More on the inner boundary condition of the outer solution

3.4 More on the inner boundary condition of the outer solution

In this section, we discuss the inner boundary condition of the outer solution in more detail. As we have seen in Section 3.3, the boundary condition on $ξℓ$ is that it is regular at $z → 0$ , at least to $𝒪(𝜖)$ , while in the full non-linear level, the horizon boundary is at $z = 𝜖$ . We therefore investigate to what order in $𝜖$ the condition of regularity at $z = 0$ can be applied.

Let us consider the general form of the horizon solution. With $x = 1 − z∕𝜖$ , it is expanded in the form

The lowest order solution ${0} ξℓ (x)$ is given by the polynomial (75

). Apart from the common overall factor, it is schematically expressed as

Thus, ${0} ξℓ$ does not have a term matched with $nℓ$ , but it matches with $jℓ$ . We have ${0} ℓ −ℓ −ℓ ξℓ = z 𝜖 ∼ 𝜖 jℓ$ . A term that matches with $nℓ$ first appears in ${1} ξℓ$ . This can be seen from the horizon solution valid to $𝒪 (𝜖)$ , Equation (79

). The second term in the square brackets of it produces a term $𝜖 (z ∕𝜖)− ℓ− 1 = 𝜖ℓ+2z− ℓ− 1 ∼ 𝜖ℓ+2n ℓ$ . This term therefore becomes $𝒪(𝜖2ℓ+2∕z2ℓ+1)$ higher than the lowest order term $− ℓ 𝜖 jℓ$ . Since $ℓ ≥ 2$ , this effect first appears at $6 𝒪 (𝜖)$ in the post-Minkowski expansion, while it first appears at $𝒪 (v13)$ in the post-Newtonian expansion if we note that $𝜖 = 𝒪(v3)$ and $z = 𝒪 (v)$ . This implies, in particular, that if we are interested in the gravitational waves emitted to infinity, we may simply impose the regularity at $z = 0$ as the inner boundary condition of the outer solution for the calculation up to 6PN order beyond the quadrupole formula.

The above fact that a non-trivial boundary condition due to the presence of the black hole horizon appears at $2ℓ+2 𝒪 (𝜖 )$ in the post-Minkowski expansion can be more easily seen as follows. Since $ℓ jℓ = 𝒪 (z)$ as $z → 0$ , we have $ℓ+1 −iz∗ X ℓ → 𝒪 (𝜖 )e$ , or $trans ℓ+1 A ℓ = 𝒪 (𝜖 )$ , where $z∗ = z + 𝜖ln(z − 𝜖)$ . On the other hand, from the asymptotic behavior of $jℓ$ at $z = ∞$ , the coefficients $Aiℓnc$ and $Arℓef$ must be of order unity. Then, using the Wronskian argument, we find

Thus, we immediately see that a non-trivial boundary condition appears at $𝒪 (𝜖2ℓ+2)$ .

It is also useful to keep in mind the above fact when we solve for $ξℓ$ under the post-Minkowski expansion. It implies that we may choose a phase such that $Ainc ℓ$ and $Aref ℓ$ are complex conjugate to each other, to $2ℓ+1 𝒪 (𝜖 )$ . With this choice, the imaginary part of $X ℓ$ , which reflects the boundary condition at the horizon, does not appear until $2ℓ+2 𝒪 (𝜖 )$ because the Regge–Wheeler equation is real. Then, recalling the relation of $ξℓ$ to $X ℓ$ , Equation (71 ), $Im (ξ(nℓ))$ for a given $n ≤ 2ℓ + 1$ is completely determined in terms of $(r) Re (ξℓ )$ for $r ≤ n − 1$ . That is, we may focus on solving only the real part of Equation (84 ).