Quantum field theory on rotating black holes

本项目研究旋转黑洞时空上量子场的性质。在弯曲时空的量子场论框架内工作，黑洞几何是纯经典的，在这个固定的背景上有一个量子场传播。这个项目有两个方面：定义量子场的状态；并通过计算真空极化和应力-能量张量等适当算子的重整化期望值来研究这些状态的物理性质。在黑洞的时空中，人们经常对哈特尔-霍金态（HH）感兴趣，它描述了黑洞霍金温度下热平衡的量子场。对于非旋转黑洞，HH状态尊重时空的潜在对称性，并且在事件视界内外的任何地方都是规则的，这使得它成为一种方便计算的状态。然而，旋转克尔黑洞上的玻色子场没有类似的HH态。对于克尔上的费米子场，存在一种状态，它具有HH状态的一些特性，但它在光速表面上发散（在光速表面上，观察者以与视界相同的角速度进行刚性旋转，将以光速行进）。一个自然的问题是，如果考虑一个旋转的黑洞，但它的时空不包括光速表面，那么类hh状态是否存在（并且是规则的）。这个问题已经在两种情况下得到了肯定的回答：被镜子包围的四维克尔黑洞和三维BTZ黑洞的人工场景。BTZ黑洞的关键特征是它不像Kerr那样渐近平坦，而是渐近反德西特（adS）。adS的边界是类时的，其作用类似于围绕着渐近平坦黑洞的一面镜子。这个项目的重点是HH态是否存在于超过三维时空的旋转渐近adS黑洞上。四维Kerr- ads黑洞度规比Kerr度规要复杂得多（其计算是出了名的具有挑战性），所以我们将考虑五维黑洞。在五个时空维度中，有两个潜在的旋转轴，而不是在四维中有一个。将两个轴的角动量设置为相等，会得到一个更对称的时空度量，这将更易于分析。我们将在这个背景下考虑一个量子标量场，重点关注黑洞是渐近的adS并且没有光速表面的情况。该项目将从在此背景下执行标量场的规范量子化开始，寻求定义标准量子态（Boulware和Unruh以及HH）的类似物。项目的其余部分将研究这些状态的属性。计算高维黑洞时空上的重整化期望值是一个尚处于起步阶段的课题，因此预计需要开发新的解析和数值技术。我们将从最简单的期望值，真空极化开始，并致力于重新规范化应力-能量张量算子的计算。

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.