A Penrose-type inequality with angular momentum and charge for axisymmetric initial data

Abstract

A lower bound for the ADM mass is established in terms of angular momentum, charge, and horizon area in the context of maximal, axisymmetric initial data for the Einstein-Maxwell equations which satisfy the weak energy condition. If, on the horizon, the given data agree to a certain extent with the associated model Kerr-Newman data, then the inequality reduces to the conjectured Penrose inequality with angular momentum and charge. In addition, a rigidity statement is also proven whereby equality is achieved if and only if the data set arises from the canonical slice of a Kerr-Newman spacetime.M. Khuri acknowledges the support of NSF Grant DMS-1708798.

A Penrose-type inequality with angular momenta for black holes with 3-sphere horizon topology

“…Note that if the initial data set (M, g, k) has the same Weyl coordinate functions as the associated Myers-Perry black hole, i.e., β(0, z) = β M P (0, z), then this result reduces to a proof of the Penrose inequality conjecture (1.5). Our result represents a generalization of the Penrose-type inequality with angular momentum established recently for three-dimensional axisymmetric initial data sets [21]. It should be noted that the Riemannian Penrose inequality, which holds up to dimension 7, tacitly assumes a minimal surface boundary of spherical topology.…”

“…The above-mentioned authors rigorously proved a result which is closely related to the desired result [21,Theorem 1.1]. Namely, they produced a lower bound for the mass in terms of the angular momentum, area of the outermost minimal surface, and an extra term involving a certain integral over the horizon of a quantity associated to the axisymmetric initial data.…”

“…Refinements to the Penrose inequality have been established rigorously when the spacetime carries additional conserved charges, such as electric charge [20]. Very recently a Penrose type inequality involving angular momentum J, was established by Khuri-Sokolowsky-Weinstein for 3-dimensional, asymptotically flat, axisymmetric, maximal initial data sets with outermost minimal boundary [21]. The axial symmetry prevents gravitational waves from carrying angular momenta away to infinity, A. Alaee and H. Kunduri acknowledge the support of an AMS-Simons travel grant and NSERC Grant RGPIN-2018-04887 respectively.…”

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

“…The inequality has been extended to include charge and angular momentum and proven under certain conditions, but all of these require some kind of non-negativity for the scalar curvature. See for example [8,14,16,17].…”

Geometric Inequalities for Quasi-Local Masses

“…The Penrose inequality has been established in the case of maximal data by Bray [8] and Huisken-Ilmanen [34], and charge was added in [40,44]. The inclusion of angular momentum is much more difficult and has not yet been established, although see [3,38] for partial results. Here we will establish Penrose-like inequalities involving angular momentum and charge for quasi-local masses.…”