On a spacetime positive mass theorem with corners

TL;DR

This paper extends the positive mass theorem to initial data sets with corners, establishing jump conditions for positivity, introducing a new quasilocal mass concept, and providing conditions for non-existence of certain fill-ins.

Contribution

It generalizes the positive mass theorem to singular initial data with corners and introduces a new positive quasilocal mass in the spacetime context.

Findings

Established jump conditions ensuring positive total spacetime mass.

Introduced a new quasilocal mass concept that is positive.

Provided conditions preventing certain spacetime fill-ins.

Abstract

In this paper we consider the positive mass theorem for general initial data sets satisfying the dominant energy condition which are singular across a piecewise smooth surface. We find jump conditions on the metric and second fundamental form which are sufficient for the positivity of the total spacetime mass. Our method extends that of Hirsch-Kazaras-Khuri to the singular case (which we refer to as initial data sets with corners) using some ideas from Hirsch-Miao-Tsang. As such we give an integral lower bound on the spacetime mass and we characterise the case of zero mass. Our approach also leads to a new notion of quasilocal mass which we show to be positive, extending the work of Shi-Tam to the spacetime case. Moreover, we give sufficient conditions under which spacetime Bartnik data sets cannot admit a fill-in satisfying the dominant energy condition. This generalises the work of Shi-Wang-Wei-Zhu and Shi-Wang-Wei to the spacetime setting.