Studies of Singularities, Black Holes and Gravitational Radiation

这个项目将研究引力的两个方面：(1)引力坍塌和(2)引力辐射。(1)当物体的引力强到足以捕获光线时，该物体就变成了黑洞。在黑洞内部，物体在自身引力的影响下继续变得越来越小，直到它变成一个具有无限密度和无限引力场的点，称为奇点。我将计算出在引力崩塌中形成的奇点的性质，特别是任何接近奇点的观察者所感受到的力。(2)就像无线电发射机中的电流产生无线电波一样，移动的质量产生重力波。就像无线电接收器对无线电波的存在做出反应一样，探测引力波的工作也在进行中。引力辐射的一个有趣的方面是所谓的引力波记忆：即使在波过去之后，引力波探测器也会发生永久性的变化。我将研究这种引力波记忆的原因和性质，推测黑洞内部的奇点由两部分组成：位于黑洞中心的类空奇点和取代黑洞内视界的零奇点。我将使用两种不同的数值方法来研究这两种奇点。在此之前，我曾对类太空奇点进行过模拟；然而，这些模拟揭示了空间尺度非常小的结构(称为“尖峰”)的存在，而模拟并未解决这一问题。我将使用自适应网格细化技术重新审视这些模拟，这应该有助于解决尖峰。我将使用奇点附近的场方程的主导项，将这些模拟的结果与对尖峰行为的解析近似的结果进行比较。我将使用爱因斯坦场方程的“双零”公式来研究零奇点。这里，不同于通常由具有恒定时间t的类空表面所构成的层状时空，而是由一对具有恒定零坐标u和v的零曲面构成的层状结构。然后，爱因斯坦场方程式和比安奇恒等式就变成了零面和Weyl张量的展开和剪切方程。我将对这些方程进行数值模拟，看看Weyl张量是否会在(过去的)黑洞内视界上爆炸。我将使用各种方法研究各种来源的引力波记忆。来自中微子的记忆将使用完全非线性的爱因斯坦-弗拉索夫方程来处理。膨胀宇宙中的记忆将用宇宙学微扰理论来处理。真空情况下的记忆将用二阶微扰理论来处理。我还将使用一种新的真空轴对称程序来研究临界引力塌缩。此外，我还将继续用数值方法研究爱因斯坦-艾特理论、反De-Sitter时空中的引力崩塌和带电黑洞。

This project will study two aspects of gravity: (1) gravitational collapse and (2) gravitational radiation. (1) When an object's gravity becomes strong enough to trap light, the object becomes a black hole. Inside the black hole the object continues to become smaller and smaller under the influence of its own gravity until it becomes a point with infinite density and infinite gravitational field called a singularity. I will work out the properties of the singularities formed in gravitational collapse, and in particular the forces felt by any observer who approaches the singularity. (2) Just as the electric currents in a radio transmitter make radio waves, so moving masses make gravity waves. And just as radio receivers react to the presence of radio waves, so there is an ongoing effort to detect gravitational waves. One interesting aspect of gravitational radiation is something called gravitational wave memory: even after the wave has passed, there is a permanent change in the gravitational wave detector. I will study the causes and properties of this gravitational wave memory.It is conjectured that the singularity inside a black hole consists of two parts: a spacelike singularity at the center of the black hole and a null singularity that takes the place of the black hole's inner horizon. I will study both types of singularities using two different numerical methods. Previously, I have performed simulations of spacelike singularities; however these simulations revealed the presence of structure with a very small spatial scale (called "spikes") that was not resolved by the simulations. I will revisit these simulations using the technique of adaptive mesh refinement, which should serve to resolve the spikes. I will compare the results of these simulations to those of an analytic approximation to spike behavior using what are expected to be the leading terms of the field equations near the singularity. I will study null singularities using a "double null" formulation of the Einstein field equations. Here instead of the usual foliation of spacetime by spacelike surfaces of constant time t, one has a foliation by pairs of null surfaces of constant null coordinates u and v. The Einstein field equations and Bianchi identities then become equations for the expansion and shear of the null surfaces and the Weyl tensor. I will perform numerical simulations of these equations to see whether the Weyl tensor blows up on (what used to be) the inner horizon of the black hole. I will study gravitational wave memory from a variety of sources using a variety of methods. Memory from neutrinos will be treated using the full nonlinear Einstein-Vlasov equations. Memory in an expanding universe will be treated using cosmological perturbation theory. Memory for the vacuum case will be treated in second order perturbation theory. I will also study critical gravitational collapse using a new vacuum axixymmetric code. In addition, I will continue, using numerical methods, my studies of Einstein-Aether theory, gravitational collapse in anti de-Sitter spacetime, and charged black holes.