论文标题
一般相对论中的边界减少了什么？
What is a reduced boundary in general relativity?
论文作者
论文摘要
边界的概念在一般相对性的多个分支中起着重要作用，例如，爱因斯坦方程的变异原理，事件范围和黑洞的明显范围，是捕获的表面的形成。另一方面，在被称为几何测量理论的数学分支中，很久以前就发现了另一种概念，即有限的晶体集合的降低边界。因此，本文提出了有限距离集合的定义及其在一般相对性中减少的边界。此外，在文献中第一次，在欧几里得施瓦茨柴尔德几何几何学的相关情况下，明确评估了几何测量理论的基本积分公式。这项研究为重力物理学的几种概念的测量方法做好了准备，并补充了几何见解。此外，这样的研究表明，应该考虑到应对有限的渗透率riemannian几何形状来评估欧几里得量子重力的振幅的可能性，该几何形状与其减少边界上的分配数据相匹配。作为可能的应用，对基本公式进行了分析，最终导致黑洞的内在量子机械熵进行校正。
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.
