THE GENERAL THEORY OF RELATIVITY

Written for students in the USC Self-paced Astronomy courses

NOTE: This Unit assumes you have studied Unit 56.

The Learning Objectives and references are in the Self-Paced Study Guide

Essay on the General Theory of Relativity

by John L. Safko

A. General Principle of Covariance (or Only the Tides are Real)

Consider yourself in an elevator. You cannot see outside, so you must determine the nature of the surrounding universe by local experiments. You let go of a coin and it falls to the bottom of the elevator. Aha!, you say, I am at rest on Earth. But, you could be in a spaceship that is accelerating and far from any other object. This is shown in Fig. 57-1.

 

Fig. 57-1:
Locally being at rest on the Earth's surface is equivalent to being in a uniformly accelerated spaceship.

Consider the opposite case. You float from the floor and the coin does not fall when you release it. Aha!, you say again, I am in space far from any other body. But, you could be freely falling towards the Earth as shown in Fig. 57-2.

 

Fig. 57-2:
Locally freely falling towards the Earth is equivalent to being at rest with respect to the distant stars far from any gravitating body.

We see that gravity is different than other forces. You can make gravity completely disappear in small regions by freely falling. This means that a free fall frame is a perfectly good inertial frame. The only way we can detect the difference is to look for tidal forces which arise if the gravitational field is not perfectly uniform. But for any real gravitational field we can always make the region we consider (our elevator in this case) small enough so we cannot detect the tidal forces. So:

"For sufficiently small regions, the special theory of relativity is correct!!"

Einstein used these ideas to conclude that the laws of physics should be independent of the coordinate system used. Another way of saying this is that the laws of physics are generally covariant. Of course it is obvious that some coordinate systems may make a physical situation easier to describe and predict than others. All frames, however, are equally valid.

B. Gravity as Curved Spacetime

A region of spacetime where, in the old Newtonian view, gravitational forces exist, can thus be broken into smaller regions where an inertial frame is defined and the special theory of relativity works. There is no single inertial frame for the entire region.

Consider our elevator example again, only this time cut small windows in the elevator so light can pass through the elevator. Since we cannot tell the difference between being at rest in empty space or freely falling in a gravitational field, the light will pass through on a straight line, as shown in Fig. 57-3. Uniform acceleration would make the light appear to be on a curved path, since the elevator moves as the light passes through. But, we have argued that uniform acceleration is the same as being at rest in a gravitational field. Light passing through an elevator at rest in a gravitational field would appear to be on a curved path as shown in Fig. 57-4. The conclusion is that a gravitational field deflects light.

 

Fig. 57-3:
If you are at rest in a box, far from any gravitating body, light should pass through your box on a straight line. So light passing through the same box, if it were freely falling towards the Earth, would also appear to move on a straight line.

 

Fig. 57-4:
If you were uniformly accelerated in the illustrated box, you would expect light passing through your box to be deflected. This means that, if you were at rest in a gravitational field, the light would also be deflected. So a gravitational field deflects light.

Gravity has a local property -- if you freely fall, you no longer feel the effects of gravity. Gravity also has a global property. In the presence of gravity, two freely falling bodies will separate or approach each other (the tides). The only way we can reconcile the local property and the global property of gravity is to give up the geometry we assumed for spacetime. What does this mean?

Consider the following question. What do we mean when we say a line is straight? We probably mean the shortest path. The only way we could determine this is to use light to define a "straight" line. But we have seen that light may travel on curved paths. This forces us to generalize the idea of a "straight" line to what is called a geodesic and to generalize the geometry of spacetime. An example of a geodesic is a great circle on a sphere, such as a line of constant longitude. Another example would be a line of constant Right Ascension on the celestial sphere. A geodesic is the path that has the shortest distance between two given points.

The spacetime of the General Theory must locally have the same properties of the Special Theory. Light must travel on a null geodesic. A material object travels on a timelike path. If it is in free-fall, this path is a timelike geodesic. (See Unit 56.)

On a larger scale, gravity bends or distorts spacetime. These geodesics are the shortest path in this distorted spacetime. The interval of the Special Theory, ds, which was written as

ds2 = (c dt)2 - [(dx)2 + (dy)2 + (dz)2],

now becomes, in general,

ds2 = gtt (c dt)2 + 2 gtx dx dt + 2 gty dy dt + 2 gtz dz dt + gxx (dx)2 + 2 gxy dx dy + 2 gxz dx dz + gyy (dy)2 + 2 gyz dy dz + gzz (dx)2,

where the "g" terms could all be functions of t, x, y, and z. This very messy expression is just all possible combinations of the changes in the coordinates, taken two at a time, multiplied by functions of the coordinates. This is called a quadratic form.

For the special theory, the coordinates can be chosen so to make the "g"s all constant and zero except gtt = 1, gxx = gyy = gzz = -1. The "g" quantities are called the metric of the space in that coordinate system. The principle of equivalence forces us to conclude that although the "g"s are different in another coordinate system, they are describing the same spacetime. So the "g"s are describing something independent of the coordinate system -- the geometry of spacetime.

 

C. The General Theory of Relativity

When Einstein realized the preceding, he was able to take over an existing mathematical structure. This structure was the theory of Reimannian geometry which, until that time, was thought to be an abstract mathematical structure with no physical uses. With the mathematical tools of Riemannian geometry, Einstein was able to formulate a theory that predicted the behavior of objects in the presence of gravity, electromagnetic, and other forces. This theory is called the General Theory of Relativity. Avoiding the mathematical details, the theory gives relations, called the field equations, that say:

properties of the geometry = properties of the non-gravitational forces present. . . . . .(57-1)

 

In order to express the appropriate properties of the non-gravitational forces, we must also use the geometry on the right hand side of this relation.

Many physical theories are linear. By this we mean that if you add two sources, the resulting solution is the sum of the two solutions produced by the sources independently. The General Theory is highly non-linear since the geometric properties needed are non-linear and the geometry also appears on the right hand side of Eq. 57-1, as we define the needed properties of the non-gravitational forces. The results of non-linear theories can not be predicted by considering only small effects. For a non-linear theory the sum of two sources may produce a resultant solution which bears no resemblance to the individual solutions from the sources considered one at a time.

John Wheeler has described the results of solving Eq. 57-1 by saying

"Matter tells spacetime how to bend and spacetime returns the complement by telling matter how to move."

The General Theory is geometrical, that suggests drawing pictures to show what is happening to the geometry. Your text shows some such pictures near the end of Chapter 19. Figs. H 19-28 and 19-31 show what are called embedding diagrams. Two of the space dimensions are shown. The third dimension is not a space dimension. It is an attempt to show how the geometry differs from Euclidean. The bending is a measure of the curvature or distortion of space from flat. These diagrams can be very helpful in understanding what is happening, but don't let them mislead you.

