series of appendices.

Unfortunately, after dispensing with physics, Muller delves into philosophy, a discussion that hardly rises above the university-bar level. For example, he takes for granted that free will is not compatible with determinism. This has been debunked in philosophy, for instance by Daniel Dennett in the 1991 Consciousness Explained (Little, Brown), or this year by Sean Carroll in The Big Picture (R. P. Crease Nature 533, 34; 2016). Instead, Muller opts for the manifestly non-scientific idea of a non-physical soul with causal powers over the quantum-mechanical wavefunction.

This is pretty far-out, but is just a side note. Muller's main thesis is that the expansion of the Universe “is continually creating not only new space but new time”. That is a good soundbite, but cosmologists debate whether the starting point — the idea of creating new space — is itself meaningful. Since writing the book, Muller has expanded on his ideas more mathematically, and applied them to this year's observations of gravitational waves (R. A. Muller and S. Maguire. Preprint at http://arxiv.org/abs/1606.07975; 2016). Kudos to him for proposing an idea that may be testable. Very few popular or professional physics books bother to make an argument, summarizing the state of the art instead. Unfortunately, I don't buy Muller's argument: whether or not 'now' is a non-problem, Muller's idea is a non-solution in my view.

Both Gleick and Muller want us to realize that time is central to our experience — that having a now is what constitutes having an experience at all. Even if travelling into the past is a fantasy, the physics of time encompasses almost everything that physicists study. Perhaps understanding its flow will give us a more complete picture of our changing Universe.

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Time is the fourth dimension .One cannot travel back in the 4th dimension .Hence the 'Future Humans'  use the 5th dimension in which they can travel from one point in time to another without taking the long path of travelling back in time.

Just like the 2D paper is bent to get from one point to the other without travelling along the longer path - This is nothing but travelling in the third dimension.

(In this image imagine 2D to be time)

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Black Holes, Wormholes, and the Origin of a Universe

So often, major progress in science comes when the orthodox paradigm clashes with a new set of ideas or some new piece of experimental evidence that won’t fit into the prevailing theories.—Paul Davies[i]

American scientist John Wheeler coined the term black hole in 1969. Black holes appear black because no light can escape them, otherwise would appear as any other star.[ii] Light waves emanating from one are unable to escape its event horizon since the gravity it produces is far too strong. Light forms an endless corpuscle around its immediate vicinity, forever teetering on the edge.

Singularities are infinite points in mathematical equations. The singularity of a black hole is a point of infinite density that draws in then obliterates anything in its vicinity. Matter, light, and gravity are no exception to the consequence of its overwhelming power. Spaghettification is the term physicists use to describe the stretching that would occur to an object as it approaches a black hole. One could imagine what a person would resemble as it gets sucked into one: at first, a long, thick strand of spaghetti. Within a singularity, all three-dimensional physics break down.

Throughout the 1920s and 1930s, Einstein did not accept the notion of a black hole. It was too bizarre to fit into his commonsense notions of how the Universe ought to behave. He submitted a detailed paper in defense of this position. A revision in theory maintained his humility as an imperfect being, which proves even the most prestigious of scientists can promote incorrect ideas.

The gravitational field at the singularity of a black hole is so strong it appears to stop the flow of time. The wave crests of light seem infinitely long here, disallowing time as a measurable circumstance. Is this perspective an actual effect on the arrow or flow of time as we measure it or nothing more than a holographic projection to those outside a black hole looking in?

For Earth to form a black hole, it would have to be compressed from its present size to one inch in diameter and for the Sun, from one million miles in diameter to four miles. The Sun is much denser than Earth.

Supermassive black holes lurk at the centers of galaxies with masses at millions, perhaps billions of suns. Galactic black holes grow larger as stars fall into them. It is possible there are more black holes than number of visible stars in the entire Universe. Some aspects of reality are a lot stranger than the strangest of fiction, and only a fraction of it is visible from Earth’s menial perspective.

Rotating black holes might form other strange entities, called wormholes. Wormholes are theoretical tunnels connecting different regions of space and time or one universe to another. Like black holes, time has no meaning inside a wormhole. If a traveler could traverse one and come out at a distant point within the Universe, no time would have elapsed. The trip would be instantaneous.

Stephen Hawking believes wormholes might lead out of the Universe altogether into what he calls a baby universe. Each of these new universes expands into its own space-time continuum with its own three-dimensional realm, separate and distinct from any other. Since these new universes are unattainable, their existence forever remains in the textbooks of philosophy and theoretical physics, never to be embraced by science.

Hawking is a bit critical toward philosophers and asserts some of them are “not in touch with the present frontier of physics.”[iii] Regardless of how they may have treated him in the past, this statement is unwarranted and unfair. Science has advanced not only because of experimental evidence and mathematic formulae but also because of the contemplations and ideas of philosophers over the centuries. Hawking is a bit of a philosopher himself, dabbling in the fringes of science with his ideas of other universes.

Philosophy lays the groundwork for science. Without it, science halts and progress stagnates. There can be no science without philosophy no matter the space or time between each concept.

Philosophers of physics may continue debating issues of relativity and quantum mechanics, but only because they are aware of how science will revise these ideas over and again. Experiment and observation are reliable until one fails to recheck his approach or until more information becomes available. In rare instances, a scientist will go so far as to “fudge the data.” Why would a scientist do such a thing? Perhaps there is financial motivation to receive more research grants or a personal drive to succeed and not look foolish due to conflicting data.

A possible interpretation for black holes include, but are not limited to, an Einstein-Rosen bridge as a funnel between two black holes in the same universe or a black hole and white hole connecting one universe to another. Quasars are candidates for white holes because they appear as black holes that expel matter and particles rather than one that pulls them in.

If other universes exist, there would be trillions branching out from ours, alone, for that is the number of black holes many cosmologists believe inhabit the Universe. Each of these universes, in turn, branch off into another trillion, ad infinitum. However large the number, there would never be enough branching universes to support the many-worlds interpretation for each possible quantum event. There are not enough black holes or quasars in a finite Universe to account for an infinite number of possible histories from which to diverge. (The next section provides a more detailed resolution to this apparent inconsistency.)

Was the big bang nothing more than a mega-quasar whose black-hole counterpart was part of a parallel universe? Some physicists refer to these initial branches as parent universes. A baby universe attached to this one would result from a black hole in this space-time continuum branching off into a white hole there. Perhaps particles are added to our Universe in this manner from a parallel one on a continuous basis. In this model, the big bang would be a new beginning, not the ultimate one.

Supermassive black holes and quasars at the centers of galaxies may be the only type of bridges that exist between universes. They could be the source of a new galaxy spawning in a parallel universe, each related to a galaxy in this one.

If one considers the general interpretation of a multiverse, one must consider the possibility of a primordial universe from which all others spawned. If so, the realm of empty hyperspace must be infinite. A true vacuum of hyperspace would be the originator of all universes and existence in its most basic form. It would contain an infinite supply of raw interdimensional energy fluxing in and out of multi-dimensional existence at a constant rate.

Physicists theorize elementary particles flux in and out of existence all the time, which is how the Universe began. This manifestation is implied as occurring in an already-existing, three-dimensional container. But what are the properties of the preexisting continuum that would harbor the ability for fluxes to occur in the first place? Many cosmologists ignore preconditions of the Universe, including some of those who adhere to multiverse interpretations. The inability to quantify a potential value should not dismiss the concept altogether, nor provide reason for ignoring it.

Perhaps because of some freak mutation, perfect nothingness, over time, became imperfect, three-dimensional “somethingness.” A portion of true vacuum became transformed, and a false vacuum appeared as something physical and chaotic. Multiple dimensions of reality, where once were none, began to coalesce and take shape.

Multiple universes may exist beyond any realm of observation yet are observable in the sense they too are three-dimensional. Such universes are parallel to our own. They harbor a distinct three-dimensional continuum unobservable from our local perspective. To bridge the observational gap, intelligent beings might seek a gateway, such as a wormhole, and learn to traverse it.

The nothingness, or hyperspace between each universe, must exist for there to be a separation or distinction between one another. Just because physicists are unable to observe or measure its properties does not mean this medium is nonexistent. Logic demands a preexisting state to all universes, before the big bang of any, and within an infinite realm of hypertime.

Energy transforms itself in all possible manners. Observable energy converts to mass and mass converts to observable energy since they are different forms of the same thing. No type of mass or energy disintegrates into thin air. If this is true on a multi-dimensional scale, then so is the concept of infinite preexistence.

[i] Davies, Paul; Gribbin, John. The Matter Myth. New York: Simon & Schuster, 1992, p. 23.

[ii] Jeffries, David. Science Frontiers: Black Holes. New York: Crabtree Publishing Co., 2006, p. 28.

[iii] Hawking, Stephen. Black Holes and Baby Universes. New York: Bantam Books, 1993, p. 42.

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Extra dimensions
Superstring theory implies that there are more than our four dimensions. Our universe may consist of a wall or membrane that exists in additional dimensions. The line on the surface of the cylinder (below right) and the flat plane represent our three-dimensional universe. All known particles and forces are trapped within that space-except for gravitation. Gravity (red lines) spreads out in all dimensions.

1.) Warped Extra Dimensions. This theory — pioneered by the aforementioned Lisa Randall along with Raman Sundrum — holds that gravity is just as strong as the other forces, but not in our three-spatial-dimension Universe. It lives in a different three-spatial-dimension Universe that’s offset by some tiny amount — like 10^(–31) meters — from our own Universe in the fourth spatial dimension. (Or, as the diagram above indicates, in the fifth dimension, once time is included.) This is interesting, because it would be stable, and it could provide a possible explanation as to why our Universe began expanding so rapidly at the beginning (warped spacetime can do that), so it’s got some compelling perks.

What it should also include are an extra set of particles; not supersymmetric particles, but Kaluza-Klein particles, which are a direct consequence of there being extra dimensions. For what it’s worth, there has been a hint from one experiment in space that there might be a Kaluza-Klein particle at an energy of about 600 GeV, or about 5 times the mass of the Higgs. Although our current colliders have been unable to probe those energies, the new LHC run should be able to create these in great enough abundance to detect them… ifthey exist.
Image credit: J. Chang et al. (2008), Nature, from the Advanced Thin Ionization Calorimeter (ATIC).

The existence of this new particle, however, is by no means a certainty, as the signal is just an excess of observed electrons over the expected background. Still, it’s worth keeping in mind as the LHC eventually ramps up to full energy; almost any new particle that’s below 1,000 GeV in mass should be within range of this machine.

And finally…

2.) Large Extra Dimensions. Instead of being warped, the extra dimensions could be “large”, where large is only large relative to the warped ones, which were 10^(–31) meters in scale. The “large” extra dimensions would be around millimeter-sized, which meant that new particles would start showing up right around the scale that the LHC is capable of probing. Again, there would be new Kaluza-Klein particles, and this could be a possible solution to the hierarchy problem.

But one extra consequence of this model would be that gravity would radically depart from Newton’s law at distances below a millimeter, something that’s been incredibly difficult to test. Modern experimentalists, however, are more than up to the challenge.

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Warped 5-D Space-time

An interesting application of warped 5^th dimension has been developed by Lisa Randall. In this model, the 5^th dimension is located in between two 3-D branes. It is found that the extra dimension is severely warped in the form of anti de Sitter space with positive curvature by the presence of positive energy Gravitybrane and negative energy Weakbrane even though the branes themselves are completely flat. The strength of gravity depends on the position of the 5^th dimension. As shown in Figure 10ua (in term of graviton's probability function), it can be very strong on the Gravitybrane but becomes feeble on the Weakbrane where all the forces and particles in the Standard Model are confined. Only the gravitons can move anywhere in the branes and in the bulk. This model

Why is gravity so weak? The traditional answer is because the fundamental scale of the gravitational interaction (i.e. the energy at which gravitational effects become comparable to the other forces) is up at the Planck scale of around 10¹⁹ GeV - far higher than the other forces. However, that only raises another question: what is the origin of this huge disparity between the fundamental scale of gravity and the scale of the other interactions?

A possible explanation currently gaining ground in theoretical circles is that the fundamental scale of gravity is not really up at the Planck scale, it just seems that way. According to this school of thought, what is actually happening is that gravity, uniquely among the forces, acts in extra dimensions. This means that much of the gravitational flux is invisible to us locked into our three dimensions of space and one of time.

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Where Are the Extra Dimensions?

As to the correctness of the prediction, that’s another matter. On the surface, it appears to be dead wrong. There are "obviously" only 3 space dimensions, not 9. But the situation is considerably more subtle (and delightful) than this. Perhaps the extra 6 dimensions are simply "small" enough to have escaped our notice. To see how this might be possible, imagine that, instead of 3-dimensional beings living in a 3-dimensional space, we are 2-dimensional beings living in a 2-dimensional space, as shown in the figure. All of our movements are confined to up-down and left-right; there is no such thing as forward-backward. One day, a magician arrives who makes our world 3-dimensional by slightly "thickening" it in the forward-backward direction. If the additional freedom of motion this affords us were sufficiently small compared to the size of our bodies, our new freedom would be imperceptible to our senses. String theorists have in mind something like 10-35 m, which is clearly in the "sufficiently small" category, even for the most precise experiments we could imagine!




The first figure (left) shows a 2-dimensional being living in a 2-dimensional space

Moreover, such extra spatial dimensions would be "curled up." In our previous example, we had only one extra dimension: movement forward or backward a short distance. If we represent this dimension of movement as a short line segment, we can curl it up by connecting the two ends of the segment, thus forming a circle. Now when we move forward, instead of encountering a boundary – the "end of space" – we simply re-emerge into the same space from the other end – a finite-sized space, but with no boundary! With two extra dimensions, we begin with a two-dimensional square instead of a one-dimensional line segment (see figure). If we connect its front and back edges, we get a cylinder; then connecting the remaining left and right edges gives us a torus (donut), as shown.




Curling up a 2-dimensional space into a torus.

Again, we have a finite-sized space with no boundary. With 6 extra dimensions, there are many ways to curl them up; the figure below attempts to illustrate the potential complexity.

Again, we have a finite-sized space with no boundary. With 6 extra dimensions, there are many ways to curl them up; the figure below attempts to illustrate the potential complexity.

How does one make sense of more than 3 dimensions of space (a concept often employed in theoretical physics concepts eg - In string theory)?

