A PDE proof of the Penrose inequality for perturbations of Schwarzschild initial data

Abstract

The classical Penrose inequality relates the mass of an asymptotically flat spacetime to the total area of its black holes. We present a PDE-based proof of the Penrose inequality for a special class of perturbations of Schwarzschild initial data. The proof is based around linearising solutions to a version of the Jang equation which is modified to deal with warped product metrics. We discuss the possibility of generalising the argument to a wider class of perturbations.

Nonexistence of solutions to the coupled generalized Jang equation/zero divergence system

“…We also mention that (1.10) was recently investigated in the case where div(ϕ q) was linearized [22]. In this case, under some very restrictive assumptions on the initial data, solutions were shown to exist, though of course this does not prove the Penrose inequality in general.…”