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Relativity, in physics, the problem of whether and how physical laws and measurements change when considered by observers in various states of motion. Specifically the term appears in the work of the German physicist Albert Einstein, whose special theory of relativity (1905) and general theory of relativity (1916) are major milestones in the history of modern physics.
Special Theory of Relativity :
Einstein put forward the idea that there were no absolute speeds in the universe. He stated that whether we are moving at a constant speed relative to something else, or that the something else is moving at a constant speed relative to us, is not relevant. Either point of view must be acceptable.
Being in a state of constant speed causes no forces to be acting on you. Hence you feel nothing even though you are zipping through the Milky Way at 900,000 km/hr. You only feel something when you accelerate or decelerate to another speed. Just as the free-float frames of reference were special in General Relativity, the frames of reference where one travels at constant speeds are special in Special Relativity.
Einstein put forward two postulates that are encapsulated
in what is called
Light is special, in that all observers in constant motion relative to somebody else will see exactly the same speed for light in vacuum.
This postulate screws up space and time totally. If everyone is to see light as going 1 billion km/hr then everyone has to be measuring different space lengths and different time durations to get the velocity of light to come out so that everyone gets the the same 1 billion km/hr value.
Nobody can perform any experiment to show any differences in physical phenomena when they are traveling at constant speed relative to somebody else.
The laws of physics stay exactly the same when the observer is in constant motion, no matter how fast that motion is. If something blows up in a constantly moving frame of reference then all constantly moving frames of references will see the same thing blow up.
These two postulates essentially imply that whenever you move relative to someone else you enter into a distinct temporal universe. You will measure time differently from the other person and you will measure lengths differently from the other person. Things, however, will conspire in such a way that everybody will measure exactly the same value for the speed of light. Things will also conspire to never allow causes to precede effects. Special relativity preserves the causality of all situations.
The speed of light is a physically unattainable speed for
any object that has a mass. To accelerate a mass closer and closer to the speed
of light you need more and more energy. This can be interpreted as the mass of
the object getting larger and larger with speed.
special relativity theory
We calculate an object's momentum by multiplying its mass by how fast (and in what direction) it moves. Momentum cannot be created or destroyed, whether for a single object or a system of objects that interact with one another, unless another force comes into play. We can apply this tenet equally well to understanding the physics of billiards, collisions between asteroids, or the motions of a hundred billion stars in our Milky Way. Without force the total momentum of any system never goes up or down. Once begun, motion continues forever. This is the reason constant motion is the natural state of anything in the universe. This is the reason everything moves.
But is everything in the universe really moving? The newspaper on your coffee table, the tree in your yard, or the building you live in--don't they all stand still around you as you sit reading this book? In fact, they are still--to you. It's all relative.
Galileo and Newton both realized that the measurement of motion depends completely on your frame of reference. Suppose you see a unicyclist ride past at a certain speed. To him you move backward at that same speed, even though you think you're standing still on the sidewalk. If he is juggling at the same time, he sees the balls bob straight up and down from his hands. However, you see the balls move along forward arcs in space as he pedals past. Both viewpoints, or "frames of reference," are equally valid.
Similarly, let's say two cars approach each other on the road, each moving 50 miles per hour. If you stand on the sidewalk, in Earth's frame of reference, you see each car doing 50. But the driver of each vehicle sees the other car zoom toward him at 100 miles per hour. Velocities simply add together in the world of classical relativity as elucidated by Galileo and Newton. That's all well and good unless the cars somehow accelerate to millions of miles per hour. Then, this kind of relativity would fail. When we deal with the realm of superhigh speed, relativity takes on a special form.
In the late nineteenth century, Newton's laws of motion began to break down for objects that move very fast. The American physicists Albert Michelson and Edward Morley tried to add the speed of Earth's revolution around the Sun to the speed of a light beam using a sensitive light-measuring device called an interferometer. They were searching for signs of the "ether," an invisible and unmoving substance believed by physicists of the day to pervade the universe and carry waves of light. To their great surprise, the combined speed of Earth's motion and the ray of light was always exactly the same as that of light alone. Light did not seem to follow the known rules of Newtonian motion.
