Perturbative evolution of conformally flat initial data for a single boosted black hole

The conformally flat families of initial data typically used in numerical relativity to represent boosted black holes are not those of a boosted slice of Schwarzschild spacetime. If such data are used for each black hole in a collision, the emitted radiation will be partially due to the relaxation of the individual holes to boosted Schwarzschild form. We attempt to compute this radiation by treating the geometry for a single boosted conformally flat hole as a perturbation of a Schwarzschild black hole, which requires the use of second order perturbation theory. In this we attempt to mimic a previous calculation we did for the conformally flat initial data for spinning holes. We find that the boosted black hole case presents additional subtleties, and although one can evolve perturbatively and compute radiated energies, it is much less clear than in the spinning case how useful for the study of collisions are the radiation estimates for the spurious energy in each hole. In addition to this we draw some lessons on which frame of reference appears as more favorable for computing black hole collisions in the close limit approximation.