[论文解读] Asymptotically flat black holes and gravitational waves in three-dimensional massive gravity

Cédric Troessaert, David Tempo+1 more|arXiv (Cornell University)|Dec 30, 2015

Black Holes and Theoretical Physics参考文献 96被引用 3

一句话总结

本文针对具有纯二次拉格朗日量的三维大质量引力理论，构造了精确解，引入了两类渐近平坦的黑洞：类Kerr-Schild的pp波与破坏球对称性的形变‘黑洞花朵’。关键结果是，尽管存在时变的引力辐射，这些黑洞花朵仍具有有限能量，并通过无中心荷的BMS₃荷遵守热力学第一定律。

ABSTRACT

Different classes of exact solutions for the BHT massive gravity theory are constructed and analyzed. We focus in the special case of the purely quadratic Lagrangian, whose field equations are irreducibly of fourth order and are known to admit asymptotically locally flat black holes endowed with gravitational hair. The first class corresponds to a Kerr-Schild deformation of Minkowski spacetime along a covariantly constant null vector. As in the case of General Relativity, the field equations linearize so that the solution can be easily shown to be described by four arbitrary functions of a single null coordinate. These solutions can be regarded as a new sort of pp-waves. The second class is obtained from a deformation of the static asymptotically locally flat black hole, that goes along the spacelike (angular) Killing vector. Remarkably, although the deformation is not of Kerr-Schild type, the field equations also linearize, and hence the generic solution can be readily integrated. It is neither static nor spherically symmetric, being described by two integration constants and two arbitrary functions of the angular coordinate. In the static case it describes "black flowers" whose event horizons break the spherical symmetry. The generic time-dependent solution appears to describe a graviton that moves away from a black flower. Despite the asymptotic behaviour of these solutions at null infinity is relaxed with respect to the one for General Relativity, the asymptotic symmetries coincide. However, the algebra of the conserved charges corresponds to BMS$_{3}$, but devoid of central extensions. The "dynamical black flowers" are shown to possess a finite energy. The surface integrals that define the global charges also turn out to be useful in the description of the thermodynamics of solutions with event horizons.

研究动机与目标

• 构造具有纯二次拉格朗日量的三维大质量引力理论的精确解，重点关注渐近平坦时空。
• 分析场方程，其为四阶方程，但在特定形变下线性化，从而实现精确积分。
• 探讨新黑洞解的物理性质，包括其热力学与守恒荷。
• 研究渐近结构与对称代数，特别是BMS₃代数中无中心扩展的问题。
• 基于曲面积分与Wald熵公式，为黑洞花朵建立热力学框架。

提出的方法

• 通过沿协变常数零向量对闵氏时空进行Kerr-Schild形变，构造解，导致线性化场方程。
• 沿静态渐近局部平坦黑洞的类空（角向）Killing向量进行形变，同样导致线性化场方程。
• 使用超势形式化方法，通过曲面积分计算守恒荷，从而评估质量与熵。
• 在null无穷远与视界处计算守恒荷，推导热力学第一定律。
• 采用Wald熵公式，结合分歧面的双法向量，计算黑洞花朵的熵。
• 分析渐近对称代数，表明其为无中心扩展的BMS₃代数，尽管渐近行为被放宽。

实验结果

研究问题

• RQ1 能否在具有纯二次拉格朗日量的三维大质量引力理论中构造出描述渐近平坦黑洞与引力荷的精确解？
• RQ2 场方程在Kerr-Schild与非Kerr-Schild形变下是否线性化，从而实现精确积分？
• RQ3 这些解的渐近对称代数是什么？与广义相对论相比有何异同？
• RQ4 时变解（如发射引力子的‘黑洞花朵’）是否具有有限能量并保持一致的热力学性质？
• RQ5 能否通过守恒曲面积分荷恢复这些黑洞解的热力学第一定律？

主要发现

• 本文构造了两类精确解：由Kerr-Schild形变得到的类pp波解与由类空Killing向量形变得到的‘黑洞花朵’解，二者均具有线性化场方程。
• Kerr-Schild型形变的通用解由四个单零坐标函数描述，构成一类新的pp波。
• 黑洞花朵解破坏球对称性，由两个积分常数与两个角坐标函数描述。
• 尽管具有时变性与向外的引力辐射，null无穷远的总能量保持恒定，表明在Bondi意义下无引力辐射‘新闻’。
• 守恒全局荷为有限值，与静态黑洞一致，质量 $ M = rac{b^2}{32G} $ 与角动量 $ J = 0 $，即使在时变情况下亦成立。
• 通过曲面积分恢复热力学第一定律：$ dM = TdS $，其中 $ ilde{k}_{ heta}^{[ur]} = rac{b heta b}{32 heta G} $，熵由Wald公式导出。