An integral equation for spacetime curvature in general relativity
An integral equation for spacetime curvature in general relativity
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广义相对论中时空曲率的积分方程
DOI：
10.4310/sdg.2005.v10.n1.a5
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发表时间：
2005
期刊：
影响因子：
--
通讯作者：
V. Moncrief
中科院分区：
文献类型：
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作者：
V. Moncrief
证明杨-米尔斯场的整体存在性的一个关键步骤，是在弯曲的、四维的、整体双曲的背景时空中传播，是导出和约化一个由杨-米尔斯场方程的任意解的曲率所满足的积分方程。本文给出了广义相对论爱因斯坦场方程真空解的曲率所满足的积分方程的相应推导。所得公式表示曲率在一个点的一个“直接”积分过去光锥从该点，一个所谓的“尾巴”积分在该锥的内部和两个额外的积分在一个球的初始数据超曲面和其边界。初始数据面中的尾部贡献和球上的积分是由于惠更斯原理在一般弯曲的四维时空中传播的结果。然而，通过应用斯托克斯定理和部分积分引理，人们可以重新表达这些“违反惠更斯”的贡献，纯粹是在圆锥本身和该圆锥与初始数据表面的二维交点上的积分。此外，通过利用杨-米尔斯论证中使用的平行传播规范条件（或克朗斯特罗姆规范条件）的推广，人们可以明确地用曲率来表示标架场和连接1形式。虽然整体存在性对于广义相对论来说肯定是错误的，但人们期望由此产生的积分方程可能在分析爱因斯坦演化过程中曲率的传播、聚焦和（有时）爆炸方面是有用的，从而揭示了整体存在性的自然替代猜想，即彭罗斯的宇宙审查猜想。
A key step in the proof of global existence for Yang-Mills fields, propagating in curved, 4-dimensional, globally hyperbolic, background spacetimes, was the derivation and reduction of an integral equation satisfied by the curvature of an arbitrary solution to the Yang-Mills field equations. This article presents the corresponding derivation of an integral equation satisfied by the curvature of a vacuum solution to the Einstein field equations of general relativity. The resultant formula expresses the curvature at a point in terms of a ‘direct’ integral over the past light cone from that point, a so-called ‘tail’ integral over the interior of that cone and two additional integrals over a ball in the initial data hypersurface and over its boundary. The tail contribution and the integral over the ball in the initial data surface result from the breakdown of Huygens’ principle for waves propagating in a general curved, 4-dimensional spacetime. By an application of Stokes’ theorem and some integration by parts lemmas, however, one can re-express these ‘Huygens-violating’ contributions purely in terms of integrals over the cone itself and over the 2-dimensional intersection of that cone with the initial data surface. Furthermore, by exploiting a generalization of the parallel propagation, or Cronstrom, gauge condition used in the Yang-Mills arguments, one can explicitly express the frame fields and connection one-forms in terms of curvature. While global existence is certainly false for general relativity one anticipates that the resulting integral equation may prove useful in analyzing the propagation, focusing and (sometimes) blow up of curvature during the course of Einsteinian evolution and thereby shed light on the natural alternative conjecture to global existence, namely Penrose’s cosmic censorship conjecture.