Gravity and Strings

7.4 The Euclidean path-integral approach 209
where K0 is calculated by substituting the vacuum metric into the expression for K.
The path integral is now to be calculated in the (semiclassical) saddle-point approximation
(from now on we set = c = G (4)
N = 1 for simplicity)
Z = e − ˜SEH(on-shell) . (7.61)
The classical solution used to calculate the on-shell Euclidean action above is the Euclidean
Schwarzschild solution that we now discuss.
7.4.1 The Euclidean Schwarzschild solution
The Euclidean Schwarzschild solution solves the Einstein equations with Euclidean metric
(in our case (−, −, −, −)). It can be obtained by performing a Wick rotation τ = it
of the Lorentzian Schwarzschild solution. If we use Kruskal–Szekeres (KS) coordinates
{T, X,θ,ϕ},wehavetodefine the Euclidean KS time T = iT. This Wick rotation has important
effects. The relation between the Schwarzschild coordinate r and T, X coordinates
was
(r/RS − 1)e r
R S = X 2 − T 2 . (7.62)
The l.h.s. is bigger than −1 and that is why the X, T coordinates also cover the BH
interior. However, in terms of T ,
(r/RS − 1)e r
R S = X 2 + T 2 > 0, (7.63)
and the interior r < RS of the BH is not covered by the Euclidean KS coordinates. On the
other hand, the relation between the Schwarzschild time t and X, T ,
X + T
X − T
= e t
R S , (7.64)
becomes
X − iT
X + T = e−2i Arg(X+iT ) iτ
− R = e S . (7.65)
Since Arg(X + iT ) takes values between 0 and 2π (which should be identified), for
consistency (to avoid conical singularities) τ must take values in a circle of length 8π M
[436, 969]. The period of the Euclidean time can be interpreted as the inverse temperature
β which coincides with the known Hawking temperature. This is the reason why we can
use this metric to calculate the thermal partition function.
The result is a Euclidean metric with periodic time that covers only the exterior of the
BH (region I of the KS diagram). The X, T part of the metric describes a semi-infinite
“cigar” (times a 2-sphere) that goes from the horizon to infinity with topology R2 × S2 .
Knowing the result beforehand, we could just as well have used Schwarzschild coordinates,
which cover smoothly the BH exterior, and proceeded in this much more economical
way [546]: given a static, spherically symmetric BH with regular horizon at r = 0, the r–τ
part of its Euclidean metric can always be put in the form
−dσ 2 = f (r)dτ 2 + f −1 (r)dr 2 ∼ f ′ (0)rdτ 2 + 1
f ′ (0)r dr2 , (7.66)