2-surfaces of constant mean curvature in physical spacetimes

Abstract

We study the mean curvature H of r = constant surfaces in spacelike hypersurfaces of the Kerr spacetimes. These surfaces are compared to those which are defined by constant mean curvature. As a result of the calculations, the Boyer--Lindquist r-coordinate turns out to be close to a radial coordinate defined by H = constant surfaces. Additionally, constant mean curvature surfaces are determined in numerically calculated spacetimes of rotating neutron stars. In order to visualize the intrinsic geometry of the surfaces they are embedded into Euclidean space. With regard to the solution of time-dependent problems in numerical relativity a radial H = constant coordinate is employed as a new spatial gauge condition in the 3+1 formalism. This leads to a linear elliptic equation, which uniquely determines the shift vector on each spacelike slice.