Brown-York energy in stationary spacetimes

One usually defines the Brown-York energy for a 2-surface embedded in a spacelike 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 the proper reference.