by Steve Bryson
As I have said many times in this class, we cannot directly visualize the curvature of the space we are in. There is another way of saying this--we cannot make any measurements of the curvature of spacetime that in any way feels like a measurement of curvature. Now of course by feeling gravity we are measuring the curvature of spacetime all the time, but it does not look like curvature. The physicist has the same problem. What physical quantities that are related to curvature are the most convenient for the physicist to measure?
There are many ways to perceive and measure curvature, but the most precise is to see how the way we measure the spacetime distance between two points is warped by the curving of the space. This is because the curvature tensor can be completely expressed in terms of the spacetime distance element. Then the problem of solving Einstein's field equations of gravitation becomes a problem of first expressing those equations in terms of the distance element and then solving for that distance element. Usually one does this by first specifying the distribution of matter in a region of spacetime. Solving for the distance element means finding that spacetime distance element which, when inserted into Einstein's field equations, will match the distribution of matter and energy in that equation.
Actually, the equations of gravitation are never (to my knowledge) actually solved this way. It is simply too difficult a problem to try to exactly match a specified distribution of matter with the correct distance element. What physicists actually do is make very simplifying assumptions about the distribution of matter. In particular they may specify that the matter takes, for example, the form of a rotating sphere that does not change with time (this particular assumption leads you to what is known as the Kerr solution). This kind of the assumption severely limits the possible mathematical forms that the distance element may take and so makes the finding of the correct distance element easier.
Let me describe a real life example. This is the solution for a distribution of matter that is a nonrotating, nonchanging sphere of matter. In this case, the matter looks the same no matter what your angle of view. The distance of a point in space from the center of the sphere is the only quantity that makes a difference. It turns out that this implies that the space around this sphere is only curved in the spatial direction pointed towards the center of the sphere and in the time direction. Thus at a point in spacetime near the sphere one can choose coordinate axes so that there is no curvature in two of the spatial directions. This makes it much easier to solve for the formula describing the distance measurement between two points.
This is a very useful example because it is a good approximate model for the distribution of matter in a star. Stars are approximately spherical, and they rotate and change shape very slowly. Thus the distance element in this simple example will describe the space around a star. This is called the Schwarzschild solution. Physicists use the Schwarzschild distance element to calculate what the straightest lines in the space around the sun are. Sure enough, the straightest line pointed in certain directions in the curved space around the form almost closed ellipses! This is the path followed by the Earth in it's orbit around the sun. The Earth is following the closest thing there is to a straight line in the space curved by the sun. Thus the Earth in it's orbit is testing General relativity all of the time!
There are other tests of general relativity that are more interesting in that they predict things that were not previously understood. One of them is the motion of the planet Mercury around the sun. I mentioned above that the straight lines that form orbits are almost closed ellipses. This is simply how it works out in this curved space. According to the Newtonian theory of gravity, orbits do form closed ellipses. It was observed for many years before the advent of general relativity that Mercury's orbit does not form a closed ellipse in a way that could not be explained by the gravitational influence of the other planets (according to Newtonian gravity). General relativity, however, exactly predicts the observed lack of closure in Mercury's orbit.
The other two classical tests of general relativity involve how light is effected by gravity. According to Newtonian gravity, if light is a particle with an effective mass generated by the energy of the light (remember that according to special relativity energy has an equivalent amount of mass (E=mc2)) then gravity should effect light in a certain way. According to general relativity light (like everything else) simply moves along straight lines in a curved space and so will be effected by gravity in a way that disagrees with Newtonian gravity. When you measure the effects, general relativity gives the correct prediction.
Recall the spacetime diagrams from when we were talking about special relativity. Light plays a very special role in spacetime, in that the speed of light is always the same no matter what your point of view. In particular, even if you are moving relative to me we will both measure a light beam to be travelling with the same speed.
Put in spacetime diagram terms, this means that the tilt of light's world line will be the same no matter what point of view in spacetime we take. Now we can manipulate our units of measurement so that on our diagram this tilt is that of a line at 45 degrees. Let us agree to do this, so that our spacetime diagram looks like this:
We call the two lines representing the world lines of light beams going opposite directions the light cone. The light cone from the center will always look the same in any point of view. If, however, you were to look at the light cone at a point away from your center in a curved space then that light cone may be tilted. Here is a drawing of the light cone structure of the spacetime around a star according to the Schwarzschild solution:
The effect of the curvature on the light cone is to cause the entire light cone to tilt (from the point of view that we take far away from the star) towards the star. This helps us to visualize the curvature around the star.
It is only the surface of the star that stops the light cone from turning over completely. One can therefore ask what would happen if the star became smaller and smaller. Then the space close to the surface of the star would become more and more curved and the light cones would tilt more and more over. When the star reaches a small enough radius (3 kilometers in the case of the sun) the light cone at it's surface has tilted completely over (remember that this is from our point of view far away from the star). This special radius is called the Schwarzschild radius. Here it is visually:
If the star is smaller than it's Schwarzschild radius, then the space at the surface is so curved over that the entire interior of the light cone is pointed at the star! Now as nothing can ever travel to a point in spacetime outside of it's own light cone, this means that any object starting from a point inside the Schwarzschild radius will find the surface of the star in it's future no matter how it moves! Another way of saying this is that for someone inside the Schwarzschild radius of a star all paths move towards the star. Even to stay at the same distance from the star would require our unfortunate friend to travel outside his own light cone. This would require him to travel faster than light, which you cannot do due to the structure of spacetime.
