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。在量子引力的半经典近似下，时空将被视为纯粹经典，量子场在此背景下传播。该项目的重点是量子场对时空度量的反作用。这种反作用是由爱因斯坦方程的半经典版本控制的。左边（几何）是经典的，右边（物质）是应力-能量张量算子的期望值。因此，在求解这些方程以找到时空度量之前，首先需要计算应力能量张量算子（以下表示为 SET）的期望值。该项目的重点是渐近反德西特黑洞的 SET 计算以及随后的反反应研究。因此，该项目分为两部分。首先需要找到 SET。该数量需要重整化，该项目将使用哈达玛重整化的成熟方法。黑洞时空重整化集合的计算是出了名的具有挑战性。然而，在过去几年中，新技术的开发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，该技术利用了黑洞时空的基本对称性。最近，使用这种方法，发现了四维渐近平坦静态球对称黑洞的 RSET。该项目的第一部分是将这种方法扩展到渐近反德西特（adS）黑洞。将研究最简单的量子场类型，即量子标量场。扩展坐标方法涉及将 Hadamard 重整化反项写为双模和，这使得重整化能够逐个模式地实现。标量场模式函数必须进行数值计算。研究渐近 adS 黑洞的一个优点是，预计场模式将涉及 Heun 函数，可以使用 mathematica 轻松计算。如果减去足够多的重整化反项，所得模态和会足够快地收敛，从而可以在合理的时间尺度内计算它们。由于需要开发新的方法，最初的重点将是 adS 中的四维、静态、球对称黑洞，而文献中明显没有对其 RSET 的计算。一旦计算出 RSET，该项目的第二部分将关注量子场对黑洞几何结构的反作用。首先，我们将研究一个由纯 adS 组成、没有黑洞的玩具模型。纯 adS 上的热状态 RSET 的半解析结果多年来一直为人所知，因此这将是计算反反应的代码的有用测试场。然后可以将该代码扩展到更复杂的黑洞情况，其中 RSET 只能通过数字得知。至少在最初，将以微扰方式研究反反应，作为对黑洞几何结构的小修正。这将使半经典爱因斯坦方程线性化。使用精确数值 RSET 得到的结果将与 RSET 解析近似得到的结果进行比较。计划解决以下问题：当包含逆反应时，黑洞温度如何变化？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有何影响？半经典近似会在任何时候崩溃吗？