Gravity and Strings

8.6 The Euclidean electric RN solution and its action 243
In these coordinates it is evident that the horizon r =−∞is at an infinite distance in
the r direction and a constant-time slice of this spacetime looks like an infinite tube whose
r = constant sections are 2-spheres of constant radius R. Itdoes not make much sense
to talk about the period of τ that makes the Wick-rotated metric on the horizon regular
because it is regular for any period. The same applies to flat Euclidean spacetime. The
temperature cannot be uniquely assigned in this formalism. The reason could be the fact
that both Minkowski spacetime and the RB solution can be considered vacua of the theory.
As a conclusion of this discussion, then, if we compactify the Euclidean time with some
arbitrary period β, the topology is not R 2 × S 2 as in the non-extreme case, but R × S 1 × S 2 .
The factor R × S 1 , with the topology of a cylinder, corresponds to R 2 −{0}, the τ−r plane
with the point at the origin (the event horizon, which is at an infinite distance) removed.
The Euclidean RN solution has, therefore, two boundaries: at infinity (as in the non-extreme
case) and at the horizon. One way to check this fact is to calculate the Euler characteristic
of the Euclidean ERN solution using the Gauss–Bonnet theorem adapted to manifolds with
boundaries. The Euler characteristic χ is a topological invariant whose value is an integer
and the Gauss–Bonnet theorem states that the integral of the 4-form
1
32π 2 ɛabcd R ab ∧ R cd
(8.115)
overafour-dimensional compact manifold M is precisely χ.Ifthe manifold has a boundary
∂M, then χ(M) is given by the integral over M of the above 4-form plus the integral over
the boundary of a 3-form [236, 347],
χ(M) = 1
32π 2
ɛabcd R
M
ab ∧ Rcd − 1
32π 2
ɛabcd
∂M
2θ ab ∧ R cd − 4
3 θ ab ∧ θ c e ∧ θ ed ,
(8.116)
where θ ab is the second fundamental 1-form on ∂M, that can be constructed as explained
in [347]. The contribution of the boundary integral is crucial in order to have χ = 2inthe
non-extreme case, corresponding to the topology R 2 × S 2 .Inthe extreme case, only by
taking into account the boundary at the horizon does one obtain χ = 0, the correct value
for the topology R × S 1 × S 2 [445].
This is going to have important consequences in what follows.
Once we have determined the period, we are ready to calculate the partition function using
the Euclidean path-integral formalism in the saddle-point approximation. We are going
to do it as in the Schwarzschild case, using the Lorentzian action and solution but taking
into account the periodicity of the Euclidean time and the fact that the Euclidean solution
covers only the exterior of the horizon.
In = c = G (4)
N = 1 the Einstein–Maxwell system with boundary terms is
SEM[gµν, Aµ] = 1
d
16π
4 x |g| R − 1
4 F 2 + 1
d
8π
3 (K − K0), (8.117)
and, using the definition of Fµν and integrating by parts, we rewrite it in the form
SEM[gµν, Aµ] = 1
d
16π
4 x |g|[R + 1
2 Aν∇µF µν ]
+ 1
d
8π
3 [(K − K0) + 1
4nµF µν Aν]. (8.118)