High-order post-newtonian contributions to gravitational self-force effects in black hole spacetimes

The explicit analytical computation of first-order metric perturbations in black hole spacetimes is described in the case of a perturbing mass moving on an equatorial circular orbit. The perturbation equations can be separated into an angular part and a radial part. The latter satisfies a single inhomogeneous radial Schrödingerlike equation with a Dirac-delta singular source term, whose solutions are built up through Green's function techniques. Various types of approximate analytical homogeneous solutions (and corresponding Green's functions) can be constructed: Post-Newtonian solutions (expanded in powers of 1=c), Mano-Suzuki-Takasugi solutions (expanded in series of hypergeometric functions), Wentzel-Kramers- Brillouin (WKB) solutions (large l expansion). The perturbed black-hole metric constructed by suitably combining these different kind of solutions can then be used to compute, in analytical form, gauge-invariant quantities. These include several "potentials" entering the effective-one-body formalism (shortly reviewed here). The latter formalism is a new way of describing the gravitational interaction of two masses which has played a crucial role in the recent detection of gravitationalwaves.