Analytic black hole perturbation approach to gravitational radiation

TL;DR

This paper surveys analytic methods to perform post-Newtonian expansions of gravitational waves from a small mass in orbit around a black hole, within the black-hole perturbation framework. It contrasts two approaches based on the Teukolsky equation: (i) transforming to a Regge-Wheeler–type equation for easier physical interpretation, and (ii) the Mano–Suzuki–Takasugi (MST) method that directly solves the Teukolsky equation in a systematic, high-order manner. The authors detail how horizon and near-zone effects are treated, provide explicit PN results for a variety of orbital configurations (circular, eccentric, inclined) around Schwarzschild and Kerr black holes, and quantify horizon absorption (including superradiant behavior). Collectively, the work offers a rigorous analytic toolkit for building accurate gravitational-wave templates in regimes where full numerical relativity is challenging, with clear implications for detectors like LIGO, VIRGO, GEO600, TAMA, and especially LISA. The methodologies enable high-precision waveform modeling and deepen understanding of curvature and horizon effects in strong gravity.

Abstract

We review analytic methods to perform the post-Newtonian expansion of gravitational waves induced by a particle orbiting a massive compact body, based on the black hole perturbation theory. There exist two different methods of the post-Newtonian expansion. Both are based 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 were introduced by Mano, Suzuki and Takasugi relatively recently, the Teukolsky equation is directly used in its original form. The former has an advantage that it is intuitively easy to understand how various curved space effects come into play. However, it becomes increasingly complicated when one goes on to higher and higher post-Newtonian orders. In contrast, the latter has an advantage that a systematic calculation to higher post-Newtonian orders is relatively easily implementable, but otherwise 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 clearly seen. We also review some results of calculations of gravitational radiation emitted by a particle orbiting a black hole.