The concept of boundary plays an important role in several branches of general relativity, e.g. the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped surfaces. On the other hand, in a branch of mathematics known as geometric measure theory, the usefulness has been discovered long ago of yet another concept, i.e. the reduced boundary of a finite-perimeter set. This paper proposes therefore a definition of finite-perimeter sets and their reduced boundary in general relativity. Moreover, a basic integral formula of geometric measure theory is evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry for the first time in the literature. This research prepares the ground for a measure-theoretic approach to several concepts in gravitational physics, supplemented by geometric insight. Moreover, such an investigation suggests considering the possibility that the in–out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole.

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什么是广义相对论的简化边界？

边界的概念在广义相对论的几个分支中发挥着重要作用，例如爱因斯坦方程的变分原理、黑洞的视界和视界、被困表面的形成。另一方面，在称为几何测度理论的数学分支中，很久以前就发现了另一个概念的有用性，即有限周长集的约化边界。因此，本文提出了有限周长集的定义及其在广义相对论中的简化边界。此外，文献中首次在欧几里得史瓦西几何的相关案例中明确评估了几何测度论的基本积分公式。这项研究为引力物理学中几个概念的测量理论方法奠定了基础，辅以几何洞察力。此外，这样的研究建议考虑欧几里得量子引力的输入-输出幅度应该在有限周长黎曼几何上进行评估的可能性，这些几何与在其简化边界上的指定数据相匹配。作为一种可能的应用，对最终导致修正黑洞固有量子力学熵的基本公式进行分析。