Notes on Relativity and Cosmology - Physics Department, UCSB

244 CHAPTER 9. BLACK HOLES
Well, our diagram is exactly the same on the right as on the left, so there seems
to be a symmetry. In fact, you can check that the Schwarzschild metric is
unchanged if we replace t by −t. So, both directions must behave identically.
If any calculation found that the worldline bends to the left, then there would
be an equally valid calculation showing that the worldline bends to the right.
As a result, the freely falling worldline will not bend in either direction and will
remain at a constant value of t.
Now, how long does it take to reach r = 0? We can compute the proper time
by using the freely falling worldline with dt = 0. For such a worldline the metric
yields:
dτ 2 = −ds 2 =
dr 2
R s /r − 1 =
r
R s − r dr2 . (9.18)
Integrating, we have:
τ =
√ r
dr
R s
R s − r . (9.19)
∫ 0
It is not important to compute this answer exactly. What is important is to
notice that the answer is finite. We can see this from the fact that, near r ≈ R s
the integral is much like √ dx
x
near x = 0. This latter integral integrates to √ x
and is finite at x = 0. Also, near r = 0 the integral is much like x R s
dx, which
clearly gives a finite result. Thus, our observer measures a finite proper time
between r = R s and r = 0 and the throat does collapse to zero size in finite
time.
9.3.2 The Singularity
This means that we should draw the line r = 0 as one of the hyperbolae on our
digram. It is clearly going to be a ‘rather singular line’ (to paraphrase Sherlock
Holmes again), and we will mark it as special by using a jagged line. As you
can see, this line is spacelike and so represents a certain time. We call this line
the singularity. Note that this means that the singularity of a black hole is not
a place at all!
The singularity is most properly thought of as being a very special time, at
which the entire interior of the black hole squashes itself (and everything in it)
to zero size. Note that, since it cuts all of the way across the future light cone
of any events in the interior (such as event A below), there is no way for any
object in the interior to avoid the singularity.