So anything that falls closer to the center of a star than it's Schwarzschild radius will necessarily eventually hit the star--including light. Any object that becomes smaller than it's own Schwarzschild radius is called a black hole. It need not have started as a star, but stars are the only objects in nature that we know of that can shrink to a size small enough to become a black hole.
I should mention that there is no Schwarzschild radius inside a star, for the matter inside the star keeps the curvature there well behaved.
What is the experience of passing through the Schwarzschild radius of a star? It is nothing special at all. For the person falling into the star, there is no Schwarzschild radius. This is because this person always carries his own point of view and in his point of view his light cone always points up. The Schwarzschild radius is only a feature of the comparison between the point of view that we have far from the star and the point of view near the star. This is not just a simple comparison, for it has very physical effects. One effect of very large curvature near a star is that time runs slower near a star than farther away. Thus if you traveled very near (but not through!) the Schwarzschild radius of a black hole and returned I would have aged much more than you would have. In fact, if I watched you approach the black hole, I would see your time slow down more and more as you got closer and closer to the Schwarzschild radius. You, on the other hand, would think that nothing particularly strange was happening to yourself, but you would see the rest of the universe speeding up.
What if you let yourself fall through? As I mentioned before, you would fall through the Schwarzschild radius and very quickly (much less than one second after you passed through the Schwarzschild radius) hit whatever it is in the center. I, however, am sitting at the safety of a great distance and would see you slowing down more and more as you approached the Schwarzschild radius. In fact, I would never see you pass through the Schwarzschild radius. Due to the curvature of the space made by the star it would take an infinite amount of my time for you to reach the Schwarzschild radius. In my way of measuring time you would never get there. Yet in your way of measuring time you get there rather quickly. Thus we live in a universe where something can happen to someone which never happens at all as far as someone else is concerned. I consider it a good thing that you could never come out of the black hole again and confront me with an event that, as far as I am concerned, has not yet happened.
Once something is inside the Schwarzschild radius of a black hole, it can never get out. This means that someone inside the Schwarzschild radius can never send anything out to the outside universe. Thus we, sitting off in the distance, can never receive any message from within the Schwarzschild radius. In particular, we can never receive any information about the black hole itself. This is why John Wheeler coined the term 'black hole' for this kind of object. This radius is also known as the event horizon. As we can not receive any messages from within the black hole, we cannot know the particular structure of the hole. The only things that we can know about the objects inside the Schwarzschild radius are the mass, electromagnetic charge, and total amount of rotation in the matter, as all of these things contribute to the spacetime curvature around the object. Further, as the world lines of all objects inside the Schwarzschild radius eventually hit the surface of the star, one can show that the matter inside the black hole must take a spherical shape. This is called the "Black holes have no hair theorem" and was first proved by Stephen Hawking and Roger Penrose.
Hawking and Penrose were the main researchers who studied the overall properties of the curvature of spacetime, introducing very pretty and abstract mathematical geometric techniques. They proved several so-called singularity theorems, which say that under very general and physically likely circumstances spacetime will curve in such a way that all of the dimensions of spacetime will be squished down to a point. Such a point is called a singularity. Some of the singularity theorems refer specifically to black holes and imply that once a star has become smaller than it's own Schwarzschild radius, then it must inevitably collapse to a point taking it's surrounding spacetime with it. This is something that physicists do not like, as it implies that there are points in spacetime where the structure of space somehow breaks down. It is generally assumed, but by no means known, that when the black hole becomes small enough general relativity breaks down and is no longer true due to quantum mechanical effects. Thus as the black hole becomes small enough general relativity no longer applies and as no one knows what does apply all bets are off.
It would be very interesting, therefore, to try to observe what happens to a collapsing star inside it's own Schwarzschild radius. Now we simply cannot do this because of the nature of the Schwarzschild curvature. Other solutions of Einstien's equations, though, do offer the possibility of creating spacetime singularities not enclosed by an event horizon (the event horizon is the generalization of the Schwarzschild radius to these other solutions). If we are looking at a sphere of matter that is electrically charged or spinning (or both), then the spacetime curvature around the star is somewhat more complicated (though not so complicated as to prevent figuring out these solutions). These solutions are called the Reisner-Nordstrom solution and the Kerr solution respectively. Here if the charge or spin is not too large there is an event horizon surrounding the black hole, but if the charge or spin are large enough then the events horizons vanish and you have a singularity of spacetime for your examination.
So if we try to create a highly charged black hole, say by dumping in lots of protons to give the hole a large positive charge, perhaps we can get a glimpse of the singularity. The trouble is that as the hole becomes more and more charged it becomes harder and harder to dump the protons in due to the electrostatic repulsion of like charges (both the hole and the protons are positively charged). It turns out that long before the hole would be charged enough to make the event horizon vanish it would be effectively impossible to keep dumping protons in. No one has been able to invent even a theoretical way to keep dumping protons in.
Similarly, for a spinning black hole the spacetime curvature is such that as objects approach the black hole they start spinning with the hole (this is called dragging of inertial frames). Say we try to make the hole spin faster by throwing things in in such a way as to add to the spin of the total system. As the hole spins faster and faster approaching objects get dragged faster and faster. Long before the spin of the hole is fast enough to eliminate the event horizon any object approaching the hole would be flung back out into space by the dragging before it could fall in.
While the above analysis depends on estimates of what we can actually do, the fact that they do not seem to come even close to cancelling the event horizons has led physicists to postulate what is called 'the cosmic censorship hypothesis': There are no singularities in the universe that are not surrounded by event horizons. In other words, there are no naked singularities. At this point in time, despite large effort by very good physicists, this is still a hypothesis. No one knows if it is true or false and it remains one of the most important outstanding problems in general relativity.