Gravity and Strings

202 The Schwarzschild black hole
The above solution for X µ (ξ) leads to an energy–momentum tensor whose only nonvanishing
component is T00 ∼ δ (3) (x). However, on recalculating carefully 21 the components
of the Einstein tensor for the Schwarzschild metric, we find that all the diagonal
components, not only G00, are different from zero at the origin:
G00 =− W
sin 2 θ 4π RSδ (3) −1 W
(r), Grr =
sin 2 θ 4π RSδ (3) (r),
2 r
Gθθ =−
sin 2 θ 2π RSδ (3) (r), Gϕϕ = sin 2 (7.40)
θ Gθθ.
This is related to the spacelike nature of the Schwarzschild singularity, as expected. In
the cases in which we will be able to identify the source of a solution with a particle (or a
brane) the singularity of the metric will be non-spacelike.
7.3 Thermodynamics
We have seen in previous sections that, classically, according to the Einstein equations,
there are two magnitudes in a Schwarzschild BH, the area A and the surface gravity κ, that
behave in some respects like the entropy S and the temperature T of a thermodynamical
system. From this point of view the constancy of κ over the event horizon would be the
“zeroth law of BH thermodynamics” and the never-decreasing nature of A would be the
“second law of BH thermodynamics.” In a thermodynamical system S, T ,and the energy
E are related by the first law of thermodynamics:
dE = TdS. (7.41)
To take the thermodynamical analogy any further, it is necessary to prove that κ and A
are also related to the analog of the energy E by a similar equation. The natural analog for
the energy is the BH mass M (times c 2 ), and, thus, it is necessary to have (the factor of G (4)
N
appears for dimensional reasons)
dM ∼ 1
G (4)
N
κdA. (7.42)
21 In this calculation one has to be careful to keep singular (δ-like) contributions that are non-zero only at a
certain point. These contributions come in two forms. One is the standard four-dimensional identity
1
∂i ∂i
|x| =−4πδ(3) (x), i = 1, 2, 3, (7.37)
adapted to spherical coordinates
∂r r 2
1
∂r =−
r
4π
sin θ δ(3) (r), (7.38)
and the other one is
∂r r 1
=
r
4π
sin θ δ(3) (r), (7.39)
both of which can be checked by partial integration. Here δ (3) (x) = δ (3)
(r). The latter is defined by
drdϕdθδ (3) (r) = 1. The result obtained coincides with the one obtained by more rigorous methods in
[71, 72].