This puzzle lasted for nearly two decades. Then, in 1905 a theory that explained the startling result arose from the mind of 26-year-old Albert Einstein, a German physicist who worked by day as a patent officer in Switzerland. His mathematical treatise, innocently titled "On the Electrodynamics of Moving Bodies," presented a revolutionary idea that would become known as the special theory of relativity. Einstein asserted that the speed of light--186,282 miles per second--remains constant and can never be exceeded. Further, he said, speed is independent of how quickly an observer might move. Passengers on a spaceship traveling at 186,281 miles per second would still measure their headlight beams streaming away at the full speed of light. Observers on the ground would see the beams moving at exactly the same speed.
Special Relativity :
Classical physics owes its definitive formulation to the British scientist Sir Isaac Newton. According to Newton, when one physical body influences another body, this influence results in a change of that body's state of motion, its velocity; that is to say, the force exerted by one particle on another results in the latter's changing the direction of its motion, the magnitude of its speed, or both. Conversely, in the absence of such external influences, a particle will continue to move in one unchanging direction and at a constant rate of speed. This statement, Newton's first law of motion, is known as the law of inertia.
As motion of a particle can be described only in relation to some agreed frame of reference, Newton's law of inertia may also be stated as the assertion that there exist frames of reference (so-called inertial frames of reference) with respect to which particles not subject to external forces move at constant speed in an unvarying direction. Ordinarily, all laws of classical mechanics are understood to hold with respect to such inertial frames of reference. Each frame of reference may be thought of as realized by a grid of surveyor's rods permitting the spatial fixation of any event, along with a clock describing the time of its occurrence.
According to Newton, any two inertial frames of reference are related to each other in that the two respective grids of rods move relative to each other only linearly and uniformly (with constant direction and speed) and without rotation, whereas the respective clocks differ from each other at most by a constant amount (as do the clocks adjusted to two different time zones on Earth) but go at the same rate. Except for the arbitrary choice of such a constant time difference, the time appropriate to various inertial frames of reference then is the same: If a certain physical process takes, say, one hour as determined in one inertial frame of reference, it will take precisely one hour with respect to any other inertial frame; and if two events are observed to take place simultaneously by an observer attached to one inertial frame, they will appear simultaneous to all other inertial observers. This universality of time and time determinations is usually referred to as the absolute character of time. The idea that a universal time can be used indiscriminately by all, irrespective of their varying states of motion--that is, by a person at rest at his home, by the driver of an automobile, and by the passenger aboard an airplane--is so deeply ingrained in most people that they do not even conceive of alternatives. It was only at the turn of the 20th century that the absolute character of time was called into question as the result of a number of ingenious experiments described below.
As long as the building blocks of the physical universe were thought to be particles and systems of particles that interacted with each other across empty space in accordance with the principles enunciated by Newton, there was no reason to doubt the validity of the space-time notions just sketched. This view of nature was first placed in doubt in the 19th century by the discoveries of a Danish physicist, Hans Christian Orsted, the English scientist Michael Faraday, and the theoretical work of the Scottish-born physicist James Clerk Maxwell, all concerned with electric and magnetic phenomena. Electrically charged bodies and magnets do not affect each other directly over large distances, but they do affect one another by way of the so-called electromagnetic field, a state of tension spreading throughout space at a high but finite rate, which amounts to a speed of propagation of approximately 186,000 miles (300,000 kilometres) per second. As this value is the same as the known speed of light in empty space, Maxwell hypothesized that light itself is a species of electromagnetic disturbance; his guess has been confirmed experimentally, first by the production of lightlike waves by entirely electric and magnetic means in the laboratory by a German physicist, Heinrich Hertz, in the late 19th century.
