Analytic Black Hole Perturbation Approach to Gravitational Radiation

Abstract

We review the analytic methods used to perform the post-Newtonian expansion of gravitational waves induced by a particle orbiting a massive, compact body, based on black hole perturbation theory. There exist two different methods of performing the post-Newtonian expansion. Both are based on the Teukolsky equation. In one method, the Teukolsky equation is transformed into a Regge-Wheeler type equation that reduces to the standard Klein Gordon equation in the flat-space limit, while in the other method (which was introduced by Mano, Suzuki, and Takasugi relatively recently, the Teukolsky equation is used directly in its original form. The former’s advantage is that it is intuitively easy to understand how various curved space effects come into play. However, it becomes increasingly complicated when one goes to higher and higher post-Newtonian orders. In contrast, the latter’s advantage is that a systematic calculation to higher post-Newtonian orders can be implemented relatively easily, but otherwise, it is so mathematical that it is hard to understand the interplay of higher order terms. In this paper, we review both methods so that their pros and cons may be seen clearly. We also review some results of calculations of gravitational radiation emitted by a particle orbiting a black hole.

Experimental mathematics meets gravitational self-force

“…Here [Eqs. (157), (158), (168), and (170) in Sasaki and Tagoshi [56], noting that κ = 1 for Schwarzschild] The K −ν−1 R −ν−1 C /(K ν R ν C ) term is likely the origin of the e 2ν ℓm log(2/R) contribution to the Υ C2 ℓm simplification (just as it is for the V ℓm simplification in [38]), since K −ν−1 R −ν−1 C /(K ν R ν C ) ∼ (2v 2 ) 2ν {gamma function terms} (cf. Eqs.…”

“…where ν is the renormalized angular momentum introduced in the MST formalism [54,56]. (Here we denote its dependence on ℓ and m explicitly, which is usually not done in the literature, though we suppress its dependence on R, even though we displayed the analogous dependence on v in [38].)…”

Sources of Gravitational Waves: Theory and Observations

“…The reader is referred to the original articles, reviews [72][73][74][75][76][77] and books [78][79][80][81] for more details.…”

The Confrontation Between General Relativity and Experiment

“…For non-rotating holes, the analytical expansions have been carried to the impressive level of 22PN order, or 22 beyond the quadrupole approximation [168], and for rotating Kerr black holes, to 20PN order [356]. All results of black hole perturbation agree precisely with the 1 → 0 limit of the PN results, up to the highest PN order where they can be compared (for reviews of earlier work see [285,219,351]). …”

“…This field of "numerical relativity" has become a mature branch of gravitational physics, whose description is beyond the scope of this review (see [247,176,35] for reviews). Another is black-hole perturbation theory (see [285,219,351,43] for reviews).…”