How quantum fields affect black hole space-times

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This project will study the effect of quantum fields on black hole geometries. Working in a semi-classical approximation to quantum gravity, the space-time will be taken to be purely classical, with quantum fields propagating on this background. The focus of the project is the back-reaction of the quantum fields on the space-time metric. This back-reaction is governed by the semi-classical version of Einstein's equations; the left-hand-side of which (the geometry) is classical, while the right-hand-side (the matter) is the expectation value of the stress-energy tensor operator. Before one can solve these equations to find the space-time metric, one therefore first needs to compute the expectation value of the stress-energy tensor operator (hereafter denoted by the SET). The focus of this project will the computation of the SET on asymptotically anti-de Sitter black holes and the consequent study of the back-reaction. The project therefore falls in two parts. First the SET needs to be found. This quantity requires renormalization, and the project will use the well-established approach of Hadamard renormalization. The calculation of the renormalized SET on a black hole space-time is notoriously challenging. However, over the past few years new techniques have been developed which render this computation much more tractable. In particular, the project will employ the "extended coordinates" technique which exploits the underlying symmetries of the black hole space-time. Very recently, using this approach, the RSET on a four-dimensional, asymptotically flat, static and spherically symmetric black hole has been found. The first part of the project will be to extend this method to asymptotically anti-de Sitter (adS) black holes. The simplest type of quantum field, a quantum scalar field, will be studied. The extended coordinates method involves writing the Hadamard renormalization counter-terms as a double-mode sum, which enables renormalization to be effected mode-by-mode. The scalar field mode functions have to be computed numerically. One advantage of working on asymptotically adS black holes is that it is anticipated that the field modes will involve Heun functions, which can be readily computed using mathematica. Providing one subtracts sufficiently many renormalization counter-terms, the resulting mode sums converge sufficiently quickly that they can be computed within a sensible time-scale. As new methodology will need to be developed, the initial focus will be four-dimensional, static, spherically symmetric black holes in adS, for which a computation of the RSET is notably absent from the literature.Once the RSET has been computed, the second part of the project will be concerned with the back-reaction of the quantum field on the black hole geometry. To begin with, a toy model consisting of pure adS with no black hole will be studied. Semi-analytic results for the RSET for a thermal state on pure adS have been known for many years, so this will be a useful testing ground for the code computing the back-reaction. This code can then be extended to the more complicated black hole case, where the RSET will only be known numerically. At least initially, the back-reaction will be studied in a perturbative manner, as a small correction to the black hole geometry. This will enable the semi-classical Einstein equations to be linearized. Results using the exact numerical RSET will be compared with those arising from an analytic approximation to the RSET. It is planned to address the following questions: How does the black hole temperature change when back-reaction is included? Does the event horizon expand or shrink? What is the effect of changing the boundary conditions applied to the field on the adS boundary? Does the semi-classical approximation break down at any point?