The next step Einstein wanted to take was to completely eliminate the right hand side and express the entirety of physics as geometry. A theory expressing physics in terms of only one "object" (field) is called a unified theory. Except in special cases, neither Einstein nor anyone else has yet been able to find a theory that unifies gravity with the three other known natural forces.

Before we consider the experimental evidence that is consistent with the General Theory and some of the surprising predictions of the theory, let us briefly consider what has happened to our view of the nature of physical reality as we have taken the cosmic voyage. We developed the Newtonian world view, generalized it to the static spacetime of the Special Theory of Relativity and have now described the dynamic spacetime of the General Theory. These developments may seem very revolutionary, but they are evolutionary. The General Theory contains as a subset of its solutions the Special Theory. The General and Special theories contain as a subset of their solutions the solutions of the Newtonian theory. We have not given up concepts; we have only generalized them.

D. Tests of the General Theory

When first proposed, the general Theory of Relativity had no direct experimental underpinning. Now, many of us use equipment that could not work without using the General Theory. For example, the Global Positioning System (GPS) must use the predictions of the Special and General Theories. The GPS allows you, for a few hundred dollars, to buy a hand held instrument which can display your longitude, latitude and altitude to within 16 meters. (Or, for more money, you can get even better accuracy.) It also gives the time to within a few billionths of a second. The GPS consists of a set of 24 Earth-orbiting satellites with one or more atomic clocks. The entire system can only work if the predictions of the Special and the General Theories of Relativity are correct for weak gravitational fields.

 

D1. The Original Tests

Originally, three tests of the General Theory were proposed, whose results the theory seemed to properly predict. Between the early 1920's and the early 1960's, little experimental work occurred except to refine the measurements on the three experimental tests. Only in the last few years has the experimental side of general relativity blossomed. We will first discuss the three tests, which are called the classical tests of relativity, and then consider some recent developments.

The three classical tests of the general theory are

  1. the precession of the perihelion of Mercury
     
  2. the deflection of light
     
  3. the gravitational red shift of light. (This tests only one aspect of the theory.)
     

Ideally, the orbit of a single planet about a star is an ellipse fixed in space. The presence of other planets changes (perturbs) this orbit as was discussed in Unit 17. For all natural orbits in the solar system. these changes are small. In the case of Mercury, we can consider the perturbed orbit as an ellipse which slowly precesses (rotates in its own plane) as is shown in Fig. 57-5 or in the text in Fig. H 19-31a. The orbit of Mercury is observed to precess about 5,600 seconds of arc per century. Since Mercury orbits the Sun about 700 times in a century, this is a small change in the orbit per orbital period. Newtonian physics could predict all this precession except 43 seconds of arc per century. The 43 seconds of arc was what the General Theory of Relativity predicts.

 

Fig. 57-5:
The orbit of Mercury is perturbed by the presence of the other planets and be small effects predicted by the General Theory. The observed precession of the perihelion is about 5,600 seconds of arc per century. After subtracting the Newtonian perturbations caused by the other planets, 43 seconds of arc remain. This is the amount predicted by the General Theory.

The second prediction is the deflection of light in a gravitational field as shown in the text Fig. H 19-31d. Since the effect is so small in the solar system, it can only be detected for light that just grazes the Sun. This is shown in Fig. 57-6. Solar eclipse expeditions took photographs that verified this prediction. Nowadays, with radio telescopes, we can measure this effect very accurately since the Sun occults several quasars and pulsars each year.

 

Fig. 57-6:
The deflection of light passing near the Sun. The figure highly exaggerates the 1".75 predicted by the General Theory.

The third prediction of the theory is light should lose energy as it climbed out of a gravitational field as shown in the text Fig. H 19-31b. This was verified in the spectra of some red stars; however, there was a lot of noise in the experimental data. The astronomical results are shown schematically in Fig. 57-7. In the late 1950's the effect was accurately verified by measuring the wavelength of light as it traveled up or down a tower on Earth.

 

Fig. 57-7:
The gravitational red shift of light was first measured in the spectra of cool red dwarf stars. Accurate measurements were made on the Earth's surface by sending light up and down a tower. The Mossbauer Effect, which allows the frequency to be measured very accurately, was used.

 

D2. Modern Experimental Tests

With the developments produced in the space age, there have been many new tests posed for the general theory of relativity. The theory seems to be meeting the tests carried out so far. Among the tests are details of the motion of the Moon as the Earth-Moon orbits the Sun, the time delay in light signals passing near the Sun; the motion of binary stars as they produce gravitational radiation, and the apparent existence of black holes in stellar and galactic systems. Among the proposed tests are the actual detection of gravitational radiation from supernovae and the predicted precession of gyroscopes in Earth orbit.

 

D2a. Gravitational Time Delay

The General Theory not only predicts a deflection of light as the light passes near a gravitating body, it also predicts that it should take the light longer to pass through the region near the star. The geometrical reason for this is shown in the text Fig. H 19-31c. This gravitational time delay was first measured in 1968 by I. Shapiro using radar signals reflected from the surfaces of Venus and Mercury. Since Mercury and Venus were near opposition when the experiments were done, the signals passed near the surface of the Sun, giving the greatest relativistic effects. Using a later launch of a Mariner probe, a transmitter on the probe bounced a signal off of the planets' surfaces as well as sending a signal directly to the receivers on Earth. Since the position of the planets were known better than the position of the Vikings, this improved the accuracy. When the Viking probes landed on Mars, the results were even more precise. Since then, the experiment has been repeated with other space probes and with the signals from the few pulsars that are occulted by the Sun. The pulsar timing signals, like the signals from the space probes, arrive slightly later than they would have if the Sun were not present. The results are in agreement with the predictions of the General Theory..

 

D2b. Gravitational Radiation

Another prediction of the General Theory is that moving a mass should produce gravitational radiation, just as moving an electric charge produces electromagnetic radiation (light). Unless the masses are moved at relativistic speeds the radiation produced is very weak. The least radiation produced by gravity is a quadrapole radiation, as shown in Fig. 57-8, rather than the dipole which can be produced by electromagnetic sources. This occurs because only positive mass seems to exist. Electric charges come with either positive or negative sign, allowing a simpler radiation pattern.

 

Fig. 57-8:
 

  1. The dipole distortion of a ring of charges through which a plane electromagnetic wave is passing.
     
  2. The quadrapole distortion of a ring of matter through which a plane gravitational wave is passing.

 

Gravitational radiation can be detected, in principle, on Earth by detecting sub-nuclear, but coherent, displacements in a massive block of material. A number of such detectors have been built. So far, what they have detected has not been confirmed as gravitational radiation. This is not surprising since the radiation should be very weak. They should have detected the gravitational radiation produced by Supervova 1987a in the Small Magellanic Cloud. (Shown in Fig. 57-9 as it first appeared and as it appears in 1995) However, all the groups had shut down for repair and improvements at the time the supernova occurred. They had shut down at the same time to avoid the possibility that one would detect a signal without independent verification by another group. It remains to be seen if this procedure will be followed in the future.

