The Penrose inequality for perturbations of the Schwarzschild initial data

Abstract

We show that in the conformally flat case the Penrose inequality is satisfied for the Schwarzschild initial data with a small addition of the axially symmetric traceless exterior curvature. In this class the inequality is saturated only for data related to special sections of the Schwarzschild spacetime.

The Penrose inequality for nonmaximal perturbations of the Schwarzschild initial data

“…This theorem generalizes partially that in [11], but now an analysis of nongeneric case is much more difficult. Note that data which satisfy conditions 1-3 of Definition 3.1 depend on one free function X.…”

“…A review of results on existence theorems in different settings can be found in [15]. As in [11], we assume in this paper that g ij = δ ij and the initial surface is…”

“…As in [11], the angular momentum J enters into a structure of the function ω coming from regularity assumption about A ij ,…”

“…Thus, one can first solve (15) and then extract information about total energy and area of the MOTS from equation ( 16). Considerations from Section 2 in [11] can be almost literally repeated under the asymptotical flatness conditions of the form…”

“…where E, p and J are, respectively, energy, momentum and angular momentum of initial data. In paper [11] we studied inequality (2) in the case of axially symmetric conformally flat maximal data on R 3 with removed ball of radius m/2. The internal boundary was a marginally outer trapped surface (MOTS).…”