Aspects of Quasi-local energy for gravity coupled to gauge fields

Abstract

We study the aspects of quasi-local energy associated with a 2−surface Σ bounding a space-like domain Ω of a physical 3+1 dimensional spacetime in the regime of gravity coupled to a gauge field. The Wang-Yau quasi-local energy together with an additional term arising due to the coupling of gravity to a gauge field constitutes the total energy (QLE) contained within the membrane Σ = ∂Ω. We specialize in the Kerr-Newman family of spacetimes which contains a U(1) gauge field coupled to gravity and an outer horizon. Through explicit calculations, we show that the total energy satisfies a weaker version of a Bekenstein type inequality QLE > Q 2 2R for large spherical membranes, Q is the charge and R is the radius of the membrane. Turning off the angular momentum (Reissner Nordström) yields QLE > Q 2 2R for all constant radii membranes containing the horizon and in such case the charge factor appearing in the right-hand side exactly equals to that of Bekenstein's inequality. Moreover, we show that the total quasi-local energy monotonically decays from 2M irr + V Q (M irr is the irreducible mass, V Q is the electric potential energy) at the outer horizon to M (M is the ADM mass) at the space-like infinity under the assumption of a small angular momentum of the black hole.

Einstein-Yang-Mills equations in the double null framework

“…It was proven in [29] that such quasi-local energy while evolving along the incoming null direction reproduces the Bel-Robinson energy and the matter stress-energy tensor (at different orders of course) at the limit of approaching vertex. Motivated by this result, it is only natural to study the evolution of this quasi-local energy in the outgoing null direction and observe the behavior in the focusing regime (note an expression of the quasi-local energy was obtained in [30] in the presence of a gauge field). We expect an alternate notion of trapped surface formation through the study of this quasi-local energy.…”