As another illustration of the application of stochastic gravity we now consider the backreaction and fluctuations in black hole spacetimes. Backreaction refers to the quantum effects of matter fields such as vacuum polarization, quantum fluctuations and particle creation on the spacetime structure and dynamics. Studying the dynamics of quantum fields in a fixed background spacetime, Hawking found that black holes emit thermal radiation with a temperature inversely proportional to their mass [156, 213, 294, 358]. When the backreaction of the quantum fields on the spacetime dynamics is included, one expects that the mass of the black hole decreases as thermal radiation at higher and higher temperatures is emitted. The reduction of the mass of a black hole due to particle creation is often referred to as the black hole ‘evaporation’ process. Backreaction of Hawking radiation [9, 18, 142, 166, 167, 380, 381, 382] could alter the evolution of the background spacetime and change the nature of its end state, more drastically so for Planck-size black holes.

Backreaction is a technically challenging but conceptually rewarding problem. Progress is slow in this long standing problem, but it cannot be ignored because existing results from test-field approximations or semiclassical analysis are not trustworthy when backreaction becomes strong enough as to alter the structure and dynamics of the background spacetime. At the least one needs to know how strong the backreaction effects are, and under what circumstances the existing predictions make sense. Without an exact quantum solution of the black-hole-plus-quantum-field system or at least a full backreaction consideration including the intrinsic and induced effects of metric fluctuations, much of the long speculation on the end-state of black hole collapse – remnants, naked singularity, baby universe formation or complete evaporation (see, e.g., [12, 47, 130, 168, 184, 250, 306, 318, 319, 375, 376]) – and the information loss issue [157, 158, 160, 286] (see, e.g., [288, 307] for an overview and recent results from quantum information [40, 333]) will remain speculation and puzzles. This issue also enters into the extension of the well-known black-hole thermodynamics [23, 25, 26, 171, 172, 214, 224, 253, 254, 337, 342, 346, 363, 364] to nonequilibrium conditions [105] and can lead to new inferences on the microscopic structure of spacetime and the true nature of Einstein’s equations [216] from the viewpoint of general relativity as geometro-hydrodynamics and gravity as emergent phenomena. (See the non-traditional views of Volovik [356, 357]and Hu [185, 186, 190] on spacetime structure, Wen [139, 240, 372, 373] on quantum order, Seiberg [326], Horowitz and Polchinsky [173] on emergent gravity, Herzog on the hydrodynamics of M-theory [164] and the seminal work of Unruh and Jacobson [215, 351] leading to analog gravity [15, 16, 320].)

8.1 General issues of backreaction

8.1.1 Regularized energy-momentum tensor

8.1.2 Backreaction and fluctuation-dissipation relation

8.1.3 Noise and fluctuations – the missing ingredient in older treatments

8.2 Backreaction on black holes under quasi-static conditions

8.2.1 The model

8.2.2 CTP effective action for the black hole

8.2.3 Near flat case

8.2.4 Near-horizon case

8.2.5 Einstein–Langevin equation

8.2.6 Comments

8.3 Metric fluctuations of an evaporating black hole

8.3.1 Evolution of the mean geometry of an evaporating black hole

8.3.2 Spherically-symmetric induced fluctuations

Case 1: Assuming that there is a relation between fluctuations

Case 2: When no such relation exists and the consequences

8.3.3 Summary and prospects

8.4 Other work on metric fluctuations but without backreaction

8.1.1 Regularized energy-momentum tensor

8.1.2 Backreaction and fluctuation-dissipation relation

8.1.3 Noise and fluctuations – the missing ingredient in older treatments

8.2 Backreaction on black holes under quasi-static conditions

8.2.1 The model

8.2.2 CTP effective action for the black hole

8.2.3 Near flat case

8.2.4 Near-horizon case

8.2.5 Einstein–Langevin equation

8.2.6 Comments

8.3 Metric fluctuations of an evaporating black hole

8.3.1 Evolution of the mean geometry of an evaporating black hole

8.3.2 Spherically-symmetric induced fluctuations

Case 1: Assuming that there is a relation between fluctuations

Case 2: When no such relation exists and the consequences

8.3.3 Summary and prospects

8.4 Other work on metric fluctuations but without backreaction

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