How quantum fields affect black hole space-times

这个项目将研究量子场对黑洞几何的影响。在量子引力的半经典近似下工作，时空将被认为是纯经典的，量子场在这种背景下传播。该项目的重点是量子场在时空度规上的反向反应。这种反向反应是由半经典版本的爱因斯坦方程控制的；其左侧(几何)是经典的，而右侧(物质)是应力-能量张量算符的期望值。因此，在求解这些方程以求出时空度规之前，首先需要计算应力-能量张量算符(下称集合)的期望值。本项目的重点是渐近反de Sitter黑洞集合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。首先，需要找到集合。这个量需要重整化，该项目将使用公认的阿达玛重整化方法。黑洞时空中的重整化集的计算是出了名的具有挑战性。然而，在过去的几年里，新的技术已经发展起来，使得这种计算变得更加容易处理。特别是，该项目将采用“扩展坐标”技术，该技术利用黑洞时空的基本对称性。最近，利用这种方法，我们发现了一个四维、渐近平坦、静态和球对称黑洞上的RSET。该项目的第一部分将是将该方法扩展到渐近反德西特(ADS)黑洞。我们将研究最简单类型的量子场--量子标量场。扩展坐标方法包括将Hadamard重整化反项写成双模和，这使得重整化能够逐个模式地实现。标量场模函数必须进行数值计算。渐近研究AdS黑洞的一个优点是，预计场模将涉及Heun函数，这可以很容易地使用数学计算。如果减去足够多的重整化反项，得到的模式和收敛得足够快，以至于可以在合理的时间尺度内计算它们。由于将需要开发新的方法，最初的焦点将是ADS中的四维静态球对称黑洞，对于这些黑洞，文献中明显缺乏对RSET的计算。一旦RSET被计算出来，项目的第二部分将关注量子场对黑洞几何的反向反应。首先，将研究一个由没有黑洞的纯广告组成的玩具模型。多年来，关于纯ADS上热态的RSET的半解析结果已为人所知，因此这将是计算反反应的代码的有用的试验场。然后，这个代码可以扩展到更复杂的黑洞情况，在那里RSET将只在数字上已知。至少在一开始，反反应将以微扰的方式进行研究，作为对黑洞几何结构的小修正。这将使半经典的爱因斯坦方程线性化。使用精确的数值RSET的结果将与解析近似的RSET的结果进行比较。它计划解决以下问题：当包括反向反应时，黑洞温度是如何变化的？事件视界是扩大还是缩小？更改应用于ADS边界上的场的边界条件会有什么影响？半经典近似会在任何时候崩溃吗？

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?