Stability of the Spacetime Penrose Inequality in Spherical Symmetry

TL;DR

This paper proves a stability result for the spacetime Penrose inequality in spherical symmetry, showing that initial data close to Schwarzschild conditions implies the spacetime is nearly Schwarzschild in several geometric senses.

Contribution

It establishes a quantitative stability statement for the spacetime Penrose inequality in spherical symmetry using the generalized Jang equation approach.

Findings

Initial data with ADM mass near the half area radius are close to Schwarzschild in volume preserving intrinsic flat distance.

Static potentials and extrinsic curvature are close to Schwarzschild values in appropriate norms.

The results provide a stability estimate linking mass, horizon area, and geometric closeness to Schwarzschild spacetime.

Abstract

We formulate and prove the stability statement associated with the spacetime Penrose inequality for $n$-dimensional spherically symmetric, asymptotically flat initial data satisfying the dominant energy condition. We assume that the ADM mass is close to the half area radius of the outermost apparent horizon and, following the generalized Jang equation approach, show that the initial data must arise from an isometric embedding into a static spacetime close to to the exterior region of a Schwarzschild spacetime in the following sense. Namely, the time slice is close to the Schwarzschild time slice in the volume preserving intrinsic flat distance, the static potentials are close in $L_{loc}^2$, and the initial data extrinsic curvature is close to the second fundamental form of the embedding in $L^2$.