Horizon surface gravity in corotating black hole binaries

For binary systems of corotating black holes, the zeroth law of mechanics states that the surface gravity is constant over each component of the horizon. Using the approximation of a conformally flat spatial metric, we compute sequences of quasi-equilibrium initial data for corotating black hole binaries with irreducible mass ratios in the range 10:1-1:1. For orbits beyond the innermost stable one, the surface gravity is found to be constant on each component of the apparent horizon at the sub-percent level. We compare those numerical results to the analytical predictions from post-Newtonian theory at the fourth (4PN) order and from black hole perturbation theory to linear order in the mass ratio. We find a remarkably good agreement for all mass ratios considered, even in the strong-field regime. In particular, our findings confirm that the domain of validity of black hole perturbative calculations appears to extend well beyond the extreme mass-ratio limit.