Vacuum Spacetimes with Future Trapped Surfaces

TL;DR

This paper constructs specific vacuum initial data for Einstein's equations on an asymptotically flat manifold with a regular center, containing a region foliated by future trapped surfaces, leading to future null incompleteness of the resulting spacetime.

Contribution

It demonstrates the explicit construction of vacuum initial data with trapped surfaces, advancing understanding of conditions leading to singularities in general relativity.

Findings

Initial data with trapped surfaces can be constructed explicitly.

Such initial data guarantees future null incompleteness via Penrose's theorem.

The data is defined on a topologically simple, asymptotically flat manifold.

Abstract

In this article we show that one can construct initial data for the Einstein equations which satisfy the vacuum constraints. This initial data is defined on a manifold with topology $R^3$ with a regular center and is asymptotically flat. Further, this initial data will contain an annular region which is foliated by two-surfaces of topology $S^2$. These two-surfaces are future trapped in the language of Penrose. The Penrose singularity theorem guarantees that the vacuum spacetime which evolves from this initial data is future null incomplete.