# Spacetime positive mass theorems for initial data sets with noncompact boundary

## TL;DR

This paper establishes positive mass theorems for initial data sets with noncompact boundaries in both asymptotically flat and hyperbolic spacetimes, under dominant energy conditions, highlighting differences between these settings.

## Contribution

It introduces a new definition of energy-momentum vectors at spatial infinity for such data sets and proves positive mass inequalities assuming the manifold is spin.

## Key findings

- Positive mass inequalities are proven under suitable energy conditions.
- A rigidity statement is established for the flat case when the energy-momentum vector is lightlike.
- The paper compares features of asymptotically Euclidean and hyperbolic settings with boundary conditions.

## Abstract

In this paper, we define an energy-momentum vector at the spatial infinity of either asymptotically flat or asymptotically hyperbolic initial data sets carrying a non-compact boundary. Under suitable dominant energy conditions (DECs) imposed both on the interior and along the boundary, we prove the corresponding positive mass inequalities under the assumption that the underlying manifold is spin. In the asymptotically flat case, we also prove a rigidity statement when the energy-momentum vector is lightlike. Our treatment aims to underline both the common features and the differences between the asymptotically Euclidean and hyperbolic settings, especially regarding the boundary DECs.