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 ξβ„“ is that it is regular at z → 0, at least to π’ͺ(πœ–), while in the full non-linear level, the horizon boundary is at z = πœ–. We therefore investigate to what order in πœ– the condition of regularity at z = 0 can be applied.

Let us consider the general form of the horizon solution. With x = 1 − zβˆ•πœ–, it is expanded in the form

ξβ„“ = ξ{β„“0}(x) + πœ– ξ{β„“1}(x ) + πœ–2ξ{β„“2}(x) + ... (91 )
The lowest order solution {0} ξβ„“ (x) is given by the polynomial (75View Equation). Apart from the common overall factor, it is schematically expressed as
( )β„“[ ( )β„“−2] ξ{0}= z- 1 + c1πœ–-+ ...+ cβ„“− 2 πœ–- . (92 ) β„“ πœ– z z
Thus, {0} ξβ„“ does not have a term matched with nβ„“, but it matches with jβ„“. We have {0} β„“ −β„“ −β„“ ξβ„“ = z πœ– ∼ πœ– jβ„“. A term that matches with nβ„“ first appears in {1} ξβ„“. This can be seen from the horizon solution valid to π’ͺ (πœ–), Equation (79View Equation). The second term in the square brackets of it produces a term πœ– (z βˆ•πœ–)− β„“− 1 = πœ–β„“+2z− β„“− 1 ∼ πœ–β„“+2n β„“. This term therefore becomes π’ͺ(πœ–2β„“+2βˆ•z2β„“+1) higher than the lowest order term − β„“ πœ– jβ„“. Since β„“ ≥ 2, this effect first appears at 6 π’ͺ (πœ–) in the post-Minkowski expansion, while it first appears at π’ͺ (v13) in the post-Newtonian expansion if we note that πœ– = π’ͺ(v3) and z = π’ͺ (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 2β„“+2 π’ͺ (πœ– ) in the post-Minkowski expansion can be more easily seen as follows. Since β„“ jβ„“ = π’ͺ (z) as z → 0, we have β„“+1 −iz∗ X β„“ → π’ͺ (πœ– )e, or trans β„“+1 A β„“ = π’ͺ (πœ– ), where z∗ = z + πœ–ln(z − πœ–). On the other hand, from the asymptotic behavior of jβ„“ at z = ∞, the coefficients Aiβ„“nc and Arβ„“ef must be of order unity. Then, using the Wronskian argument, we find

inc ref --|Atrβ„“ans|2--- 2β„“+2 |A β„“ | − |A β„“ | = |Ainc| + |Aref| = π’ͺ (πœ– ). (93 ) β„“ β„“
Thus, we immediately see that a non-trivial boundary condition appears at π’ͺ (πœ–2β„“+2).

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


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