### 8.2 CTP effective action for the black hole

We first derive the CTP effective action for the model described in Sec. (7.1). Using the metric (156)
(and neglecting the surface terms that appear in an integration by parts) we have the action for the scalar
field written perturbatively as
where the first and second order perturbative operators and are given by
In the above expressions, is the -order term in the perturbation of the scalar curvature
, and and denote a linear and a quadratic combination of the perturbation, respectively:
From quantum field theory in curved spacetime considerations discussed above we take the following action
for the gravitational field:
The first term is the classical Einstein–Hilbert action, and the second term is the counterterm in four
dimensions used to renormalize the divergent effective action. In this action ,
, and is an arbitrary mass scale.
We are interested in computing the CTP effective action (163) for the matter action and when the field
is initially in the Hartle–Hawking vacuum. This is equivalent to saying that the initial state of the field
is described by a thermal density matrix at a finite temperature . The CTP effective
action at finite temperature for this model is given by (for details see [54, 55])

where denote the forward and backward time path of the CTP formalism, and is the
complete matrix propagator ( and take values: , , and correspond
to the Feynman, Wightman, and Schwinger Green’s functions respectively) with thermal boundary
conditions for the differential operator . The actual form of cannot
be explicitly given. However, it is easy to obtain a perturbative expansion in terms of , the -order
matrix version of the complete differential operator defined by and , and ,
the thermal matrix propagator for a massless scalar field in Schwarzschild spacetime. To second order
reads
Expanding the logarithm and dropping one term independent of the perturbation , the CTP
effective action may be perturbatively written as
In computing the traces, some terms containing divergences are canceled using counterterms introduced in
the classical gravitational action after dimensional regularization.