Staticity Theorem for Higher Dimensional Generalized Einstein-Maxwell System

TL;DR

This paper establishes a generalized staticity theorem for higher-dimensional Einstein-Maxwell systems, deriving formulas for variations of physical quantities and extending black hole thermodynamics to these complex settings.

Contribution

It introduces a new staticity theorem for higher-dimensional Einstein-Maxwell systems with boundary conditions, extending classical results to more complex gauge fields.

Findings

Derived formulas for variations of mass, angular momentum, and energy.

Established a generalized first law of black hole thermodynamics.

Proved conditions under which solutions are static in higher dimensions.

Abstract

We derive formulas for variations of mass, angular momentum and canonical energy in Einstein (n-2)-gauge form field theory by means of the ADM formalism. Considering the initial data for the manifold with an interior boundary which has the topology of (n-2)-sphere we obtained the generalized first law of black hole thermodynamics. Supposing that a black hole evevt horizon comprisesw a bifurcation Killing horizon with a bifurcate surface we find that the solution is static in the exterior world, when the Killing timelike vector field is normal to the horizon and has vanishing electric or magnetic fields on static slices.