Gravity and Strings

210 The Schwarzschild black hole
near the horizon. Defining another radial coordinate ρ such that gρρ = 1, we obtain
−dσ 2 ∼ f ′ (0)/2 2 dτ 2 + dρ 2 ≡ ρ 2 dτ 2 + dρ 2 . (7.67)
Now this metric is just the 2-plane metric in polar coordinates if τ ′ ∈ [0, 2π]. Otherwise
it is the metric of a cone and has a conical singularity at ρ = 0 (the horizon). Then, τ ∈
[0,β = 4π/f ′ (0)].
In practice we do not even need the Euclidean Schwarzschild metric. We need only
the information about the period of the Euclidean time (temperature) and the fact that the
BH interior disappears (the integration region) and we can simply replace − ˜SEH(on-shell)
by +iSEH(on-shell) because it gives the same result once we take into account the above
two points. Thus, in our calculation we will use the Lorentzian Schwarzschild metric in
Schwarzschild coordinates using these observations.
7.4.2 The boundary terms
The Euclidean Schwarzschild solution, being a solution of the vacuum Einstein equations,
has R = 0everywhere (the singularity r = 0 together with the whole BH interior is not
included) and only the boundary term contributes to the on-shell action. We are going to
calculate its value in this section.
The only boundary of the Euclidean Schwarzschild metric, with the time compactified
on a circle of length β, isr→∞.(If we gave the Euclidean time a different periodicity,
there would be another boundary at the horizon, but there is no reason to do this.) This
boundary is then the hypersurface r = rc when the constant rc goes to infinity. A vector
normal to the hypersurfaces r − rc = 0isnµ ∼ ∂µ(r − rc) = δµr, and, normalized to unity
(nµn µ =−1 because it is spacelike) with the right sign to make it outward-pointing, is, for
a generic spherically symmetric metric Eq. (7.22),
nµ =− δµr
√
−n2 =−√−grr δµr. (7.68)
The four-dimensional metric gµν induces the following metric hµν on the hypersurface
r − rc = 0:
ds 2 (3) = hµνdx µ dx ν = gttdt 2 − r 2 d 2
(2)
r=rc
. (7.69)
The covariant derivative of nµ is
∇µnν =− √
−grr δµrδνr∂r ln √ −grr − Ɣµν r , (7.70)
and the trace of the extrinsic curvature of the r − rc = 0hypersurfaces is (the Christoffel
symbols can be found in Appendix F.1)
K = h µν ∇µnν =
The regulator K0 can be found form this expression to be
1 1
√ 2
−grr
∂r ln gtt + 2/r
r=rc
. (7.71)
K0 = (2/r)| r=r0
. (7.72)