Introduction

Gravitational Wave Physics and Astronomy

The detection of gravitational waves by LIGO and Virgo has opened a new window onto the Universe! This entirely new tool provides us not only with a direct experimental test of Einstein's general relativity, but also with a new approach to observing the Universe. However, because gravitational-wave signals are extremely weak, predicting theoretical waveforms in advance is mandatory. Once these waveforms are known, matched filtering techniques can be used to facilitate signal extraction and interpretation in current and planned gravitational-wave detectors.

Coalescing Compact Object Binaries and Extreme Mass-Ratio Inspirals: EMRIs

When considering gravitational-wave observatories on Earth and in space, the coalescence of compact binary systems is among the most important phenomena to observe. This merger process can be divided into three stages: inspiral, merger, and ringdown. During the inspiral stage, the binary's orbital separation gradually decreases as it emits gravitational waves. In the merger phase, the two compact objects coalesce to form a highly distorted remnant, which eventually settles into a stationary Kerr black hole during the ringdown phase.

Extreme mass-ratio inspirals (EMRIs) consist of a stellar-mass compact object captured by a supermassive black hole in a galactic nucleus. EMRIs are dominated by the inspiral phase, during which they complete a vast number of orbital cycles dependent on the mass ratio. Consequently, highly accurate and computationally efficient waveform models are essential to track their long-term evolution. 

Black Hole Perturbation: BHP and Gravitational Self Force: GSF

Given the disparate mass scales, black hole perturbation theory is a powerful tool for modeling EMRIs. Gravitational perturbations of a Schwarzschild black hole were first studied by Regge and Wheeler over 60 years ago, and subsequently developed by Zerilli. Later, a master equation for perturbations of a Kerr black hole was derived by Teukolsky. An elegant presentation of this subject can also be found in the classic text The Mathematical Theory of Black Holes by Chandrasekhar. 

In the language of the general relativistic two-body problem, an EMRI describes the dynamics of a relatively small object in the strongly curved spacetime of a massive, rotating (Kerr) black hole. As such, we can apply black hole perturbation theory based on an expansion in the small mass ratio. At "zeroth order", the motion is geodesic on the background spacetime. At "first order", the small body induces a perturbation that backreacts on itself via a "gravitational self-force", driving a gradual inspiral and generating gravitational waves. This expansion must be carried to at least "second order" in order for waveform models to accurately extract EMRI parameters.

Gravitational self-force theory was first formulated by some of BHPC members: Mino, Sasaki and Tanaka in 1996. (Quinn and Wald independently derived it around the same time; hence, the first-order equations of motion are referred to as the MiSaTaQuWa equation).  Since then, remarkable progress has been made in our understanding of self-force physics. For a thorough entry point to the field and its state of the art, readers are invited to consult the review by, e.g., Barack and Pound, as well as the CAPRA community website.