Both Maxwell and Hertz were puzzled and profoundly disturbed by the question of what might be the carrier of the electric and magnetic fields in regions free of any known matter. Up to their time, the only fields and waves known to spread at a finite rate had been elastic waves, which appear to the senses as sound and which occur at low frequencies as the shocks of earthquakes, and surface waves, such as water waves on lakes and seas. Maxwell called the mysterious carrier of electromagnetic waves the aether, thereby reviving notions going back to antiquity. He attempted to endow his aether with properties that would account for the known properties of electromagnetic waves, but he was never entirely successful. The aether hypothesis, however, led two U.S. scientists, Albert Abraham Michelson and Edward Williams Morley, to conceive of an experiment (1887) intended to measure the motion of the aether on the surface of the Earth in their laboratory. On the reasonable hypothesis that the Earth is not the pivot of the whole universe, they argued that the motion of the Earth relative to the aether should result in slight variations in the observed speed of light (relative to the Earth and to the instruments of a laboratory) travelling in different directions. The measurement of the speed of light requires but one clock, if, by use of a mirror, a pencil of light is made to travel back and forth so that its speed is measured by clocking the total time elapsed in a round trip at one site; such an arrangement obviates the need for synchronizing two clocks at the ends of a one-way trip. Finally, if one is concerned with variations in the speed of light, rather than with an absolute determination of that speed itself, then it suffices to compare with each other round-trip-travel times along two tracks at right angles to each other, and that is essentially what Michelson and Morley did. To avoid the use of a clock altogether, they compared travel times in terms of the numbers of wavelengths travelled, by making the beams travelling on the two distinct tracks interfere optically with each other. (If the waves meet at a point when both are in the same phase--e.g., both at their peak--the result is visible as the sum of the two in amplitude; if the peak of one coincides with the trough of the other, they cancel each other and no light is visible. Since the wavelengths are known, the relative positions of the peaks give an exact measure of how far one wave has advanced with respect to the other.) This highly precise experiment, repeated many times with ever-improved instrumental techniques, has consistently led to the result that the speed of light relative to the laboratory is the same in all directions, regardless of the time of the day, the time of the year, and the elevation of the laboratory above sea level.
The special theory of relativity resulted from the acceptance of this experimental finding. If an Earth-bound observer could not detect the motion of the Earth through the aether, then, it was felt, probably any observer, regardless of his state of motion, would find the speed of light the same in all directions.
From P. Davies, 'About Time':
Betty is going to leave Earth in the year 2000 and travel by rocket ship to a star eight light-years away (as measured in Earth's frame of reference) at a speed of 240,000 kilometers per second. To keep the sums simple, I shall neglect the periods the ship spends accelerating and braking (i.e., treat these periods as instantaneous), and also assume Betty doesn't spend any time sightseeing when she reaches the star. To achieve 80 percent of the speed of light in a negligible time implies an enormous acceleration, which would be fatal to a real human, but this is incidental to the argument. I could easily include a more realistic treatment of the acceleration, but at the price of making the arithmetic more complicated; the overall conclusions would be unaffected.
First let me compute the total duration of the journey as predicted by Einstein for each twin. At 80 percent of the speed of light. it takes ten years to travel eight light-years, so Ann, on Earth, will find that Betty returns in Earth year 2020. Betty, on return, agrees that it is Earth year 2020, but insists that only twelve years have elapsed for her, and her rocket clock - a standard atomic clock carefully synchronized before takeoff with Betty's identical clock on Earth - confirms this assertion: it reads 2012.
Now suppose we equip our twins with powerful telescopes so that they can watch each other's clocks throughout the journey and see for themselves what is going on. Ann's Earth clock ticks steadily on, and Betty looks back at it through her telescope as she speeds away into space. According to Einstein, Betty should see Ann's clock running at 60 percent of the rate of her own clock. In other words, during one hour of rocket time, Betty is supposed to see the Earth clock advance only thirty-six minutes. In fact, she sees it going even slower than this. The reason concerns an extra effect, not directly connected with relativity, that is usually left out of discussions of thetwins paradox. It is vital to include the extra effect if you want to make sense of what the twins actually see.
Let me explain what causes this extra slowing. When Betty looks back I at Earth, she does not see it as it is at that instant, but as it was when the light left Earth some time before. The time taken for light to travel from Earth to the rocket will steadily increase as the rocket gets farther out in space. Thus Betty will see events on Earth progressively more delayed, because of the need for the light to traverse an ever-widening gap between Earth and rocket. For example, after one hour's flight as measured from Earth, Betty is 0.8 light-hours (48 light-minutes) away, so she sees what was happening on Earth forty-eight minutes earlier, for the light, which conveys the images of Earth to Betty, to reach her at that point in the journey. In particular, Ann's clock would appear to Betty-I'm referring to its actual visual appearance-to be slow anyway, irrespective of the theory of relativity. After two hours' flight, the Earth clock would appear to Betty to lag even more behind. This "ordinary" slowing down of clocks. and events generally, as seen by a moving observer, is called the "Doppler effect," named for a Swedish physicist who first used it to describe a property of sound waves. By adding the Doppler effect to the time-dilation effect, you get the combined slowdown factor.
Ann will also see Betty's rocket clock slowed by the Doppler effect, because light from the rocket takes longer and longer to get back to Earth. She will in addition see Betty's clock slowed by the time-dilation effect. By symmetry, the combined slowdown factor of the other clock should be the same for both of them.
