As another illustration of the application of stochastic gravity we consider fluctuations and backreaction in black hole spacetimes. The celebrated Hawking effect of particle creation from black holes is constructed from a quantum field theory in a curved spacetime framework. The oft-mentioned ‘black hole evaporation’ referring to the reduction of the mass of a black hole due to particle creation must entail backreaction considerations. Backreaction of Hawking radiation [118, 13, 297, 298, 299, 135, 136, 8] could alter the evolution of the background spacetime and change the nature of its end state, more drastically so for Planck size black holes. Because of the higher symmetry in cosmological spacetimes, backreaction studies of processes therein have progressed further than the corresponding black hole problems, which to a large degree is still concerned with finding the right approximations for the regularized energy-momentum tensor [176, 233, 210, 6, 7, 5, 134] for even the simplest spacetimes such as the spherically symmetric family including the important Schwarzschild metric (for a summary of the cosmological backreaction problem treated in the stochastic gravity theory, see [169]). The latest important work is that of Hiscock, Larson, and Anderson [134] on backreaction in the interior of a black hole, where one can find a concise summary of earlier work. To name a few of the important landmarks in this endeavor (this is adopted from [134]), Howard and Candelas [145, 144] have computed the stress-energy of a conformally invariant scalar field in the Schwarzschild geometry. Jensen and Ottewill [177] have computed the vacuum stress-energy of a massless vector field in Schwarzschild spacetime. Approximation methods have been developed by Page, Brown, and Ottewill [230, 28, 29] for conformally invariant fields in Schwarzschild spacetime, Frolov and Zel’nikov [99] for conformally invariant fields in a general static spacetime, and Anderson, Hiscock, and Samuel [6, 7] for massless arbitrarily coupled scalar fields in a general static spherically symmetric spacetime. Furthermore the DeWitt–Schwinger approximation has been derived by Frolov and Zel’nikov [97, 98] for massive fields in Kerr spacetime, by Anderson, Hiscock, and Samuel [6, 7] for a general (arbitrary curvature coupling and mass) scalar field in a general static spherically symmetric spacetime and Anderson, Hiscock, and Samuel have applied their method to the Reissner–Nordström geometry [5]. Though arduous and demanding, the effort continues on because of the importance of backreaction effects of Hawking radiation in black holes. They are expected to address some of the most basic issues such as black hole thermodynamics [174, 235, 281, 16, 17, 18, 267, 175, 287, 286, 275, 183, 271, 139, 140, 205, 204] and the black hole end state and information loss puzzles [231].

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 [150, 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 [10, 9] 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, 243, 245, 244]. 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 [147]. 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 [230]), which, though reasonably accurate for the stress tensor, fails poorly for the noise kernel [245, 244]. 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.

8.1 The model

8.2 CTP effective action for the black hole

8.3 Near flat case

8.4 Near horizon case

8.5 The Einstein–Langevin equation

8.6 Discussions

8.6.1 Black hole backreaction

8.6.2 Metric fluctuations in black holes

8.2 CTP effective action for the black hole

8.3 Near flat case

8.4 Near horizon case

8.5 The Einstein–Langevin equation

8.6 Discussions

8.6.1 Black hole backreaction

8.6.2 Metric fluctuations in black holes

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