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8.1 General issues of backreaction

Backreaction studies of quantum field-processes in cosmological spacetimes have progressed further than the corresponding black-hole problems, partly because of the relative technical simplicity associated with the higher symmetry of relevant cosmological-background geometries. (For a summary of the cosmological backreaction problem treated in stochastic-gravity theory, see [208Jump To The Next Citation Point].) How the problem is set up and approached, e.g., via effective action, in these previously-studied models can be carried over to black-hole problems. In fact, since the interior of a black hole can be described by a cosmological model (e.g., the Kantowski–Sachs universe for a spherically symmetric black hole), some aspects even convey directly. The latest important work on this problem is that of Hiscock, Larson and Anderson [165Jump To The Next Citation Point] on backreaction in the interior of a black hole, in which one can find a concise summary of earlier work.

8.1.1 Regularized energy-momentum tensor

The first step in a backreaction problem is to find a regularized energy-momentum tensor of the quantum fields using reasonable techniques, since the expectation value of this serves as the source in the semiclassical Einstein equation. For this, much work started in the 1980s (and still ongoing sparingly) is concerned with finding the right approximations for the regularized energy-momentum tensor [6Jump To The Next Citation Point, 7Jump To The Next Citation Point, 8Jump To The Next Citation Point, 165Jump To The Next Citation Point, 217, 260Jump To The Next Citation Point, 292]. Even in the simplest spherically symmetric spacetime, including the important Schwarzschild metric, it is technically quite involved. To name a few of the important landmarks in this endeavor (this is adopted from [165]), Howard and Candelas [177Jump To The Next Citation Point, 178Jump To The Next Citation Point] have computed the stress-energy of a conformally-invariant scalar field in the Schwarzschild geometry; Jensen and Ottewill [218] have computed the vacuum stress-energy of a massless vector field in Schwarzschild. Approximation methods have been developed by Page, Brown, and Ottewill [41, 42, 287Jump To The Next Citation Point] for conformally-invariant fields in Schwarzschild spacetime, by Frolov and Zel’nikov [120] for conformally-invariant fields in a general static spacetime, and by Anderson, Hiscock and Samuel [7Jump To The Next Citation Point, 8Jump To The Next Citation Point] 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 [118, 119] for massive fields in Kerr spacetime, and by Anderson, Hiscock and Samuel [7, 8Jump To The Next Citation Point] for a general (arbitrary curvature coupling and mass) scalar field in a general, static, spherically-symmetric spacetime. And they have applied their method to the Reissner–Nordström geometry [6]. Though arduous and demanding, the effort continues on because of its importance in finding the backreaction effects of Hawking radiation on the evolution of black holes and the quantum structure of spacetime.

Here we wish to address the black hole backreaction problem with new insights and methods provided by stochastic gravity. (For the latest developments, see, e.g., [183Jump To The Next Citation Point, 187Jump To The Next Citation Point, 207, 208].) 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 semiclassical-gravity framework. In particular one needs to consider both the dissipation and the fluctuation aspects in the backreaction of particle creation and vacuum polarization.

In a short note Hu, Raval and Sinha [199Jump To The Next Citation Point] first used the stochastic gravity formalism to address the backreaction of evaporating black holes. A more detailed analysis is given by the recent work of Hu and Roura [201, 202Jump To The Next Citation Point]. For the class of quasi-static black holes, the formulation of the problem in this new light was sketched out by Sinha, Raval, and Hu [332Jump To The Next Citation Point]. We follow these two latter works in the stochastic gravity theory approach to the black-hole fluctuations and backreaction problems.

8.1.2 Backreaction and fluctuation-dissipation relation

From the statistical field-theory perspective provided by stochastic gravity, one can understand that backreaction effect is the manifestation of a fluctuation-dissipation relation [48, 49, 103, 104, 274, 365]. This was first conjectured by Candelas and Sciama [76Jump To The Next Citation Point, 324Jump To The Next Citation Point, 325Jump To The Next Citation Point] for a dynamic Kerr black hole emitting Hawking radiation and Mottola [268Jump To The Next Citation Point] for a static black hole (in a box) in quasi-equilibrium with its radiation via linear-response theory [33, 234, 235, 236, 238]. This postulate was shown to hold for fully dynamical spacetimes. From the cosmological-backreaction problem Hu and Sinha [206Jump To The Next Citation Point] derived a generalized fluctuation-dissipation relation relating dissipation (of anisotropy in Bianchi Type I universes) and fluctuations (measured by particle numbers created in neighboring histories).

While the fluctuation-dissipation relation in linear-response theory captures the response of the system (e.g., dissipation of the black hole) to the environment (in these cases the quantum matter field), linear-response theory (in the way it is commonly presented in statistical thermodynamics) cannot provide a full description of self-consistent backreaction on at least two counts:

First, because it is usually based on the assumption of a specified background spacetime (static in this case) and state (thermal) of the matter field(s) (e.g., [268Jump To The Next Citation Point]). The spacetime and the state of matter should be determined in a self-consistent manner by their dynamics and mutual influence. Second, the fluctuation part represented by the noise kernel is amiss, e.g., [10, 11]. This is also a problem in the fluctuation-dissipation relation proposed by Candelas and Sciama [76Jump To The Next Citation Point, 324Jump To The Next Citation Point, 325Jump To The Next Citation Point] (see below). As demonstrated by many authors [73Jump To The Next Citation Point, 206Jump To The Next Citation Point] backreaction is intrinsically a dynamic process. The Einstein–Langevin equation in stochastic gravity overcomes both of these deficiencies.

For Candelas and Sciama [76Jump To The Next Citation Point, 324Jump To The Next Citation Point, 325Jump To The Next Citation Point], 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 [199Jump To The Next Citation Point] 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.

8.1.3 Noise and fluctuations – the missing ingredient in older treatments

From this reasoning, we see that the vacuum expectation value of the stress-energy bitensor, known as the noise kernel, is the necessary new ingredient in addition to the dissipation kernel, and that stochastic gravity as an extension of semiclassical gravity is the appropriate framework for backreaction considerations. The noise kernel for quantum fields in Minkowski and de Sitter spacetime has been carried out by Martin, Roura and Verdaguer [257Jump To The Next Citation Point, 259, 316], and for thermal fields in black-hole spacetimes and scalar fields in general spacetimes by Campos, Hu and Phillips [69Jump To The Next Citation Point, 70Jump To The Next Citation Point, 304Jump To The Next Citation Point, 305Jump To The Next Citation Point].


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