Finding apparent horizons and other two-surfaces of constant expansion

Apparent horizons are structures of spacelike hypersurfaces that can be determined locally in time. Closed surfaces of constant expansion (CE surfaces) are a generalisation of apparent horizons. I present an efficient method for locating CE surfaces. This method uses an explicit representation of the surface, allowing for arbitrary resolutions and, in principle, shapes. The CE surface equation is then solved as a nonlinear elliptic equation. It is reasonable to assume that CE surfaces foliate a spacelike hypersurface outside of some interior region, thus defining an invariant (but still slicing-dependent) radial coordinate. This can be used to determine gauge modes and to compare time evolutions with different gauge conditions. CE surfaces also provide an efficient way to find new apparent horizons as they appear e.g. in binary black hole simulations.