String Theory Demystified

CHAPTER 14 Black Holes 251
Recalling that the entropy of an excited string is proportional to the product of its
mass and length [Eq. (14.34)], this allows us to write the entropy in terms of the
mass of the black hole and the gravitational constant using [Eqs. (14.39) and
(14.37)]. This gives a result proportional to Eq. (14.33):
D
A
D S mBGD G
D
−2
1
−3−3 = ∝
4
D
(14.41)
This calculation was not formal by any means. It just relied on some basic
considerations from string theory, but it gave the correct result modulo a constant.
Now let’s turn to the fi ve-dimensional black hole example.
The structure of the fi ve-dimensional geometry is as follows. We take a circular
dimension of radius R denoted by S 1 and a four torus T 4 so that
5 4 1
T = T × S
As mentioned above, the black hole actually has two string components. One is an
actual string (a D1-brane) which wraps around S 1 and so has winding modes which
will contribute to its mass. The D5-brane wraps around S 1 and also has Kaluza-
Klein modes quantized on the circle T 5 .
The starting point is the metric given by
where:
( 3 )
2 −2/
3 2 1/ 3 2 2 2
ds =− λ dt + λ dr + r dΩ
(14.42)
λ = + ⎛
3 ⎡ ri
⎞
∏ ⎢1
⎝
⎜
⎠
⎟
i=
1 ⎣ r
2
⎤
⎥
⎦
(14.43)
A result from superstring theory that we simply take as a given because it’s beyond
the scope of our discussion that the BPS condition is satisfi ed. What this means
for us is that charges are additive. The upshot of this is we can write the mass of
the black hole as
M M M M
= + +
1 2 3