Quantum field theory on rotating black holes

本项目主要研究旋转黑洞时空中量子场的性质。在弯曲时空的量子场论框架内工作，黑洞几何是纯经典的，并且有一个量子场在这个固定的背景上传播。该项目包括两个方面：为量子场定义态；通过计算适当算符的重整化期望值，如真空极化和应力-能量张量，研究这些态的物理性质。在黑洞时空中，人们通常对Hartle-Hawking(HH)态感兴趣，它描述了在黑洞的Hawking温度下处于热平衡状态的量子场。对于非旋转黑洞，HH态尊重时空的基本对称性，并且在视界内外处处都是规则的，这使得它成为一种便于计算的状态。然而，旋转克尔黑洞上的玻色场不存在类似的HH态。对于克尔上的费米子场，存在一种状态，它具有HH态的一些属性，但它在光速表面上发散(在该表面上，观察者以与事件视界相同的角速度刚性旋转，将以光速运行)。一个自然的问题是，如果一个黑洞在旋转，但其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)。这个问题在两种情况下得到了肯定的回答：一种是人为的四维克尔黑洞被镜子包围的场景，另一种是三维的BTZ黑洞。BTZ黑洞的主要特征是它不像克尔那样是渐近平坦的，而是渐近反德西特(ADS)的。ADS边界具有时间性质，其作用方式类似于围绕渐近平坦黑洞的一面镜子。这个项目的重点是在超过三个时空维度的旋转、渐近Add黑洞上是否存在HH态。四维Kerr-Ads黑洞度规比Kerr度规(其计算是出了名的具有挑战性)要复杂得多，所以我们将转而考虑五维黑洞。在五个时空维度中，有两个潜在的旋转轴，而不是四个维度中的一个。将两个轴周围的角动量设置为相等会产生具有更多对称性的时空度规，这将更易于分析。我们将在此背景下考虑量子标量场，重点讨论黑洞是渐近ADS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正则量子化开始，寻求定义标准量子态(Boulware和Unruh以及HH)的类似物。该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处于起步阶段的课题，因此预计将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，并致力于重整化应力-能量张量算符的计算。

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.