Let me now compute the combined slowdown factor, first from Ann's point of view, then Betty's. To do so, I shall focus on the great event of Betty's arrival at the star. The outward journey takes ten years as measured on Earth. However, Ann will not actually see the rocket reach the star in the year 2010, because by this stage Betty is eight light-years away. Since it will take light a further eight years to get back to Earth, it will not be until the year 2018 that Ann gets to witness visually Betty's arrival at the star.
What is the time of the arrival event as registered on Betty's clock? Einstein's formula tells us that Betty's clock runs at 0.6 the rate of the clock on Earth, so ten years of Earth time implies six years in the rocket. The rocket clock therefore stands at six years on Betty's arrival at the star. So, when Ann gets to witness this arrival in 2018, the rocket clock says 2006. Thus, as far as the visual appearance of the rocket clock is concerned, Ann sees only six years having elapsed in her eighteen years-i.e., Betty's rocket clock has been running at one-third the rate of Ann's Earth clock. Now, Ann is perfectly capable of untangling the time-dilation and Doppler effects, and computing the "actual" rate of Betty's clock, having factored out the effect of the light delay. She will find the answer to be 0.6, in accordance with Einstein's formula. Thus Ann deduces (but does not actually see) that throughout Betty's outward journey Betty's clock was running at thirty-six minutes to Ann's hour.
From Betty's perspective, things are the other way about. She agrees, of course, that her rocket clock stands at 2006 when she arrives at the star, but what does she see the Earth clock registering at that moment? We know that in the Earth's frame of reference the arrival event occurs at 2010, but, because the star is eight light-years away, the light that actually reaches the rocket at that moment will be from eight years previously-i.e., 2002. So Betty will look back at Earth, on arrival at the star, and see the Earth clock registering 2002. Her clock says 2006. Therefore as far as the actual appearance of the Earth clock is concerned, it records two years having elapsed for Betty's six years. Thus Betty concludes that the Earth clock has been running at one-third the rate of her own rocket clock for the outward part of the journey. This is the same factor that Ann perceived Betty's clock to be slowed by, so the situation is indeed perfectly symmetric. Again, Betty can untangle the Doppler effect from the time-dilation effect and deduce that Ann's clock has "really" been running at 0.6 the rate of her own.
Without delay, Betty embarks on the return journey. Because Betty is approaching rather than receding from Earth the light-delay (i.e., Doppler) effect now works in opposition to the time-dilation effect. The former causes events to appear speeded, although time dilation still works to tell slow them down. Let's put the numbers in. First, what does Ann see as Betty speeds back towards Earth? Since we are agreed that Betty returns to Earth in the year 2020, and Ann actually sees Betty reach the star in 2018, the return part of the journey will appear to Ann, viewing the approach of the rocket from Earth, to be compressed into just two years of Earth time. We have already determined that, when, in 2018, Ann sees Betty's clock at the halfway point, it registers 2006, and that when Betty returns to Earth it will register 2012. So, for the two Earth years during which Ann sees the rocket traveling back, she will witness the rocket clock progress through the remaining six years. In other words, on the return leg of the journey Ann sees Betty's clock running three times faster than her own, Earthbound, clock. This is a key point: during the return journey the rocket clock appears from Earth to be speeded up, not slowed down. The Doppler effect beats the time-dilation effect. Again, Ann can untangle the time-dilation and light-delay effects and deduce that the rocket clock is "really" running at 0.6 the rate of her clock-i.e., although the rocket clock looks to Ann to be speeded up, she deduces that it is "really" running slow at exactly the same reduced rate (0.6) as it was on the outward journey. So, although the visual appearance of the rocket clock is quite different for the two legs of the journey, the time-dilation factor of 0.6 remains the same throughout.
Finally, let me examine the return journey as observed by Betty, in the rocket. She has experienced six years for the outward trip, and she experiences another six years for the return, reaching Earth in 2012 as registered on her own clock. During the return journey, however, Betty also observes the clock on Earth. She saw it (actually, visually) standing at 2002 at the moment she reached the star. We know she will get home I in 2020, so Betty will see the Earth clock progress through eighteen years during the six years aboard the rocket. Thus the Earth clock appears to the same factor as that by which Ann saw Betty's clock speeded up- there is complete symmetry on the return part of the journey too. Betty can again factor out the light-delay effect and deduce that the Earth clock is "really" running slow-at 0.6 of the rate of her rocket clock.
