Gravitational energy-momentum flow in binary systems

We investigate the gravitational energy-momentum distribution in the space-time of two black holes in circular orbit, in the context of the teleparallel equivalent of general relativity. This field configuration is important because gravitational waves are expected to be emitted in the final stages of inspiral and merger of binary black holes. We address an approximate solution of Einstein's field equations that describes two non-spinning black holes that circle each other in the $xy$ plane, obtain the total energy of the space-time and verify that the gravitational binding energy is negative. We show that gravitational radiation is emitted as long as the separation between the holes decreases in time. If the black holes are spinning and circle each other, it has been found in the literature that, during the pre-merger inspiral, they bob up and down sinusoidally. The understanding of this phenomenon requires the understanding of the gravitational energy-momentum flow in the space-time of binary black holes. For the time dependent metric tensor of a general binary black hole system, the non-vanishing of the gravitational momentum may explain the bobbing of spinning black holes.