We follow the strategy outlined by Sinha, Raval and Hu [332] 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 [380
, 381
, 382
] semiclassical equations for radially perturbed
quasi-static black holes. For the near-horizon case one cannot obtain the full details yet, because the
Green’s function for a scalar field in the Schwarzschild metric comes only in an approximate form,
e.g., Page’s approximation [287
], which, though reasonably accurate for the stress tensor, fails
badly for the noise kernel [305
]. In addition, a formula is derived in [332] expressing the CTP
effective action in terms of the Bogoliubov 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.
In this model the black-hole spacetime is described by a spherically-symmetric static metric with line element of the following general form written in advanced-time Eddington-Finkelstein coordinates as
where Let us consider linear perturbations of a background Schwarzschild metric
,
To facilitate later comparisons with our program we briefly recall York’s work [380, 381, 382]. (See also
the work by Hochberg and Kephart [166] for a massless vector field, by Hochberg, Kephart, and York [167]
for a massless spinor field, and by Anderson, Hiscock, Whitesell, and York [9] for a quantized massless
scalar field with arbitrary coupling to spacetime curvature.) York considered the semiclassical Einstein
equation,
We first derive the CTP effective action for the model described in the previous section. Using the
metric (182) (and neglecting the surface terms that appear in an integration by parts) we have the action
for the scalar field written perturbatively as
We are interested in computing the CTP effective action (189) 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 [69
, 70
])
At this point we divide our considerations into two cases. In the far field limit represent perturbations
about flat space, i.e.,
. The exact “unperturbed” thermal propagators for scalar fields are
known, i.e., the Euclidean propagator with periodicity
. Using the Fourier-transformed form (those
quantities are denoted with a tilde) of the thermal propagators
, the trace terms of the form
can be written as [69
, 70
]
Using the property , it is easy to see that the kernel
is
symmetric and
is antisymmetric in its arguments; that is,
and
.
The physical meanings of these kernels can be extracted if we write the renormalized CTP effective
action at finite temperature (195) in an influence-functional form [45
, 133
, 196
, 197
].
, the
imaginary part of the CTP effective action, can be identified with the noise kernel and
,
the antisymmetric piece of the real part, with the dissipation kernel. Campos and Hu [69
, 70
]
have shown that these kernels identified as such indeed satisfy a thermal fluctuation-dissipation
relation.
If we denote the difference and the sum of the perturbations , defined along each branch
of
the complex time path of integration
, by
and
, respectively,
the influence-functional form of the thermal CTP effective action may be written to second order in
as
In the above and subsequent equations, we denote the coupling parameter in four dimensions
by
, and consequently
means
evaluated at
.
is the
complete contribution of a free massless quantum scalar field to the thermal graviton polarization
tensor [39
, 89
, 311
, 312
], and it is responsible for the instabilities found in flat spacetime at finite
temperature [39
, 89
, 138, 311
, 312
]. Note that the addition of the contribution of other kinds of
matter fields to the effective action, even graviton contributions, does not change the tensor
structure of these kernels, and only the overall factors are different to leading order [311
, 312
].
Equation (203
) reflects the fact that the kernel
has thermal, as well as non-thermal,
contributions. Note that it reduces to the first term in the zero temperature limit (
),
Finally, as defined above, is the noise kernel representing random fluctuations of thermal
radiance and
is the dissipation kernel, describing the dissipation of energy of the gravitational
field.
In this case, since the perturbation is taken around the Schwarzschild spacetime, exact expressions for the
corresponding unperturbed propagators are not known. Therefore, apart from the approximation
of computing the CTP effective action to certain order in perturbation theory, an appropriate
approximation scheme for the unperturbed Green’s functions is also required. This feature manifested itself
in York’s calculation of backreaction as well, where, in writing
on the right-hand side of
the semiclassical Einstein equation in the unperturbed Schwarzschild metric, he had to use an
approximate expression for
in the Schwarzschild metric, given by Page [287
]. The additional
complication here is that, while to obtain
as in York’s calculation, the knowledge of only the
thermal Feynman Green’s function is required, to calculate the CTP effective action one needs the
knowledge of the full matrix propagator, which involves the Feynman, Schwinger and Wightman
functions.
