Here we wish to address the black hole
backreaction problem with new insights provided by stochastic
semiclassical gravity. (For the latest developments see, e.g., the
reviews [151, 154
, 168, 169]). It is not our
intention to seek better approximations for the regularized
energy-momentum tensor, but to point out new ingredients lacking in
the existing framework based on semiclassical gravity. In
particular one needs to consider both the dissipation and the
fluctuations aspects in the backreaction of particle creation or
vacuum polarization.
In a short note [164] Hu, Raval, and
Sinha discussed the formulation of the problem in this new light,
commented on some shortcomings of existing works, and sketched the
strategy [264
] behind the
stochastic gravity theory approach to the black hole fluctuations
and backreaction problem. Here we follow their treatment with focus
on the class of quasi-static black holes.
From the new perspective provided by statistical
field theory and stochastic gravity, it is not difficult to
postulate that the backreaction effect is the manifestation of a
fluctuation-dissipation relation [85, 86, 220, 35, 34, 288]. This was first
conjectured by Candelas and Sciama [60, 258
, 259
] for a dynamic Kerr
black hole emitting Hawking radiation, and by Mottola [217
] for a static black
hole (in a box) in quasi-equilibrium with its radiation via linear
response theory [191, 24, 192, 195, 193]. However,
these proposals as originally formulated do not capture the full
spirit and content of the self-consistent dynamical backreaction
problem. Generally speaking (paraphrasing Mottola), linear response
theory is not designed for tackling backreaction problems. More
specifically, if one assumes a specified background spacetime
(static in this case) and state (thermal) of the matter field(s) as
done in [217
], one would get a
specific self-consistent solution. But in the most general
situation which a full backreaction program demands of, the
spacetime and the state of matter should be determined by their
dynamics under mutual influence on an equal footing, and the
solutions checked to be physically sound by some criteria like
stability consideration. A recent work of Anderson, Molina-Paris,
and Mottola [9, 10] on linear
response theory does not make these restrictions. They addressed
the issue of the validity of semiclassical gravity (SCG) based on
an analysis of the stability of solutions to the semiclassical
Einstein equation (SEE). However, on this issue, Hu, Roura, and
Verdaguer [165] pointed out the importance
of including both the intrinsic and induced fluctuations in the
stability analysis, the latter being given by the noise kernel. The
fluctuation part represented by the noise kernel is amiss in the
fluctuation-dissipation relation proposed by Candelas and
Sciama [60
, 258
, 259
] (see below). As
will be shown in an explicit example later, the backreaction is an
intrinsically dynamic process. The Einstein-Langevin equation in
stochastic gravity overcomes both of these deficiencies.
For Candelas and Sciama [60, 258
, 259
], the classical
formula they showed relating the dissipation in area linearly to
the squared absolute value of the shear amplitude is suggestive of
a fluctuation-dissipation relation. When the gravitational
perturbations are quantized (they choose the quantum state to be
the Unruh vacuum) they argue that it approximates a flux of
radiation from the hole at large radii. Thus the dissipation in
area due to the Hawking flux of gravitational radiation is
allegedly related to the quantum fluctuations of gravitons. The
criticism in [164
] is that their’s is
not a fluctuation-dissipation relation in the truly statistical
mechanical sense, because it does not relate dissipation of a
certain quantity (in this case, horizon area) to the fluctuations
of the same quantity. To do so would
require one to compute the two point function of the area, which,
being a four-point function of the graviton field, is related to a
two-point function of the stress tensor. The stress tensor is the
true “generalized force” acting on the spacetime via the equations
of motion, and the dissipation in the metric must eventually be
related to the fluctuations of this generalized force for the
relation to qualify as a fluctuation-dissipation relation.
