On a spacetime positive mass theorem with corners

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.

arXiv comment49 pages, references updated, topology conclusion updated

References in corpus (18)

• Quasilocal Energy and Conserved Charges Derived from the Gravitational Action
• The Gravitational Hamiltonian, Action, Entropy, and Surface Terms
• Positivity of Quasilocal Mass
• Quasilocal mass in general relativity
• Isometric embeddings into the Minkowski space and new quasi-local mass
• On the positive mass theorem for manifolds with corners
• A comment on Liu and Yau's positive quasi-local mass
• Scalar curvature and singular metrics
• On geometric problems related to Brown-York and Liu-Yau quasilocal mass
• Positive scalar curvature with skeleton singularities
• Equality in the Spacetime Positive Mass Theorem
• Scalar curvature and harmonic one-forms on three-manifolds with boundary
• On the geometry and topology of initial data sets with horizons
• A localized spacetime Penrose inequality and horizon detection with quasi-local mass
• Mass of asymptotically flat $3$-manifolds with boundary
• Spacetime Harmonic Functions and Applications to Mass
• Lorentzian positive mass theorem for spacetimes with distributional curvature
• Dihedral rigidity for cubic initial data sets