How quantum fields affect black hole space-times

量子场如何影响黑洞时空

• 批准号: 2745699
• 金额: --
• 依托单位: University of Sheffield
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2022
• 资助国家: 英国
• 起止时间: 2022 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2745699
• 关键词: How; quantum; fields; affect; black

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?

该项目将研究量子场对黑洞几何形状的影响。在半古典近似量子的量子重力中工作，时空将被视为纯粹的经典，并在此背景下传播量子场。该项目的重点是量子场在时空度量标准上的后反应。这种反应由爱因斯坦方程的半古典版本约束。其（几何）的左侧是经典的，而右侧（物质）是应力 - 能量张量操作员的期望值。因此，在解决这些方程式以找到时空度量标准之前，因此首先需要计算应力 - 能量张量操作员的期望值（以下用该集合表示）。该项目的重点将计算集合在渐近反DE保姆黑洞上，并对后反应进行研究。因此，该项目分为两个部分。首先，需要找到集合。该数量需要重新规定，该项目将采用哈达姆重生的良好方法。众所周知，在黑洞时空上重新归一化的设置的计算是具有挑战性的。但是，在过去的几年中，已经开发了新技术，从而使该计算更加可行。特别是，该项目将采用“扩展坐标”技术，该技术利用了黑洞时空的基础对称性。最近，使用这种方法，发现了四维，渐近平坦，静态和球形对称黑洞的rset。该项目的第一部分是将此方法扩展到渐近的反DE保姆（AD）黑洞。将研究最简单的量子场，即量子标量场。扩展的坐标方法涉及将Hadamard重量化反向处理作为双模式总和，这使得可以逐示为模式。标量场模式函数必须数值计算。在渐近广告黑洞上工作的一个优点是，预计该场模式将涉及Heun函数，可以使用Mathematica轻松计算。提供一个足够多的重新归一化反对处理，结果模式总和足够快地收敛，以便可以在明智的时间尺度内计算它们。由于需要开发新的方法，因此初始焦点将是四维，静态的，球面上的对称黑洞，在该广告中明显不存在对rset的计算。一旦计算，该项目的第二部分将与黑洞量角的量定场对量子域的反应有关。首先，将研究一个由没有黑洞的纯广告组成的玩具模型。纯AD上热态的RSET的半分析结果已知多年了，因此这将是计算后反应的代码的有用测试基础。然后可以将此代码扩展到更复杂的黑洞案例，其中只能以数值为单位。至少最初，将以扰动的方式研究后反应，作为对黑洞几何形状的较小校正。这将使半古典的爱因斯坦方程能够线性化。使用精确的数值RSET的结果将与由分析近似与RSET产生的结果进行比较。计划解决以下问题：包括背反应时黑洞温度如何变化？事件视野会扩展还是收缩？改变对广告边界上田地应用的边界条件的影响是什么？半古典近似是否在任何时候分解？