Excitation of the Free Oscillation of a Schwarzschild Black Hole by the Gravitational Waves from a Scattered Test Particle

Abstract

In order to find conditions of excitation of the free oscillation of a Schwarzschild black hole by gravitational waves, we have analyzed the gravitational waves radiated by a test particle of mass µ and orbital angular momentum µLZ starting at infinity with finite velocity √1-ε-2c and scattered by a Schwarzschild black hole of mass M (≫µ). Using the generalized Regge-Wheeler equation, the formalism of computation of the energy of the wave going into the horizon, the energy of the ingoing wave on the horizon and the outgoing wave at infinity are given. It is found that if the energy of the particle at infinity (εµc2) is greater than 1.1µc2, the ingoing wave can excite the free oscillation of the black hole. For large values of ε (\gtrsim2), even if the periastron of the particle is located considerably outside the barrier of the Regge-Wheeler potential, the quasi-normal mode is excited. This is because high frequency waves are radiated inward that can get over the barrier.