It should also be possible to detect changes in optical path lengths as gravitational radiation passes through an interferometer causing one arm to expand more than the other. One such interferometer is shown in Fig. 57-10. A set of devices, the LIGO (Laser Interferometry Gravity Observatory) project, are under construction and testing in California and Louisiana. Hopefully they will be functional by 2005.

     Laser Interferometry Gravity Observatories

Fig. 57-10:
A sketch of one of the proposed Laser Interferometry Gravity Observatories. The ground was broken for construction in 1995.

Another possible method of detecting gravitational radiation is to examine the behavior of the source of that radiation. The argument is that in a close binary star system we would expect gravitational radiation to occur and the orbit of the stars about each other to decay as the gravitational radiation removes energy from the system. Several such systems have been studied whose models seem to be in agreement with theory. There are, however, two star systems, DI Her and AS Cam, whose behavior seems inconsistent with the theory. Since the observed data must be used to fit a model of these systems, it is not clear what is occurring. Future studies will resolve these problems either in favor of or against the general theory.

 

D2c. Gyroscope Precession

Both the special and the general theories predict that the axis of a rotating body that is orbiting another body should precess. The general theory predicts a slightly larger precession. The effect on the Earth's, or other planetary, axis is masked by irregularities in the rotation and classical precession of these axes.

One possible way of detecting this effect is to put a carefully shielded superconducting gyroscope in Earth orbit. Such a gyroscope is currently under construction at Stanford. It was originally scheduled for launch in 1986. This schedule has slipped to a current launch date after 2000.

 

E. Black Holes and Stellar Collapse

One of the more esoteric predictions of the general theory is the existence of black holes. A black hole is an object who mass is so large that light cannot escape from its surface. A simple argument for the existence of black holes is given in the next paragraph. As we shall see, the black holes predicted by the general theory are much more complicated.

 

E1. A classical argument for the formation of a Black Hole

In Units 3 and 13 we discussed the idea of escape velocity. For a body of mass M and size given by a radius R, the minimum velocity for a small body to escape from the surface is given by

vescape = square root of(2GM / R). . . . . . .(57-2)

Laplace argued in the late 1700's, that if a star had enough mass for its size and if the speed of light is finite, we would expect that if we increased the mass of a star sufficiently light could not escape from its surface. Using c to represent the speed of light, M the mass of the star and R the radius of the star, Laplace was arguing that

c = vescape = square root of(2GM / R) . . . . .(57-3)

gives the mass to size ratio needed for a black hole. Combining Eqs. 57-1 and 57-2 gives the condition that a star traps all its emitted light when

 

(2GM) / (Rc2) = 1. . . . . . . (57-4)

 

E2. The Exterior of a Spherical Black Hole

When we solve the exact spherically symmetric solution for a non-rotating source of gravity in the full General Theory of Relativity, we obtain the same result for the formation of a black hole. This R is called the Schwarzchild radius. For reasons we will discuss, the surface surrounding the source at R is called the event horizon.

If we use the mass of the Sun in Eq. 57-4, we find that a black hole with one solar mass would have a radius of 2.9 km. We can then divide Eq. 57-2 by itself using the solar values in the second case and expressing masses in solar masses and radii in km to get

R = 2.9 km (M / MSun). . . . . . .(57-5)

For example, a 3.0 solar mass star would have a Schwarzchild radius of

R = 2.9 km x 3.0 = 8.7 km.

The black hole found in the general theory is much more complicated than Laplace's argument would suggest. It is not possible to measure the diameter, and hence the radius, by sticking a meter stick from one side to the other. The surface is a one way membrane that can only be entered, not exited. Even light cannot exit from inside this surface (see text Fig. H 19-27). So you could push in as long a stick as you can get and it will never get out, nor could you pull it back out.

This is expected, since you would have to move faster than light to exit from a distance closer than R from the center. Since even light cannot escape, no information about what is happening inside the surface can be communicated to the outside; that is, no events can be observed inside this surface. This is why we say the Schwarzchild radius defines an event horizon.

We define the radius, R, as the number defined by the area of the surface which just contains the event horizon. If we call this area Ao, then R is defined by

Ao = 4(Pi)R2.

Another property of the event horizon, which could not have been anticipated by Laplace, is the behavior of time near the horizon as as determined by different observers. To discuss this we must be careful to distingush a measurement with a given set of coordinates from what is "seen". We have already considered this problem in the Special Theory (Unit 56). Consider a freely-falling body which starts from rest far from the black hole and falls radially towards it.

The appropriate coordinates for a distant observer to use are called Schwarzchild coordinates. In these coordinate both space displacement and time displacement are distorted by the geometry near the horizon such that an infinite coordinate time will pass before the falling object will reach the horizon. Thus, a distant observer measures a slowing of the body as it reaches the event horizon. To this distant observer the body would, according to her coordinates, never cross the event horizon in a finite time.

An observer at rest near the horizon would measure the velocity of the falling body to be nearly the speed of light as it passes. An observer on the body will feel nothing unusual, except for possible tidal effects, as he crosses the event horizon it a finite time according to his clock and measures his relative speed as less than light.

This apparent paradox can be resolved by examining the full mathematical structure of the theory. What the distant observer says is an infinite time is only a finite time for an observer at rest near the horizon and for the observer on the falling body. As in Unit 56, note the difference between a measurement and "seeing."

 

E3. Collapse of a Spherical Star

Suppose we consider the spherical collapse of a non-rotating star. This is a highly simplified case, but it will show some of the features of a rotating, not quite spherical star. As a star collapses, the emitted light is red shifted and non-radial paths are curved to a greater and greater extent until the star reaches the diameter at which light emitted tangentially goes in circular orbit about the star. Your text shows this in Fig. H 19-31. This tangentially emitted light is trapped in circular orbit about the star. The light so trapped is called the photon sphere. As the star continues to collapse, less and less of the non-radial light emitted from the surface can escape. At the Schwarzchild radius even the radial light does not escape as shown in Figs. 57-11 and 57-12.

 

Fig. 57-11:
A sketch showing the formation of an event horizon as a star collapses to form a black hole Time is vertically upwards.

 

Fig 57-12:
A schematic of a black hole showing the singularity, the event horizon and the photon sphere. The photon sphere is the distance from a black hole that light emitted tangentially is just able to make a circular path. Any closer to the black hole and tangentially emitted light will spiral into the black hole.

If the black hole was formed by a collapsing star, the star, which is now interior to the event horizon, continues to collapse to the center, forming a singularity. Assuming the tidal effects outside the event horizon were not too large, an observer on a freely falling body would cross the event horizon without difficulty and in a finite time according to her clock. She would then have only a short amount of time before she comes too close to the center of the black hole and is crushed by the tidal forces as is shown in Fig. 57-13.

