Pragmatic approach to gravitational radiation reaction in binary black holes

The study the relativistic orbital motion of two black holes in the extreme mass ratio regime, i.e. $m/M\ll1$, beyond the first order approximation is of crucial importance to accurately compute effects of gravitational radiation such as the inspiral phase and second order perturbations. The trajectory of the particle is determined by the geodesic equation on the perturbed massive black hole spacetime. The particle itself generates the perturbations and this leads to a problem that needs regularization. Here we study perturbations around a Schwarzschild black hole and use Moncrief gauge invariant formalism. We decompose the perturbations into $\ell-$multipoles to show that for each given $\ell$ metric coefficients are $C^0$ at the location of the particle. The sum over $\ell$, however, diverges. We bring this sum to a $\zeta-$function regularization scheme and show that this is tantamount to subtract the $\ell\to\infty$ piece to each multipole. We explicitly perform the numerical computation of the corrected geodesics.