A note on the rigidity of marginally outer trapped 2-spheres

TL;DR

This paper investigates the rigidity properties of marginally outer trapped 2-spheres in matter-filled spacetimes with positive cosmological constant, establishing conditions under which these spheres attain maximal area bounds and illustrating the results with specific spacetime examples.

Contribution

It extends rigidity results for area-maximizing marginally outer trapped 2-spheres to the context of dynamical horizons in spacetimes with matter and positive cosmological constant.

Findings

Stable marginally outer trapped 2-spheres have area bounds related to energy conditions.

Achieving the area bound implies specific geometric rigidity.

Examples include canonical horizons in Vaidya and Nariai spacetimes.

Abstract

As discussed in the paper, in a matter-filled spacetime, perhaps with positive cosmological constant, a stable marginally outer trapped 2-sphere must satisfy a certain area inquality. Namely, its area must be bounded above by $4\pi/c$, where $c > 0$ is a lower bound on a natural energy momentum term. In this note we consider the rigidity that results for stable, or weakly outermost, marginally outer trapped 2-spheres that achieve this upper bound on the area. The "canonical" dynamical horizon in Vaidya spacetime and certain spacelike hypersurfaces in Nariai spacetime provide illustrations of the main results. These results may be viewed as spacetime analogues of the rigidity results of Bray, Brendle and Neves [10] concerning area minimizing 2-spheres in Riemannian 3-manifolds with scalar curvature having positive lower bound.