We can lift him out of anywhere, they try to confine him, and put him any where else. For the flat landers, It would appear that the person disappeared from his confinement and reappeared outside, some sort of a miracle.

2. We can make happen things that looks really mysterious to folks in the flat lands, but its just the advantage of our perspective which enables us in doing them.

If we lift a flat lander out of his world, flip him and put him back to his world, his heart would be in the right side instead of left. This would seem an impossible act for people in flat land.

 

Some Other feats possible.

We can reach this conclusion that for a lower dimensional being, A higher dimensional being may appear to be having God like Abilities. Is that just Co - Incidence or does that imply something indirectly. I leave this question for open debate.

Now Let me come back to this question.

Hinton’s Cube( The Tesseract)

Charles Howard Hinton was an English Mathematecian who devoted his life to develop ingenious methods by which the average person could see four dimensional objects. Hinton knew the fact that one cannot visualize four dimensional objects in its entirety. However he reasoned that it is possible to visualize the cross section of the unravelling of a four dimensional object.

As an analogy consider the case of flat landers.

For a flat lander , He cannot directly visualize a 3-D cube. However he can visualize a set of six squares arranged in the form of a cross, which forms the cube in 3 Dimensions. For the flat lander the hinges between different squares is rigid. It is impossible for him to arrange the square into the 3-D cube. However for a 3-D being we can easily bend the squares in position to form a cube.

 

For the flat lander it may appear that at first there were six squares and then it vanished to form a single square.

Similiarly for a 3-D being , we wont be able to visualize a hypercube in 4-D , but one can unravel a hypercube into its lower components,which in this case are normal 3-D cubes. In turn we can arrange these cubes into a 3-D cross- a Tesseract. It is impossible for us to wrap these cubes to form a hypercube. But for a 4-D being he can easily place these cubes in position to form the Hypercube. For our Eyes It would appear that the six cubes disappeared to form a single cube in our dimensions. The rest is not conceivable to our senses. Salvador Dali’s painting ‘Christus Hypercubus’ was inspired by Hinton’s cube.

Another method to visualize higher dimensional objects is to observe the shadow that it casts on lower dimensions. For Example a hollow cube appears to the flat lander as a square within a square. If we rotate the cube in this condition, for the flat lander it would appear that motions impossible to his realm is occurring.

Two points to watch when you are drawing this tilting of hypersurf

Second, if one follows the usual convention of drawing light lines at 45^o, then the angle of the observer's worldline to the vertical will be the same as the angle of the hypersurface of simultaneity to the horizontal.

Concepts in Spacetime

Having established the basics of spacetime in previous discussions, we can now turn our attention to some of concepts used describe spacetime. First, we possibly need to reflect on the fact that spacetime is a 4-dimensional union of the classical concept of 3-dimensional space plus absolute time, as inferred by the Galilean transforms.

So what new concepts emerge from the union of space and time into a single 4-dimensional continuum?

Possibly, the key issue to consider is the concept of the ‘spacetime interval’, which we shall initially, and possibly misleading, describe as the ‘distance’ between two events in 4-dimensional spacetime. We might also try to visualise this ‘distance’ in terms of the diagram below, which plots ‘time interval [t]’ on the vertical axis and ‘spatial distance [d]’ on the horizontal axis, where the different units of [t] and [d] are unified via the relationship [d=ct] with [c] being the speed of light.

So, with reference to the diagram above, event [A] at the origin of the axes is separated in spacetime from events [B] and [C]. Just by way of general observation, we may note that the time component of the ‘distance’ between [A-B] is greater than its spatial component. In contrast, the spatial component of the ‘distance’ between [A-C] seems to be greater than its time component. However, in contradiction to what the diagram might suggest, the axial components of time [t] and space [d] do not combine in some form of vector addition analogous to Pythagoras’ theorem. Therefore, the following bullets will try to clarify some of the basic rules that do apply:

• The spatial distances [d] is still constructed from its 3-dimensional [xyz] components via [d²=x²+y²+z²], i.e. this aspect does conform to Pythagoras’ theorem.
  
   
• However, the ‘spacetime interval [s]’ is a 4-dimensional measure that is defined by [s²=t²-d²], where the square of the components [t] and [d] are subtracted, not summed.

In essence, the diagram above is a simplistic example of a 2-dimensional spacetime diagram, which features the ‘light-cone’ formed when plotting [c] in terms of a series of time [t] versus distance [d] values. While it is not possible to fully illustrate 4-dimensional spacetime, we can extend the previous diagram by extending the representation of space along the x-axis to the xz-plane, while still retaining time along the vertical axis. In the 3-dimensional spacetime diagram below, the path of 3 chronological events, i.e. [-A, A, +A] are shown, where [-A] represents some event in the past, which is able to affect [A] because it originates within the past light cone. In a similar fashion, [A] is able to affect [+A] in the future, because [+A] is inside the light cone of [A]. Typically, the speed of light is normalised to [c=1], which then allows an offset in space to be equate to an offset in time, where [c] acts as a conversion factor between the units of distance and the units of time. On this basis, anything travelling at the speed of light [c] moves along the surface of the light cone at an angle 45^o to the origin.  

However, the purpose of the diagram is to try to further illustrate some of the spacetime concepts at work. By way of a reference, let us assume that the red-orange dots, in the diagram above, corresponds to a clock travelling a distance of 3 light-years in 5 years at a constant velocity [v=0.6]. Relativity tells us that the time on the moving clock [+A] must be slower than a stationary clock that remains at [A]. However, if we generalise this statement, it means that any path through spacetime must conceptually carry its own clock, which measures the `proper time [τ] ` in the frame of motion. The proper time [τ] on our moving clock can also be calculated using the following equation:

  [1]      c² τ ²   = c ² t ²  – (x ²  + y ²  + z ² )

However, [1] can be simplified even further in the context of the previous 2-D spacetime diagram by restricting the spatial dimensions to just [x] and normalising [c=1]: 

  [2]      τ = √(t ²  – x ² )

While [2] is easier to work with, especially when trying to draw diagrams restricted to two-dimensions, i.e. [x] and [t], it is still equivalent to the Lorentz transform shown below, although this is not always immediately obvious: 

  [3]       

Rather than going through the mathematical derivation to prove the equivalence of [2] and [3], we might simply plug in the figures from the diagram into both equations and compare the results. Let us start with [2]:

  [4]      τ = t' = √(t ²  – x ² ) = √(5 ²  – 3 ² ) = √(25 – 9) = √16 = 4 years

When we turn our attention to [3], we immediately realise that we need to know the velocity [v]. However, in this example,  the velocity [v=0.6c] and so we might realise that by plotting the values of [t] and [x] on the spacetime diagram allows a line to be drawn from the origin, which creates an angle with the axis. By definition, the slope of this line corresponds to the velocity [v=x/t], which is also reflected in the ratio [v/c]. As such, it points to the geometric solution of [2], implicit in [3]:

  [5]       

Therefore, [2] and [3] both determine the proper time [t], i.e. the time on our moving clock [+A], which would register 4 years as opposed to the 5 years on the stationary clock at [A].

Higher dimensional objects

This article shall give you your first glimpse at a 4-dimensional object.

Wait! Is that even possible? How does one look at a 4-D object in this world? Didn’t the previous article explain how one cannot see the higher-dimensions in a lower-dimensional world?

Correct, you won’t really see the 4th dimension; but I can still show what it would look like in this world. Just like you can see a 2-D photograph of a 3-D object you can see a 3-D visualization of a 4-D object.

First things first, let me explain how we arrived at the conclusion of how a 4-D object would look like. Obviously no one has ever seen a 4-D object, so this is pretty much theoretical (but interesting!).

 

0-D and 1-D objects

0-D object

Start off with a 0-dimensional object – a point (shown in red). It has (obviously) 1 point only. No edges and no surfaces (or planes).

Points : 1
Edges : 0
Planes : 0

1-D object

To create a 1-dimensional object out of this, we take two of the 0-dimensional objects and join them with an edge (shown in green). This 1-dimensional object now has 2 points, 1 edge and 0 planes.

Points : 2 (2 X 1 points from each 0-D object)
Edges : 1
Planes : 0



2-D object

Now to form a 2-dimensional object, we take two of the 1-dimensional objects we created earlier (shown in red), and place them side by side. We use 2 new edges (shown in green) to join the end points of the objects.

This 2-D object now encloses a new plane within it (shown in blue).

Points : 4 (2 X 2 points from each 1-D object)
Edges : 4 (2 X 1 edge from each 1-D object + 2 new edges)
Planes : 1



3-D object

Continuing the same trend, we now take 2 of the 2-D objects created earlier (shown in red) and place them one over the other (at a little distance from each other).

 

This time, we add 4 new edges (shown in green) to join the points of the 2-D objects. In addition to the 2 planes we initially had from the 2-D objects, the new edges now enclose 4 new planes (shown in blue). If the initial planes formed the floor and ceiling of the box, these 4 new planes form the wall.

Points : 8 (2 X 4 points from each 2-D object)
Edges : 12 (2 X 4 edge from each 2-D object + 4 new edges)
Planes : 6 (2 X 1 plane from each 2-D object + 4 new planes)



4-D object

This is how we arrive at the formation of a 4D cube (it’s not called a cube though).

                

We take two of the 3-D objects (shown in red & blue) and place them one inside the other. We use 8 new edges (shown in green) to connect the points of the inner object to the outer object. What do we get?

Finally, a 4-D object! Note that the 8 new edges we added now form an additional 12 new planes (click here to count the 12 new planes formed)

This 4-D object is very commonly known as a ‘hypercube’ or ‘tesseract’. It is simply a 4-dimensional cube.

Points : 16 (2 X 8 points from each 3-D object)
Edges : 32 (2 X 12 edge from each 3-D object + 8 new edges)
Planes : 24 (2 X 6 plane from each 3-D object + 12 new planes)

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The constancy and finiteness of the speed of light provides some very interesting questions regarding simultaneity. For example, consider the diagram below:

Figure 1: Flatland Photoshoot

The camera lies above Flatland and is taking a series of photographs (like a video camera). In order to get a spacetime representation of the scene, one could stack the individual photographs. If stacked on a tabletop with the pictures facing up, time is also in the 'up' direction.

Figure 2: Photograph Spacetime

The key idea here is that, since the square is farther from the camera than the triangle, it takes longer for light to get from the square to the camera than from the triangle to the camera. What this means is that the light entering the camera contains photons that left the triangle a time t ago, and photons that left the square a time t' ago. Since the square is farther than the triangle, t' is greater than t. In other words, when the camera takes a snapshot of the scene, the camera's image of the square is an older image than the camera's image of the triangle. Pictures do not show the square and triangle at the same time!

There are two ways to get slices of simultaneity from the stack of photogrpahs shown above. It has already been shown that each photograph contains an image of the triangle and an older image of the square.

We imagine our worldline in this spacetime diagram. Then, as David Park wrote, "our consciousness crawls along our worldline as a spark burns along a fuse" (in J.T. Fraser et al., eds., The Study of Time, pg. 113). As it crawls up our worldline we discover new slices of spacetime.

Postle included a continuous block of spacetime between the two different ways of slicing it. Quantum Mechanics calls into question whether such a concept is valid.

Imagine we take one of the piles of frames of the movie and shuffle it. The correlation between our consciousness and what it perceives remains the same. So -- would we notice any difference? I don't have any good way to approach a discussion of this question, but it is one that has fascinated me for years.

Louis de Broglie wrote a famous commentary on the worldview of the theory of relativity:

"In space-time, everything which for each of us constitutes the past, the present, and the future is given in block, and the entire collection of events, successive for us, which form the existence of a material particle is represented by a line, the world-line of the particle .... Each observer, as his time passes, discovers, so to speak, new slices of space-time which appear to him as successive aspects of the material world, though in reality this ensemble of events constituting space-time exist prior to his knowledge of them." -- in Albert Einstein: Philosopher-Scientist, pg. 114.

The View From Outside

Einstein's idea -- later demonstrated by experiment -- that space and time are relative provoked a revolution in science. No longer were space and time independent of the things in them. In fact, neither are space and time independent of each other -- they form a "continuum," an inseparable mesh.

This "space-time continuum" -- our universe -- has four dimensions (length, width, height and duration). Objects do not move through space and time; they extend, unmoving, through the space-time continuum.

Flatland

Since it is not easy to visualize four dimensions, scientists have often imagined a fictional two-dimensional world to compare to our own.

Imagine a two-dimensional world -- much like a flat sheet of paper. On this sheet are little two-dimensional people. These 2D people have only two-dimensional senses; they can see their world, but not above or below. Everything in their world has length and width, but no height.

Now imagine that time is our flatlanders' third dimension (what would normally be height). The 2D person experiences himself as moving through time, but someone who could see the third dimension -- time -- would see him as a fixed, unchanging shape, like a series of snapshots connected together.

          ,

Our space-time universe may be something like this. From a view outside time, everything that ever happened or will happen may be eternally frozen like a sculpture.

Mathematical truths


In elementary school we learn basic truths of geometry: parallel lines never cross, and so forth.

One thing that is not mentioned in geometry class is that these truths are not necessarily true in our universe. Two perfectly straight, parallel lines could cross in our universe. Why? Our space-time continuum has a geometry different from the one taught in elementary school.

Imagine a flat wooden board. If you have a ruler, you can start at one spot on the board, and draw straight lines with your ruler, to make a grid pattern. The grid would form perfect squares.

Now imagine that the board isn't flat -- it's been left out in the rain too many times, and is now warped. Now, if you try to make a grid on it with your ruler, you won't be able to make perfect squares, because the wood is warped up and down. Your "straight lines" are straight on the wood, but because the wood itself is curved up and down, your straight lines curve up and down too.

Our universe is like this -- warped and curved. In fact, what we call gravity is a warping of space-time. Objects normally travel in a straight line if left alone, but the planets travel in a curved path around the sun. Why? The sun's mass warps the space-time around it. The planets are travelling in a straight line -- but that straight line is curved in space-time itself.

Gravity warps not only lines in space, but time as well -- the stronger gravity is, the slower time goes! This effect has been measured on earth. A highly accurate clock at the top of a mountain (where gravity is weaker) will run faster than the same type of clock at the bottom of a mountain (where gravity is stronger).


Reality revisited

Not only are space and time relative to how fast you are going, but they form an inseparable, warped, four-dimensional mesh that doesn't play by the rules of elementary school geometry.