The crucial point to be extracted from all this is that during the periods when the rocket is traveling at a fixed speed Ann deduces that Betty's clock is running slow and Betty deduces that Ann's clock is running slow. On the outward part of the journey, each actually sees the other's clock running (even more) slowly, but on the return part of the journey each sees the other's clock speeded up. The deductions and experiences all fit together consistently, and refute the claim that there is any paradox attached to the statement that "each clock runs slow relative to the other.
For those readers who have waded through this arithmetic, it contains a hidden conclusion about distances. If you use the fact that in Betty's frame of reference Earth recedes at 0.8 of the speed of light, and the journey to the star takes just six rocket years, then the distance to the star as measured by Betty must be 0.8 x 6 = 4.8 light-years. Thus, although Ann measures the star to be eight light-years away, Betty measures the distance to the star to be only 4.8 light-years. The distance is shrunk by the same factor (0.6) as that by which time is dilated.
what is the Time?
Einstein wrote when his friend Besso died, "For us believing physicists, the distinction between past, present, and future is illusion, however persistent."
Here is another spacetime diagram, this time from D. Postle, Fabric of the Universe, pg. 106:
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.
To avoid the two
logical problems given above, it is mandatory to allow for the existence of
multiple worlds. It turns out that the theory of physics, called quantum
mechanics, that governs all physical phenomena on the smallest scales can be
interpreted literally as a theory of parallel universes. Hence, quantum
theory is quite consistent with the idea that time travel can exist without
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.
Dogen Zenji seemed to have a similar view 800 years ago. "It is believed by most that time passes; in actual fact it stays where it is. This idea of passing may be called time, but it is an incorrect idea, for since one only sees it as passing, one cannot understand that it stays just where it is. In a word, every being in the entire world is a separate time in one continuum." -- Shobogenzo.
Spacetime, in physical science, single concept that recognizes the union of space and time, posited by Albert Einstein in the theories of relativity (1905, 1915).
Common intuition previously supposed no connection between space and time. Physical space was held to be a flat, three-dimensional continuum--i.e., an arrangement of all possible point locations--to which Euclidean postulates would apply. To such a spatial manifold, Cartesian coordinates seemed most naturally adapted, and straight lines could be conveniently accommodated. Time was viewed independent of space--as a separate, one-dimensional continuum, completely homogeneous along its infinite extent. Any "now" in time could be regarded as an origin from which to take duration past or future to any other time instant. Within a separately conceived space and time, from the possible states of motion one could not find an absolute state of rest. Uniformly moving spatial coordinate systems attached to uniform time continua represented all unaccelerated motions, the special class of so-called inertial reference frames. The universe according to this convention was called Newtonian.
By use of a four-dimensional space-time continuum, another well-defined flat geometry, the Minkowski universe (after Hermann Minkowski), can be constructed. In that universe, the time coordinate of one coordinate system depends on both the time and space coordinates of another relatively moving system, forming the essential alteration required for Einstein's special theory of relativity. The Minkowski universe, like its predecessor, contains a distinct class of inertial reference frames and is likewise not affected by the presence of matter (masses) within it. Every set of coordinates, or particular space-time event, in such a universe is described as a "here-now" or a world point. Apparent space and time intervals between events depend upon the velocity of the observer, which cannot, in any case, exceed the velocity of light. In every inertial reference frame, all physical laws remain unchanged.
A further alteration of this geometry, locally resembling the Minkowski universe, derives from the use of a four-dimensional continuum containing mass points. This continuum is also non-Euclidean, but it allows for the elimination of gravitation as a dynamical force and is used in Einstein's general theory of relativity (1915). In this general theory, the continuum still consists of world points that may be identified, though non-uniquely, by coordinates. Corresponding to each world point is a coordinate system such that, within the small, local region containing it, the time of special relativity will be approximated. Any succession of these world points, denoting a particle trajectory or light ray path, is known as a world line, or geodesic. Maximum velocities relative to an observer are still defined as the world lines of light flashes, at the constant velocity c.
