Such a program of research has been pursued rigorously by Hu and Roura (HR) [200, 202] In contrast to the claims made before, they find that even for states regular on the horizon the accumulated fluctuations become significant by the time the black-hole mass has changed substantially, but well before reaching the Planckian regime. This result is different from those obtained in prior studies, but in agreement with earlier work by Bekenstein . The apparent difference from the conclusions drawn in the earlier work of Hu, Raval and Sinha , which was also based on stochastic gravity, will be explained later. We begin with the evolution of the mean geometry.
Backreaction of the Hawking radiation emitted by the black hole on the dynamics of spacetime geometry has been studied in some detail for spherically-symmetric black holes [18, 260]. For a general spherically-symmetric metric there always exists a system of coordinates in which it takes the formapparent horizon, at which the expansion of the outgoing radial null geodesics vanishes and which separates regions with positive and negative expansion for those geodesics, at those radii that correspond to (odd degree) zeroes of the metric component. We mark the location of the apparent horizon by , where satisfies the equation .
The non-zero components of the semiclassical Einstein equation associated with the metric in Equation (214) become
Solving Equations (215)–(217) is not easy. However, one can introduce a useful adiabatic approximation in the regime, in which the mass of the black hole is much larger than the Planck mass, which is, in any case, a necessary condition for the semiclassical treatment to be valid. What this entails is that when (remember that we are using Planckian units) for each value of one can simply substitute by its “parametric value” – by this we mean the expectation value of the stress-energy tensor of the quantum field in a Schwarzschild black hole with a mass corresponding to evaluated at that value of . This is in contrast to its dynamical value, which should be determined by solving self-consistently the semiclassical Einstein equation for the spacetime metric and the equations of motion for the quantum matter fields. This kind of approximation introduces errors of higher order in ( is a dimensionless parameter that depends on the number of massless fields and their spins and accounts for their corresponding grey-body factors; it has been estimated to be of order ), which are very small for black holes well above Planckian scales. These errors are due to the fact that is not constant and that, even for a constant , the resulting static geometry is not exactly Schwarzschild because the vacuum polarization of the quantum fields gives rise to a non-vanishing .
The expectation value of the stress tensor for a Schwarzschild spacetime has been found to correspond to a thermal flux of radiation (with ) for large radii and of order near the horizon [8, 75, 177, 178, 287]. This shows the consistency of the adiabatic approximation for : the right-hand side of Equations (215)–(217) contains terms of order and higher, so that the derivatives of and are indeed small. We note that the natural quantum state for a black hole formed by gravitational collapse is the Unruh vacuum, which corresponds to the absence of incoming radiation far from the horizon. The expectation value of the stress-tensor operator for that state is finite on the future horizon of Schwarzschild, which is the relevant one when identifying a region of the Schwarzschild geometry with the spacetime outside the collapsing matter for a black hole formed by gravitational collapse.
One can use the component of the stress-energy conservation equation
We now consider metric fluctuations around a background metric that corresponds to a given solution of semiclassical gravity. Their dynamics are governed by the Einstein–Langevin equation [73, 192, 206, 248]
As explained earlier, the symmetrized two-point function consists of two contributions: intrinsic and induced fluctuations. The intrinsic fluctuations are a consequence of the quantum width of the initial state of the metric perturbations; they are obtained in stochastic gravity by averaging over the initial conditions for the solutions of the homogeneous part of Equation (221), distributed according to the reduced Wigner function associated with the initial quantum state of the metric perturbations. On the other hand, the induced fluctuations are due to the quantum fluctuations of the matter fields interacting with the metric perturbations; they are obtained by solving the Einstein–Langevin equation using a retarded propagator with vanishing initial conditions.
In this section we study the spherically-symmetric sector, i.e., the monopole contribution, which corresponds to , in a multipole expansion in terms of spherical harmonics , of metric fluctuations for an evaporating black hole. Restricting one’s attention to the spherically-symmetric sector of metric fluctuations necessarily implies a partial description of the fluctuations because, contrary to the case for semiclassical-gravity solutions, even if one starts with spherically-symmetric initial conditions, the stress-tensor fluctuations will induce fluctuations involving higher multipoles. Thus, the multipole structure of the fluctuations is far richer than that of spherically-symmetric semiclassical-gravity solutions, but this also means that obtaining a complete solution (including all multipoles) for fluctuations, rather than the mean value, is much more difficult. For spherically-symmetric fluctuations only induced fluctuations are possible. The fact that intrinsic fluctuations cannot exist can be clearly seen if one neglects vacuum-polarization effects, since Birkhoff’s theorem forbids the existence of spherically-symmetric free metric perturbations in the exterior vacuum region of a spherically-symmetric black hole that keep the ADM mass constant. (This fact rings an alarm in the approach taken in  to the black-hole fluctuation problem. The degrees of freedom corresponding to spherically-symmetric perturbations are constrained by the Hamiltonian and momentum constraints both at the classical and quantum level. Therefore, they will not exhibit quantum fluctuations unless they are coupled to a quantum matter field.) Even when vacuum-polarization effects are included, spherically-symmetric perturbations, characterized by and , are not independent degrees of freedom. This follows from Equations (215)–(217), which can be regarded as constraint equations.
