Classical and quantum black holes

The project will study the behaviour of quantum fields on black hole background spacetimes, focussing on a charged scalar field on a charged Reissner-Nordstrom black hole. Low-frequency classical modes for a charged scalar field experience superradiant scattering off a charged black hole. The aim of the project is to elucidate the effect of this charge-superradiance phenomenon for the definition and properties of quantum states. A similar process of superradiant scattering occurs for low-frequency scalar field waves scattering off a rotating Kerr black hole, and it is known that there is no analogue of the Hartle-Hawking state for a quantum scalar field on a Kerr black hole background. Unfortunately, the complexity of the Kerr metric means that computing renormalized expectation values on this background is extremely challenging. By looking at the simpler spherically symmetric Reissner-Nordstrom black hole, it should be possible to progress further and reveal more of the underlying physics. It is hoped that this simpler toy model will also shed some light on the Kerr case.

The plan of the project is as follows. To begin, classical modes of a charged scalar field on a Reissner-Nordstrom black hole background will be studied, and their norm computed. These modes will form an orthonormal basis for canonical quantization, keeping the background black hole metric fixed and classical. Canonical quantization will be employed to define the analogues of the standard Boulware, Hartle-Hawking and Unruh vacua (if they exist). It is anticipated that the low-frequency modes which exhibit charge superradiance will require careful treatment in this process. The properties of those states that can be defined will then be studied by computing expectation values of operators such as the vacuum polarization and stress-energy tensor. Of particular interest for the physical interpretation of these states will be the regularity of these expectation values. First of all, differences in expectation values between two quantum states will be found, since these do not require renormalization, but still reveal some physical properties of the states. The most challenging part of the project will be to find renormalized expectation values. New methodologies for these kinds of computations have recently been developed by Breen and Taylor, but only for a neutral scalar field. The plan is to extend their approach to a charged scalar field. This will involve implementing Hadamard renormalization and then delicate numerical computations.