Analytic treatment of black-hole gravitational waves at the algebraically special frequency

Physical Review D

Volumn 62, Issue 6, 2000, Pages 0640091-06400916

  Van Brink, Alec Maassen Den ^a,b  

^a CHINESE UNIVERSITY OF HONG KONG   (Hong Kong)

^b INSTITUTE OF THEORETICAL PHYSICS   (China)

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EID: 4244189313     PISSN: 05562821     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article

References (54)

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4. For consistency with the general treatment of Eq. (1.1), we use an observer-centered definition of "outgoing into the horizon"; however, several other authors use a hole-centered definition "outgoing from the horizon," in which case on the left the notions of outgoing and incoming are reversed. The last remaining ambiguity, arising since Eq. (1.1) involves or only and since "outgoing" at to obviously is equivalent to "incoming" at -a) (see further Sec. II A), is resolved in the standard manner by choosing Imco0 (modes with real o> will not occur).
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10. Thus, the tilde notation is defined differently for the generator than for other functions. Below, / and g will be defined differently again, namely, as the normalized outgoing waves in the partner according to Eq. (2.1)
11. However, no confusion arises.
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16. At least the possibility that f, could itself be a QNM can be rejected even without considering its concrete form (1.5)(1.7): fi being an "ordinary" QNM, i.e., one not coinciding with other types of modes and hence increasing at both ends, translates into f, being an NM. However, below Eq. (1.4) this has been ruled out already.
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22. In fact, whether the phenomenon of higher-order modes is restricted to the NIA is still an open question.
23. Since g(x, -ca) cannot have a real zero for a purely repulsive potential, in this case-which includes Eqs. (1.3) and (1.4)the definition a(ia) = Sg(x,(a)fg(x,- (a) is not only correct for generic .t, but in fact for all real x. Compare with an analogous situation in [20].
24. v, individually; the result (6.12) is then found to be correct.
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26. A fourth case, the odd man out w = 0, is known to be exceptional [9] but does not concern us here.
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30. One can express the sum for y3 in terms of the incomplete gamma function γ(N- 1,N); however, doing so has no advantage here.
31. Computer algebra should also be useful to calculate the Leaver series [14] for/fan expansion in (r - 1)/r], which converges more rapidly than the Born series, and for all 1rε Namely, for rational r and rational-complex ω one can then use exact integer arithmetic at least for the first few terms, which largely cancel [44].
32. 2(0, which indeed has a logarithmic branch point at r= 1, the integrand having the same nonzero residue there which caused 73 =£0 in Eq. (3.1). With the general solution of the ZE at ft" thus at one's disposal, it is straightforward to study, e.g., its additional singularity at 2nr+3 = 0. However, this will not be taken up here.
33. V(x e R) e R, so the range must be symmetric in arg.v.
34. 3iπ4|x| is unusable not because of its asymptotic part, but because it passes x=iπ on the wrong side.
35. In contrast, V(x→-ε) only has β= π/2, so that we could not have determined /(fi) by continuation nor, of course, ruled out anomalous points for/on the NIA.
36. -N/2 be analytic near r=ε [cf. Eq. (6.1)], comparison with Eq. (5.1) at once reproduces γ = 75. The criterion is thus seen to be a shorthand for the asymptotic matching of the main text.
37. g(ω is found exactly, not the whole function.
38. ζ, g(xΩ) has a single real zero, so that the Ansatz leads to a real pole in h(x,Ω). However, this will cause no problems.
39. 1,(ω) = 0, consistent with /I=2.
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41. Curiously, thus the truncated series does not involve U(3,5,z) and U(4,5,z) although these are single-valued as well. The changes in character of U occur at the zeros of the third coefficient in U(k- 1,5,z)-(2A- 5 + z)U(k,5,z) + k(k-4) X U(k+ l,5,z) = 0, a relation which can also be used when doing the sum in Eq. (6.4) by induction, as outlined in the main text.
42. K.W. Mak and K. Young (in progress).
43. 2, since S in general is not real unless a) is. Likewise, Ref. [13] uses "complex conjugated" for some object pairs which are only so for real ω.
44. 2 our Eq. (7.21) is equivalent to Seidel's more complicated Eq. (8).
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53. K. Young (private communication).
54. x) are different.