8.6 Discussions

8.6.1 Black hole backreaction

As remarked earlier, except for the near-flat case, an analytic form of the Green function is not available. Even the Page approximation [230], which gives unexpectedly good results for the stress-energy tensor, has been shown to fail in the fluctuations of the energy density [245, 244]. 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 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 (174). Thus, this offers a new route to arrive at York’s semiclassical Einstein 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 our treatment will also yield a non-local dissipation term arising from the fourth term in Equation (174) in the CTP effective action which is absent in York’s treatment. This difference is 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 pursued here. 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 effective action 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 approach is that related to the fluctuation-dissipation relation. In the quantum Brownian motion analog (see, e.g., [32, 111, 161, 162] 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 [54, 55] and we anticipate that it should also exist between the noise and dissipation kernels for the general case, as it is a categorical relation [32, 111, 161, 162, 150]. 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 [207]. The existence of a fluctuation-dissipation relation for the black hole case has been discussed by some authors previously [60, 258, 259, 217]. In [164], Hu, Raval, and Sinha have described how this approach and its results differ from those of previous authors. The fluctuation-dissipation relation reveals an interesting connection between black holes interacting with quantum fields and non-equilibrium statistical mechanics. Even in its restricted quasi-static form, this relation will allow us to study nonequilibrium thermodynamic properties of the black hole under the influence of stochastic fluctuations of the energy-momentum tensor dictated by the noise terms.

There are limitations of a technical nature in the specific example invoked here. For one we have to confine ourselves to small perturbations about a background metric. For another, as mentioned above, there is no reliable approximation to the Schwarzschild thermal Green’s function to explicitly compute the noise and dissipation kernels. This limits our ability to present explicit analytical expressions for these kernels. One can try to improve on Page’s approximation by retaining terms to higher order. A less ambitious first step could be to confine attention to the horizon and using approximations that are restricted to near the horizon and work out the Influence Functional in this regime.

Yet another technical limitation of the specific 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, we have essentially confined our analysis to a Hartle-Hawking thermal state of the field. This analysis does not directly extend to a more general class of states, for example to the case where the initial state of the field is in the Unruh vacuum. Thus, we will not be able to comment on issues of the stability of an isolated radiating black hole under the influence of stochastic fluctuations.

8.6.2 Metric fluctuations in black holes

In addition to the work described above by Campos, Hu, Raval, and Sinha [54, 55, 164, 264] and earlier work quoted therein, we mention also some recent work on black hole metric fluctuations and their effect on Hawking radiation. For example, Casher et al. [64] and Sorkin [266, 268] have concentrated on the issue of fluctuations of the horizon induced by a fluctuating metric. Casher et al. [64] consider the fluctuations of the horizon induced by the “atmosphere” of high angular momentum particles near the horizon, while Sorkin [266, 268] calculates fluctuations of the shape of the horizon induced by the quantum field fluctuations under a Newtonian approximation. Both group of authors come to the conclusion that horizon fluctuations become large at scales much larger than the Planck scale (note that Ford and Svaiter [94] later presented results contrary to this claim). However, though these works do deal with backreaction, the fluctuations considered do not arise as an explicit stochastic noise term as in our treatment. It may be worthwhile exploring the horizon fluctuations induced by the stochastic metric in our model and comparing the conclusions with the above authors. Barrabes et al. [14, 15] have considered the propagation of null rays and massless fields in a black hole fluctuating geometry, and have shown that the stochastic nature of the metric leads to a modified dispersion relation and helps to confront the trans-Planckian frequency problem. However, in this case the stochastic noise is put in by hand and does not naturally arise from coarse graining as in the quantum open systems approach. It also does not take backreaction into account. It will be interesting to explore how a stochastic black hole metric, arising as a solution to the Einstein–Langevin equation, hence fully incorporating backreaction, would affect the trans-Planckian problem.

Ford and his collaborators [94, 95, 294] have also explored the issue of metric fluctuations in detail and in particular have studied the fluctuations of the black hole horizon induced by metric fluctuations. However, the fluctuations they have considered are in the context of a fixed background and do not relate to the backreaction.

Another work originating from the same vein of stochastic gravity but not complying with the backreaction spirit is that of Hu and Shiokawa [166], who study effects associated with electromagnetic wave propagation in a Robertson-Walker universe and the Schwarzschild spacetime with a small amount of given metric stochasticity. They find that time-independent randomness can decrease the total luminosity of Hawking radiation due to multiple scattering of waves outside the black hole and gives rise to event horizon fluctuations and fluctuations in the Hawking temperature. The stochasticity in a background metric in their work is assumed rather than derived (from quantum field fluctuations, as in this work), and so is not in the same spirit of backreaction. But it is interesting to compare their results with that of backreaction, so one can begin to get a sense of the different sources of stochasticity and their weights (see, e.g., [154] for a list of possible sources of stochasticity).

In a subsequent paper Shiokawa [261] showed that the scalar and spinor waves in a stochastic spacetime behave similarly to the electrons in a disordered system. Viewing this as a quantum transport problem, he expressed the conductance and its fluctuations in terms of a nonlinear sigma model in the closed time path formalism and showed that the conductance fluctuations are universal, independent of the volume of the stochastic region and the amount of stochasticity. This result can have significant importance in characterizing the mesoscopic behavior of spacetimes resting between the semiclassical and the quantum regimes.