How quantum fields affect black hole space-times

该项目将研究量子场对黑洞几何形状的影响。在量子引力的半经典近似下，时空将被视为纯经典的，量子场在此背景下传播。该项目的重点是量子场对时空度规的反作用。这种反作用由爱因斯坦方程的半经典版本控制;其中左手边（几何）是经典的，而右手边（物质）是应力-能量张量算符的期望值。因此，在求解这些方程以找到时空度规之前，首先需要计算应力-能量张量算符的期望值（以下用SET表示）。本项目的重点是对渐近反德西特黑洞的SET的计算以及由此产生的反反应的研究。因此，该项目分为两个部分：福尔斯。首先需要找到SET，这个量需要重整化，这个项目将使用已经建立的Hadamard重整化方法。在黑洞时空中计算重整化SET是众所周知的挑战。然而，在过去的几年中，新的技术已经开发，使这种计算更容易处理。特别是，该项目将采用“扩展坐标”技术，利用黑洞时空的基本对称性。最近，利用这种方法，一个四维的，渐近平坦的，静态的和球对称的黑洞上的RSET已被发现。该项目的第一部分将扩展这种方法的渐近anti-de Sitter（adS）黑洞。我们将研究最简单的量子场，即量子标量场。扩展坐标方法涉及将阿达玛重整化反项写成双模和，这使得重整化能够逐模进行。标量场模函数必须用数值计算。研究渐近adS黑洞的一个优点是，可以预期场模式将涉及Heun函数，这可以很容易地使用Approximatica计算。如果减去足够多的重正化反项，得到的模式和收敛得足够快，可以在合理的时间尺度内计算。由于需要开发新的方法，最初的重点将是四维，静态，球对称的黑洞在adS，其中的RSET的计算是显着缺乏从literary.Once RSET已计算，该项目的第二部分将关注的反作用的量子场的黑洞几何。开始，我们将研究一个由纯adS组成的没有黑洞的玩具模型。对于纯adS的热状态的RSET的半解析结果已经知道很多年了，所以这将是计算反反应的代码的有用的测试基础。然后，这个代码可以扩展到更复杂的黑洞情况下，在那里RSET将只知道数字。至少在最初，反反应将以微扰的方式进行研究，作为对黑洞几何形状的一个小修正。这将使半经典爱因斯坦方程线性化。使用精确的数值RSET的结果将与从RSET的解析近似所产生的结果进行比较。计划解决以下问题：当包括反反应时，黑洞温度如何变化？视界是扩大还是缩小？改变应用于adS边界上的场的边界条件有什么影响？半经典近似在任何一点上都失效了吗？

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?