 

Fig. 57-13:
The event horizon of a Schwarzchild black hole may present no immediate problem; but, once you cross the event horizon, you have no way to go but towards the center. Getting too close to the center will result in tidal forces which can not be neglected. This will happen outside the event horizon for a small black hole. These forces are real, your feet will be accelerated relative to your head and your left and right sides will be squeezed together.

The distant gravitational field of a spherical star remains constant as this collapse occurs. Even after the star crosses the Schwarzchild radius the external curvature of spacetime remains. To a planet in orbit about the star before the collapse, the external gravity appears unchanged and its orbit is unchanged. In some sense the star becomes like the Cheshire cat in "Alice in Wonderland". The star (cat) is gone but its gravitational field (the grin) remains.

The star continues to collapse inside the Schwartzchild radius until it reaches a singularity at the center of symmetry. Some who study astrophysics suggest that at some point in this final collapse there may be new physics that prevents the singularity from forming. This may be a valid argument, but it does not prevent the event horizon from forming. A massive enough star will form the event horizon long before the density of the star reaches nuclear density. We have a good understanding of the behavior of matter at nuclear densities, so new physics will not prevent the formation of the event horizon and thus the black hole in those cases.

Since the black hole is much smaller than the star that formed it, it is possible for bodies to approach much closer to the center of the star, but to still stay outside the event horizon. We find that many of these new possible orbits are unstable. Any material body must move slower than light. Bodies on such orbits will spiral in towards the black hole and eventually be absorbed. Some orbits make many rotations about the black hole before they begin the spiring process. Orbits that existed before the collapse began are not effected; they remain as stable orbits.

At some point, either inside or outside the event horizon, the tidal effects will become large enough to tear the body apart. The most distant part of the body will be accelerated less than the part towards the center and the left and right sides will be squeezed together. These real forces will tear the body apart as was shown in Fig. 57-13 for a person. If it were atoms approaching the black hole, this tidal force makes an atom scream (radiate), just as you would. This will later be shown to provide a method of detecting a black hole.

F. Rotating Black Holes

If the collapsing star has some rotation, the nature of the collapse is changed, but the collapse not prevented. Information on the star¹s structure is radiated away. All that remains is the mass, the angular momentum, and possibly (but unlikely) the net charge. The rotating black hole is often called a Kerr black hole after the scientist who first formulated this solution to the field equations .

 

Fig. 57-14:
A schematic of the horizons and regions around a rotating black hole. There are two event horizons and two surfaces of infinite red shift. The singularity is now a ring about the axis. (after D"Inverno: Introducing Einstein's Relativity)

The resulting structure, shown in Fig 57-14, is much more complicated. There are two event horizons and two surfaces of infinite red shift. Each horizon touches one infinite reds shift surface at the axis of rotation Between the outer horizon and the outer surface of infinite red shift there exists a region called the ergosphere. In this ergosphere, a real particle must orbit in the direction of rotation of the black hole. Even light cannot travel against the rotation in the ergosphere.

Suppose you are in the ergosphere moving in the direction of rotation. Then you can enter and exit the surface of infinite red shift. If, while you are inside the ergosphere, you throw some mass towards the black hole, you can exit the outer surface with more energy than you had when you entered. This energy is provided by the rotational energy of the black hole. The black hole has less angular momentum after this interaction.

If you cross the outer event horizon, you must also cross the inner ecent horizon. You can enter but not exit. You will be forced towards the interior singularity. This singularity is not a point but is a ring in the plane perpendicular the the axis. Again the tidal forces become very large as a body approach the singularity, which would lead to the body's destruction.

The theory suggests that there are geodesics which pass through the ring and exit in another universe avoiding the singularity. This has been used as the deux ex machina in many science fiction books and movies. Studies of the formal solutions have shown that the presence of any finite sized body cuts off these possible paths. If something crosses the inner event horizon, it must eventually hit the singularity and be destroyed in the process.

G. Evidence for Black Holes

How can we detect an object which does not radiate? The answer lies in the behavior of light and matter near a black hole. If a black hole nearly lines up with a background star, we will see a displacement of the apparent position of the star or even multiple images. A very nearby black hole would even show as a black disk as illustrated in the computer generated Fig. 57-15. Black holes or other gravitating sources can even generate a ring of light if the positioning is exact.

 

Fig. 57-15:
 

  1. Left: A computer generated image of the sky in the region of Orion as seen from Earth. The three stars of nearly equal brightness make up Orion's belt.
     
  2. Right: The same region of the sky with a black hole located at the center of the drawing. The black hole's strong gravity bends the light passing near it. This causes a noticable visual distortion. Each star in (a) appears twice in (b) on each side of the black hole. Near the black hole you can see the entire sky, as light is bent around the hole. (Robert Nemiroff (GMU, NASA)

Another method that will work for black holes surrounded by infalling matter is to detect the radiation produced by the tidal effect and the bumping together of nuclei as they crowd towards the hole. (See Fig. 57-13)

Since most black holes will be rotating, the same dynamics that led to the solar system being in a plane will lead to the matter around the black hole forming a disk. This disk is named the accretion disk. Far from the black hole the matter in the accretion disk can have a stable circular orbit. Nearer the center, there are no stable orbits. The matter in that part of the accretion disk must spiral in towards the horizons. This matter "screams" as it is squeezed and distorted by the tidal effects and as it hits other matter. The net effect is a massive emission of radiation at x-ray and other wavelengths. Some of the matter will even be ejected along the axis of rotation of the black hole producing jets of relativistically moving matter. This matter can interact with matter in the surrounding interstellar medium producing radio emission and visible light. Your text shows several such examples in Fig. H 25-22 through Fig. H 25-26.

In December 1995 NASA launched the X-ray Timing Explorer (XTE) into near Earth orbit to look for x-ray pulses as brief as a microsecond. Neutron stars, white dwarfs and black holes all can produce such radiation. Even before the 1995 launch x-rays were detected by previous satellites. One of these sources was detected in the constellation of Cygnus. This source also emits gamma rays as well as visible light. It is called Cygnus X-1 and is currently believed to be a black hole in orbit about a blue super giant as shown in text Fig. H 19-29. Some of the observational data includes periodic changes in the x-ray emission, periodic Doppler shifting of the visible star, and changes in the radiation from the accretion disk. When all the observational data is considered we model the system as including a black hole of at least 3.5 solar masses and possibly as large as 15 solar masses.

When your text was written there were at least 3 other binary star systems in our galaxy that are good black hole candidates. They, along with Cygnus X-1, are listed in Table 19-2 of the text. Since then V404 Cygni has also been shown to be a probable black hole. It is most likely that others have been added to the list since this essay was written.

The Hubble space telescope has enabled astronomers to produce images of cluster centers and of other galaxies that were not possible before.

This allows us to look for supermassive black holes that might have been part of the early formation of galaxies. Astronomers look for supermassive black holes there by the following methods:

  1. A rapid increase in stellar density as the center of the galaxy is approached but without enough starlight being emitted from the very center.
     
