Lindblad resonance torques in relativistic discs - II. Computation of resonance strengths

We present a fully relativistic computation of the torques due to Lindblad resonances from perturbers on circular, equatorial orbits on discs around Schwarzschild and Kerr black holes. The computation proceeds by establishing a relation between the Lindblad torques and the gravitational waveforms emitted by the perturber and a test particle in a slightly eccentric orbit at the radius of the Lindblad resonance. We show that our result reduces to the usual formula when taking the non-relativistic limit. Discs around a black hole possess an m= 1 inner Lindblad resonance (ILR) with no Newtonian-Keplerian analogue; however, its strength is very weak even in the moderately relativistic regime (r/M∼ few tens), which is in part due to the partial cancellation of the two leading contributions to the resonant amplitude (the gravitoelectric octupole and gravitomagnetic quadrupole). For equatorial orbits around Kerr black holes, we find that the m= 1 ILR strength is enhanced for retrograde spins and suppressed for prograde spins. We also find that the torque associated with the m≥ 2 ILRs is enhanced relative to the non-relativistic case; the enhancement is a factor of 2 for the Schwarzschild hole even when the perturber is at a radius of 25M.