A Spinor Approach to Penrose Inequality

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.