Foliation dependence of black hole apparent horizons in spherical   symmetry
  Numerical studies of gravitational collapse to black holes make use of apparent horizons, which are intrinsically foliation-dependent. We expose the problem and discuss possible solutions using the Hawking quasilocal mass. In spherical symmetry, we present a physically sensible approach to the problem by restricting to spherically symmetric spacetime slicings. In spherical symmetry the apparent horizons are gauge-independent in any spherically symmetric foliation but physical quantities associated with them, such as surface gravity and temperature, are not. The widely used comoving and Kodama foliations, which are of particular interest, are discussed in detail. 
Valerio Faraoni, George F.R. Ellis, Javad T. Firouzjaee, Alexis Helou,   and Ilia Musco
gr-qc