Calculation of gravitational Euclidean action of Schwartzchild BH I am reading the paper of Gibbons and Hawking Action integrals and partition functions in quantum gravity, Phys. Rev. D 15 (1977) 2752 where they compute the gravitational action of black holes.
In particular, the action for a metric $g$ over a region $Y$ with boundary $\partial Y$ has the form
$$
I = \frac{1}{16\pi}\int_Yd^4x \sqrt{-g}R + \frac{1}{8\pi}\int_{\partial Y}d^3x \sqrt{-h}K
$$
where $R$ is the Ricci Scalar of the metric $g$, $K$ is the trace of the extrinsic curvature and $h$ is the induced metric on the boundary. There is also a constant to add in order to make the action convergent at spatial infinity. This constant is not important for my question.
As a concrete example, you can consider a Schwartzchild solution
$$
ds^2 = -f(r) dt^2 + \frac{1}{f(r)}dr^2 + r^2 d\Omega_2^2
$$
with $f(r) = (1-2m/r)$.
In their paper, Gibbons and Hawking exploit a method to compute explicitly the (euclidean) action by redefining the coordinates and taking an analytic continuation in the complex plane. They explicitly integrate on a section $Y$ where
$$
r\geq 2m
$$
My very naive question is: Why do they compute the action for $r\geq 2m$ and do not consider smaller values of $r$?
I might understand they want to avoid the singularity at $r=0$, but I don't understand why they are not integrating possibly on a subset of the region below the Schwarzschild radius, e.g. for $r\geq 2m -\epsilon$ for some small $\epsilon$.
 A: The idea of the analytic continuation is like this (I am referring to the whole  Kruskal-Szekeres diagram). Consider a Schwarzshild chart, where $t$ is a coordinate (I). We have four  such charts, and we consider two of them: the static region outside the black hole region where $t$ is a timelike coordinate (region I) and the non-stationary black hole (region II) region, where the coordinate $t$ is spacelike.
Let us first focus on the external region (I). Let put ourseves  on the $3D$ submanifold at $t=0$. This manifold can be viewed also as submanifold of another $4D$ manifold where $t$ takes imaginary values. That is just because $0= i0$. The two $3D$  manifolds: the real-time and the imaginary-time spacetimes  meet exactly on that $3D$ submanifold which is simultaneously part of both $4D$ manifolds.
On both $4D$ manifolds we can put the Schwarzshild metric (with $r>2m$) since it is analytic for complex values of $t$ and this metric becomes a common Euclidean $3D$ metric on the $t=it=0$ submanifold.
I stress that these common structures  are unique in view of the uniquenes of analytic extensions.
The same idea  works with the internal black hole region (II) and you have another euclidean 4D manifold which shares with the black hole region the $3D$ submanifold at $t=0$.
I stress that this submanifold is a vertical line from the bifurcation to the singularty.
These two $3D$ $t=0$ submanifolds meet, say, "othogonally" at the bifurcation in the Lorentzian spacetime and this means that they do not define a common  tangent space where they meet. In other words there is no common smooth submanifold  structure of the Lorentzian and Euclidean formalism where these two extensions are embedded.
Obviously you can always construct some smooth or analytic  bridge between them in the Euclidean spacetime containing both submanifolds equipped with the Schwarzschild metrics at $t=0$, and avoiding the apparent singularity at $r=2m$
as it happens with the Kruskal–Szekeres coordinates in the Lorentzian spacetime.
However every such bridge cannot be smoothly extended to the Lorentzian section, since there the two $3D$ submanifolds  do not smoothly join!.
Hence an analytic Euclidean spacetime that encompasses both regions with $r>2m$ and $r<2m$ would result, in a sense, arbitrary as it does not respect the analytic Lorentzian structure. I stress that however one may decide to renounce to incredibly rigid analytic structures, since they are only a possible way to describe physical objects.  In that case, some physical assumption would be however necessary and made explicit to replace the powerful idea of the uniqueness of analytic continuation.
The only possibility to join the two Euclidean sections, in this caese, would rely on a merely continuous structure. That sound a quite weak in my view.
Another completely physical  reason for not considering Euclidean extensions including the regions with $r>2m$ and $r<2m$ is that the meaning of $t$ in the internal region does not permit a thermodynamical interpretation as it defines a spacelike Killing vector instead of a timelike Killing vector.
