Chapter 23: Black Holes and Curved Spacetime

When a massive star collapses, it can form a neutron star (if the collapsed core has less mass than 3 Msun), or a black hole (if it's mass is more than 3 Msun). Black holes are discussed frequently in the popular press, but what are they? To understand that, we begin with a discussion of the theory that predicts black holes, the general theory of relativity.

23.1 The Principal of Equivalence

The complex theory of general relativity begins with a simple idea: an object in gravity will behave identically to the same object in a state of uniform acceleration. That is, you couldn't sense any difference between sitting as you are now, in a classroom in Harper Woods, or sitting on a specially designed space ship, far off in space, accelerating upward at 1g. (1g is an acceleration equal to the acceleration due to gravity on the surface of the Earth.) Not only would everything feel the same, but any experiment that you were to perform inside this room would behave in the same way.

Of course if you looked outside at the surroundings, or the stars, you would find a difference, but that is due to the relative motion of you with respect to the rest of the universe and not to some difference in the physical laws.

So, according to Einstein, gravity pulling downward is equivalent to (can be mimicked by, or replaced by) an acceleration upward.

The space shuttle, and other craft in orbit about the Earth, offer an interesting example of the equivalence principle in action. Astronauts on the shuttle (in orbit) feel weightless, as if they are in a region of zero gravity. But the Earth's gravity just 200 miles above the surface is not zero, it is in fact 90% of its strength on the surface. So why do astronauts not feel it?

One way to explain why they don't feel gravity is to see that the astronauts, the space shuttle, and everything else inside the shuttle are falling around the Earth with the same acceleration due to gravity. They don't feel gravity because, in their local environment, gravity doesn't have anything to push against.

Einstein's way of explaining it is that the gravity inside the shuttle is exactly cancelled by the acceleration of the shuttle in its motion around the Earth.

The Paths of Light and Matter

So far, pretty straight forward. But here's where things get weird.

Now imagine that same space shuttle orbiting around the Earth. Since its acceleration exactly cancels gravity, the astronauts on board can perform experiments that will yield the same results as if done at rest in a region of space with zero gravity. One of the astronauts fires a laser beam from the back of the shuttle to the front, as shown in Figure 23.4. At rest in a region of zero gravity, the laser beam will follow a straight line path from the back of the shuttle to the front. Therefore it must also do this in the shuttle that is orbiting around the Earth.

But the shuttle curves in its orbit around the Earth. Between the time the laser beam starts at the rear and reaches the front, the shuttle has moved a little in its orbit, so that the front is no longer in the same place, nor in the same straight line as it was originally. Yet, the light must hit dead center at the front of the shuttle! The only way this is possible is if the light bends!

23.2 Spacetime and Gravity

23.3 Tests of General Relativity

23.4 Time in General Relativity

23.5 Black Holes

Now let's apply this to black holes. First I'll discuss black holes using Newton's law of gravity, and then I'll use general relativity.

Newtonian Gravity

According to Newton's law of gravity, an object launched from the surface of the Earth must have a speed greater than the escape velocity of the Earth (about 11 km/s) in order to break free of Earth's gravity. The Sun is larger than the Earth, and its escape velocity is also considerably larger, about 618 km/s.

The escape velocity for an object is related to the strength of gravity at its surface, and this depends on both its mass and diameter. If we keep the mass of the Sun the same but shrink its diameter, say to the size of a neutron star (less than 100 km in diameter), then the gravitational pull on the surface will be much greater. This translates into a greater escape velocity, about half the speed of light. If we shrink the Sun down more, the escape velocity will get still larger, eventually exceeding the speed of light.

Since nothing can travel faster than the speed of light, no material on the surface of this star could ever escape from the star's gravitational pull. Even light will be stuck. This is a balck hole, an object from which not even light can escape.

Note that, although light is massless, we can make a more general argument that light is bent by gravity. Photons of light carry energy that depends on their frequency. This is why atoms absorb or emit light at specific frequencies, the ones that correspond to the energy difference between energy levels in the atoms. Einstein's theory of special relativity relates mass and energy through E = mc2, or m = E/c2. If we replace m in Newtons gravity with E/c2, then we have a relation that implies that light is affected by gravity.

Collapse with General Relativity

But the relation doesn't predict the correct amount of bending of light. To get it correctly, we need the general theory of relativity. The idea is somewhat similar. As the Sun collapses, the space around it curves more and more. Eventually when it has collapsed sufficiently far, space curves so much that not even light can get out, and a black hole is formed. One difference in relativity is that the black hole has no solid surface! It has a surface, called an event horizon, that defines the region from which nothing can escape. What happens inside the event horizon is completely out of our view, and according to general relativity, is completely lost to us. But as far as our theories can predict, the Sun itself will become just a point, a singularity as it is called.

If on the other hand you were to travel into a black hole, you wouldn't notice much as you passed through the event horizon. The stars would still to visible, and nothing drastic would occur to you. An observer outside the black hole would see something quite different however. To the observer, you would seem to linger forever at the event horizon of the black hole. This is because time slows down in the vicinity of massive objects, and it essentialy stops at the event horizon of a black hole.

However, that assumes that you could survive the trip into the black hole. The trip into a black hole is not pleasant. Suppose that you went into an event horizon feet first. The gravity at your feet would be so much greater than at your head that your body would be stretched. Gravity is pulling inward to a point, and this will tend to squeeze you together (can one ever be too thin?). Eventually your body will be stretched so far that it tears apart. In a short time, all the atoms of your body will be ripped apart, and then squashed back together in the singularity of the black hole.

23.6 Evidence for Black Holes

If light cannot escape from a black hole how can we detect one?

Our evidence for black holes is indirect. It basically reduces to demonstrating that an object with a mass greater than 3Msun and a very small size exists.

While the black hole itself cannot be seen, its effects on nearby matter can be seen, and can sometimes be quite dramatic. If there is a source of matter near to the black hole, then the matter will be slowly consumed by the black hole. For instance, consider a binary system consisting of a very large star and a smaller star. The large star will rapidly go through its life and form a black hole. The black hole can now pull in material from the companion star, as suggested in Fig. 23.14.

The material will not immediately fall into the black hole, but will first rotate around it, forming an accretion disk. The material in the accretion disk will heat up as it falls in the gravity well of the black hole, and like any hot matter it will emit radiation. The temperature can rise so much that the material will emit x-rays. So astronomers have searched for binary systems, where one star is invisible, and that seems to be a source for x-rays.

The mass of the invisible star can be determined from the orbit of the visible companion using Keplers law. Some of the black hole candidates found thus far are listed in table 23.1. But even this doesn't prove that the invisible star is a black hole. It could be a neutron star for example. We need to prove beyond any doubt that the size of the object precludes it from being a neutron star, such that the only remaining viable alternative is that it's a black hole.

Someday it could be possible to measure some other visible property of the candidate that would prove that it's a black hole. Stephen Hawking proposed that it's possible for black holes to emit radiation, dubbed Hawking radiation. If astronomers were to observe an object that emitted Hawking radiation, this may directly prove that it's a black hole.

© Robert Harr 2002