Time doesn't work according to common sense. There is no such moment "now" valid for the entire universe. Whether things happen "at the same time" is simply a point of view. The entire space-time continuum -- containing all moments of time -- is entirely present.

What does this mean for human concepts like cause and effect, when past and future are one inseparable whole? For free will? Why do we experience ourselves as moving through time if we're not really doing so? It seems that science's answers have only given birth to more questions.

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The tools of transformation optics were readily taken from those of Einstein’s theory of general relativity, where gravitational fields actually induce a warping and distortion of space and time.  With this connection, it was quite natural for researchers to ask whether various types of astrophysical phenomena, both real and hypothetical, could be simulated for light using exotic optical materials.

In 2007, the most spectacular of these possibilities was proposed* by a group of researchers from the U.S., the U.K. and Finland.  They suggested that it is possible to use transformation optics to design an optical wormhole — a tunnel for light between distant points in space!  A longtime staple of science fiction stories, such wormholes (also known as Einstein-Rosen bridges) would provide a hidden tunnel for light that allows it to travel from one region to another.  At first glance, as we will see, this would seem impossible, as a wormhole is an extra-dimensional region of ordinary space, and we can’t add extra dimensions to our three-dimensional space just by the use of weird materials.  Or can we?  It turns out that it is not only possible, but that the construction is far simpler than you might imagine.

But what is a wormhole?  The name comes from the actual holes that worms burrow through apples, as might appear below.

If you were an insect that lived entirely on the surface of the apple, a wormhole presents a definite advantage for travel: the path through the wormhole from A to B is much shorter than the path along the outside of the apple.  Similarly, if wormholes actually exist in our universe, they could present a shortcut between distant points in space, which is why they are especially attractive for use in science fiction.

Travel through a wormhole would likely be an exceedingly strange experience, however.  Let us restrict ourselves to a world that is wholly two-dimensional at first, such as presented in the classic 1884 book Flatland, and take a closer look at such two-dimensional wormholes.

An illustration of one is shown below.  To emphasize the two-dimensional nature of the world, I chose Pac-Man as a resident.

First of all, it can be seen that we can enter the wormhole from any direction in two-dimensional space.  That is, Pac-Man can approach it from North, South, East or West, and enter it.  The view ahead of him as he approaches the wormhole will be strange indeed; if he approaches the entrance from the direction between the two, he will actually see his own rear-end! The picture below shows how a ray of light emanating from his rear could be seen at his front.

The situation is even more potentially bizarre, as is best illustrated within the hole itself.  When Pac-Man is inside, he lies on the surface of a long cylinder.  Because light will travel a full circle around the circumference of the cylinder, he will actually see an infinite number of exact duplicates of himself, doing exactly the same thing that he is doing, on either side!  Even more surprising, he can play catch with himself, throwing a ball to one of these images, and catching it from the other side.

The easiest way to imagine constructing a wormhole for Pac-Man is to use two flat sheets of paper.  Take one sheet and cut two circular holes out of it; take the other sheet and tape two edges together to form a tube.  Now tape the two edges of the two to the two circular holes, leaving no gap, and you will crudely get Pac-Man’s wormhole illustrated above.

A similar construction can be done for a wormhole in three-dimensional space by analogy, although it is much more difficult to visualize the result.  For the 2-D case, we start with a plane and cut out two circles from it; for the 3-D case, we start with a volume and cut out two spheres from it.  For the 2-D case, we connect the two holes with a three-dimensional cylinder formed from a surface; for the 3-D case, we connect the two spherical voids with a four-dimensional cylinder formed from a volume!

If this is a hard concept to grasp, don’t worry too much about it right now!  More important for our discussion is the question of how to create a virtual 3-D wormhole for light.  Here we run into what appears to be an insurmountable problem, as can be seen by considering the 2-D wormhole model constructed above.  Using the construction above, we see that we have to create the model in three-dimensions; that is, a 2-D wormhole seems to require 3-D space to make it.  By analogy, it would seem that the creation of a 3-D wormhole requires working in 4-D space!  Since we live in a world of only three dimensions, it would seem at first that it is impossible to make a wormhole using transformation optics.

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Uzay/zamanda kestirme yollar fikri bir bilim kurgu senaryosu gibi gelebilir. Fakat solucan delikleri ile ilgili fikirler Einstein'ın hesaplamalarından türetilmiştir.Solucan deliği boru hattı/tüneli evrenin farklı noktaları arasında bizlere daha kısa yollar sunabilir. En kısa yol ve uzun yol kavramı evrenin geometrisi ile ilgilidir. Uzay/zamanda bir sapma bu yolları uzatabilirde kısaltabilirde.

Three-dimensional space

We must be familiar with the three-dimensional space, we all live in a three-dimensional space.Three dimensional space with length, width and height.

However, I want to use another way of thinking to express the three-dimensional space, only in this way can we advance to a higher dimension.

Well, now we have a newspaper. There is an ant on it. We will let the Ant King as "two-dimensional creature", I move in the two-dimensional paper. If you want him to climb to the other side from the side of the paper, then you need to walk through the entire paper. But we put this piece of paper up? Become a cylinder, a three-dimensional space in the object; then the ant you just need to walk through the joint position, to reach the destination. (right! Is the legend of the wormhole) in other words, the two-dimensional space curved, get the three-dimensional space, we can express this.

Again, in this diagram, the ants disappear from the A point, B point appears, you think, that is, the meaning of curl to generate a new dimension!

Well, start to burn the brain stage!

In the first three dimensions, we can simply understand the growth, width and height. So how do we understand the four-dimensional space?

Four dimensional space

D more than three dimensional, what is it? Is time!

Torus: The Shape of Creation

The torus, or donut shape, is the latest physicists' conception of the shape of the universe. The actual shape of this universe is impossible to represent or imagine but the torus is a close analogy. A torus can be formed by taking a square and gluing the opposite sides together to form a doughnut. Shown in the picture to the left. The equivalent to this in 3 dimensions is doing this same process but with a cube.

If you exist on the surface of torus, it seems flat locally just like the surface of the Earth. If one were to walk in any direction for long enough they would end up back where they began. This is similar to computer games in two dimensions where the spaceship goes off one side of the screen and returns on the opposite side.

This same mathematical form occurs everywhere in the natural world, from electromagnetic fields to galaxies, from atoms to apples.

In mathematics, a HYPERSPHERE is a sphere having more than three dimensions. Since the early twentieth century, physicists have used this idea of a higher-dimensional sphere to describe a universe in which time is the fourth dimension.

             

Today, cosmologists say that the universe of relativity and quantum physics can best be understood when seen as a torus, or donut shape. A universe containing black holes, white holes and "wormholes" conforms best to this model. And a torus has the same formula (2pi2r3) as the HYPERSPHERE.


As a model of the universe, the HYPERSPHERE shows how things emerge in time and are enfolded back into fabric of the universe.
The HYPERSPHERE also shows how all things in the universe are interconnected, even when they appear to be separate from one another. If you isolate point a from point b in this diagram with cut c, the two points can still be connected--without crossing the cut--by going through the center:

The vortex, which is a section of the torus, occurs throughout the natural world – from tornados, whirlpools and electromagnetic fields to the formation of galaxies. And the torus shape is not limited to vortices. An apple, a tree, even a human being all share this same "toroidal" topology. (tornado, apple, magnet, tree)     

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The Einstein-Rosen Bridge

11 January 2015  - Kelvin F.Long

With the recent box office success of the film ‘interstellar’ many people are excited about the prospects of wormholes as a means for interstellar transport. Although there is currently no evidence that such exotic objects exist in nature, it is possible that they could be artificially created, perhaps from versions of higher dimensional string theory and engineering of the fundamental space-time foam. Wormhole research is today an exciting subject with dozens of papers published in peer reviewed journals every year, but it is worthwhile to be reminded of its origins - and it starts from a surprising place.

In 1915 Albert Einstein published his General Theory of Relativity, his description of gravity that neatly defines how objects will attract one another and affect the space and time around them. Many years later the American physicist John Wheeler would coin the phrase “space tells matter how to move, and matter tells space how to curve”. Einstein described gravity as a manifestation of space-time curvature. General Relativity is a continuous field theory in contrast to the particle theory of matter which led to quantum mechanics.
Einstein was also involved in the development of quantum mechanics, the theory that describes sub-atomic particles. But he was not entirely happy with its inherent uncertainties and probabilistic character. So in 1935 he worked with Nathan Rosen to produce a field theory for electrons, using General Relativity. His paper was titled “The Particle Problem in the General Theory of Relativity” and was published in Phys.Rev.48, 73. Einstein and Rosen were investigating the possibility of an atomistic theory of matter and electricity which, excluding discontinuities (singularities) in the field making use of no other variables other than the description (metric) of general relativity and Maxwell's electromagnetic theory. One of the consequences was that the most elementary charged particle was found to be one of zero mass.

In the end, what they produced was something quite original. They started with the equations for a spherically symmetric mass distribution, already used for black holes, and known as the Schwarzschild solution. They performed a coordinate transformation to remove the region containing the curvature singularity, a discontinuity in space curvature implied by black holes and similar phenomena. The solution was a mathematical representation of physical space by a space of two asymptotically flat sheets (negative infinity to positive infinity) connected by a bridge or a Schwarzschild wormhole with a ‘throat’. This connects the two sheets and, by analogy, two separate parts of the real, three dimensional, universe.

Now this was not a traversable wormhole, for that we had to await the arrival of physicists John Wheeler in the 1950s and Kip Thorne in the 1980s. In 1987, with the encouragement of Carl Sagan for his novel “Contact” (later a feature film) Thorne and his colleague Michael Morris, were able to construct a mathematical description, a metric, to describe a spherically symmetric and static wormhole with a real, finite, circumference. This had a coordinate decreasing from negative infinity - out in minimally-curved space - to a minimum value where the throat was located and then increasing from the throat to positive infinity - in a different minimally-curved space. This solution has the distinctive feature of having no event horizon - unlike a black hole. The Thorne and Morris paper was titled “Wormholes in Space-time and their use for Interstellar Travel: A Tool for Teaching General Relativity” and was published in American Journal of Physics, Volume 56, issue 5, May 1988. This paper helped to establish wormhole research as new area of academic enquiry.

Since then many papers have been published, and indeed astronomical surveys have been conducted, to examine the furthest stars and galaxies in search of natural wormholes. None have yet been identified. But remember the origin of this field of research, the Einstein-Rosen Bridge was not a traversable wormhole and it wasn’t the author’s intention to produce one but they did produce the first mathematical description of a wormhole. They should be remembered for this. In science research often produces something quite unexpected with implications reaching far beyond the original intentions of the researchers.                                                                              

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He presents a conventional scientific viewpoint of physics in explaining the universe. It covers subjects from quantum physics to Einstein’s theory of relativity and introduces all the mathematics that, for the sake of readability, I avoided in this paper.

I will try to explain things in an approximate way to communicate basic concepts. I do this with apologies to mathematicians and physicists who find Sir Penrose’s book an easy read.


Beyond 4-Dimensions
Einstein thought in visual pictures. He was uncomfortable with ideas dealing with more than his visual four dimensions. I call this Einstein's Horizon. The well-known subjects of quantum mechanics and string theory go beyond Einstein’s visualization horizon. They both use different ideas about dimensions beyond four. Neither of these subjects can describe the mechanism of quantum entanglement.

The mathematics must go in a different direction beyond Einstein’s Horizon to explain quantum entanglement. The dimensions must include complex numbers. I will start there.

Complex Numbers


I will avoid mathematical equations. However, it is necessary to review the concepts of imaginary numbers because they are fundamental to the whole idea of going beyond Einstein’s Horizon.

I didn’t run across complex numbers, numbers made up of real and imaginary parts, until I was an undergraduate in college. Now, they are taught in junior high school. Although they are still a mystery to many–those who “hate math”–they are useful as a mathematical convenience in describing and analyzing things.

A complex number has a real and imaginary part. It is written in the form of: a + ib where “a” and “b” are real numbers and “i” is the square root of (-1).

“a” and ”b” can be added, subtracted, multiplied, and divided like normal numbers. A complex number can be viewed as a point in two dimensional space, generally called the complex plane.

I won’t try to give you a course in complex numbers. But, they are part of most people’s belief systems of being acceptable scientifically. Either you have had enough mathematics to be comfortable with equations with complex numbers, or you have heard of them as being part of the acceptable ideas of mathematics

A Theory of Higher Dimensions


Most people, including researchers in other fields such as medicine, are happy with a 4-dimensions. To go beyond Einstein’s Horizon one must understand the ideas of higher dimensions.
I hope to provide a roadmap to an understanding of higher dimensions which will allow you as well as technically trained people to understand how quantum entanglement works. I wouldn’t claim that I am an expert at many subjects to be visited on that road. First, however, we need to conceptually transport our thinking from a 4-dimensional world to one of higher dimensions.

Need For More Dimensions

In the 4-dimensional physics world, there is no explanation for transfer of information in quantum entanglement. Brilliant mathematicians, Elizabeth Rauscher, and R. L Amoroso, came up with a simple solution: add more dimensions to the physics. In their landmark paper (which is included in the “Rauscher/Amoroso page of this site), they built on the work of Einstein’s math teacher, Herman Minkowski to go beyond Einstein’s Horizon. I will discuss how Minkowsky’s work provides a bridge of belief between the 4-dimensional belief world and one in which “outlaw experiences” such as quantum entanglement are “lawful.”

Minkowski Space

Herman Minkowski was born in Lithuania in 1864 and became Einstein’s mathematics teacher at Zurich Polytechnic. After Einstein’s graduation, he was familiar with Einstein’s work from their contact in the post graduate work Einstein was doing at Zurich Polytechnic. After Einstein published his paper on Special Relativity, few people were aware of it. There were some problems with reconciling Einstein’s Special Relativity with all the independent, similar work going on in mathematics around 1905. Minkowsky was one of the first people to grasp the importance of Einstein’s theory in 1907, two years after it was published. He proceeded to expand the mathematics around the theory and sell it to other mathematicians. In 1908, Minkowsky advocated considering time as a fourth dimension.

Some people say that he nearly hijacked development of the ideas of relativity because he thought the ideas were too important to be left to physicists such as Einstein. He is quoted as saying about relativity to another physicist, “It came as a tremendous surprise, for in his student days Einstein had been a lazy dog. He never bothered about mathematics at all.”