Considering only spherical-symmetry fluctuations is a simplification but it should be emphasized that it gives more accurate results than two-dimensional dilation-gravity models resulting from simple dimensional reduction [249, 341, 349]. This is because we project the solutions of the Einstein–Langevin equation just at the end, rather than considering only the contribution of the -wave modes to the classical action for both the metric and the matter fields from the very beginning. Hence, an infinite number of modes for the matter fields with contribute to the projection of the noise kernel, whereas only the -wave modes for each matter field would contribute to the noise kernel if dimensional reduction had been imposed right from the start, as done in [289, 290, 291] as well as in studies of two-dimensional dilation-gravity models.
The Einstein–Langevin equation for the spherically-symmetric sector of metric perturbations can be obtained by considering linear perturbations of and , projecting the stochastic source that accounts for the stress-tensor fluctuations to the sector, and adding it to the right-hand side of Equations (215)–(217). We will focus our attention on the equation for the evolution of , the perturbation of :
A more serious issue raised by HR is that in most previous investigations [24, 377] of the problem of metric fluctuations driven by quantum matter field fluctuations of states regular on the horizon (as far as the expectation value of the stress tensor is concerned) most authors assumed the existence of correlations between the outgoing energy flux far from the horizon and a negative energy flux crossing the horizon. (See, however, [290, 291], in which those correlators were shown to vanish in an effectively two-dimensional model.) In semiclassical gravity, using energy conservation arguments, such correlations have been confirmed for the expectation value of the energy fluxes, provided that the mass of the black hole is much larger than the Planck mass. However, a more careful analysis by HR shows that no such simple connection exists for energy flux fluctuations. It also reveals that the fluctuations on the horizon are in fact divergent. This requires that one modify the classical picture of the event horizon from a sharply defined three-dimensional hypersurface to that possessing a finite width, i.e., a fluctuating geometry. One needs to find an appropriate way of probing the metric fluctuations near the horizon and extracting physically meaningful information. It also testifies to the necessity of a complete reexamination of all cases afresh and that an evaluation of the noise kernel near the horizon seems unavoidable for the consideration of fluctuations and backreaction issues.
Having registered this cautionary note, Hu and Roura  first make the assumption that a relation between the fluctuations of the fluxes exists, so as to be able to compare with earlier work. They then show that this relation does not hold and discuss the essential elements required in understanding not only the mathematical theory but also the operational meaning of metric fluctuations.
Since the generation of Hawking radiation is especially sensitive to what happens near the horizon, from now on we will concentrate on the metric perturbations near the horizon and consider . This means that possible effects on the Hawking radiation due to the fluctuations of the potential barrier for the radial mode functions will be missed by our analysis. Assuming that the fluctuations of the energy flux crossing the horizon and those far from it are exactly correlated, from Equation (223) we have
The stochastic equation (224) for can be solved in the usual way and the correlation function for can then be computed. Alternatively, one can obtain an equation for by first multiplying Equation (224) by and then taking the expectation value. This brings out a term on the right-hand side. For delta-correlated noise (the Stratonovich prescription is the appropriate one here), it is equal to one half the time-dependent coefficient multiplying the delta function in the correlator , which is given by in our case. Finally, changing from the coordinate to the mass function for the background solution, we obtain
The result of HR for the growth of the fluctuations in size of the black-hole horizon agrees with the result obtained by Bekenstein in  and implies that, for a sufficiently massive black hole (a few solar masses or a supermassive black hole), the fluctuations become important before the Planckian regime is reached.
This growth of the fluctuations, which was found by Bekenstein and confirmed here via the Einstein–Langevin equation, seems to be in conflict with the estimate given by Wu and Ford in . According to their estimate, the accumulated mass fluctuations over a period on the order of the black hole evaporation time () would be on the order of the Planck mass. The discrepancy is due to the fact that the first term on the right-hand side of Equation (224), which corresponds to the perturbed expectation value in Equation (221), was not taken into account in . The larger growth obtained here is a consequence of the secular effect of that term, which builds up in time (slowly at first, during most of the evaporation time, and becoming more significant at late times when the mass has changed substantially) and reflects the unstable nature of the background solution for an evaporating black hole.
As for the relation between HR’s results reported here and earlier results of Hu, Raval and Sinha in , there should not be any discrepancy, since both adopted the stochastic gravity framework and performed their analysis based on the Einstein–Langevin equation. The claim in  was based on a qualitative argument that focused on the dynamics of the stochastic source alone. If one adds in the consideration that the perturbations around the mean are unstable for an evaporating black hole, their results agree.