  2. A highly energetic source at the center with highly energetic jets.

Such objects have been found in M32, M87, and M51 with masses ranging from 3 million to 3 billion solar masses. Your text has a picture of M32 in Fig. H 24-25. There is even evidence that there is a supermassive black hole in the center of our own Milky Way galaxy. This black hole is currently not active since little matter is falling onto it.

H. Wormholes

There is another type of black hole permitted besides those produced by the collapse of a star. These are topological black holes or wormholes in which two separate regions of spacetime (or even two separate spacetimes) are connected by a path that is not in the dimensions of the spacetime. An analogy is the handle of a cup which connects two separate portions of the cup. A model of a wormhole is shown in Fig 57-16.

 

Fig. 57-16:
A wormhole connecting two disjoint portions of spacetime. The distance through the wormhole may be shorter than the normal distances. If so, a wormhole traveler could cover regions of spacetime faster than light. Most likely such an attempt would close the wormhole and destroy the traveler.

Mathematically, wormholes are solutions or approximate solutions of Einstein's field equations without sources. The exact solutions depend heavily upon some assumed symmetry. It has been mathematically shown, in many cases, that if that symmetry is disturbed, the wormhole closes and becomes a singularity. A body trying to go through the wormhole would be such a disturbance.

Much use has been made of wormholes in science fiction as a means of rapid travel between different portions of spacetime without the need for speeds exceeding light. At this time we do not know if wormholes exist, but the theory does not seem to forbid them. So, remember the old adage "What is not forbidden, will occur."

I. Consequences on Cosmology

The general theory has had major effects on which cosmological theories we consider. In the large scale, cosmologists usually consider only cosmological theories that are consistent with the General Theory. This is what led the the statements such as: "the universe could be finite and not have a surface" and "there is no region outside a finite universe." To fit a cosmological model with observation we need the value of the deceleration parameter, qo. This value can only be determined from observational data after an assumption is made about the curvature of spacetime.

We still have conceptual problems with the very early universe. There is no consistent theory that successfully unites gravity and quantum mechanics Also we have no experimental data to guide us towards such a theory. So, our understanding of the very early universe remains problematic. The material discussed in the last two chapters of your text (Chapters 26 and 27) assumes, at least on the large scale, that the General Theory of Relativity is correct.

The General Theory of Relativity

a.
What is wrong with Newton's theory of gravity?:
-
Newton's theory incorporates ``action at a distance'', in which this action propagates instantly over space (if I move my hand, the gravitational effect of this at some distant point will be instantaneous). This is inconsistent with the special theory of relativity, which says that nothing can propagate faster the the speed of light.
-
Problems with the idea of defining what an inertial frame is.

 

b.
The Postulate of the general Theory of Relativity (Principle of Equivalence):
i.
Statement of the principle of equivalence:
A non-accelerating reference frame in the presence of a uniform gravitational field is indistinguishable from a reference frame undergoing uniform acceleration.
ii.
Example: An observer in an rocket ship in outer space (no gravity) that is accelerating at 9.8 m/s2 will observer the same laws of physics as an observer in a rocket ship stationary on the Earth's surface (in the presence of a gravitation field of g = 9.8 m/s2.
iii.
Alternative statement of the principle of equivalence:
A freely falling reference frame in a uniform gravitational field is indistinguishable from an inertial frame (in the absence of gravity).

In a lecture, Einstein told the story about what happened in 1907:

 
I was sitting in a chair in the patent office at Bern when all of a sudden a thought occurred to me: `If a person falls freely he will not feel his own weight.' I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.

Einstein later referred to this thought as ``the happiest thought of my life''.

c.
Consequences of the Equivalence Principle
i.
Inertial mass and gravitational mass are the same. ``Thought Experiments'' with elevators in space.

 

ii.
Light is bent in a gravitation field, as can be deduced by another ``thought experiment''. Einstein predicted that the gravitational field of the Sun would bend the light ray from a star. To see the star, one needed to do the experiment during a total eclipse of the Sun. Actual experiment was done in 1919. The result agreed with Einstein's prediction of 1916 [the amount of bending is actually twice the value that one would predict from the equivalence principle alone. The difference is due to the curvature of space (to be discussed later).] This made him an instant celebrity.

 

iii.
Clocks run slower in a gravitational field.
-
The Doppler shift: If a source is sending out a series of pulses of light (or sound), an observer moving away from the source will receive those pulses at a lower rate than they were sent out by the source. This is because each successive pulse has to travel further to reach the observer, and therefore takes a slightly longer time to reach the observer. On the other hand, if the observer is moving toward the source, then he will receive those pulses at a higher rate than they are being sent out. This is because each successive pulse needs to travel a shorter distance than the previous pulse, and so takes less time to reach the observer.

The specific expression for the Doppler shift (for light) is:

$\displaystyle {\frac{\Delta f}{f_{\rm source}}}$ = $\displaystyle {\frac{f_{\rm observer} - f_{\rm source}}{f_{\rm source}}}$ = ±$\displaystyle {\frac{V}{c}}$ , (1)

where fsource is the number of pulses per second (frequency) that the source emits, fobserver is the number of pulses per second that the observer sees, and V is the velocity of the observer relative to the source (we have assumed in this formula that V $ \ll$ c). The plus sign us used when the observer is moving toward the source, and the minus sign is used when the observer is moving away from the source.

 

-
The Gravitational Red Shift: We consider the following thought experiment. An elevator is accelerating upward in outer space with acceleration g. Pulses of light are emitted upward from the bottom of the elevator. The situation is being observed by an observer in an inertial frame, and we assume that when the pulses are first emitted, the elevator is stationary with respect to this inertial observer. By the time the pulses reach the top of the elevator (a time t = L/c after they are emitted, where L is the height of the elevator), the velocity of the elevator has increased from V = 0 to V = gt = gL/c. Because an observer moving with the top of the elevator is moving away from the source at velocity V, this observer sees the frequency of the source shifted by
$\displaystyle {\frac{\Delta f}{f}}$ = - $\displaystyle {\frac{V}{c}}$ = - $\displaystyle {\frac{g L}{c^2}}$ (2)

By the principle of equivalence, this same result should occur in a stationary observers in a gravitational field. (Note that the above equation is correct only for gL/c2 $ \ll$ 1.)

Two experiments were done that showed agreement with this effect. One was the gravitational red shift of light emitted from the surface of the sun. The other was a change in frequency an electromagnetic wave that was emitted from the ground and detected from the top of a building. You will be given a homework problem to evaluate Equation (2) in the gravitational field of the Earth. You will find that the red shift is very small.

 

-
Gravitation time dilation (Clocks run slower in a gravitational field): From the previous discussion, we found that the person on top of the building received the pulses less frequently than the person on the ground sent them. For example, if the person on the ground sends pulses at 1 second intervals (according to a clock on the ground), then the person on the top of the building would receive the pulses at intervals of greater than one second (according to a clock on top of the building). The person on the top of the building would therefore conclude that the ground clock is running slow. Alternatively, if the person on the top of the building sent pulses at one second intervals down to the person on the ground, the person on the ground would receive the pulses at intervals of less than one second, and conclude that the top clock was running fast. We therefore conclude that clocks closer to the source of a gravitational field run more slowly. Since clocks measure time, we conclude that near the source of a gravitational field, time advances more slowly!

