5.7 Absorption of gravitational waves by a black hole

In this section, we evaluate the energy absorption rate by a black hole. The energy flux formula is given by [107]
( dE ) ∑ ∫ S2 128ωk (k2 + 4&tidle;𝜖2)(k2 + 16 &tidle;𝜖2)(2M r )5 ---hole = dω 2-ℓm-------------------2-------------+--|&tidle;ZHℓmω|2, (204 ) dt dΩ â„“m 2π |C |
where &tidle;𝜖 = κ∕(4r+), and
( ) ( ) |C |2 = (λ + 2)2 + 4a ωm − 4a2ω2 λ2 + 36aωm − 36a2 ω2 2 2 2 2 2 +(2λ + 3 )(96a ω − 48aωm ) + 144 ω (M − a ). (205 )
In calculating Z&tidle;Hℓmω, we need to evaluate the upgoing solution Rup, and the asymptotic amplitude of ingoing and upgoing solutions, Binc, Btrans, and Ctrans in Equations (19View Equation) and (20View Equation). Evaluation of the incident amplitude trans B of the ingoing solution is essential in the calculation. Poisson and Sasaki [85Jump To The Next Citation Point] evaluated them, in the case of a circular orbit around the Schwarzschild black hole, up to 𝒪(𝜖) beyond the lowest order, and obtained the energy flux at the lowest order, using the method we have described in Section 3. Later, Tagoshi, Mano, and Takasugi [98] evaluated the energy absorption rate in the Kerr case to 𝒪 (v8) beyond the lowest order using the method in Section 4. Since the resulting formula is very long and complicated, we show it here only to 𝒪 (v3 ) beyond the lowest order. The energy absorption rate is given by
( ) ( )2 [ ( ) dE- = 32- -μ- x10x5 − 1q − 3-q3 + − q − 33q3 x2 dt H 5 M 4 4 16 ( 1 13 35 1 1 ) ] + 2qB2 + --+ --κq2 + ---q2 − -q4 + --κ + 3q4κ + 6q3B2 x3 , 2 2 6 4 2 (206 )
where
[ ( ) ( ) ] -1 (0) --niq---- (0) --niq---- Bn = 2i ψ 3 + ∘1--−-q2 − ψ 3 − ∘1--−-q2 , (207 )
and ψ(n)(z) is the polygamma function. We see that the absorption effect begins at 𝒪(v5) beyond the quadrupole formula in the case q ⁄= 0. If we set q = 0 in the above formula, we have
( ) ( ) dE dE ( 8 10 ) dt- = dt- x + 𝒪 (v ) , (208 ) H N
which was obtained by Poisson and Sasaki [85].

We note that the leading terms in (dE ∕dt)H are negative for q > 0, i.e., the black hole loses energy if the particle is co-rotating. This is because of the superradiance for modes with k < 0.


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