In Minkowsky’s 4- dimensional space definition, there is something that creates difficulty for people like Einstein that like to visualize geometries. The fourth dimension, time, is an (mathematically) imaginary dimension that is measured in terms of the square root of (-1).

Einstein initially rejected Minkowsky’s mathematic approach as too complicated. I conjecture that he rejected it because it was beyond his visualization horizon.
 

The basic idea of dimensions is shown in the above diagram. Mathematically, a point in space can have no dimensions. Drag the point through space in one direction, call it the x direction, and it becomes a straight line of one-dimension. Then, drag the line in a direction perpendicular to the x direction, call it the y direction, and you produce a two-dimensional rectangle. Drag the rectangle in a direction perpendicular to the two-dimensional plane, call it the z direction, and you have a three-dimensional box. This is all easy to visualize within Einstein’s Horizon.
 

Next, add time as a dimension. If a bullet were shot through the box, we could represent its path as a line with time ticks along the path. Then, we would have visualized four-dimensions. That is also easy. However, four-dimensions are really created by dragging the three-dimensional box through a time dimension. That is harder to visualize. But, one can be comfortable with the time-ticked line in a three dimensional space if you have worked with geometry enough to understand that it is a three-dimensional representation of slices through a four-dimensional space. All this sort of visualization is on the edge of what humans can do, the edge of Einstein’s visualization horizon.

Mathematicians are not bound by what can be visualized. They work with equations and are perfectly comfortable defining all sorts of abstract spaces of any number of dimensions. String theory, initially used twenty-one dimensions, although no one had any idea how to visualize them.

When Minkowsky called time another dimension, in extending Einstein’s relativity, he threw in a twist. To make his mathematics agree with other mathematician's work in the area of relativity he had to make time an imaginary dimension: in his equations t was multiplied by “i” which as explained in a preceding section, is the imaginary square root of (-1).

This is beyond my horizon. I can’t visualize “imaginary time.” What is it?

I speculate that when Einstein saw Minkowsky’s equations which dealt with imaginary time, it was beyond Einstein’s Horizon. That’s why he rejected Minkowsky’s approach as too difficult to understand.

Eventually, he adopted Minkowsky’s 4-dimensions of space-time. That shift was important in allowing him to eventually come up with General Relativity. However, his horizon was limited to these four dimensions.

Since then, people have talked about 4-dimensional space-time and called it Minkowsky space. There are diagrams presenting visualizations of Minkowsky space. However, they are somewhat difficult to understand and unnecessary to the reasoning I am following.

Quantum mechanics moved beyond 4-dimensional spacetime and Einstein refused to go there and accept the theories. String theory, which came after Einstein’s time, moved beyond 4-dimensional spacetime. Neither of those can explain quantum entanglement. Another extension beyond 4-dimensional spacetime is needed. To convince scientists whose thinking is also limited to Einstein’s Horizon, the extension needs to done in a way to be compatible with Quantum Mechanics, Classical Physics, Electromagnetism, and Relativity.


Those Who Have Gone Beyond Einstein’s Horizon


As we have mentioned before, string theory is beyond Einstein’s Horizon. The media tells us that string theory is one of the ultimate quests in physics. Edward Witten, a theoretical physicist at the Institute for Advanced Study at Princeton, coalesced competing string theories into something he called M-theory. Some of his peers rank him as the greatest living mathematician and sometimes greater than Einstein or even Newton. He was on Time Magazine’s list of 100 most influential people of 2004. He, like a great number of other mathematicians seem to have gone in a direction that can’t explain “outlaw” phenomena like quantum entanglement. Some critics who point to the lack of experimental evidence to support string or M-theory, quip that the theories are no more than a very complicated version of sudoko.

Elizabeth Rauscher is a mathematician and physicist that didn’t go down the rabbit hole after string theory. Some of her peers call her a female Einstein. However, unlike Einstein whose major accomplishments were only early in his career, she has had a long career and her accomplishments continue to grow. She had published more than 260 papers over more than forty years, many of which were on subjects off the mainline of physics. Her 2008 paper, with Richard Amoroso, included in this site goes beyond Einstein’s Horizon.

Why Pick An 8-Dimensional Minkowski Space?


Most people who do not read a lot about physics may not realize how many different models and theories there are about reality: Loop Quantum Gravity, lattice approaches, Euclidean Quantum Gravity and Twistor Theory, to mention a few described in the peer-reviewed literature of physics. On the internet, you can also find hundreds or maybe thousands of self-proclaimed quantum physicists and other assorted people who have read physics and made up their own theory of how things work.

The 8-dimensional complex Minkowski space has been shown to be consistent with our present understanding of the equations of Newton, Einstein, electromagnetic theory and the foundations of quantum mechanics, It is advocated by credential academics in peer-reviewed journals. It has been under consideration for explaining “outlaw” phenomena for over two decades.

For instance, in 1983, Elizabeth Rauscher extended Maxwell’s equations into complex 8-dimensional Minkowsky space. In 2002, Elizabeth Rauscher and Richard Amoroso further described Maxwell’s equations in terms of complex 8-dimensional Minkowsky space.

This 8-dimensional complex Minkowski space seems to encompass the major theories of “lawful” physics:

 

• Newtonian mechanics that we all learned in high school physics,
• Maxwell’s equations that describe electromagnetics,
• Einstein’s Relativity, and
• Standard Quantum Mechanics.
•  

If quantum entanglement can be explained in terms of complex 8-dimensional Minkowski space, they can be considered “lawful.” I will first describe the 8-dimensional Minkowsky Space in metaphorical terms.

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Alice Harpole - Feb 23, 2017

A wormhole is an object that links two different regions of spacetime - Einstein’s theory of special relativity tells us that space and time are not separate, but that time is simply another dimension of four-dimensional spacetime. This means that under extreme conditions (e.g. if something is moving extremely quickly or if it’s in a very strong gravitational field), these space and time dimensions can get mixed up. Compared to stationary onlookers, time can slow down and objects can get smaller.

The standard picture of wormholes and portals that we see in sci-fi and fantasy are of gateways that you can step through and instantly emerge at a different point in space(time). In ‘reality’ (inverted commas as we don’t really know if wormholes could actually exist), wormholes are extended objects - a wormhole big enough to travel though will likely be many kilometres across, distorting the spacetime around it for many hundreds / thousands of kilometres. The inside of the wormhole will also be pretty extended - you would not be able to simply walk through and instantly emerge the other side.

This is because we think that wormholes take the form of a sort of ‘double-ended black hole’. Einstein’s theory of General Relativity tells us that all objects in the Universe distort the spacetime around them: how large this distortion is depends on how ‘compact’ or dense the object is. The image below demonstrates how different objects in space bend spacetime. A star like our Sun will bend spacetime a bit, however this effect is quite small – the light from a star passing just past the edge of the Sun on its way to us will be deflected by about 1/2000 of a degree.

Neutron stars form when larger stars collapse at the ends of their lives. They are about 1 or 2 times as massive as the Sun, but have a radius of only around 10km. This means that they are incredibly dense – if you took all the people on the Earth and squashed us all down so that we were the same density as a neutron star, we’d all fit within a ping pong ball. Their gravity will therefore distort the spacetime around them much more than the Sun does.

Illustration 2: An illustration showing how spacetime is distorted by different astronomical objects. Image from sciencenews.org

Finally, a black hole is a region where spacetime has become so distorted by gravity that the very fabric of the Universe collapses down to a single point or ‘singularity’. The gravitational forces about this point are so strong that nothing - not even light - can escape. In the image above, you can see that we can picture the spacetime bending into a bottomless pit.

You can imagine a wormhole to be like two black holes glued together just before the point where spacetime collapses to a singularity: it becomes extremely distorted, but instead of pinching off into a single point, it instead flares out again, emerging at a different region of spacetime.

Illustration 3: An illustration of the spacetime around a wormhole. For Wikipedia Creative Commons.

Wormholes were first predicted as a mathematical solution of Einstein’s theory of general relativity just a year after it was first published. However, just because the maths says they could occur, doesn’t mean that they actually exist in nature. Most of the mathematical models for wormholes are incredibly unstable and completely impractical for the kind of travel we’d hope to use them for. The maths also has very little to say about how they could actually form. So, while there is nothing we know of currently that says they couldn’t exist, there’s nothing that says they definitely do exist either.

It is possible (and indeed very probable) that there exists some physics beyond our current understanding. With a different model of how gravity works, or with some form of exotic matter, it is possible that wormholes could form and remain stable. But that wouldn’t necessarily mean that we’d be able to travel through them…

As mentioned above, wormholes link different regions of spacetime. These regions may be at different points in the same universe or, if multiple universes indeed exist, could be in different universes. If there was a wormhole near enough to the Earth that we could travel there in a reasonable length of time, and the wormhole exit point happened to be near enough to another planet, then we could use it to travel to another planet.

In order for us to be able to travel such wormholes, they would need to be:

1. Stable enough for us to pass through without it collapsing around us (some solutions are incredibly unstable to small perturbations, and we need it to be stable enough for a great big spaceship to pass through)
2. The environment in and around the wormhole must be safe enough that we can pass through it unscathed (i.e. it’s no good if we’re ripped apart by tidal forces!)
3. Short enough that would could get through in a reasonable length of time (i.e. the journey time must not take many lifetimes)

These conditions impose quite a stringent set of constraints on the possible wormhole models. In order for such wormholes to be possible, we require gravity to behave quite differently from how it is predicted to do so by general relativity and / or for the existence of something physicists like to call ‘exotic matter’ (a type of matter with weird properties that is not currently described by the Standard Model of particle physics).

Neither of these things are completely beyond the realms of possibility - many physicists believe that a large part of the Universe is made up of dark energy which has the weird properties needed to hold open a wormhole. In the future, we could develop technology that would allow us to harness this to build our own wormholes to other planets or even other universes. There are also currently a number of theories of ‘alternative gravity’ that have different predictions from general relativity. With further study of these theories and observations of objects with strong gravitational fields (including gravitational wave detections using telescopes such as LIGO), we may find that one of these models is more consistent with our observations of the Universe than general relativity.

Similarly however, it could be possible that an improved understanding of physics rules out wormholes entirely. We just don’t know.

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Note too that in the diagram the film shows a ball moving from one corner of the screen to the other. However, in the three dimensional stack, the ball now follows a three dimensional path through space-time. In four dimensional space-time, objects which we see moving in time through three dimensional space are following a four-dimensional path through space-time. On space-time diagrams, paths you draw represent objects moving through space as time passes, but we'll see more about that later in the chapter.

Further, consider an event such as "the ball reaches the far corner of the screen." That is a single event--it occurs at one moment in time and at one particular place in space. On our diagram, it is a single point (it is a spot represented by the ball which is on the upper most frame in the stack). Any single event which occurs is represented by a single point on a space-time diagram.

And so, a space-time diagram gives us a means of representing events which occur at different locations and at different times. Every event is portrayed as a point somewhere on the space-time diagram.

Now, because of relativity, different observers which are moving relative to one another will have different coordinates for any given event. However, with space-time diagrams, we can picture these different coordinate systems on the same diagram, and this allows us to understand how they are related to one another.

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WORMHOLE - video installation in public space

2013 © Dr. Thomas R. Huber

The theme of the 2008 Busan Biennale, "Expenditure", is based on Georges Bataille's notions of consumption and excess as principle agents of the creative process, hence irony and paradox become key elements in this process. This is not a calculated approach to reason but an immeasurable area, one that is created via the internal experience without any guarantee of a result or conclusion. “Voyage without Boundaries " is an exhibition that explores Bataille's notion of surplus and energy as an agent of the creative process, which is not easily understood or defined.

Sea Art Festival as a part of 2008 Busan Biennale runs under the title "Voyage without boundaries"
“Voyage without Boundaries " symbolizes a journey towards some unknown space. Modern concepts of travel are encapsulated in a linear Euclidean space: one that has a concrete beginning and end. Great ocean voyages of the past to today's travel in our global village follow routes based in some concept of space starting from some beginning to some end point. As much as Euclidean concepts have been forced to change with discoveries in non- Euclidean geometry, so space and knowledge are not arrived at by a linear and defined journey, but by experiential observation.

Intellectual voyages that discovered the lands of the non-linear and non-Euclidean have validated the artistic voyage in a unique way. This new voyage allows us to transcend those epic voyages of the past and their historic inevitabilities of exploitation and rule. "Voyage without Boundaries" is an artistic exploration that goes beyond the conventional wisdom of time and space, and allows us to form voyages of our own that have no defined inevitabilities.


This work aims to unite the examination of the wisdom of time and space and idea of global village. But it also includes the fears, wishes and fantasies. In this work are 3 themes linked: wormholes, mandalas and the story of the hollow and inhabited earth, which was considered a scientific theory until the 19th century.

The wormhole is a metapher which comes from the image of a worm, bitting his way right through an apple.
A wormhole therefore connects 2 sides of 1 space, with a tunnel. 2 places that lie on 2 different sides of the earth (e.g. Busan - NYC)and geographically are far away from each other, seen to converge through modern technology and globalisation, just as through a wormhole between them exists. I would like to create such wormhole with my work. 
Through the hole, which starts in Busan, you can see other cities all over the world, their houses and people, who also  look through it. You would also be able to see the sky and therefore into the infinity, the cosmos. It is a connection between two cities but further more it is also a connection of the infinity lying on both sides of the world. There is no beginning and no ending, just two perspectives. Travelling with the means of wormholes, although theoretically possible, is going to remain a dream. The surplus of energy needed will probably never be avaible.

The word mandala means circle, ring or plate. Originally the mandala was used to illustrate the world. Nowadays it is used to express something mental into something visual. The implementation is said to guide to an higher mental concentration. The Ego, in the buddhist sence, the origin of all suffering, is to be overcome and finally all earthly and material thinking exceeded.
Bataill was fascinated by the buddhist ideas.
The order of houses, trees etc. are to remind of such mandalas and their centralised order.

The theory of the hollow earth was scientifically approved until 19th century and had many great representatives like Edmund Halley (17TH century) from the Royal Society of Science. The Theory said, that the earth is hollow and that there were manholes lying at the poles. Many tried to find there manholes, there are several travellreports and articles abouts them. Some claimed to have reached the middle of the earth, some place with its own sun. 
The place was said to be inhibited by people living on an higher step of civilisation. This ideal of a society is similar to our modell of a global village or to the new better society in terms of Bataille...or it could be even one of his experiments.  
Todays search after the life in space is a continuation of this yearning. 

