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 , , , and is the line element on the two-sphere. Hawking radiation is described by a massless, conformally-coupled quantum scalar field with the classical action where ( is the dimension of spacetime) and is the curvature scalar of the spacetime it lives in.Let us consider linear perturbations of a background Schwarzschild metric ,

with standard line element We look for this class of perturbed metrics in the form given by Equation (179) (thus restricting our consideration only to spherically symmetric perturbations), and where with and , and where is the Hawking temperature. This particular parameterization of the perturbation is chosen following York’s notation [380, 381, 382]. Thus, the only non-zero components of are and So this represents a metric with small static and radial perturbations about a Schwarzschild black hole. The initial quantum state of the scalar field is taken to be the Hartle–Hawking vacuum, which is essentially a thermal state at the Hawking temperature, and it represents a black hole in (unstable) thermal equilibrium with its own Hawking radiation. In the far-field limit, the gravitational field is described by a linear perturbation of Minkowski spacetime. In equilibrium the thermal bath can be characterized by a relativistic fluid with a four-velocity (time-like normalized vector field) and temperature in its own rest frame .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,

with , where is the Einstein tensor for the background spacetime. The zeroth-order solution gives a background metric in empty space, i.e, the Schwarzschild metric. is the linear correction to the Einstein tensor in the perturbed metric. The semiclassical Einstein equation in this approximation therefore reduces to York solved this equation to first order by using the expectation value of the energy-momentum tensor for a conformally-coupled scalar field in the Hartle–Hawking vacuum in the unperturbed (Schwarzschild) spacetime on the right-hand side and using Equations (185) and (186) to calculate on the left-hand side. Unfortunately, no exact analytical expression is available for the in a Schwarzschild metric with the quantum field in the Hartle–Hawking vacuum that goes on the right-hand side. York, therefore, used the approximate expression given by Page [287], which is known to give excellent agreement with numerical results. Page’s approximate expression for was constructed using a thermal Feynman Green’s function obtained by a conformal transformation of a WKB-approximated Green’s function for an optical Schwarzschild metric. York then solves the semiclassical Einstein equation (188) self-consistently to obtain the corrections to the background metric induced by the backreaction encoded in the functions and . There was no mention of fluctuations or its effects. As we shall see, in the language of Section 4, the semiclassical gravity procedure, which York followed, working at the equation of motion level, is equivalent to looking at the noise-averaged backreaction effects.

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

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; 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 (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])

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.

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]

where the tensor is defined in [69, 70] after an expansion in terms of a basis of 14 tensors [311, 312]. In particular, the last trace of Equation (195) may be split into two different kernels and , One can express the Fourier transforms of these kernels as respectively.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

The first line is the Einstein–Hilbert action to second order in the perturbation . is a symmetric kernel, i.e., = . In the near flat case, its Fourier transform is given by The 14 elements of the tensor basis , , are defined in [311, 312]. The second is a local term, linear in . Only far away from the black hole it takes the form of the stress tensor of massless scalar particles at temperature , which has the form of a perfect fluid stress-energy tensor, where is the four-velocity of the plasma and the factor is the familiar thermal energy density for massless scalar particles at temperature . In the far field limit, taking into account the four-velocity of the fluid, a manifestly Lorentz-covariant approach to thermal field theory may be used [371]. However, in order to simplify the involved tensorial structure, we work in the co-moving coordinate system of the fluid, where . In the third line, the Fourier transform of the symmetric kernel can be expressed as where is a simple redefinition of the renormalization parameter given by and the tensors and are defined by respectively.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 (),

and at high temperatures the leading term () may be written as where we have introduced the dimensionless external momentum . The coefficients were first given in [311, 312] and generalized to the next-to-leading order in [39, 89]. (They are given with the MTW sign convention in [69, 70].)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

where . This is the general functional form of a fluctuation-dissipation relation and is called the fluctuation-dissipation kernel [45, 133, 196, 197]. In the present context this relation depicts the backreaction of thermal Hawking radiance for a black hole in quasi-equilibrium.

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

where In the far-field limit this equation should reduce to that obtained by Campos and Hu [69, 70]; for gravitational perturbations defined in Equation (191) under the harmonic gauge , their Einstein–Langevin equation is given by where the tensor is given by The expression for in the near-horizon limit, of course, cannot be expressed in such a simple form. Note that this differential stochastic equation includes a non-local term responsible for the dissipation of the gravitational field and a noise-source term, which accounts for the fluctuations of the quantum field. Note also that this equation, in combination with the correlation for the stochastic variable (211) determines, the two-point correlation for the stochastic metric fluctuations self-consistently.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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