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

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 ($\mathcal{QLE}$) contained within the membrane $Σ=\partialΩ$. 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 $\mathcal{QLE}> \frac{Q^{2}}{2R}$ for large spherical membranes, $Q$ is the charge and $R$ is the radius of the membrane. Turning off the angular momentum (Reissner Nordström) yields $\mathcal{QLE}> \frac{Q^{2}}{2R}$ 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.