The logic of our argument leading to the gravitational time dilation was: (1) calculate the Doppler effect (or frequency shift) in an accelerating elevator (as observed by an inertial frame), (2) use the principle of equivalence to get the gravitational red shift, (3) from the gravitational red shift, we deduce the the fact that time ``runs'' differently at different places in a gravitational field.

Important Note: Although the Doppler shift (combined with the equivalence principle) was used to derive the gravitational red shift (or gravitational frequency shift), the two concepts are not the same. The Doppler shift is a shift due to the motion of the source relative to the observer and has nothing to do with gravity. The gravitational frequency shift is a shift due to the different positions of the source and observer in the presence of a gravitional field.

 

iv.
Space-time geometry is non-Euclidean - thought experiments with a rotating reference frame (see Gamow and Stannard, Chapter 4 (and in particular, pages 47 and 48):
We imagine a rotating platform and a (non inertial) reference frame ``attached to this platform''. Measurement by someone on the platform is the same as that of a co-moving inertial observer.
-
Clocks run slower toward the source of a gravitational field.
-
Geometry is non-Euclidean. In this case the circumference c of a circle as measured by an observer in the rotating coordinate system is greater than 2$ \pi$ times the radius ( c > 2$ \pi$r). This is an indication that the space as measured in the rotating frame is curved.

 

d.
Curvature of space-time: (See the supplementary discussions ``General Theory of Relativity - Curved Space-Time'' on page  and ``Essential Points Related to Curved Space-Time'' on Page ) The curvature of space-time tells particles how to move. Particles move in ``straight lines'' in local inertial frames.
i.
Examples: Curvature of space: For a saddle, the circumference is larger than 2$ \pi$ times the radius (as in the case of the rotating disk discussed in section c.iv.), and the sum of the angles in a triangle is less than 180o. A surface with these properties is said to be negatively curved. For a sphere (meaning the surface of a sphere), the circumference is smaller than 2$ \pi$ times the radius, and the sum of the angles in a triangle is more than 180o. A surface with these properties is said to be positively curved. Although the spherical surface has constant positive curvature, in general the curvature can vary from place to place. Although the sphere and saddle, which are 2-dimensional surfaces are embedded in a higher (3) dimensional space, it is important to realize that the curvature properties of these surfaces can be completely determined from within the surfaces. Flat-landers who live in these surfaces can determine all the properties by exploring the geometry of the surface. One final comment about curved surfaces is that if small enough, a given region of space (or space-time) is approximately flat. So if we know how particles move in flat space, we can figure out how they move in curved space by effectively patching these pieces of flat space together.

 

ii.
Geodesics in Curved space: In the previous section we discussed triangles. In flat space, a triangle is made up of three straight lines. In order to define a triangle in curved space, one needs to define something like a straight line in curved space. In flat space, one can define a straight line as the shortest path between two points. One can also do this for curved space. The shortest distance between two points in curved space is called a geodesic.

On a globe, geodesics are parts of great circles. (A great circle is a circle on the Earth whose center is at the center of the Earth. The Equator is an example of a great circle. Other lines of latitude are not great circles.) Airplanes often fly along great circles. This minimizes the distance the plane must fly.

 

iii.
Parallel Transport as a measure of curved space: If you move a vector around a closed path in curved space always keeping the vector pointing in the same direction as the previous point on the path (this is called parallel transport), you will find that when you get back to the starting point, the vector will be pointing in a different direction.

 

iv.
What are coordinates? Coordinates are just a way of labeling points in space or events in space-time, for the purpose of ``talking about'' such points or events. For example, the position and time of an event in space-time (given by the values of x, y, z, and t relative to some origin) are the coordinates of that event. As another example, the coordinates of a point on the Earth are given by the latitude and longitude of that point.

 

v.
The metric as a way of measuring curved space: The metric tells you the ``distance'', or space-time interval s2 between nearby events in space-time. You can think of the metric as a machine, into which you put the coordinates of two nearby points (or events), and out of which pops the distance between those points. If you know the metric for all pairs of points, then you know everything there is to know about that space (or space-time). That is, all properties of the space or space-time can be determined from knowledge of the metric.

Curved space in two dimensions: In the case of space, (space-time is discussed below), the metric tells you the distance between any two nearby points (for example, P1 and P2 in the figure below). For Euclidean (flat) space, this distance is just given by the Pythagorean theorem

l2 = ($\displaystyle \Delta$x)2 + ($\displaystyle \Delta$y)2, (3)

where $ \Delta$x and $ \Delta$y

 

\includegraphics[width=2.8in]{Lectures/Figures/Lecture21_fig0.eps}

 

are the difference in the x and y coordinates of two nearby points, and l is the distance between these points. In curved space, we cannot always express the distance between two points l by the Pythagorean theorem, but must use a more general expression:

l2 = gxx($\displaystyle \Delta$x)2 + gxy$\displaystyle \Delta$x$\displaystyle \Delta$y + gyy($\displaystyle \Delta$y)2 , (4)

where the g's are the metric of the space. In general, the g's can have different values at different places. Note that if, gxx = gyy = 1 and gxy = 0 in Eq. (4), we just recover the Pythagorean theorem Eq. (3). What you should remember from all this is that the g's give you the prescription for determining the distance between nearby points in space (or space-time), given the coordinates of those points.

An example of the use of the metric is given by a map of the entire Earth, projected onto a flat surface. If we let longitude and latitude be our x and y coordinates, respectively, then Eq. (3) will not correctly describe the distance between two points on the Earth at all latitudes. The distance between two points on the Earth for a given $ \Delta$x is much smaller near the poles than near the equator. If you look at a globe, you can see this by noting that the lines of longitude get closer together near the poles.

The metric can be used to find the length of any line in curved space. To do this, you divide the line up into a bunch of small segments (label them 1, 2, 3, etc). Then use Equation (4) to find the length of each segment. The total length of the line will be the sum of the lengths of each segment ( ltotal = l1 + l2 + l3 +...).

A geodesic is defined as the shortest path between two points. It is the curved space analogue of a straight line. Since the metric can be used to find the length of any line, it can be used to find the geodesics. In general, the metric tells you everything there is to know about a given curved surface (or curved space).

Curved space-time: In the case of space-time, one can define the ``distance'' between two nearby events by the space-time interval s2 between these events (remember that in flat space-time, s2 = (c$ \Delta$t)2 - ($ \Delta$x)2 - ($ \Delta$y)2 - ($ \Delta$z)2, where $ \Delta$t is the time interval between the two events, and ($ \Delta$x,$ \Delta$y,$ \Delta$z) is the displacement between the events). For the paths of particles (world lines), the proper time t0 between nearby points on the world line can be obtained from the space-time interval s2 (specifically, ct0 = $ \sqrt{s^2}$ -).