Of Wormholes, Hollow Earth and Mandalas by Navina Sundaram

A bridge and a mound of sand and the sea lapping the shore and the original instinct to dig into the ground, to burrow, to ferret and to see what is on the other side.
Where is that other side? Did Alice fall through a wormhole when she fell into her Wonderland? Yet the Einstein-Rosen bridges had not in general relativity been realised when Lewis Caroll sent his Alice hurtling through time and space. „All time is eternally present. In my beginning is my end“. So maybe those space time tubes acting as short cuts between vast distances existed long before they were discovered by man.
Journeying on time-travel, the nomads of imagination plunging into a hollow earth to find new civilisations not at the other end but at its very centre, the unending yet futile search for a better, for an other humanity. The entry into brave new worlds or utopias, and what does it mirror? "Hollow men" and women in a wasteland, "headpieces filled with straw"? A Kingdom where the sun never sets? With its religious, if you will, Christian overtones of theopolitics and theoeconomics and theocolonialism making a perfect metaphor for, and dovetailing neatly into, the other present day Empire of globalisation, of the New World Order, where the dance around the golden calf of Christian lore now metamorphosed into a chase of the golden deer of Capital, this shy creature, wary of social justice, and always ready to run –away from responsibility, from rooting. Its golden and silver dung dropping through global wormholes and cyber tunnels that span the globe and connect the world – virtual and real - on which the sun never sets. Electronic data gathered by cyber coolies, sitting in miniscule cubicles, for cyber lords. Is this connectivity - all these hotlines and call centres in Calcutta or Ougadougou? And is this productivity, all this outsourcing to China or Rumania? Or is this all just a subtler form of slavery?
One man’s gasoline is another man’s hunger. Globalisation gone mad. Reflections, spitting images, digitally pixelled beamed illusions or Maya?

And the nomads of imagination now soaring -bound by no boundaries –to create art, for instance, ancient mandalas made with a crushed sand of precious or semiprecious stones. To labor for days tracing intricate patterns only to erase it on completion. Chanting Tibetan monks on one side of the world sweeping their mandala into a jar and pouring it into water, into the stream, the river, the sea. In my end is my beginning. And on the other side of the world Navajo American Indians also destroying their sand painting, their work of labour, before dawn to maintain harmony. Only the impermanant is permanent. This is the ultimate luxury in the Bataille sense of the word: the creation of art, and with its erasure the destruction of riches – a wasteful expenditure which cannot be accounted for by the principle of gain.
Come a storm, come the tide, the work is reclaimed and incorporated into the cycle of life or it is released through the cosmic hole. Thus the voyage without boundaries may begin. And on the wind swept shores of Busan one hears the creakings of self-mockery.

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Time as Another Dimension

One of the first points to make as we begin discussing space-time diagrams is that we are treating time as another dimension along with the three dimensions of space. Generally, people aren't used to thinking of time as just another dimension, but doing so allows us to truly understand how relativity works. So, how do we represent time as just another dimension?

Obviously we can't actually picture four dimensions all at once (three of space and one of time). Our minds are limited to picturing the three dimensions of space that we are used to dealing with. However, we can consider one or two dimensions of space and then use another dimension of space to represent time.

To see how this can work, consider Diagram 2-1here you see a film strip on which each frame represents a moment in time. As you watch a film, you see each moment in time presented one right after another, and this gives the impression of seeing time pass. If we cut the film up into frames then we can stack the frames flat, evenly spaced, and one on top of the other (as shown in the diagram). Then each frame is a two dimensional representation of space and as you move through the third dimension you go up the stack, and each frame you pass represents another point in time. Thus, we have a three dimensional stack which represents two dimensions of space and the third dimension represents time.

Diagram 2-1

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THE HYPERCUBE

• The hypercube is the four-dimensional analog of the cube, square, and line segment.
   
• A hypercube is formed by taking a 3-D cube, pushing a copy of it into the fourth dimension, and connecting it with cubes.
   
• Envisioning this object in lower dimensions requires that we distort certain aspects.
   
• The tesseract is a 3-D object that can be "folded up," using the fourth dimension, to create a hypercube.

You may recall that our "new" fourth dimension must introduce a quantifiable property that has not yet existed in any of the lower dimensions—this is simply a pre-requisite of a degree of freedom. Objects in four-space have a property, analogous to area and volume, that we call "hyper-volume." Possibly the most famous object with this property is the hypercube. To prepare to understand it, let's first look at how we formally construct "normal" squares and cubes.

First, to create a square in two dimensions, or a cube in three dimensions, we start with the analogous object from the dimension that is one lower. That is, we use parallel line segments, joined by perpendicular line segments, to create the square. To create the cube, we use parallel squares connected by perpendicular squares.

So, to create the hypercube, we start with a cube in 3-D space; then we create another cube at a distance equal to the side-length of the original cube along the w-axis. These two cubes can be thought of as being parallel in the same way that the opposite sides of a square or the opposite faces of a cube are parallel.

Think back: to make a square, we connected the endpoints of two parallel line segments using line segments of equal length; and to make a cube, we connected the edges of two parallel squares with squares of equal shape. So, to construct a hypercube, we will connect the faces of our parallel cubes with cubes of equal size. It should be clear that connecting all the faces of our two parallel cubes requires six "connector" cubes. Consequently, the hypercube is made up of eight regular cubes that are "glued together" such that all of their faces are attached to one another.

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The last doubts about the heliocentric model were removed years later by Isaac Newton (1643 – 1727).  Based on Galileo’s and Kepler’s works, Newton published “Principia” in 1687. In this book, Newton posed the theory of Gravity, in which the force that makes planets to move around the Sun is the same force that makes object to fall in the Earth: force of gravity. In his theory, Newton deduced gravity is a force of mutual interaction of body with mass and this force is inversely proportional to the square of the distance between objects.

The heliocentric model was established by Newton but there were some question about the gravity, for example, its action at a distance and immediately action.  Even Newton had doubts about the gravity action at a distance. How can massive objects attract each other at distance without mediation of anything?  And how can attraction force between them be immediately without a time to action?

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In order to move forward into the exciting science of energy medicine (specifically PEMF therapy) with an empowering attitude of taking responsibility for our own health; we need to see why this paradigm is flawed by first looking at the basic concepts of Newtonian physics. Then we’ll introduce the exciting new physics, namely quantum field theory, and the latest developments in science.

Isaac Newton (1642-1727) is regarded as the founder of modern Western science that dominated for at least 200 years until the early 20th century with the discovery of quantum mechanics. A whole new universe opened up in the early part of the 20th century and here we are 100 years later still gazing through the tinted windows of Newton’s physics.

Newton's law of universal gravitation states that any two bodies in the universe attract each other with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. (Separately it was shown that large spherically symmetrical masses attract and are attracted as if all their mass were concentrated at their centers.) This is a general physical law derived from empirical observations by what Isaac Newton called induction.[2] It is a part of classical mechanics and was formulated in Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on 5 July 1687. (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke made a claim that Newton had obtained the inverse square law from him – see History section below.) In modern language, the law states the following:

Every point mass attracts every single other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them:[3]

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Newtonian Gravity - Force at a Distance

•  F is the force between the masses,
• G is the gravitational constant,
• m1 is the first mass,
• m2 is the second mass, and
• r is the distance between the centers of the masses.

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Einstein's Relativity Subsumes Newton's Laws as Approximations

In 1905, Albert Einstein changed the prevailing worldview of Newtonian physics for good with the introduction of his special relativity theory, followed in 1915 by general relativity. He proved Newtonian laws of physics are by no means static, but relative to the observer and the observed. Depending on the difference in speed between the observer and the object under observation, space begins to either shrink or expand and time slows down or speeds up. Also mass increases with increasing speeds (due to increase in kinetic energy).

In Newton’s universe, there are notions of absolute space and time. Space was seen as a three dimensional stage, and time the ticking of a well-made clock. The two were separate and distinct. In Einstein’s special relativity, space and time form one 4-dimensional space-time continuum with the speed of light being the only fundamental absolute measurement. This led to the famous equation E=mc^2 showing the equivalence between matter and energy.

General relativity supplants Newton’s action at a distance with the curving of space and time. Simply put, mass curves space and curved space guides mass in a way that follows Einstein’s elegant field equations of general relativity. The force of gravity is now known to be a curving of space and time, rather than forces acting at a distance.

Einstein's Theories of Relativity are a much more refined version of Newton’s laws to take into account, relativity between observers, high speeds and intense gravitational fields (curved space). 

However, Newton’s physics was “good enough” to send a man to the moon, so it is a good approximation at speeds significantly less than the speed of light and space that is mainly flat (both conditions hold in the lunar landing triumph).

Newton's law has since been superseded by Einstein's theory of general relativity, but it continues to be used as an excellent approximation of the effects of gravity. Relativity is required only when there is a need for extreme precision, or when dealing with very strong gravitational fields, such as those found near extremely massive and dense objects, or at very close distances (such as Mercury's orbit around the sun). Though Newtons laws work well in situations with low gravity, they are still based on inaccurate assumptions and involve the incorrect force at a distance explanation.

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What is Relativity?

The principle of relativity states that measurements of motion, time, and space make sense only when we describe whom or what they are being measured relative to -- there are no absolute answers.
The same is true when one person or object is accelerating with respect to the other.
Extending this general idea further, the equivalence principle states that
The effects of gravity are exactly equivalent to the effects of acceleration.
Scientific ideas for artificial gravity utilize this concept.

Einstein's 1st Ah-ha - All Observers Measure the Speed of Light the Same.

Special relativity unlocked the secrets of the stars and revealed the fantastic quantities of energy stored deep inside the atom. But the seed of relativity was planted when Einstein was only 16 years old and asked himself a childlike question: What would a beam of light look like if you could race alongside it? According to Newton, you could catch up to any speeding object if you moved quickly enough. If you could catch up to a light wave, Einstein realized, it would look like a wave frozen in time. But even as a teenager, he knew that no one had ever seen a frozen light wave before. In fact, such a wave makes no physical sense.

When Einstein studied Maxwell’s theory of light, he found something that others missed—that the speed of light always appears the same, no matter how quickly you move. Einstein then boldly formulated the principle of special relativity: The speed of light is a constant in all inertial frames (frames that move at constant velocity).

Previously, physicists believed in the ether, a mysterious substance that pervaded the universe and provided the absolute reference frame for all motions. But experiments to measure the “ether wind” blowing past Earth found nothing. Even if Earth were by chance motionless at one moment, there should be a discernible ether.

In desperation to save Newtonian physics, some scientists suggested that the ether wind had physically compressed the meter sticks in their experiments, thus explaining the null result. Einstein showed that the ether theory was unnecessary and that space itself contracts and time slows down as you move near the speed of light.

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Special Relativity - Combining Space and Time into 4D Spacetime

spacetime = 4-D combination of space and time
5d viewing of 4D --- Like Interstellar. Each 3d cube has an entire history in time. Like a panoramic life review.

The dimension of time is related to the dimension of space as distance = (time) x (speed of light).
Space is different for different observers.
Time is different for different observers.
Spacetime is the same for everyone.


Since space has three dimensions (length, width, and height), time can be viewed as the fourth dimension. For example, to arrange arendezvous with a friend in Manhattan, you need to give four coordinates: “Meet me on the northeast corner of 5th Avenue and 42nd Street, on the 30th floor, at one o’clock.” Relativity introducedthe concept of the fourth dimension.

Imagine plotting your location on a graph, with time on the vertical axis and space on the horizontal axis. The bottom of the graph represents the past, and the top part represents the future. If you simply sit in one place and do not move, you trace a vertical line. If you start to move, you trace a vertical line that curves a bit.

No longer were space and time absolutes, as Newton thought. Space compresses and clocks tick at different speeds throughout the universe.

In Newton’s universe, there are notions of absolute space and time. Space was seen as a three dimensional stage, and time the ticking of a well-made clock. The two were separate and distinct. In Einstein’s special relativity, space and time form one 4-dimensional space-time continuum with the speed of light being the only fundamental absolute measurement. Measuring distance in time may seem strange but consider the following examples.

To Newton time is uniform through the universe. One second on Mars = One second on the earth. But to Einstein, time beats at different rates. The faster you travel, the slower times moves. Gravity also slows the passage of time.

In the time of pioneers distance was measured in time, for example getting from one city to another might have been agreed upon as a four days journey. Cosmologists use light-years to measure distance. For example our nearest star (besides the sun) is Alpha Centauri, which is about 4 light years from earth. 4 light years is the distance light travels in 4 years. Our moon is about 1 light second away and the sun is 9 light minutes away. When you see the sun, you are actually viewing it 9 minutes in the past. One of the illusions of space-time is that to see anything “out there” in space, you are always seeing it backwards in time. But because light travels so fast it is almost instantaneous for distances on earth, so  this illusion is not apparent. To give you a sense at how fast light travels, light can take 7.5 laps around the earth in ONE SECOND!

This slowing down of time is very real and must be factored into to GPS and satellites moving relative to the earth’s surface (especially when satellites orbit in the opposite direction of the earth’s rotation.

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Special Relativity Part 2 - Matter and Energy Equivalence E=mc^2

Also, energy and matter were two distinct notions in Newton’s mechanics, and there were separate conservation laws for both: the conservation of matter and the conservation of energy. Einstein with his famous equation E = mc2 forever changed this notion as well. Matter and energy are interchangeable with energy being the more fundamental unit. This is one of the most important new scientific notions in this book that we can apply to new understandings of the human body; namely that we are primarily energetic beings and secondarily physical ones!

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Einstein's Second Ah-ha - All Things Fall to the Ground with the Same Acceleration 

This famous result in physics was first discovered by Galileo who legend has it dropped two balls of different masses from the Leaning Tower of Pisa and observed they hit the ground at the same time. Whether that actually happened or not, the important point is that Galileo knew what the outcome would be.

The real triumph of this fact was seen in 1971 when David Scott, the Apollo 15 commander, dropped a feather and a hammer on the moon and both hit the ground at the same time. We cannot do that experiment on earth, because the feather experiences a air resistance due to the earth’s atmosphere (the moon has no atmosphere, so no air resistance).

The important fact is that everything falls at the same rate if air resistance can be removed.

This led Eistein to the formulate the famous Principle of Equivalence – The effects of gravity are exactly equivalent to the effects of acceleration. 

