String Theory Demystified

CHAPTER 14 Black Holes 247
hole squared. In terms of area, the entropy is 1/4 of the area of the horizon in units
of Planck length:
A
S = (14.25)
2
4 Let us compute the temperature in the case of a Schwarzschild black hole. In the
Chapter Quiz you will get a chance to try your luck fi nding the temperature of a
charged black hole. We follow a procedure outlined in a note published by P.R.
Silva. 2
We proceed as follows. We perform a Wick rotation t→iτand write the
Schwarzschild metric as
Now set
ds
p
Computing the Temperature of a Black Hole
GM
GM 2 2
1 d
dr r
r
r
2
1 2
⎛ ⎞ ⎛ ⎞
=− −
⎝
⎜
⎠
⎟ τ + −
⎝
⎜
⎠
⎟ + dΩ 2 (14.26)
2 4 2 4
12 /
⎛ GM⎞
Rdα
= − dτ
⎝
⎜1
r ⎠
⎟
−
⎛ GM⎞
dR = − d
⎝
⎜
r ⎠
⎟
2
1 2
4
12 /
4 r
and integrate. We take the limits of integration to be
This gives us two relations:
α : 0≤α ≤2π
τ : 0 ≤τ ≤β
r: G m≤ r′ ≤ r
2 4
−1
−12
/ 12 /
2πR= ( 2G m) ( r−2G m)
β
4
12
R= 22 ( G m) ( r−2G m)
4
/ 12 /
4
2 Available on the arXiv at http://arxiv.org/ftp/gr-qc/papers/0605/0605051.pdf.
4
(14.27)
(14.28)