3.1 Basic assumptions

We consider the case of a test particle with mass μ in a nearly circular orbit around a black hole with mass M ≫ μ. For a nearly circular orbit, say at r ∼ r 0, what we need to know is the behavior of in R ℓmω at r ∼ r0. In addition, the contribution of ω to in R ℓm ω comes mainly from ω ∼ m Ω φ, where 3 1∕2 Ω φ ∼ (M ∕r0) is the orbital angular frequency.

Thus, if we express the Regge–Wheeler equation (63View Equation) in terms of a non-dimensional variable z ≡ ωr, with a non-dimensional parameter 𝜖 ≡ 2M ω, we are interested in the behavior of Xin (z) ℓm ω at z ∼ ωr ∼ m (M ∕r )1∕2 ∼ v 0 0 with 𝜖 ∼ 2m (M ∕r )3∕2 0 ∼ v3, where v ≡ (M ∕r )1∕2 0 is the characteristic orbital velocity. The post-Newtonian expansion assumes that v is much smaller than the velocity of light: v ≪ 1. Consequently, we have 𝜖 ≪ v ≪ 1 in the post-Newtonian expansion.

To obtain Xin ℓm ω (which we denote below by X ℓ for simplicity) under these assumptions, we find it convenient to rewrite the Regge–Wheeler equation in an alternative form. It is

[ d2 2 d ( ℓ(ℓ + 1)) ] d [ 1 d ( )] --2-+ -----+ 1 − ----2--- ξℓ(z) = 𝜖e− iz--- -3--- eizz2ξℓ(z) , (70 ) dz z dz z dz z dz
where ξℓ is a function related to Xℓ as
−i𝜖ln(z−𝜖) X ℓ = ze ξℓ. (71 )
The ingoing wave boundary condition of ξℓ is derived from Equations (65View Equation) and (71View Equation) as
{ Ainℓcei𝜖ln𝜖z− 1e −iz + Areℓfe−i𝜖ln𝜖z−1ei(z+2𝜖lnz) for r∗ → ∞, ξℓ → trans −1 i𝜖(ln𝜖−1) ∗ (72 ) A ℓ 𝜖 e for r → − ∞.
The above form of the Regge–Wheeler equation is used in Sections 3.2, 3.3, 3.4, and 3.5.

It should be noted that if we reinstate the gravitational constant G, we have 𝜖 = 2GM ω. Thus, the expansion in terms of 𝜖 corresponds to the post-Minkowski expansion, and expanding the Regge–Wheeler equation with the assumption 𝜖 ≪ 1 gives a set of iterative wave equations on the flat spacetime background. One of the most significant differences between the black hole perturbation theory and any theory based on the flat spacetime background is the presence of the black hole horizon in the former case. Thus, if we naively expand the Regge–Wheeler equation with respect to 𝜖, the horizon boundary condition becomes unclear, since there is no horizon on the flat spacetime. To establish the boundary condition at the horizon, we need to treat the Regge–Wheeler equation near the horizon separately. We thus have to find a solution near the horizon, and the solution obtained by the post-Minkowski expansion must be matched with it in the region where both solutions are valid.

It may be of interest to note the difference between the matching used in the BDI approach for the post-Newtonian expansion [7, 12] and the matching used here. In the BDI approach, the matching is done between the post-Minkowskian metric and the near-zone post-Newtonian metric. In our case, the matching is done between the post-Minkowskian gravitational field and the gravitational field near the black hole horizon.


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