Starting with the idea of the Principle of equivalence, Einstein spent 10 years of his life developing a new theory of Gravity that is to this day, still the best theory we have on Gravity.

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Why do things fall at the same rate and why is it such a big deal?

Imagine you are standing in a stationary elevator. Your feet are firmly pressed on the floor, your head pushes on your shoulders and your stomach rests securely inside your body.

Now imagine you are in an elevator with the cord cut (don't worry, there is a big spring at the bottom of the fall). Since everything falls at the same rate, your feet no longer push on the floor, you head is no longer pushing on your shoulders, and your stomach floats freely in your body. In short you are weightless. It's as if gravity was turned off! An astronaut floating in space would feel exactly the same.

To be more precise, there are no experiments you could inside the falling elevator that would distinguish whether you were falling in the elevator or floating in space. Of course you would know the difference since you walked into the elevator, but that is not the point. The point is that the laws of physics are the same in both situations. This is why it is called the principle of equivalence.

The effects of gravity are exactly equivalent to the effects of acceleration.

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General Relativity - Explains Gravity

Special relativity was incomplete because it made no mention of acceleration or gravity. Einstein then made the next key observation: Motion under gravity and motion in an accelerated frame are indistinguishable. Since a light beam will bend in a rocket that is accelerating, a light beam must also bend under gravity.

To show this, Einstein introduced the concept of curved space. In this interpretation, planets move around the sun not because of a gravitational pull but because the sun has warped the space around it, and space itself pushes the planets. Gravity does not pull you into a chair; space pushes on you, creating the feeling of weight. Space-time has been replaced by a fabric that can stretch and bend.

General relativity can describe the extreme warping of space caused by the gravity of a massive dead star—a black hole. When we apply general relativity to the universe as a whole, one solution naturally describes an expanding cosmos that originated in a fiery big bang .

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Curved Space

To see how curved space causes the apparent attraction of gravity, imagine two planes taking off on parallel lines (think of longitude or meridians) heading towards the North pole. As they travel it appears they are moving closer to each other by some mysterious force. But this illusion is only a result of the earth being a curved space. Simillarly, what we call gravity was shown by Albert Einstein to be a result of the curving of a space, not a mysterious force from a distance.

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Bringing Space, Time, Energy and Matter all Together!

1915-1916 - General Relativity - Allowed Einstein to explain gravity in a much wider context with speeds approaching the speed of light and very massive objects

Einstein's General Theory of Relativity further unified spacetime with matter-energy in a new theory if Gravity.

Energy-mass tells spacetime how to curve and curved spacetime tells mass how to move. So Einstein's field equations now show an equivalence between curved and warping space and energy-matter density.

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General Relativity - Field Equations (Actually 10 Equations)

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Stress Energy Tensor - Shows Matter - Energy Equivalence

Mass-Energy Curves Spacetime and Curved Spacetime Guides Mass-Energy Giving Rise to Gravity

The curvature of space near a massive object (e.g. Sun) forces the light beam passing near it to bend, much like a lens.
Changes in angular separation between stars were measured to change near the Sun during the solar eclipse in 1919 by Sir Arthur Eddington.
Trajectories of light from distant stars or galaxies are bent by the gravitational field of a massive object located along the line-of-sight, producing multiple images or a ring of images (read Einstein's 1936 Science paper on gravitational lensing).

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Proof of Einstein's General Theory of Relativity

Mercury's orbit slowly precesses around the Sun.
cannot be explained by Newton's law of gravity.
Time runs slower and space is more curved on the part of Mercury's orbit that is nearer the Sun.

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First, our technology would fail. The Global Positioning System (which locates our position on Earth to within 50 feet or less) would malfunction, because the clock on the satellite does not tick at the same speed as Earth clocks. Moreover, since relativity governs the properties of electricity and magnetism, all modern electronics would come to a halt, including generators,computers, radios, and TV.

Without correcting for the effects of relativity, the GPS signals would have errors of several parts per billion, enough to make them useless.

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GRAVITY

Gravity differs from the other interactions in multiple ways. First of all, it is by far the weakest of fundamental forces. In fact, we can simply demonstrate this fact. Try to lift an object using your hand. A pencil, a glass, anything. If you have succeeded and the object is safely in the air surrounded by your palm, congratulations – you have just managed to overcome the gravitational pull of the entire Earth, whose mass is in trillions of trillions of kilograms. How can gravity be the dominant force of the universe when it is so immensely weak?

The reason is that the other three interactions, though much stronger, simply are not customized to become the prevailing force of the universe. Strong and weak forces have a very short range – they only affect objects that are far less than a billionth of a meter apart. And the last interaction, electromagnetism, only influences objects with an electric charge. The problem is that you do not find such objects very often in the macroworld – most objects are neutrally charged. So the only reason that this ridiculously frail interaction has become the motive force of the cosmos is that it simply has no competition.

The second factor that makes gravity special is that it is presumably not really a force, even though it has been viewed as such to the beginning of the 20th century. However, with the advent of Einstein’s theory of relativity, our view of gravity has changed radically. Einstein saw gravity merely as a curvature of space-time. Every object in the universe simply creates a kind of dimple in the space-time continuum and all other objects are inclined to move closer to that object.

It is like placing a heavy object into the middle of a trampoline – the entire surface of the trampoline curves downwards, and if you place a different object near its rim, it starts to roll towards the original object. This analogy, however, has an imperfection. Just like with the inflation of the universe after the Big Bang, we need to take away one dimension to comprehend the phenomenon.

The surface of a trampoline can be perceived as two-dimensional space (it has width and height, but no depth) similarly to a sheet of paper. An object placed to its middle causes its two-dimensional space to curve. Therefore, the surface of a trampoline with an object in its centre can be understood as a two-dimensional space curved in the third dimension.

However, our universe is three-dimensional, so any curvature caused by the presence of an object in our space-time occurs in the fourth dimension. That is also the reason why we can never perceive any gravitational curvature. We would need to be four-dimensional beings for the curvature to be revealed to us.

However, it does not hurt to know that nobody is sure whether this theory of gravitational space-time curvature is true. With today’s technical advancement, we are struggling to find evidence that gravity indeed curves our three-dimensional space.

But there is another view of gravity, completely different from the one I have just described. According to this view, gravity is provided by a hypothetical particle called the graviton. How? Simply said, every two objects in the universe exchange various numbers of gravitons, which causes them to attract.

To understand why there are two different perceptions of gravity today, we first need to become acquainted with the greatest problem of today’s physics – the everlasting search for the theory of everything. To achieve that, we need to travel more than a hundred years to the past, to the beginning of the 20th century, where we will witness the birth of the two greatest physical theories of today.

By the end of the 19th century, some physicists presumed that physics was already complete. They thought that everything had already been described by the old physical theories. But then came the year 1900, along with a new revolutionary theory called quantum mechanics, which proved how immensely wrong those physicists were. This theory describes the behaviour of objects from the microworld, which is completely different form the behaviour of “normal” objects.

Fifteen years later, classical physics was stabbed again by Einstein’s general theory of relativity, which utterly transformed our view of gravity and beautifully described the motion of objects at high velocities.

However, there is a tremendous problem with these two theories – each one seems to describe a completely different world. While quantum mechanics successfully uncovers the peculiarities of the microworld, general relativity brilliantly describes the motion of objects of the macroworld. But if we wish to fully comprehend our mysterious universe, we need to unify these two incompatible theories into one. Physicists have been trying to achieve that for the past hundred years, so far without much success.

And the problem with today’s view of gravity rises from here. While the description of the remaining three interactions comes from quantum mechanics, the best understanding of gravity is provided by general relativity. Physicists therefore aim to describe gravity within the framework of quantum mechanics, so that it forms a single integrated theory. This non-existent theory is called the theory of quantum gravity or simply the theory of everything.

And that is the reason why there are two different views of gravity today – one almost perfect in the framework of general relativity, which is not compatible with other interactions, the other not so perfect within quantum mechanics, which is crucial for the upcoming theory of everything, but includes these peculiar particles called gravitons, which have never been detected.

Not to worry though – luckily, there are a few things that we know about gravity with certainty. Firstly, gravity is always attractive. There is no instance of two objects gravitationally repulsing each other. Secondly, gravity propagates with the speed of light, which is the highest velocity anything can reach when traveling through space-time.
That means that if the Sun were to disappear now and stop influencing us gravitationally, it would take exactly 8 minutes and 20 seconds for us to notice it and free ourselves from the Sun’s gravitational field (at the same time, the Sun would also disappear from the sky, as the last of its light would reach our planet). Until then, the Earth would keep revolving around the non-existent Sun.

Bumps and Wiggles: An Introduction to General Relativity

 Gary Felder

This paper is a brief introduction to the ideas of Einstein's general theory of relativity, one of the cornerstones of modern physics. The development of general relativity brought about a radical change in our concepts of space and time. This paper is not a course in general relativity, but after reading it you should have at least some understanding of what the theory says, and in particular how space and time are viewed in this context. The paper is almost entirely non-mathematical, but I do assume that you are already comfortable with some Newtonian physics and at least the basic ideas of special relativity. For the latter you could start with my brother's paper: "The Day the Universe Went All Funny."

For readers with a stronger background in physics there is also a sequel paper which goes into somewhat more detail about how the laws of general relativity are formulated. That paper assumes a working knowledge of calculus and introductory physics such as you would get in the first year or so of a university physics major.

Introduction: What General Relativity is About

General relativity (GR) can be viewed in a number of different ways. I will start by briefly describing two of these viewpoints and how they relate to each other.

1) General relativity is a theory of the behavior of space and time

Prior to the 20^th century all physics theories assumed space and time to be absolutes. Together they formed a background within which matter moved. The role of a physical theory was to describe how different kinds of matter would interact with each other and, by doing so, predict their motions. With the development of special and later general relativity theory in the early 20^th century, the role of space and time in our theories of physics changed dramatically. Instead of being a passive background, space and time came to be viewed as dynamic actors in physics, capable of being changed by the matter within them and in turn changing the way that matter behaves.

In GR, spacetime becomes curved in response to the effects of matter. I will discuss below what it means for spacetime to be curved, but just to give a flavor of this idea I can note here that in a curved spacetime the laws of Euclidean geometry no longer hold: the angles of a triangle do not in general add up to 180^°, the ratio of the circumference of a circle to its diameter is in general not p, and so on. This curvature in turn affects the behavior of matter. In Newtonian physics a particle with nothing pushing or pulling it (no forces acting on it) will move in a straight line. In a curved spacetime what used to be straight lines are now twisted and bent, and particles with no forces acting on them are seen to move along curved paths.

2) General relativity is a theory of gravity

Newtonian theory, which held sway prior to the 20^th century, described gravity as a force. In other words two massive bodies like the Earth and an apple were understood to exert a pull on each other as a result of the law of gravity. If an apple started out at rest, say just as it broke off from a tree, then gravity would cause it to move towards the Earth until it collided with it. Newton's law of gravity was able to explain in detail not only the fall of apples, but also the orbit of the moon about the Earth, the motions of the planets about the sun, and much more. GR can also explain all of those things, but in a very different way. In GR, a massive body like the sun causes the spacetime around it to curve, and this curvature in turn affects the motion of the planets, causing them to orbit around the sun.

In later sections I'll discuss in more detail how GR describes results like falling apples and orbiting planets. For the most part the predictions of GR and Newtonian gravity are very similar. There are small differences that can and have been measured in the solar system, however, and to date all the data have matched the predictions of GR. Moreover, there are certain situations, like the vicinity of a black hole, where GR makes predictions drastically different from those of Newtonian theory. I will briefly discuss some of these later and talk about what evidence we have for some of the more exotic predictions of GR.

In short, GR is a theory in which gravity is described by saying that space and time are dynamics quantities that can curve in response to the effects of matter and can in turn alter the behavior of matter. Before I discuss GR in any greater depth, I need to talk more generally about the idea of curved spaces. The next section thus discusses what it means for space to be curved. In GR, however, it is not just space but spacetime that is curved, and the following section discusses that idea. Using these ideas I then describe how gravity is viewed in GR. Having thus presented the basic ideas of GR, I go on to discuss a number of applications where GR gives results very different from those of Newtonian theory. In the conclusion I discuss some of the open questions remaining in our understanding of the nature of space and time.

For the sake of completeness, I can note here one other way of describing the content of GR, which is that GR is a theory of physics in arbitrary coordinate systems. The laws of physics that were known prior to GR, most notably Newtonian physics and special relativity, were only valid in a restricted set of coordinate systems known as inertial reference frames. The laws of GR are formulated in a way that is equally valid in any reference frame. In my sequel paper I explore this idea in greater depth.

Curved Space

Suppose you and a friend stand one meter apart from each other facing the same direction and begin walking. Assuming you both walk in a straight line at the same speed you should stay exactly one meter apart. The two of you are tracing out two parallel lines. Imagine instead, however, that you walk for quite a while and notice that you are starting to drift apart. Eventually you are two meters apart, and if you look carefully you realize you're not pointed in exactly the same direction any more. You would presumably conclude that one or both of you had failed to walk in a straight line.

To test this idea you would need a definition of exactly what a straight line is. We know that one of the properties of straight lines is that if they are parallel then they stay parallel, so clearly the paths you and your friends walked on can not be straight lines. On the other hand we also know that a straight line is the shortest distance between two points. So if you were accidentally walking on a curved path then you should be able to draw a path connecting your initial and final points that is shorter than the one you actually walked along. To picture this test you can imagine that you were laying down a trail of red paint behind you as you walked. You can carefully measure your red trail with a tape measure. Then you can try to paint a shorter, blue trail connecting your initial and starting points. If your path was curved, you should be able to make a shorter path.

If you are living in a curved space, however, then you might fail. In other words it is possible that you and your friend each took the shortest possible route between your starting and ending points, and yet you ended up slowly turning away from each other. In a curved space paths that stay parallel to each other are not the paths of minimal distance and vice-versa. Since there is no path in such a space that fits all our usual notions of a straight line, mathematicians came up with another word to use for this situation. In any space, the shortest path between two points is called a geodesic.(1) In a flat space, meaning one with no curvature, the geodesics are normal straight lines that stay at a constant angle to each other. In curved spaces they are generally more complicated.

