Notes on Relativity and Cosmology - Physics Department, UCSB

230 CHAPTER 9. BLACK HOLES
However, for small r, α is much bigger. In particular, look at what happens
when r = R S . There we have α(R s ) = ∞! So, at r = R s , it takes an infinite
proper acceleration for a clock to remain static. A static person at r = R s
would therefore feel infinitely heavy. This is clearly a various special value of
the radius coordinate, r. This value is known as the Schwarzschild radius.
Now, let’s remember that the Schwarzschild metric only gives the right answer
outside of all of the matter. Suppose then that the actual physical radius of
the matter is bigger than the associated Schwarzschild radius (as is the case for
the earth and the Sun). In this case, you will not see the effect described above
since the place where it would have occurred (r = R s ) in inside the earth where
the matter is non-zero and the Schwarzschild metric does not apply.
But what if the matter source is very small so that its physical radius is less
than R s ? Then the Schwarzschild radius R s will lie outside the matter at a
place you could actually visit. In this case, we call the object a “black hole.”
You will see why in a moment.
9.2 On Black Holes
Objects that are smaller than their Schwarzschild radius (i.e., black holes) are
one of the most intriguing features of general relativity. We now proceed to
explore them in some detail, discussing both the formation of such objects and
a number of their interesting properties. Although black holes may seem very
strange at first, we will soon find that many of their properties are in quite
similar to features that we encountered in our development of special relativity
some time ago.
9.2.1 Forming a black hole
A question that often arises when discussing black holes is whether such objects
actually exist or even whether they could be formed in principle. After all, to
get R s = 2MG/c 2 to be bigger than the actual radius of the matter, you’ve
got to put a lot of matter in a very small space, right? So, maybe matter just
can’t be compactified that much. In fact, it turns out that making black holes
(at least big ones) is actually very easy. In order to stress the importance of
understanding black holes and the Schwarzschild radius in detail, we’ll first talk
about just why making a black hole is so easy before going on to investigate the
properties of black holes in more detail in section 9.2.2.
Suppose we want to make a black hole out of, say, normal rock. What would
be the associated Schwarzschild radius? We know that R s = 2MG/c 2 . Suppose
we have a big ball of rock or radius r. How much mass in in that ball? Well,
our experience is that rock does not curve spacetime so much, so let’s use the
flat space formula for the volume of a sphere: V = 4 3 πr3 . The mass of the ball