It is indeed possible to construct the full
thermal matrix propagator based
on Page’s approximate Feynman Green’s function by using identities
relating the Feynman Green’s function with the other Green’s
functions with different boundary conditions. One can then proceed
to explicitly compute a CTP effective action and hence the
influence functional based on this approximation. However, we
desist from delving into such a calculation for the following
reason. Our main interest in performing such a calculation is to
identify and analyze the noise term which is the new ingredient in
the backreaction. We have mentioned that the noise term gives a
stochastic contribution
to the Einstein-Langevin
equation (14
). We had also stated
that this term is related to the variance of fluctuations in
, i.e, schematically, to
.
However, a calculation of
in the
Hartle-Hawking state in a Schwarzschild background using the Page
approximation was performed by Phillips and Hu [244, 245
, 241
], and it was shown
that though the approximation is excellent as far as
is concerned, it gives unacceptably large errors for
at the horizon. In fact, similar errors
will be propagated in the non-local dissipation term as well,
because both terms originate from the same source, that is, they
come from the last trace term in Equation (169
) which contains terms
quadratic in the Green’s function. However, the Influence
Functional or CTP formalism itself does not depend on the nature of
the approximation, so we will attempt to exhibit the general
structure of the calculation without resorting to a specific form
for the Green’s function and conjecture on what is to be expected.
A more accurate computation can be performed using this formal
structure once a better approximation becomes available.
The general structure of the CTP effective action
arising from the calculation of the traces in equation (169) remains the same. But
to write down explicit expressions for the non-local kernels one
requires the input of the explicit form of
in the Schwarzschild metric, which is not available
in closed form. We can make some general observations about the
terms in there. The first line containing
does not have an explicit Fourier representation as
given in the far field case, neither will
in the second line representing the zeroth order
contribution to
have a perfect fluid form.
The third and fourth terms containing the remaining quadratic
component of the real part of the effective action will not have
any simple or even complicated analytic form. The symmetry
properties of the kernels
and
remain intact, i.e., they are even and odd in
, respectively. The last term in the CTP effective
action gives the imaginary part of the effective action and the
kernel
is symmetric.
Continuing our general observations from this CTP
effective action, using the connection between this thermal CTP
effective action to the influence functional [272, 43] via an equation in
the schematic form (17
), we see that the
nonlocal imaginary term containing the kernel
is responsible for the generation of the stochastic
noise term in the Einstein-Langevin equation, and the real
non-local term containing kernel
is
responsible for the non-local dissipation term. To derive the
Einstein-Langevin equation we first construct the stochastic
effective action (27
). We then derive the
equation of motion, as shown earlier in Equation (29
), by taking its
functional derivative with respect to
and equating it
to zero. With the identification of noise and dissipation kernels,
one can write down a linear, non-local relation of the form