You may at this point be having trouble picturing exactly what is happening. If you picture two paths that start out parallel to each other and end up pointing away from each other then it seems that they must be bending, and they couldn't possibly be geodesics. This problem stems from the fact that our brains are designed to think in terms of flat geometry, so we can not picture a curved space any more than we can picture a four dimensional space. Fortunately there is a trick we can use to imagine more clearly how this works.

Imagine very small ants walking on the surface of a globe. Two ants start out at different points on the equator heading due south. The paths they are walking along are initially exactly parallel. If neither ant turns then they will continue heading south until they reach the South Pole. In other words they can start out moving on parallel paths, walk straight the whole way, and yet end up at the same point. On the surface of a sphere, the geodesics do not stay parallel to each other. Such a space, where parallel lines tend to curve inwards towards each other, is said to have positive curvature. To picture a negative curvature space, where parallel lines curve outwards, you could put the two ants on the surface of a saddle. The sign of the curvature is determined by whether parallel lines bend towards or away from each other, and the magnitude of it (how big a number it is) is determined by how quickly they do so. A space with large, positive curvature, for example, is like the surface of a very small sphere.

We've just used a trick called embedding, which means describing the properties of a curved space by considering it to be a curved surface in some higher dimensional flat space. This trick is very useful because it allows us to use our natural ability to picture flat spaces when trying to understand curved spaces. There is a danger, however, of taking the embedding too literally. If you picture a geodesic on the surface of a sphere you might be inclined to think that it's only a sort of fake geodesic. If I really want to find the shortest path between the equator and the pole I can make a straight line that goes through the interior of the sphere. I only called the ant's path a geodesic because I had put on the limitation that it wasn't allowed to move off the surface. That's true for the embedded space, but it's not true for a real curved space. A particular two dimensional curved space might have the same mathematical properties as the surface of a sphere, but that doesn't mean that it actually has to exist in some flat, three dimensional space. In fact not all curved spaces can be embedded in this way. So I would urge you to use embedding as a tool for picturing the properties of curved spaces, but always remember that it is just a tool. In general curvature is simply an intrinsic property of a space.

To wrap up this section, I want to note one more property of geodesics, which is that to an observer in the curved space they will appear straight on short enough scales. Imagine that instead of an ant on a globe our traveler is once again you, now moving on the surface of the Earth. Let's assume for simplicity that the Earth is a perfect sphere with no mountains or canyons. If you can only see a few miles in any direction then the surface of the Earth appears flat; you would need to see much farther than that to notice the curvature. In this seemingly flat space you can draw geodesics and they will look to you like straight lines. If you started laying out one foot rulers end to end you could circle the whole Earth and at each point it would seem to you that you were marking out a straight line, yet if you marked out two such lines that started parallel you would eventually find them converging.

Spacetime

In Newtonian physics space and time were viewed completely separately. Ask most people how many dimensions our world has, for example, and if they understand the question they will most likely answer three. In relativity theory, however, it is conceptually simpler to view time as a fourth dimension.

(Three different world lines representing travel at different constant speeds. t is time and x distance.)

We can't picture a 4D world, so instead let's imagine that we are one dimensional beings. In other words we live and move only on a line. In that case we can picture spacetime as a 2D surface, where the horizontal direction is space and the vertical direction is time.

The motion of a particle in this 2D spacetime traces out a curve, called a world line.

Spacetime diagrams such as the one above are critical to relativity, so I would urge you to spend a little time playing with them in your mind now to get comfortable with them. For instance, try to answer the following questions before you read on. (I give the answers in the next paragraph.) What's the world line of a particle at rest? What's the world line of a particle moving with constant speed in one direction? How would you describe the motion of a particle with the world line shown below?

The answers: The world line of a particle at rest is a vertical line. As time (t) moves forward it always stays at the same position (x). The world line of a particle moving at constant velocity is a tilted line. In each interval of time, say each second, the position of the particle changes by some constant amount. Finally, the world line shown above describes a particle that's oscillating, like a mass on a spring. As time progresses the particle moves back and forth periodically. Once again, I would urge you to make sure you are comfortable with these ideas before reading on.

Viewing space and time this way allows us to formulate physics in a new way. Consider, for example, Newton's first law, which states that an object with no forces acting on it will move in a straight line at a constant speed. Another way to say this is that the world line of a free object (one with no forces on it) is a straight line.

Be careful not to confuse an object's motion in space (its trajectory) with its motion in spacetime (its world line). The latter contains more information than the former. For example, saying that the world line of an object is a straight line tells you not only that its trajectory is a straight line, but also that it is moving with constant speed. As another exercise, try to picture the world line of a particle accelerating in a straight line. Try to picture the world line of a particle moving in a circle at constant speed. (This last exercise requires two spatial dimensions, and hence a 3D spacetime.)

In Newtonian mechanics the notion of spacetime is unnecessary. You are free to think of space and time as separate things and formulate Newton's laws in terms of the motion of particles or think of them as unified and formulate those laws in terms of world lines. In fact the former description is simpler and easier to work with. In relativity, however, (both special and general) it is necessary to view spacetime in a unified way. In GR this unified spacetime is curved by the effects of gravity. Newton's first law continues to hold in GR, but in a generalized form: A free particle will move along a geodesic. In the presence of gravity, however, that geodesic will in general be more complicated than a simple straight line.

Gravity

Consider two objects initially at rest. They could be planets, stars, or elementary particles. We will assume that they are far enough away from anything else that they feel no influence from anything but each other. Moreover we'll assume that they are exerting no non-gravitational forces on each other.

In a Newtonian picture these objects will exert a mutual gravitational attraction, causing them to accelerate towards each other until they eventually collide. In GR the same effect will occur, but the description will be very different. Because gravity is not a force in GR, and we said the objects neither exert nor feel any non-gravitational forces, the objects should act like free particles, moving along geodesics. (Such an object experiencing no non-gravitational forces is said to be in "free fall," like an object falling towards the Earth with no other forces on it.) In a flat spacetime—no gravity—the geodesics would be straight lines. In particular, since we specified that the objects started out at rest, their world lines would be vertical lines. In other words they would always stay the same distance from each other.

When we consider the effects of gravity, however, we know that the objects will warp the spacetime around them. Recall that in a curved space, parallel lines do not always stay parallel. In this particular curved spacetime, the geodesics followed by the objects start out parallel but converge over time. Thus the objects eventually collide. Qualitatively the result is the same as predicted by Newton's theory, but the underlying description is radically different.

To show how this works more explicitly, I'm going to assume for simplicity that one body is much heavier than the other. For example this could be a description of the sun and the Earth. That way we can ignore the gravitational effects of the smaller body and just consider what spacetime looks like around a single, massive object. The spacetime diagram for this situation is shown below with a couple of geodesics drawn in.

The yellow rectangle in this diagram is the sun itself. Of course the space around the sun is really three-dimensional, but the spatial dimension in this diagram is just a line going directly outward from the sun. I've labeled the spatial axis "r" (for radius) rather than "x" to remind you that I'm only showing one of the three spatial directions. This means the geodesics I've shown are for particles moving directly towards or away from the sun. The full spacetime diagram would also have geodesics corresponding to stable orbits around the sun like those of the planets. (Try to picture what those would look like.) Bearing in mind that only one spatial direction is shown, think about why the sun should appear as a rectangle in the diagram.

The red geodesic shows that an object initially at rest will curve in towards the sun. Even an object initially moving away from the sun could fall back in if it were moving slowly enough. The blue geodesic, however, is for a particle starting out at the same place but with an initial outward velocity large enough that it will never fall back. Such an object is said to have escape velocity.

This description is all very interesting, but so what? If the measurable results are the same as they were in Newtonian theory, why invent this new, more complicated theory? The answer is that the measurable results are not the same. Qualitatively the behavior described above is the same in both theories, but the exact details come out slightly different. In particular, you can prove that when you have weak gravitational fields and objects moving much slower than light, the predictions of GR are very close to those of Newtonian theory. That had to be true for GR to be a viable theory because we know that to high accuracy Newtonian theory had worked to predict the effects of gravity in many situations. When you violate those conditions, however, the predicted results begin to diverge.

Without giving you the math behind GR I can't describe in any detail why or how these predictions diverge, but I can give you some examples of the new predictions of the theory. For instance, I said above that Newton's theory of gravity accounted for the orbits of the planets. Strictly speaking, however, there was a tiny discrepancy between theory and observation in the orbit of Mercury, the planet closest to the sun (where the sun's gravity is strongest). Mercury is observed to precess, meaning the long axis of its ellipse revolves slowly around the sun, at a rate greater than predicted by Newton's theory. Specifically, the axis revolves faster than the Newtonian prediction by one degree every 8300 years! GR predicts this precession exactly.

More interesting than such numerical predictions, however, are some of the qualitatively new effects that can occur in GR. In the next section I describe a few of them.

Consequences of GR

In the following four subsections I describe four predictions of GR that are qualitatively different from anything in Newtonian physics. These predictions (explained below) are the existence of light cones, black holes, gravity waves, and the big bang model of the universe. If you wish you can skip any or all of these sections without losing anything else, except that the section on black holes requires the section on light cones.

Light Cones

The existence of light cones, which I will define below, is a prediction of special relativity even in the absence of gravity. I include it here because it continues to hold in GR, where it has some very strange consequences. (See for instance the section on black holes below.) The basic idea behind light cones is the fact that nothing can travel faster than the speed of light. That means that if I send a signal out from the point x=0 at t=0 it can't reach the point x=7 until a time equal to or greater than 7/c, where c is the speed of light. For instance if my friend is seven light years away, she can't possibly get my signal before at least seven years have passed. It doesn't matter whether I send the signal out by radio waves, smoke signals, or carrier pigeon; 7 years after I send the signal is the absolute earliest time my friend could see it.

(From this point on I will omit units when writing distances and times. All times will be in years and all distances will be in light-years. In these units the speed of light, c, is equal to one.)

 
Spacetime diagram showing the paths of signals sent out from the origin, ie x=0, t=0

To see this another way, consider the spacetime diagram above for a spacetime with no gravity. Mathematically this is the limit where general relativity reduces to special relativity. The bold lines represent the paths light beams would follow if I emitted them at the origin (x=0, t=0) either to the right or left. The shaded region in between these two lines represents all the points in spacetime that I could send a signal to. These points are said to be inside my (future) light cone. For example the point (x=7 light years, t=4 years) is outside the light cone. Nothing that I do at (x=0, t=0) can possibly have an effect on what happens at (x=7, t=4) or vice-versa. Note that this light cone is particular to a specific point in space and time, namely (x=0, t=0). If I were to draw the light beams emanating from this same point in space (x=0) at a different time, for example, they would demarcate a different region of spacetime.

 
Spacetime diagram showing the paths of signals sent to the origin

In addition to the future light cone, there is also a past light cone associated with each point in spacetime. In the diagram above the bold lines at negative times represent light beams that would reach me at (x=0, t=0) coming from either direction. You should take a moment to convince yourself that the region in between those two lines represents all the points in spacetime that can have affected me. For example my friend at x=7 can send a signal to me at t=-9 telling me what to do at time t=0, but if she sends the signal at t=-3 it will be too late for that signal to reach me by t=0.

Finally, for those of you wondering what any of this has to do with cones, the diagram above shows a spacetime diagram with two spatial dimensions instead of one. The figure in this diagram represents the paths of light beams moving in all possible directions. This figure is the aptly named "light cone" for the point (x=0, y=0, t=0).

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An example of a light cone, the three-dimensional surface of all possible light rays arriving at and departing from a point in spacetime. Here, it is depicted with one spatial dimension suppressed.

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Black Holes

Consider the gravitational field around a massive object such as the sun. In GR, "gravitational field" refers to the spacetime curvature induced by a set of objects, so we're really talking about what space and time are like near the sun. Recall the spacetime diagram I showed before for the vicinity of a massive object, where geodesics were bent inwards towards the object. I've reproduced that diagram here, only this time showing light cones instead of world lines. (The light cones continue past where I've shown them, but I've only shown small portions of them to keep the figure less cluttered. I'm also only showing the future light cones emanating from various points.)

 
Light cones in the vicinity of a massive object such as the sun

Light, like everything else, is bent towards the object. This means that the light cones are now somewhat bent, but the bending is very slight. Recall that the future light cone shows the limits of the possible world lines of particles. Particles moving away from the sun can not outrace the outgoing light beams. Since these are bent slightly towards the sun in the spacetime diagram, a signal sent outward to my friend three light-minutes away from me will take longer than three minutes to reach her. Conversely a signal sent to my other friend who is floating three light-minutes closer to the sun than I am could take less than three minutes to reach him. This doesn't violate the rule that nothing can move faster than light because the signal must still lag behind (or exactly match) a light beam moving towards the sun.

What happens if we consider something denser than the sun? For example a neutron star is an object a few times more massive than the sun but only about ten miles across. Very close to the surface of such an object the gravitational field is very strong, and geodesics are all very strongly bent, as shown in the diagram below.

 
Light cones in the vicinity of a neutron star

What happens when an object becomes even denser than a neutron star? Consider a hypothetical object with the mass of the sun but a radius of only half a mile. I show below the spacetime diagram for such an object.

 
Light cones in the vicinity of a black hole. The dashed line shows the event horizon (defined below)

The light cones very near the object are so strongly bent that a light beam shined directly outwards can not escape from the gravitational field of the object. Recall from above that the light cone emanating from a particular point in spacetime defines the region to which one can send a signal. If you are sitting close enough to the object shown above, then any light beam, carrier pigeon, or rocket ship you try to send out—including yourself—can only move closer to the center of the object.

An object so dense that it acts like this is called a black hole because even light can not escape from it. The place where the light cones start to turn inwards is called the event horizon. I've indicated it with a dashed line in the diagram above. That's the point of no return; if you cross inside the event horizon you can never get back out again. Remember that I'm only showing radial distance in these diagrams, so the event horizon looks like a point in space, and thus a line in spacetime. In the full three-dimensional space the event horizon is not just a point but rather a sphere surrounding the black hole.

On the event horizon, the outer edge of the light cone is vertical. Recall that the two branches of the light cone represent the paths taken by light emitted in either direction. That means that if you were exactly at the event horizon and shined a beam of light directly outward the light would eternally hover at this same radius. If you were inside the event horizon, the outer edge of the light cone would actually be tipped inwards. In other words whether you shined light inward or outward it would still approach the center of the black hole. The motion of any object, such as yourself, is bounded by the light cone. Thus if you are inside the event horizon you must approach the center.

