Rigidity of marginally outer trapped 2-spheres

TL;DR

This paper establishes an area inequality for stable marginally outer trapped 2-spheres in matter-filled spacetimes and explores the rigidity and geometric structure of those achieving the bound, linking to spacetime embeddings.

Contribution

It proves a new area inequality for marginally outer trapped spheres and demonstrates rigidity results, including a splitting theorem and spacetime embedding properties.

Findings

Stable marginally outer trapped 2-spheres have an area bound of 4π/c.

Achieving the bound implies a splitting of the initial data set.

Such initial data sets embed into Nariai spacetime.

Abstract

In a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inequality. Namely, as discussed in the paper, its area must be bounded above by $4\pi/c$, where $c > 0$ is a lower bound on a natural energy-momentum term. We then consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. In particular, we prove a splitting result for 3-dimensional initial data sets analogous to a result of Bray, Brendle and Neves [10] concerning area minimizing 2-spheres in Riemannian 3-manifolds with positive scalar curvature. We further show that these initial data sets locally embed as spacelike hypersurfaces into the Nariai spacetime. Connections to the Vaidya spacetime and dynamical horizons are also discussed.