Analytical Perturbative Construction of Initial Data for Binary Black Holes up to Third Order in Spin and Momentum

We analytically solve the constraints in General Relativity for two black holes with arbitrary momenta and spin up to third order in these parameters. We compute the location and geometry of the apparent horizon, which depend on the spins, momenta, relative orientation angles, and the separation between the black holes, and present the result in a coordinate-independent form. We also extract the ADM mass and the irreducible mass and verify their consistency. The final expressions are depicted in a coordinate-independent form. The results can be easily extended to any number of black holes and used to complement numerical relativity simulations.