The total proper time along a world line can then be obtained by adding up the proper times between nearby points on the world line. Remember that the proper time along a world line is just the time elapsed on a clock that travels along that world line.

In general relativity, an object in ``free fall'' (i.e., in the absence of forces other than gravity) follows a geodesic in curved space-time (remember that a geodesic is the closest thing to a straight line in curved space or space-time). It turns out that a geodesic connecting two (time-like) events is the path with the longest proper time. This is consistent with the result of the twin-paradox, where the twin moving at constant velocity (this would be free-fall in the absence of gravity) ages more than the accelerating twin.

In summary, the curvature of space-time ``tells'' matter how to move (objects move along geodesics). The question we now need to answer is: what determines the curvature of space-time?

 

e.
Einstein Field Equations: The distribution and flow of matter and energy determines the structure of space-time. (By definition, ``matter'' consists of particles with mass. Remember that mass is a form of energy.) This is described by the Einstein field equation:
R$\scriptstyle \mu$$\scriptstyle \nu$ - $\displaystyle {\textstyle\frac{1}{2}}$g$\scriptstyle \mu$$\scriptstyle \nu$R + $\displaystyle \Lambda$g$\scriptstyle \mu$$\scriptstyle \nu$ = - $\displaystyle {\frac{8 \pi G}{c^{4}}}$T$\scriptstyle \mu$$\scriptstyle \nu$. (5)

You don't have to memorize this equation! The following description is just to give you a flavor of the physical meaning of different parts of this equation: The left side of this equation represents properties of space-time. g$\scriptstyle \mu$$\scriptstyle \nu$ is the metric tensor, which gives the space-time interval for two events that are ``close'' to each other in space-time. R$\scriptstyle \mu$$\scriptstyle \nu$ and R represent different aspects of the curvature of space-time, and $ \Lambda$ is the cosmological constant. The right side of the equation represents the ``sources'' of the fields in a similar way that charges and currents are ``sources'' for electromagnetic fields. T$\scriptstyle \mu$$\scriptstyle \nu$ is called the energy-momentum tensor (a tensor is similar to a vector, but has more components). This energy-momentum tensor describes the distribution of energy (which includes mass) as well as the flow of energy.

In electromagnetism, the charges and the flow of charge create the field (remember that charges create electric fields, and moving, or ``flowing'' charges create magnetic fields). Similarly, the gravitational fields are created by momentum-energy and the flow of momentum-energy. In other words, momentum-energy plays the role in general relativity that charge plays in electromagnetism.

To sum up, we could say that matter tells space-time how to bend, and the curvature of space-time tells matter how to move.

 

f.
Solutions to the Field Equations (in brief): There are three solutions of the field equations that are particularly interesting:
i.
The Schwarzschild solution: This is the solution for the situation where the mass is concentrated at the origin. It gives the gravitational field around stars, and also describes properties of black holes.
ii.
The Friedman Solution (for the structure of the Universe): This is the solution for a isotropic homogeneous Universe in which the mass is uniformly distributed throughout. It makes predictions about the origin and ultimate fate of the Universe.
iii.
Gravitational Waves: Just as Maxwell's ``field equation'' predicted electromagnetic waves, the Einstein field equations predict gravitational waves.

 

g.
Experimental (or observational) Tests of General Relativity: For iii., iv., and v. below, see the Supplementary Discussion ``General Relativity and Cosmology'' on page .
i.
Bending of starlight by the Sun (discussed above).
ii.
Gravitational Red Shift:
- light emitted by atoms in the Sun's atmosphere (discussed above).
- terrestrial experiments where electromagnetic waves are emitted at one altitude and detected at another altitude (discussed above).
iii.
Precession of the perihelion of Mercury.
iv.
Gravitational lensing of distance stars by large mass distributions in and around galaxies.
v.
Loss of energy in binary pulsar systems due to emission of gravitational waves. Currently there is a big effort under way to build a detector (or antenna) for gravitational waves.

 

h.
The Schwarzschild solution and black holes : The Schwarzschild solution describes the situation for a static spherically symmetric mass distribution. A key parameter of the is the Schwarzschild radius defined by
Rs = $\displaystyle {\frac{2GM}{c^2}}$ (6)

where G is the universal constant of gravitation and M is the total mass. The Schwarzschild radius is the radius that the mass would have to have such that the escape velocity at its surface is the speed of light. For the Earth and Sun, the radius R > > Rs, and the situations deviates only slightly from Newtonian gravity (Rs for the Sun is about 3 km and for the Earth is about 1 cm). What happens when R < Rs. In this case we have a black hole. Black holes have a number of interesting properties:

 

i.
Black holes have an event horizon. This event horizon is a distance r = Rs from the center. Nothing can get out from the event horizon, not even light. No information can pass from inside the event horizon to the outside.
ii.
Inside the event horizon, all matter must fall towards the center, even light. The distance from the center becomes time-like. As we ``age'', r decreases. After a finite time we will end up at r = 0, where the density of matter is infinite. Such a point is referred to as a ``singularity''.

 

iii.
When an object falls inside a black hole, an observer from the outside never sees the object cross the horizon. From the outside observers point of view, the object gets closer and closer to the horizon, but never goes through. This is because time slows down an infinite amount at the event horizon, compared to points far from the black hole (this is an extreme form of the gravitational time dilation). In addition, the object gets redder and redder (as viewed by the outside observer) as it approaches the horizon (it also gets dimmer). This is an extreme form of the gravitational red shift. Eventually the object becomes so faint and red that it can't be observed anymore.
iv.
From the point of view of the falling object, nothing remarkable happens when the object falls through the event horizon.
v.
There is a minimum radius at which a quantity M of (approximately spherical) mass of ordinary matter can exist in the universe. This radius is equal to the Schwarzschild radius. If you try to pack a given amount of mass into a radius less than this, a black hole will form and swallow up the mass.
Example: After a star stops burning fuel (Hydrogen to Helium), it collapses in on itself. If the mass of the star is sufficiently large (about two times the mass of the sun), the star will eventually collapse to black hole.

 

vi.
Example: A black hole with a mass M of 3 billion times the mass of the sun has a radius of about 9 billion km. This radius is larger than the radius of the Solar System. The density (mass per unit volume) of this black hole is very low - about 2x10-3 grams/cm3. Such a black hole has been (almost certainly) observed in the core of the galaxy M87 (50 million light-years away from us). If a traveler fell through the event horizon of such a black hole, he would live for a maximum of about 3 hours before being ripped apart by tidal forces at the ``singularity'' at the center.

 

i.
Cosmology:  Cosmology is the study of the properties and history of the Universe as a whole.
i.
Distances of stars and Galaxies: Distance to nearest star $ \sim$ 4 light years. Size of galaxy $ \sim$ 100,000 light-years. Distance to nearest galaxy $ \sim$ 3 million light-years. Size of Universe $ \sim$ billions of light-years.