What would that look like to you? If you shined a light beam away from the black hole would you see it turn around and start rushing towards the center? No. The light would be approaching the center, but you would be approaching it even faster. From your point of view the outgoing light beam would be moving away from you at exactly the speed of light. This is a general property of GR. No matter how strange a given spacetime might be, a local observer will always see light beams moving in all directions at exactly the same speed. This is the sense in which GR preserves the special relativity rule that says nothing can move faster than light. Your coordinate speed might be faster than c, but you will never see yourself catch up to a light beam. (2)

One could easily write an entire paper about the strange behavior of black holes, but here I will simply note a few of their more important properties. First, consider the massive object in the center. That object is the source of the gravitational field, but each particle in the object is itself living in the spacetime we've been describing. That means that it can not stay at rest but must move towards the center. In other words once an object has become dense enough to form a black hole it must keep collapsing and getting denser. If gravity were a force you might imagine some repulsive force strong enough to overcome it even in these extreme circumstances. As it is, however, no conceivable force in the universe could possibly resist this pull because spacetime itself is so strongly bent that there are no paths leading out.

In fact you can calculate the maximum time that a particle could possibly take to fall in to the center of the black hole, assuming it was being pushed outwards with an arbitrarily strong force, and the result is finite. Thus the object that formed the black hole originally will keep collapsing down and within a finite time it will have all collapsed into a single point in the center. Such a point of infinite density is called a singularity. (3)

To sum up, massive, dense objects exert strong gravitational fields, warping the spacetime around them. For an object of any given mass, there is a characteristic radius called the Schwarzschild radius with the property that if the object gets squeezed into a sphere of that radius or smaller it will become a black hole. If that happens it will continue to shrink until a short time later it forms a singularity. The Schwarzschild radius for an object of mass m is 2G m/c², where G is Newton's gravitational constant. For example the Earth, in order to become a black hole, would have to be squeezed into a sphere less than 5 mm across.

The event horizon of a black hole is a sphere at the Schwarzschild radius. Note that as the object inside the black hole collapses its mass remains constant, so the size of the event horizon remains fixed at the Schwarzschild radius 2G m/c². That means that if we were to observe the formation of a black hole we would see an object (typically a dying star) collapse down to within its Schwarzschild radius and disappear. The theory tells us that the object would continue to collapse after that, but without going in ourselves to take a look we can never directly observe that fact. From the outside, once a black hole has formed you can't tell what's going on inside it. This property is usually referred to by the whimsical phrase: Black holes have no hair.

Gravitational Lensing

When light moves from one medium to another—say from air to glass—it gets bent. This fact forms the basis for the idea of a lens, which can be used to focus light.

GR tells us that light rays (and everything else) get bent in the vicinity of a massive object. This means that a sufficiently massive object can act just like a lens, focusing the light from sources behind it. In fact this observation provided one of the earliest tests of GR. When the light from other stars passes near the sun it gets bent, which changes the apparent positions of the stars. (In effect the area of the sky behind the sun has been magnified to look bigger than it is.)

This effect is very hard to observe, first because the bending of the light rays is very small, but even more so because we can't normally see starlight close to the sun. We only see stars at night when we're looking away from the sun. The only exception to this is during a solar eclipse when the moon momentarily blocks the sun's light. Einstein first published the theory of GR in 1916. In 1919 there was a total solar eclipse and a team of scientists led by Arthur Eddington traveled to West Africa to observe the eclipse and check whether the starlight was deflected. The stars did appear to be slightly moved relative to their nighttime positions in exactly the way predicted by GR. This observation was the first crucial test of Einstein's theory.

This kind of magnification of a region of space is called weak gravitational lensing. If the object causing the lensing is massive enough, however, you can even bend the light so much that you see multiple images of another object behind it.

The production of multiple images in this way is called strong gravitational lensing, and has been observed dramatically in many instances by the Hubble Space Telescope among others.

Gravity Waves

In the theory of electricity and magnetism we know that a changing electromagnetic field can cause additional changes in the electromagnetic field in the surrounding space. (If you don't know this take my word for it. If you can't follow this paragraph you should still be able to understand the rest of the section.) The result is (in some circumstances) an electromagnetic wave that propagates through space reproducing itself continually. The original source of an electromagnetic field is charged matter, so if you take an electron and shake it back and forth to produce an oscillating field, you will start a wave that will radiate outwards into space.

In GR the same effect occurs for gravitational fields. Ripples in spacetime can induce other ripples in spacetime in such a way that you get a wave propagating through space. Thus if you take a heavy mass and shake it back and forth vigorously enough you will cause a spacetime ripple that will radiate outwards.

What would such a ripple look like? Say you were holding a spherical object as a gravity wave passed through where you were standing. The space around you would alternately stretch in two different directions, causing the sphere to get elongated first along one axis and then the other.

 
The effect of a gravity wave passing through a circular object

Such an effect could be detected in principle by measuring the time it takes a light beam to travel from one side of the sphere to another in different directions. In the presence of a gravity wave, the relative times in the two directions would oscillate.

In fact just such an experiment has been built recently. Instead of a sphere it uses two long perpendicular lines. These lines are actually evacuated chambers four kilometers long. Lasers are continually fired back and forth along both chambers. At the intersection of the two lines is a detector that can detect minute changes in the relative light travel times along the two paths. (For those familiar with optics the detector is actually a laser interferometer.) A regularly oscillating change would be the hallmark of a gravity wave. This experiment is called LIGO, or Laser Interferometer Gravitational Wave Observatory.

 
The Washington State LIGO observatory
(LIGO also includes a similar observatory in Louisiana)

To date neither LIGO nor any other experiment has ever detected a gravity wave. The reason for that is simply that gravity is weak. Even in the vicinity of the sun the bending of spacetime by gravity is a small effect. Moreover to get a gravity wave it's not enough to have a massive object; something has to shake that object back and forth very vigorously. What could produce such violent behavior?

There are several candidate sources for gravity waves. The most dramatic, and potentially easiest to detect, are closely orbiting massive objects. Imagine for example two very dense objects like neutron stars or black holes orbiting around each other at a very small distance. From the point of view of a distant observer each of these massive objects would be swinging back and forth in the sky as they moved through their orbits. The result would be the emission of a strong gravity wave.

Gravity waves, like electromagnetic waves, carry energy. That means that the orbiting objects mentioned above would steadily lose energy as they emitted these waves. When an object in orbit loses energy it tends to fall inwards towards the center of its orbit. Thus the two objects would spiral in towards each other. This effect would be very gradual at first, but the closer they got the faster the oscillations would be. That means the rate of energy loss from gravity waves would increase, causing them to spiral in faster and faster until they collided. It is hoped that the last stages of an inspiraling pair of massive objects would produce gravity waves strong enough for us to detect on Earth.

Another possible source for gravity waves is the early universe. In the first few seconds after the big bang, the universe was an extremely hot, dense soup of elementary particles. At the very beginning, a tiny fraction of a second after the big bang, that soup would have been so dense that any ripples propagating through it could have moved enough matter around to emit strong gravity waves. If such waves could ever be detected they might give us direct observational data about processes occurring in the universe within its first fraction of a second of existence!

The Expanding Universe

From the time of Newton until the development of GR most physicists assumed that the universe was essentially unchanging, or static. Of course things change on small scales—people are born and die, moons and planets move around, etc—but it was generally believed that the universe as a whole had always looked more or less like it does today.

Shortly after Einstein developed the theory of GR, he and others thought of applying it to the question of cosmology, the study of the large scale structure of the universe. The equations of GR describe the nature of space and time, so in a sense it was natural to ask what those equations said about the nature of things on the largest scales. The answer is that the equations have no solutions that are static on large scales. More specifically, the equations of GR predict that the universe must either be expanding or contracting.

This behavior essentially comes from the attractive nature of gravity. If you were to have a universe where all the stars were at rest relative to each other, their mutual gravity would cause them to start moving towards each other. In Newtonian physics, it was assumed that the universe went on forever and thus the attraction felt by any given star would be equally balanced on all sides. In GR you can show, however, that even in such a case the space as a whole will contract and the distances between the stars will shrink. (4) The universe could start out expanding, and depending on how fast it was expanding it might continue to do so or it might eventually stop and start contracting. It could never stay still, however.

A more complete description of what it means to say the universe is expanding or contracting would be beyond the scope of this paper. For a longer discussion of that issue see my paper on the Big Bang Model: The Expanding Universe. That paper also discusses the big bang itself and some of the evidence for the Big Bang Model in more detail than I do below. Here I simply note a few of the key ideas behind the model.

The conclusion that the universe couldn't be static seemed so implausible to Einstein that he attempted to modify the theory in order to allow static solutions. His modifications didn't work, however, and it remained an inescapable conclusion of the theory that the universe could not be static. In 1929, thirteen years after the publication of GR, Edwin Hubble observed that all distant galaxies appeared to be moving directly away from us in exactly the way predicted by GR for an expanding universe.

Conclusion—Open Questions

The theory of GR has brought about one of the most dramatic upheavals ever to occur in our understanding of the universe. Space and time, long considered to be a simple fixed background for all events, are now seen as dynamic, curving and changing in response to the matter and energy within them. Gravity is no longer viewed as a force but rather as a description of the geometry of the universe.

Nonetheless, while GR may be a beautiful theory, the ultimate judge of its value is not its aesthetic appeal but its ability to predict the results of experiments. Since the theory was first developed there have been a number of high precision tests of its predictions. I have already mentioned the precession of Mercury, the bending of starlight near the sun and galaxy images near large clusters, and the evidence for the expansion of the universe. Other pieces of evidence for the theory include a change in the speed of clocks near gravitational sources, the observation of objects believed to be black holes in the centers of galaxies (including our own), and more. Thus far in every case where an experiment or observation has been done to test a prediction of GR, the theory has been shown to be correct.

Meanwhile the early twentieth century saw another revolution in physics, the development of quantum mechanics. This theory completely changed our understanding of matter and energy. In quantum mechanics a particle is not seen as a simple dot existing at a particular place, but as a fuzzy wave existing as a collection of probabilities for where it could be and how it could be moving. A description of quantum mechanics would be beyond the scope of this paper, but for a general introduction see my paper with Kenny Felder on the topic: Quantum Mechanics: The Young Double-Slit Experiment.

Taken together these two theories—GR and quantum mechanics—form our best current understanding of the physical laws of the universe. The problem is that you can't take them together. Every attempt that has been made so far to reconcile the geometric view of spacetime in GR with the fuzziness of quantum mechanics has led to contradictions. The search for a single theory that could bring these two pieces together—a theory of quantum gravity—occupied Einstein for much of his life and is still one of the greatest outstanding challenges in science today.

There is currently one favored candidate called string theory, which essentially reworks quantum mechanics by treating particles as small strings rather than points. This simple idea has dramatic consequences for the theory, and it seems that it may be able to resolve the contradictions of quantum gravity. Unfortunately it's very difficult to do calculations in string theory, so the theory is still untested.

Regardless of whether or not string theory is correct, many questions certainly still remain. The nature of geometry on small scales where quantum mechanics is important is not understood. It's likely that some of our basic concepts of space, time, and causality may need to be radically changed on those scales. The history of the universe for the past fourteen billion years or so is pretty well understood, but what happened before that is not. Was there a big bang? What, if anything, happened before it? What will be the ultimate fate of the universe in the future?

In short, what Newton said about himself hundreds of years ago remains true of us today:

"I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me."

-Sir Isaac Newton

Footnotes

¹Technically the definition of a geodesic is more complicated than this, and it doesn't always correspond to a path of minimal distance. For our purposes in this paper, however, this definition is all we will need.

²Technically light beams inside matter don't move at the maximum speed c, so what I'm saying about light is only true in a vacuum. That fact is simply due to the nature of light and has nothing to do with the properties of spacetime. Whether you're inside matter or inside a vacuum the maximum speed is c, roughly 300 million meters per second.

³As we know from special relativity, the time it takes for a sequence of events to occur can be different from the point of view of different observers. When I say it takes a finite time for an object to reach the center of the black hole, I am talking about time as experienced by that object. In other words, if you fell into a black hole wearing an indestructible stopwatch, that stopwatch would still have some finite reading (depending on the size of the black hole) when you reached the center.

⁴It turns out that even in Newtonian physics a static universe would be unstable and the stars would eventually start collapsing in towards each other in different regions. This fact wasn't appreciated when the theory was developed, though.

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 “Hyperspace”, “subspace” and similar constructs are entirely in the realm of science-fiction, with no basis in reality.

Wormholes do have a basis in reality, but it is a tenuous one: While general relativity does allow for wormhole solutions, creating a wormhole, especially a stable wormhole, seems to require conditions that do not exist in our universe, such as matter with negative mass-energy density (so-called “exotic” matter).

Similarly, warp drives have an equally tenuous basis in reality, as such solutions are known in general relativity, but once again, realizing such a solution does not seem possible without exotic matter.

In any case, even if these solutions one day become reality, we are presently _very_ far from achieving them. Our faster spacecraft to date achieved, approximately, 0.01% of the speed of light or less. To use an imperfect historical analogy, it’s as if you were asking an ancient civilization that hasn’t even invented the sailing ship yet about the possibility of jet travel between continents. Except that the technology gap between sailing ships and jet airplanes is, in many ways, less than the technology gap between present-day spacecraft and hypothetical faster-than-light travel.

I understand it is conceptually valid to visualize the fabric of space-time as residing within a higher-dimensional “hyper-space”. This is called “embedding”.

To travel through any such “hyper-space” would require somehow “disengaging” mass-energy from our space-time fabric such that it was released “out” into the surrounding embedding hyper-spatial “Bulk”.

That would be like “Spherius” taking “A Square” out of 2D “Flatland” into the surrounding 3D realm. Fascinating to consider, but of dubious scientific practicality. As pancakes only lie flat on the griddle, but droop and flop about when picked up off of the same, it’s logical that any matter somehow extracted from our space-time and deposited “out “in any surrounding higher-dimensional “Bulk” would be vulnerable to catastrophic structural failure. And as pancakes, if they fold up and pleat, are difficult to put back flat on the griddle, so any hypothetical re-entry into our space-time from hyper-space could be very problematic, requiring some marvelous means of, for want of worthier words, “gently laying the pancake back flat on the griddle, right-side up (*not* upside down !!!), with no wrinkles”.

 Wormhole Theory infographic

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