P.d.e.'s which imply the Penrose conjecture

TL;DR

This paper proposes a new approach to the Penrose conjecture by reducing it to the Riemannian Penrose inequality using solutions to certain geometric systems, introducing a generalized identity and new concepts of horizons.

Contribution

It introduces the generalized Schoen-Yau identity and a method to reduce the Penrose conjecture to known cases, along with proposing generalized horizons and a broader conjecture.

Findings

Derived a new generalized Schoen-Yau identity.

Proposed a reduction of the Penrose conjecture to the Riemannian case.

Suggested new definitions of apparent horizons and trapped surfaces.

Abstract

In this paper, we show how to reduce the Penrose conjecture to the known Riemannian Penrose inequality case whenever certain geometrically motivated systems of equations can be solved. Whether or not these special systems of equations have general existence theories is therefore an important open problem. The key tool in our method is the derivation of a new identity which we call the generalized Schoen-Yau identity, which is of independent interest. Using a generalized Jang equation, we propose canonical embeddings of Cauchy data into corresponding static spacetimes. In addition, our techniques suggest a more general Penrose conjecture and generalized notions of apparent horizons and trapped surfaces, which are also of independent interest.