Brown–York energy in stationary spacetimes

Abstract

One usually defines the Brown–York energy for a 2-surface embedded in a space-like 3-slice as an integration of the mean curvature of the 2-surface isometrically embedded into the 3-slice, with a proper reference 3-space. We demonstrate that this naive definition is ill for stationary spacetimes. As an example, we investigate the Kerr–Newman spacetime in detail. We show that the naive definition of the Brown–York energy is not a component of the Brown–York boundary stress tensor, and thus deviates from the original idea of Brown and York. Furthermore, we present the exact form of the Brown–York energy for the Kerr–Newman spacetime with proper reference.