

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.

