Semiclassical backreaction on asymptotically anti–de Sitter black holes

Abstract

We consider a quantum scalar field on the classical background of an asymptotically anti-de Sitter black hole and the backreaction the field's stress-energy tensor induces on the black hole geometry. The backreaction is computed by solving the reduced-order semiclassical Einstein field equations sourced by simple analytical approximations for the renormalized expectation value of the scalar field stress-energy tensor. When the field is massless and conformally coupled, we adopt Page's approximation to the renormalized stress-energy tensor, while for massive fields we adopt a modified version of the DeWitt-Schwinger approximation. The latter approximation must be modified so that it possesses the correct renormalization freedom required to ensure the semiclassical equations are consistent. Equipped with these approximations, the reduced-order field equations are easily integrated and the first-order (in ℏ) corrections to the metric are obtained. We also compute the corrections to the black hole event horizon, surface gravity, and minimum temperature as well as corrections to the photon sphere and quadratic curvature invariants. We pay particular attention to the temperature profiles of the semiclassical black holes compared with their classical counterparts, pointing out some interesting qualitative features produced by the backreaction. These results ought to provide reasonable approximations to the first-order (one-loop) quantum backreaction on the geometry of asymptotically anti-de Sitter black holes when the exact numerical stressenergy tensor sources the semiclassical equations.

Mode-sum prescription for the renormalized stress energy tensor on black hole spacetimes

“…An issue that arose during York's calculation of the backreaction induced by a conformal field was that the perturbation was unbounded for large r and the backreaction had to be computed in an artificial box and matched to an asymptotically flat solution. However, the backreaction induced by the large-mass approximation to the RSET does not suffer this pathology [21,23]. What about the intermediate field masses?…”

“…Turning now to the calculation of the backreaction. One can use the exact numerical calculation of the RSET to solve a reduced-order version of the semiclassical equations (see [20,21] for details of the reduction of order prescription) perturbed about the classical Schwarzschild background. This is achieved by assuming the perturbation respects the symmetry of the background and then expressing the perturbed metric components in the form:…”

A mode-sum prescription for the renormalized stress energy tensor on black hole spacetimes

“…An issue that arose during York's exposition of the backreaction induced by a conformal field was that the perturbation was unbounded for large r and the black hole had to be solved in an artificial box and matched to an asymptotically flat solution. However, the back-reaction induced by the DeWitt-Schwinger large-mass approximation to the RSET does not suffer this pathology [19,21]. What about the intermediate field masses?…”

“…One can use the exact numerical calculation to solve a reduced-order version of the semi-classical equations (see [18,19] for details of the reduction of order prescription) perturbed about the classical Schwarzschild background. This is achieved by assuming the perturbation respects the symmetry of the background and then expressing the perturbed metric components in the form:…”

Practical derivations of fermion and gauge boson reduction formulae in curved spacetimes

“…scalar decay calculations [4,[8][9][10], scalar decay in to fermions in Robertson-Walker spacetimes [11], and solutions for free fermions in assorted spacetimes [12,13]. There are also recent results for backreaction [14][15][16][17][18][19]. The calculations so far have relied on conformally coupled, massless particles.…”