Absorption of gravitational waves by a black hole

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 ]

where $&tidle;𝜖 = κ∕(4r+)$ , and

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 (19

) and (20

). Evaluation of the incident amplitude $trans B$ of the ingoing solution is essential in the calculation. Poisson and Sasaki [85

] 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

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

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 $k < 0$ .