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.
ON THE MEASURE-THEORETIC FOUNDATIONS OF THE SECOND LAW OF THERMODYNAMICS
