Quantum field theory on rotating black holes

This project is concerned with the properties of quantum fields on rotating black hole space-times. Working within the framework of quantum field theory on curved space-time, the black hole geometry is purely classical and there is a quantum field propagating on this fixed background. There are two aspects to the project: defining states for the quantum field; and studying the physical properties of these states by computing renormalized expectation values of appropriate operators, such as the vacuum polarization and stress-energy tensor. Often on black hole space-times one is interested in the Hartle-Hawking (HH) state, which describes a quantum field in thermal equilibrium at the Hawking temperature of the black hole. For a non-rotating black hole, the HH state respects the underlying symmetries of the space-time and is regular everywhere on and outside the event horizon, which renders it a convenient state for computations. However, there is no analogous HH state for a bosonic field on a rotating Kerr black hole. For a fermion field on Kerr, there is a state which has some of the properties of a HH state but it diverges on the speed-of-light surface (the surface on which an observer rigidly-rotating with the same angular speed as the event horizon will be travelling at the speed of light). A natural question is then whether a HH-like state exists (and is regular) if one considers a black hole which is rotating but whose space-time does not include a speed-of-light surface. This question has been answered in the affirmative in two cases: the somewhat artificial scenario of a four-dimensional Kerr black hole surrounded by a mirror and the three-dimensional BTZ black hole. The key feature of the BTZ black hole is that it is not asymptotically flat like Kerr but, instead, asymptotically anti-de Sitter (adS). The adS boundary is timelike and acts in a similar way to a mirror surrounding an asymptotically flat black hole. The focus of this project is whether the HH state exists on rotating, asymptotically adS black holes in more than three space-time dimensions. The four-dimensional Kerr-adS black hole metric is considerably more complicated than the Kerr metric (on which computations are notoriously challenging), so we will consider instead five-dimensional black holes. In five space-time dimensions, there are two potential axes of rotation rather than one in four dimensions. Setting the angular momentum about both axes to be equal yields a space-time metric with more symmetries, which will be more amenable to analysis. We will consider a quantum scalar field on this background, focussing on the case where the black hole is asymptotically adS and there is no speed-of-light surface. The project will be begin by performing the canonical quantization of a scalar field on this background, seeking to define analogues of the standard quantum states (Boulware and Unruh as well as HH). The remainder of the project will investigate the properties of these states. Computing renormalized expectation values on higher-dimensional black hole space-times is a subject in its infancy, so it is anticipated that new analytic and numerical techniques will need to be developed. We will begin with the simplest expectation value, the vacuum polarization and aim to work towards a computation of the renormalized stress-energy tensor operator.