Global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces

TL;DR

This paper investigates the global and uniqueness properties of stationary and static spacetimes with outer trapped surfaces, demonstrating that such initial data lead to black hole spacetimes and establishing a uniqueness theorem for static cases.

Contribution

It proves that under certain conditions, the maximal Cauchy development forms a black hole spacetime and establishes a uniqueness theorem for static initial data with outer trapped boundaries.

Findings

Future Cauchy development is a black hole spacetime under null energy condition.

All Killing prehorizons are embedded in static initial data with outer trapped boundary.

Uniqueness of static initial data sets with outer trapped boundary is established.

Abstract

Global properties of maximal future Cauchy developments of stationary, m-dimensional asymptotically flat initial data with an outer trapped boundary are analyzed. We prove that, whenever the matter model is well posed and satisfies the null energy condition, the future Cauchy development of the data is a black hole spacetime. More specifically, we show that the future Killing development of the exterior of a sufficiently large sphere in the initial data set can be isometrically embedded in the maximal Cauchy development of the data. In the static setting we prove, by working directly on the initial data set, that all Killing prehorizons are embedded whenever the initial data set has an outer trapped boundary and satisfies the null energy condition. By combining both results we prove a uniqueness theorem for static initial data sets with outer trapped boundary.