Notes on Relativity and Cosmology - Physics Department, UCSB

234 CHAPTER 9. BLACK HOLES
√
dr r
ds = √
1 − Rs /r = dr. (9.12)
r − R s
I have rewritten this formula in this way because we only want to study what
happens near the Schwarzschild radius. In other words, we are interested in the
behavior when r − R s is small. To examine this, it is useful to introduce the
quantity ∆ = r − R s . We can then write the above formula as: ds = √ r
∆ d∆.
Integrating, we get
s =
∫ ∆
0
√ r
d∆. (9.13)
∆
This integral is hard to perform exactly since r = R s + ∆ is a function of ∆.
However, since we are only interested in small ∆ (for our local comparison), r
doesn’t differ much from R s . So, we can simplify our work and still maintain
sufficient accuracy by replacing r in the above integral by R s . The result is:
s ≈ √ R s
∫ ∆
0
d∆
√
∆
= 2 √ R s ∆. (9.14)
Let us use this to write α for the black hole (let’s call this α BH ) in terms of the
proper distance s. From above, we have
c 2
α BH =
2 √ R s
1 − R s /r r
√ 2
= c2 r R s
2 r − R s r 2
= c2 1 r R
√ √r s
2 ∆ r 2
c 2
≈
2 √ = c2
∆R s s . (9.15)
Note that this is identical to the expression for α near an acceleration horizon.
It worked! Thus we can conclude:
Near the Schwarzschild radius, the black hole spacetime is just the same as
flat spacetime near an acceleration horizon.
The part of the black hole spacetime at the Schwarzschild radius is known as
the horizon of the black hole.
9.2.4 Going Beyond the Horizon
We are of course interested in what happens when we go below the horizon of
a black hole. However, the connection with acceleration horizons tells us that
we will need to be careful in investigating this question. In particular, so far