Analysis of black hole stability.

黑洞稳定性猜想是非线性偏微分方程分析中的一个具有挑战性的问题，也是数学相对论中的一个重大公开问题。由于我和其他人的工作，在过去的十年里，在这个问题上取得了巨大的进展。拟议的赠款将部分资助一名博士后研究助理(PDRA)，该助理将对保持最近的进展速度至关重要。广义相对论是一种几何引力理论，其中宇宙由一组四维的时空点和一个必须满足爱因斯坦方程的不定内积来描述。物理学家认为，黑洞将在我们对理论物理的理解中发挥关键作用，将吸收宇宙后期的所有物质，并且是已知存在于大多数星系中心的巨大质量的物体。科尔的爱因斯坦方程的显式解族是由质量和角动量参数表示的，它们描述了当角动量相对于质量很小时的黑洞。对于零质量，解简化为Minkowski解，对于角动量为零，简化为Schwarzschild解。对于具有适当渐近行为的解，Kerr族是爱因斯坦方程唯一的定常黑洞解族。尽管物理学家认为，所有黑洞都会像纳维尔-斯托克斯方程一样，以渐进的方式接近克尔解，这一点毋庸置疑，但在物理基础上的预期与可以证明的结果之间存在着巨大的差距。因此，人们对证明Kerr解的渐近稳定性很感兴趣：猜想K：如果一组初始数据与产生Kerr解的数据非常接近，则相应的解最终将逼近Kerr解。如果没有对克尔解的衰变速度的估计，这个猜想就不太可能得到证明。对于可以作为爱因斯坦方程模型的非线性波动方程，众所周知，对于足够小的初始数据和足够弱的非线性，初始数据的小确保非线性的影响直到中间时刻都很小，从而允许解以与线性波动方程的解相同的速度衰减。然后，从中后期，由于当非线性项较小时，非线性项小于线性项(但当线性项较大时，非线性项大于线性项--这是相关非线性项的性质)，所以非线性的影响保持小且减小，使得非线性方程的解表现为线性方程的解。爱因斯坦方程的分析具有挑战性，因为它是非线性的和几何的，而且它既有有限时间曲率发散的解，也有全局光滑的解。只有当解具有高度对称性时，才有可能对发散解进行彻底的研究。数学相对论研究的一个里程碑式的结果是证明了平坦空间(称为Minkowski空间)是稳定的。这是建立在对波、麦克斯韦和线性化的爱因斯坦方程的衰变估计的基础上的。在Schwarzschild的情况下，这些方程也被研究过。在一般的克尔情况下，波动方程的衰变估计已经得到证明，我预计我和我的合作者将在这一拟议的赠款期间开始之前完成对麦克斯韦方程的分析。这笔拨款的目的是继续这项计划，并聘请一名博士后研究员在克尔背景下研究线性化的爱因斯坦方程。这应该会提供重要的进展，帮助数学相对论社区解决克尔稳定性猜想。

The black hole stability conjecture is a challenging problem in the analysis of nonlinear partial differential equations and is one of the major open problems in mathematical relativity. Because of work by me and others, there has been great progress on this problem in the last decade. The proposed grant would partially fund a post-doctoral research assistant (PDRA), who will be essential to maintaining the recent pace of progress. General relativity is a geometric theory of gravity, in which the universe is described by a four-dimensional set of space-time points and by an indefinite inner-product, which must satisfy the Einstein equations. Physicists believe that black holes will play a crucial role in our understanding of theoretical physics, will absorb all matter in the late stages of the universe, and are the enormously massive objects known to exist at the centre of most galaxies. Kerr's family of explicit solutions to the Einstein equation are parametrised by mass and angular momentum, and they describe black holes when the angular momentum is small relative to the mass. For zero mass, the solution reduces to the Minkowski solution, and for angular momentum zero to the Schwarzschild solution.For solutions having the appropriate asymptotic behaviour, the Kerr family is the unique family of stationary, black hole solutions of the Einstein equations. Although physicists believe that there can be no reasonable doubt that all black holes will asymptotically approach a Kerr solution, as with the Navier-Stokes equation, there is an enormous gap between what is expected on physical grounds and what can be proved. Hence, there is great interest in proving the asymptoticstability of the Kerr solutions:Conjecture K: If a set of initial data is very close to one that generates a Kerr solution, then the corresponding solution will eventually approach a Kerr solution. It is unlikely that that this conjecture can be proved without estimates on the rate of decay to the Kerr solution. For a nonlinear wave equation, which can serve as a model for the Einstein equation, it is known that for sufficiently small initial data and a sufficiently weak nonlinearity, the smallness of the initial data guarantees that the influence of the nonlinearity is small up to intermediate times, allowing the solution to decay at the same rate as a solution to the linear wave equation. Then, from intermediate to late times, since the nonlinear term is smaller than the linear terms when they are small (but larger than the linear terms, when the linear terms are large -this being the nature of the relevant nonlinear terms), the influence of the nonlinearity remains small and diminishing, so that the solution to the nonlinear equation behaves like solutions to the linear equation. Analysis of the Einstein equation is challenging because it is nonlinear and geometric and because it has both solutions for which the curvature diverges in finite time and globally smooth solutions. A thorough investigation of divergent solutions has been possible only when solutions have a high degree of symmetry. One of the landmark results in the study of mathematical relativity was the proof that the flat space (known as Minkowski space) is stable. This built on decay estimates for the wave, Maxwell, and linearised Einstein equations. In the Schwarzschild case, these equations have also been studied. In the general Kerr case, decay estimates for the wave equation have been proved, and I anticipate that my collaborators and I will have completed our analysis of the Maxwell equation by the start of this proposed period of the grant. The purpose of this grant is to continue this program, and to employ a post-doctoral researcher to investigate the linearised Einstein equation in the Kerr context. This should provide important progress that will help the mathematical relativity community resolve the Kerr stability conjecture.