Energy, Momentum and Angular Momentum of Gravitational Waves Induced by a Particle Plunging into a Schwarzschild Black Hole

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Abstract
Using the generalized Regge-Wheeler equation, we have computed the energy, the linear momentum and the angular momentum of gravitational waves generated by a particle of mass µ and orbital angular momentum µLz plunging into a Schwarzschild black hole of mass M(≫µ). It is found that the maximum value of the radiated linear momentum is 4.5 ×10-2(µ/ M)µc, which suggests the recoil velocity is 120 km/ sec for µ=M/ 10. Although the total energy (ΔE) and the angular momentum (ΔJ) radiated diverge in the limit of Lz →4(GM/ c2)c, the ratio ΔE/ ΔJ remains almost constant ∼0.15 (c/ (GM/ c2)) for Lz ≳(GM/ c2)c. We have also calculated the energy from a ring plunging into a black hole and found that a rotating one always emits less gravitational waves than a non-rotating one. This suggests the collapse of a rotating star may not be always effective for the generation of gravitational rediation.