As remarked earlier, except for the near-flat
case, an analytic form of the Green function is not available. Even
the Page approximation [231], which gives
unexpectedly good results for the stress-energy tensor, has been
shown to fail in the fluctuations of the energy density [245, 241]. Thus, using
such an approximation for the noise kernel will give unreliable
results for the Einstein-Langevin equation. If we confine ourselves
to Page’s approximation and derive the equation of motion without
the stochastic term, we expect to recover York’s semiclassical
Einstein equation if one retains only the zeroth order
contribution, i.e, the first two terms in the expression for the
CTP effective action in Equation (174). Thus, this offers a
new route to arrive at York’s semiclassical Einstein equations. Not
only is it a derivation of York’s result from a different point of
view, but it also shows how his result arises as an appropriate
limit of a more complete framework, i.e, it arises when one
averages over the noise. Another point worth noting is that our
treatment will also yield a non-local dissipation term arising from
the fourth term in Equation (174
) in the CTP effective
action which is absent in York’s treatment. This difference is
primarily due to the difference in the way backreaction is treated,
at the level of iterative approximations on the equation of motion
as in York, versus the treatment at the effective action level as
pursued here. In York’s treatment, the Einstein tensor is computed
to first order in perturbation theory, while
on the right-hand side of the semiclassical Einstein
equation is replaced by the zeroth order term. In the effective
action treatment the full effective action is computed to second
order in perturbation, and hence includes the higher order
non-local terms.
The other important conceptual point that comes
to light from this approach is that related to the
fluctuation-dissipation relation. In the quantum Brownian motion
analog (see, e.g., [32, 111
, 161
, 162
] and references
therein), the dissipation of the energy of the Brownian particle as
it approaches equilibrium and the fluctuations at equilibrium are
connected by the fluctuation-dissipation relation. Here the
backreaction of quantum fields on black holes also consists of two
forms - dissipation and fluctuation or noise - corresponding to the
real and imaginary parts of the influence functional as embodied in
the dissipation and noise kernels. A fluctuation-dissipation
relation has been shown to exist for the near flat case by Campos
and Hu [54
, 55
] and we anticipate
that it should also exist between the noise and dissipation kernels
for the general case, as it is a categorical relation [32, 111, 161, 162, 151]. Martin and Verdaguer
have also proved the existence of a fluctuation-dissipation
relation when the semiclassical background is a stationary
spacetime and the quantum field is in thermal equilibrium. Their
result was then extended to a conformal field in a conformally
stationary background [207]. The existence
of a fluctuation-dissipation relation for the black hole case has
been discussed by some authors previously [60, 258, 259, 217]. In [164
], Hu, Raval, and
Sinha have described how this approach and its results differ from
those of previous authors. The fluctuation-dissipation relation
reveals an interesting connection between black holes interacting
with quantum fields and non-equilibrium statistical mechanics. Even
in its restricted quasi-static form, this relation will allow us to
study nonequilibrium thermodynamic
properties of the black hole under the influence of stochastic
fluctuations of the energy-momentum tensor dictated by the noise
terms.
There are limitations of a technical nature in the specific example invoked here. For one we have to confine ourselves to small perturbations about a background metric. For another, as mentioned above, there is no reliable approximation to the Schwarzschild thermal Green’s function to explicitly compute the noise and dissipation kernels. This limits our ability to present explicit analytical expressions for these kernels. One can try to improve on Page’s approximation by retaining terms to higher order. A less ambitious first step could be to confine attention to the horizon and using approximations that are restricted to near the horizon and work out the Influence Functional in this regime.
Yet another technical limitation of the specific example is the following. Although we have allowed for backreaction effects to modify the initial state in the sense that the temperature of the Hartle-Hawking state gets affected by the backreaction, we have essentially confined our analysis to a Hartle-Hawking thermal state of the field. This analysis does not directly extend to a more general class of states, for example to the case where the initial state of the field is in the Unruh vacuum. Thus, we will not be able to comment on issues of the stability of an isolated radiating black hole under the influence of stochastic fluctuations.
In addition to the work described above by
Campos, Hu, Raval, and Sinha [54, 55, 164, 264] and earlier
work quoted therein, we mention also some recent work on black hole
metric fluctuations and their effect on Hawking radiation. For
example, Casher et al. [64] and
Sorkin [267
, 268
] have concentrated
on the issue of fluctuations of the horizon induced by a
fluctuating metric. Casher et al. [64] consider the
fluctuations of the horizon induced by the “atmosphere” of high
angular momentum particles near the horizon, while
Sorkin [267, 268] calculates
fluctuations of the shape of the horizon induced by the quantum
field fluctuations under a Newtonian approximation. Both group of
authors come to the conclusion that horizon fluctuations become
large at scales much larger than the Planck scale (note that Ford
and Svaiter [94
] later presented
results contrary to this claim). However, though these works do
deal with backreaction, the fluctuations considered do not arise as
an explicit stochastic noise term as in our treatment. It may be
worthwhile exploring the horizon fluctuations induced by the
stochastic metric in our model and comparing the conclusions with
the above authors. Barrabes et al. [14, 15] have
considered the propagation of null rays and massless fields in a
black hole fluctuating geometry, and have shown that the stochastic
nature of the metric leads to a modified dispersion relation and
helps to confront the trans-Planckian frequency problem. However,
in this case the stochastic noise is put in by hand and does not
naturally arise from coarse graining as in the quantum open systems
approach. It also does not take backreaction into account. It will
be interesting to explore how a stochastic black hole metric,
arising as a solution to the Einstein-Langevin equation, hence
fully incorporating backreaction, would affect the trans-Planckian
problem.
Ford and his collaborators [94, 95, 294] have also explored the issue of metric fluctuations in detail and in particular have studied the fluctuations of the black hole horizon induced by metric fluctuations. However, the fluctuations they have considered are in the context of a fixed background and do not relate to the backreaction.
Another work originating from the same vein of
stochastic gravity but not complying with the backreaction spirit
is that of Hu and Shiokawa [166], who study effects
associated with electromagnetic wave propagation in a
Robertson-Walker universe and the Schwarzschild spacetime with a
small amount of given metric stochasticity. They find that
time-independent randomness can decrease the total luminosity of
Hawking radiation due to multiple scattering of waves outside the
black hole and gives rise to event horizon fluctuations and
fluctuations in the Hawking temperature. The stochasticity in a
background metric in their work is assumed rather than derived
(from quantum field fluctuations, as in this work), and so is not
in the same spirit of backreaction. But it is interesting to
compare their results with that of backreaction, so one can begin
to get a sense of the different sources of stochasticity and their
weights (see, e.g., [154] for a list of
possible sources of stochasticity).
In a subsequent paper Shiokawa [261] showed that the scalar and spinor waves in a stochastic spacetime behave similarly to the electrons in a disordered system. Viewing this as a quantum transport problem, he expressed the conductance and its fluctuations in terms of a nonlinear sigma model in the closed time path formalism and showed that the conductance fluctuations are universal, independent of the volume of the stochastic region and the amount of stochasticity. This result can have significant importance in characterizing the mesoscopic behavior of spacetimes resting between the semiclassical and the quantum regimes.