 

ii.
The Cosmological Principle: On a large scale the Universe is homogeneous and isotropic. All places in the universe ``look'' the same, and all directions in the universe are the same. The implication of this is that the universe has no boundary. It is either infinite with (approximately) uniform distribution of matter (no ``edge'' to the matter distribution), or it is finite and ``folds back on itself''. An example of a finite unbounded universe in two dimensions is the surface of a sphere.

 

iii.
Hubble's Law: Hubble discovered in 1929 that all galaxies are receding from ours at a speed proportional to their distance. To make his measurements, Hubble needed to know both the speed that galaxies are receding from us and their distance. Hubble used the Doppler shift to measure the speed, which requires knowledge of the frequency emitted by the source. To do this, he used the fact that each element (Hydrogen, Helium, etc) emits light at a set of discreet well known frequencies that characterize that element. By comparing these frequencies to those of the same elements on Earth, Hubble saw that the light emitted from galaxies is shifted to a lower frequency, or red shifted. [We say light is red shifted if it is shifted to lower frequencies, or toward the red end of the spectrum, and we say that light is blue shifted if it is shifted to higher frequencies, or toward the blue end of the spectrum.] Hubble's observation of this red shift is direct evidence that the galaxies are moving away from us, and that the universe is expanding.

The distances to galaxies are much harder to measure, and depend on estimates of the absolute luminosity (the rate that the star radiates energy) of certain kinds of stars. Knowing the luminosity and apparent brightness as viewed from Earth allows one to determine the distance.

A plot of this relationship is shown in the figure below.

 

 

 

Each point in the plot corresponds to a particular galaxy. The horizontal axis is the distance of the galaxy from us, and the vertical axis is the speed at which the galaxy is moving away from us. The straight line indicates that the speed of any galaxy is roughly proportional to the galaxy's distance. This relationship can be expressed by the equation v = H0R, where v is the relative speed, and R is the distance between galaxies. H0 is called the Hubble Constant. The fact that the distance of galaxies from us is proportional to their velocity leads to the conclusion that at some time in the past all galaxies were in the same place. This can be understood my imagining two travelers who start out from a particular point at the same time, but one is traveling twice as fast as the other. After a given amount of time, you will find the fast traveler twice as far from the origin as the slow traveler. This is exactly the linear relationship discovered by Hubble. If the universe were expanding at a constant rate, then the age would be t = R/v = 1/H0 $ \approx$ 14.3 billion years. The idea that the universe started out some finite time in the past at very high (infinite?) density and expanded to it's present size is called the ``big bang'' model.

Hubble's observations are consistent with the Cosmological principle. A universe where every galaxy is receding from every other galaxy at a speed v = H0R where R is the distance between any pair of galaxy's is consistent with both the cosmological principle and Hubble's observations. Analogy of a closed expanding universe with points on a balloon that is being inflated.

 

iv.
In 1922 Friedman published solutions to Einstein's equations for a homogeneous isotropic universe (these solutions are known as the Friedman Solutions): In all his solutions, the Universe has a beginning after which it undergoes a period of expansion. In addition there is a critical density $ \rho_{0}^{}$ of the Universe that determines the evolution and ultimate fate of the Universe. The parameter $ \Omega$ = $ \rho$/$ \rho_{0}^{}$ gives the ratio of the actual density to the critical density. There are three situations:
-
If $ \Omega$ < 1 (the density is below the critical density), then the Universe is infinite and will keep expanding forever and will approach a fixed rate of expansion.
-
If $ \Omega$ > 1 (density above the critical density), then the universe is closed and finite, and will eventually stop expanding and contract again to a ``big crunch''.
-
If $ \Omega$ = 1 (density exactly equal to the critical density), then the Universe is infinite and will expand forever, but the rate of expansion will continue to decrease getting closer and closer to zero.

All these models are consistent with Hubble's observations. We don't yet know the ultimate fate of the Universe, because we don't know the density of the Universe.

[It is interesting that the behavior of the Universe as a function of time can be explained by a Newtonian model (whether the universe is finite or infinite cannot be explained by Newtonian physics). If you throw up an object from a planet with a given velocity v, one can ask if the object will every return. If the mass of the planet is sufficiently small then v will be larger than the escape velocity and the object will never return (this corresponds to $ \Omega$ < 1 above) The gravitational force is not strong enough to pull the object back to the planet. If the mass of the planet is sufficiently large, then v will be smaller than the escape velocity, and the object will eventually fall back down (this corresponds to $ \Omega$ > 1 above). If the planet's mass is such that the velocity is exactly equal to the escape velocity, then the object will never come back, but its velocity will get closer and closer to zero as it gets further and further from the planet (this corresponds to $ \Omega$ = 1 above).]

 

v.
The cosmic background radiation: All objects at a temperature above absolute zero ``glow''. i.e. they emit electro-magnetic radiation. This radiation is called black body radiation. The average wavelength of this radiation depends only on the temperature. A ``red-hot'' poker is an example of black body radiation. The surface of the Sun also emits black-body radiation with an average wavelength (yellow) corresponding to the temperature of its surface of about 6000 K (6000 degrees C above absolute zero temperature).

One prediction of the big bang model is that in the past, the universe was much hotter than it is today. As it expanded it cooled. At some time in the past (before there were stars or galaxies) the temperature of the Universe was about the temperature of the surface of the Sun. At temperatures above this, the atoms in the Universe were ionized (that is, the electrons were not bound to the atoms, but could wander around freely in space), and the universe was opaque, just as the Sun is opaque (the Sun is a ball of ionized gas). Black-body radiation with a wavelength corresponding to this temperature filled the Universe. As the Universe expanded, it cooled below this temperature, and the electrons recombined with the atoms. When this happened, the Universe went from opaque to transparent. According to the big bang model, this black-body radiation should still be present today, although shifted to a much larger wavelength due to the expansion of the Universe. This residual black-body radiation was predicted by George Gamow (who also coined the term ``big bang'', and wrote the book ``Mr. Tompkins in Wonderland'').

In 1965, the black-body radiation was discovered (by accident) by Penzias and Wilson at Bell Labs. The average wavelength corresponded to a temperature of about 3 degrees above absolute zero. This discovery provided further support for the big bang model.

 

vi.
Dark matter: There are two reasons to believe that the visible matter in the universe (stars) accounts for only a small fraction of the total matter.
-
There are strong theoretical reasons to believe that $ \Omega$ = 1 for the Universe. The amount of visible matter is about a tenth of this. This means the about 90% of the matter in the universe is invisible.
-
By studying the orbits of stars about the center of galaxies, we find that there is not enough visible matter in the galaxies to account for the properties of the orbits.

Nobody knows what this dark matter is made of. The ``dark-matter problem'' is one of the biggest unsolved problems in cosmology.

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