All this can be qualitatively understood as follows. Consider an evaporating black hole with initial mass and suppose that the initial mass is perturbed by an amount . The mean evolution for the perturbed black hole (without taking into account any fluctuations) leads to a mass perturbation that grows like , so that it becomes comparable to the unperturbed mass when , which coincides with the result obtained above. Such a coincidence has a simple explanation: the fluctuations of the Hawking flux, which are on the order of the Planck mass, slowly accumulated during most of the evaporating time, as found by Wu and Ford, and gave a dispersion of that order for the mass distribution at the time when the instability of the small perturbations around the background solution start to become significant.
For conformal fields in two-dimensional spacetimes, HR shows that the correlations between the energy flux crossing the horizon and the flux far from it vanish. The correlation function for the outgoing and ingoing null-energy fluxes in an effectively two-dimensional model is explicitly computed in [290, 291] and is also found to vanish. On the other hand, in four dimensions the correlation function does not vanish in general and correlations between outgoing and ingoing fluxes do exist near the horizon (at least partially).
For black-hole masses much larger than the Planck mass one can use the adiabatic approximation for the background mean evolution. Therefore, to lowest order in one can compute the fluctuations of the stress tensor in Schwarzschild spacetime. In Schwarzschild, the amplitude of the fluctuations of far from the horizon is of order () when smearing over a correlation time of order , which one can estimate for a hot thermal plasma in flat space [69, 70] (see also  for a computation of the fluctuations of far from the horizon). The amplitude of the fluctuations of is, thus, of the same order as its expectation value. However, their derivatives with respect to are rather different: since the characteristic variation times for the expectation value and the fluctuations are and , respectively, is of order , whereas is of order . This implies an additional contribution of order due to the second term in Equation (218) if one radially integrates the same equation applied to stress-tensor fluctuations (the stochastic source in the Einstein–Langevin equation). Hence, in contrast to the case of the mean value, the contribution from the second term in Equation (218) cannot be neglected when radially integrating, since it is of the same order as the contributions from the first term, and one can no longer obtain a simple relation between the outgoing energy flux far from the horizon and the energy flux crossing the horizon.
What then? Without this convenience (which almost all earlier researchers have taken for granted), to get a more precise depiction we need to compute the noise kernel near the horizon. However, as shown by Hu and Phillips earlier  when they examine the coincidence limit of the noise kernel and confirmed by the careful analysis of HR using smearing functions , the noise kernel smeared over the horizon is divergent and so are the induced metric fluctuations. Hence, one cannot study the fluctuations of the horizon as a three-dimensional hypersurface for each realization of the stochastic source because the amplitude of the fluctuations is infinite, even when restricting one’s attention to the sector. Instead, one should regard the horizon as possessing a finite effective width due to quantum fluctuations. In order to characterize its width one must find a sensible way of probing the metric fluctuations near the horizon and extracting physically-meaningful information, such as their effect on the Hawking radiation emitted by the black hole. How to probe metric fluctuations is an issue at the root base, which needs be dealt with in all discussions of metric fluctuations.
The work of HR , based on the stochastic-gravity program, found that the spherically-symmetric fluctuations of the horizon size of an evaporating black hole become important at late times, and even comparable to its mean value when , where is the mass of the black hole at some initial time when the fluctuations of the horizon radius are much smaller than the Planck length (remember that for large black-hole masses this can still correspond to physical distances much larger than the Planck length, as explained in Section 4). This is consistent with the result previously obtained by Bekenstein in .
It is important to realize that, for a sufficiently massive black hole, the fluctuations become significant well before the Planckian regime is reached. More specifically, for a solar-mass black hole, they become comparable to the mean value when the black-hole radius is on the order of 10 nm, whereas for a supermassive black hole with , that happens when the radius reaches a size on the order of 1 mm. One expects that in those circumstances the low-energy effective-field theory approach of stochastic gravity should provide a reliable description.
Due to the nonlinear nature of the backreaction equations, such as Equation (222), the fact that the fluctuations of the horizon size can grow and become comparable to the mean value implies non-negligible corrections to the dynamics of the mean value itself. This can be seen by expanding Equation (222) (evaluated on the horizon) in powers of and taking the expectation value. Through order we get
Finally, we remark on the relation of this finding to earlier well-known results. Does the existence of significant deviations for the mean evolution mentioned above invalidate the earlier results by Bardeen and Massar based on semiclassical gravity in [18, 260]? First, those deviations start to become significant only after a period on the order of the evaporation time, when the mass of the black hole has decreased substantially. Second, since fluctuations were not considered in those references, a direct comparison cannot be established. Nevertheless, we can compare the average of the fluctuating ensemble. Doing so exhibits an evolution that deviates significantly when the fluctuations become important. However, if one considers a single member of the ensemble at that time, its evolution will be accurately described by the corresponding semiclassical gravity solution until the fluctuations around that particular solution become important again, after a period on the order of the evaporation time associated with the new initial value of the mass at that time.
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