Whereas the geodesics of a Minkowski continuum (without mass-point accelerations) are straight lines, those of a general relativistic, or Riemannian, universe containing local concentrations of mass are curved; and gravitational fields can be interpreted as manifestations of the space-time curvature. However, one can always find coordinate systems in which, locally, the gravitational field strength is nonexistent. Such a reference frame, affixed to a selected world point, would naturally be in free-fall acceleration near a concentrated mass. Only in this region is the concept well defined--i.e., in the neighbourhood of the world point, in a limited region of space, for a limited duration. Its free-fall toward the mass is due either to an externally produced gravitational field or to the equivalent, an intrinsic property of inertial reference frames. Mathematically, gravitational potentials in the Riemannian space can be evaluated by the procedures of tensor analysis to yield a solution of the Einstein gravitational field equations outside the mass points themselves, for any particular distribution of matter.
The first rigorous solution, for a single spherical mass, was carried out by a German astronomer, Karl Schwarzschild (1916). For so-called small masses, the solution does not differ appreciably from that afforded by Newton's gravitational law; but for "large" masses the radius of space-time curvature may approach or exceed that of the physical object, and the Schwarzschild solution predicts unusual properties. Astronomical observations of dwarf stars eventually led U.S. astrophysicists J.R. Oppenheimer and H. Snyder (1939) to postulate super-dense states of matter. These, and other hypothetical conditions of gravitational collapse, were borne out in later discoveries of pulsars and neutron stars. They also have a bearing on black holes thought to exist in interstellar space. Other implications of space-time are important cosmologically and to unified field theory.
Einstein's theory of general
relativity and curved space-time
A dramatic example of non-inertial frames can been seen in accelerating rockets.
When a rocket accelerates, an occupant feels "gravity".
Consider a rocket with a small "window" in one side and a major league pitcher with amazing accuracy.
An outside observer sees a horizontally thrown ball travel in a straight line. Since the ship is moving upward while the ball travels horizontally, the ball strikes the wall somewhat below a point opposite the window.
To an inside observer, the path of the ball bends as if in a gravitational field.
From this thought experiment we realize that we cannot tell the difference between an accelerating frame and one that is at rest and has a gravitational pull. This is the "equivalence principle" and is the corrnerstone to Einstein's theory.
Now replace the ball with a light beam. Again, to the observer in the rocket it will apear to have it's path "bent."
This tells us that gravity "bends" light
beams. Stars, including our sun are massive enough to bend light from
But, wait a minute... doesn't light travel
in a stright line? How can this be?
Perhaps space is somehow "curved" If this is the case then light still travels in a stright line, but now the idea of "striaght" has to change since our surface is no longer flat. Think about how we define parallel lines on a flat surface and one a sphere.
Einstein's theory of special relativity talks about how space and time are connected. This theory deals with objects and reference frames moving at constant speeds. His theory of general relativity deals with accelerating objects (which includes gravity). The mathematics of special relativity isn't too bad, but the mathematics of general realtivity is a matter for a graduate level course. Both theories are well establish experimentally.
Principle of Equivalence
Mathematical aspects of relativity
Einstein's theory of general relativity describes the gravitational field in terms of the curvature of spacetime. The precise form of the curvature is determined through Einstein's equations which form a set of 12 coupled hyperbolic partial differential equations. Because of the complexity of these equations exact solutions can only be found by looking for spacetimes with a high degree of symmetry. An example of such a spacetime is the Schwarzschild solution which describes a spherically symmetric black hole with a gravitational singularity at its heart. However when the Schwarzschild solution was first discovered most physicists believed that the existence of the singularity was the result of the high degree of symmetry. By using methods from differential geometry and topology Roger Penrose showed that singularities are general features of gravitational collapse and do not require spherical symmetry. This work was then extended by Hawking and Penrose to establish a number of singularity theorems which showed that gravitational singularities occur in many situations.
A feature of the singularity theorems is that they do not directly show the existence of black holes. Instead they show that spacetime is geodesically incomplete so that the worldline of an observer comes to an end and cannot be extended. The obstruction to extending the worldline is some kind of singularity, but this might be rather mild and need not correspond to a black hole. Research at Southampton is trying to improve on the singularity theorems to show that they predict the existence of genuine gravitational singularities. This work involves studying the structure and geometry of weak singularities. One can then apply Einstein's equations to a larger class of spacetimes so that points that were previously regarded as weak singularities can now be regarded as interior points of a more general spacetime. This work allows one to develop a new definition of gravitational singularity in which singularities are regarded as obstructions to the evolution of Einstein's equations rather than obstructions to extending geodesics. Mathematically this involves developing a new mathematical theory of non-linear generalised functions, which is essentail due to the inherent non-linearity of the gravitational field equations.
The main references on this research topic are
and Time Warps Cont...
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