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 (15
). 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 [304, 305
] 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 (195
), 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 (195) 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 L
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 respectively even and odd in
. 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 [58, 343] via an equation in the schematic
form (24), 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 (34
). We then derive the equation of motion, as shown earlier in (36
), 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
In this section we show how a semiclassical Einstein–Langevin equation can be derived from the previous thermal CTP effective action. This equation depicts the stochastic evolution of the perturbations of the black hole under the influence of the fluctuations of the thermal scalar field.
The influence functional previously introduced in Equation (23
) can be
written in terms of the the CTP effective action
derived in Equation (200
) using
Equation (24
). The Einstein–Langevin equation follows from taking the functional derivative of the
stochastic effective action (34
) with respect to
and imposing
. This leads to
As we have seen before and here, the Einstein–Langevin equation is a dynamical equation governing the dissipative evolution of the gravitational field under the influence of the fluctuations of the quantum field, which, in the case of black holes, takes the form of thermal radiance. From its form we can see that, even for the quasi-static case under study, the backreaction of Hawking radiation on the black-hole spacetime has an innate dynamical nature.
For the far-field case, making use of the explicit forms available for the noise and dissipation kernels,
Campos and Hu [69, 70
] formally prove the existence of a fluctuation-dissipation relation at all
temperatures between the quantum fluctuations of the thermal radiance and the dissipation of the
gravitational field. They also show the formal equivalence of this method with linear-response theory for
lowest order perturbations of a near-equilibrium system and how the response functions, such as the
contribution of the quantum scalar field to the thermal graviton polarization tensor, can be
derived. An important quantity not usually obtained in linear-response theory, but of equal
importance, manifest in the CTP stochastic approach is the noise term arising from the quantum and
statistical fluctuations in the thermal field. The example given in this section shows that the
backreaction is intrinsically a dynamic process described (at this level of sophistication) by the
Einstein–Langevin equation. By comparison, traditional linear response theory calculations cannot
capture the dynamics as fully and thus cannot provide a complete description of the backreaction
problem.
As remarked earlier, except for the near-flat case, an analytic form of the Green’s function is not available.
Even the Page approximation [287], which gives unexpectedly good results for the stress-energy tensor, has
been shown to fail in the fluctuations of the energy density [305
]. 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’s 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 (200
). Thus, this offers a new route to
arrive at York’s semiclassical Einstein’s 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 a non-local dissipation term arises from the fourth term in Equation (200
) in the CTP
effective action, which is absent in York’s treatment. This difference exists 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 in the
influence-functional approach. 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 influence-functional 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 new approach is that related to the
fluctuation-dissipation relation. In the quantum Brownian motion analog (e.g., [45, 133
, 196
, 197
] 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 [69
, 70
] and is expected to exist between the noise and dissipation
kernels for the general case, as it is a categorical relation [45, 133, 183, 196, 197]. 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 [257].
As discussed earlier, the existence of a fluctuation-dissipation relation for the black-hole case
has previously been suggested by some authors previously [76, 268, 324, 325]. This relation
and the relevant physical quantities contained therein, such as the black-hole susceptibility
function, which characterizes the statistical mechanical and dynamical responses of a black
hole interacting with its quantum field environment, will allow us to study the nonequilibrium
thermodynamic properties of the black hole and through it, perhaps, the microscopic structure of
spacetime.
There are limitations of a technical nature in the quasi-static case studied, as mentioned above, i.e., there is no reliable approximation to the Schwarzschild thermal Green’s function to explicitly compute the noise and dissipation kernels. Another technical limitation of this 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, our analysis is essentially confined to a Hartle–Hawking thermal state of the field. It does not directly extend to a more general class of states, for example to the case in which the initial state of the field is in the Unruh vacuum. To study the dynamics of a radiating black hole under the influence of a quantum field and its fluctuations a different model and approach are needed, which we now discuss.
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