From this reasoning, we see that the
stress-energy bi-tensor and its vacuum expectation value known as
the noise kernel are the new ingredients in backreaction
considerations. But these are exactly the centerpieces in
stochastic gravity. Therefore the correct framework to address
semiclassical backreaction problems is stochastic gravity theory,
where fluctuations and dissipation are the equally essential
components. The noise kernel for quantum fields in Minkowski and de
Sitter spacetime has been carried out by Martin, Roura, and
Verdaguer [207, 209, 254], and for
thermal fields in black hole spacetimes and scalar fields in
general spacetimes by Campos, Hu, and Phillips [54
, 55
, 244
, 245
, 241
]. Earlier, for
cosmological backreaction problems Hu and Sinha [167
] derived a
generalized expression relating dissipation (of anisotropy in
Bianchi Type I universes) and fluctuations (measured by particle
numbers created in neighboring histories). This example shows that
one can understand the backreaction of particle creation as a
manifestation of a (generalized) fluctuation-dissipation relation.
As an illustration of the application of
stochastic gravity theory we outline the steps in a black hole
backreaction calculation, focusing on the manageable quasi-static
class. We adopt the Hartle-Hawking picture [127] where the black
hole is bathed eternally - actually in quasi-thermal equilibrium -
in the Hawking radiance it emits. It is described here by a
massless scalar quantum field at the Hawking temperature. As is
well-known, this quasi-equilibrium condition is possible only if
the black hole is enclosed in a box of size suitably larger than
the event horizon. We can divide our consideration into the far
field case and the near horizon case. Campos and Hu [54, 55
] have treated a
relativistic thermal plasma in a weak gravitational field. Since
the far field limit of a Schwarzschild metric is just the perturbed
Minkowski spacetime, one can perform a perturbation expansion off
hot flat space using the thermal Green functions [108]. Strictly
speaking the location of the box holding the black hole in
equilibrium with its thermal radiation is as far as one can go,
thus the metric may not reach the perturbed Minkowski form. But one
can also put the black hole and its radiation in an anti-de Sitter
space [133], which contains
such a region. Hot flat space has been studied before for various
purposes (see, e.g., [116
, 249
, 250
, 72
, 27
]). Campos and Hu
derived a stochastic CTP effective action and from it an equation
of motion, the Einstein-Langevin equation, for the dynamical effect
of a scalar quantum field on a background spacetime. To perform
calculations leading to the Einstein-Langevin equation, one needs
to begin with a self-consistent solution of the semiclassical
Einstein equation for the thermal field and the perturbed
background spacetime. For a black hole background, a semiclassical
gravity solution is provided by York [297
, 298
, 299
]. For a
Robertson-Walker background with thermal fields, it is given by
Hu [148]. Recently, Sinha, Raval,
and Hu [264
] outlined a strategy
for treating the near horizon case, following the same scheme of
Campos and Hu. In both cases two new terms appear which are absent
in semiclassical gravity considerations: a nonlocal dissipation and
a (generally colored) noise kernel. When one takes the noise
average, one recovers York’s [297
, 298
, 299
] semiclassical
equations for radially perturbed quasi-static black holes. For the
near horizon case one cannot obtain the full details yet, because
the Green function for a scalar field in the Schwarzschild metric
comes only in an approximate form (e.g., Page
approximation [231
]), which, though
reasonably accurate for the stress tensor, fails poorly for the
noise kernel [245
, 241
]. In addition a
formula is derived in [264
] expressing the CTP
effective action in terms of the Bogolyubov coefficients. Since it
measures not only the number of particles created, but also the
difference of particle creation in alternative histories, this
provides a useful avenue to explore the wider set of issues in
black hole physics related to noise and fluctuations.
Since backreaction calculations in semiclassical gravity have been under study for a much longer time than in stochastic gravity, we will concentrate on explaining how the new stochastic features arise from the framework of semiclassical gravity, i.e., noise and fluctuations and their consequences. Technically the goal is to obtain an influence action for this model of a black hole coupled to a scalar field and to derive an Einstein-Langevin equation from it. As a by-product, from the fluctuation-dissipation relation, one can derive the vacuum susceptibility function and the isothermal compressibility function for black holes, two quantities of fundamental interest in characterizing the nonequilibrium thermodynamic properties of black holes.