A Spinor Approach to Penrose Inequality

TL;DR

This paper develops a spinorial framework to prove a Penrose inequality relating the ADM energy-momentum and the areal radius of trapped surfaces, extending positive energy theorems using conformal invariants and spinor techniques.

Contribution

It introduces a refined spinorial approach and a new inequality involving the Sen-Witten operator to establish a Penrose inequality under the dominant energy condition.

Findings

Established a Penrose type inequality using spinor methods.

Proved a Kato-Yau inequality for the Sen-Witten operator.

Connected the spinorial identity with Yamabe-type conformal invariants.

Abstract

Consider an asymptotically Euclidean initial data set with a smooth marginally trapped surface (possibly a union of future and past multi-connected components) as inner boundary. By a further development of the spinorial framework underlying the positive energy theorem, a refined Witten identity is worked out and in the maximal slicing case, a close connection of the identity with a conformal invariant of Yamabe type is revealed. A Kato-Yau inequality for the Sen-Witten operator is also proven from a conformal geometry perspective. Guided by the Hamiltonian picture underlying the spinorial framework, a Penrose type inequality is then proven to the effect that given the dominant energy condition, the ADM energy-momentum is, up to a non-zero constant less than unity, bounded by the areal radius of the marginally trapped surface. To establish the Penrose inequality in full generality, it is then sufficient to show that the norm of the Sen-Witten spinor, subject to the APS boundary condition imposed on a suitably defined outermost marginally trapped surface, is bounded below by that attained in the Schwarzschild metric.