Introduction

Gravitational Wave Physics and Astronomy

The gravitational wave detection by LIGO and Virgo opens a new window on the Universe! This entirely new tool provides us with not only a direct experimental test of Einstein's general relativity but also a new approach to observe our universe. However, because gravitational wave signals are very weak, so to use them as a new tool of observation, it is mandatory to predict the theoretical waveforms in advance . Once we know them, we may appeal to the matched filtering technique in order to facilitate signal extraction and interpretation in current and  planned gravitational-wave detectors.

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

When we look at gravitational wave observatories on Earth and in space, the merger of two compact object binaries is one of the most important things to watch out for. This process of merging can be broken down into three stages: inspiral, merger and ringdown. During the inspiral stage, the binary's orbital motion gradually reduces the separation between the two compact objects, as they emit gravitational waves. Then, in the merger phase, the two compact objects come together to form a highly distorted compact object. If the remnant object is a black hole, the merged black hole eventually becomes a statinary Kerr black hole in the ringdown phase.

Extreme mass-ratio inspirals (EMRIs) are binaries composed of a neutron star or a stellar mass black hole captured by a much larger black hole in a galactic core. EMRIs are all about the inspiral phase, which is where they build up a huge number of cycles (in a way that depends on the ratio of the two black holes' masses). This means we need really accurate waveform models that are both thorough and efficient to understand how things change over time. 

Black Hole Perturbation: BHP and Gravitational Self Force: GSF

Given the disparate mass scales, the black hole perturbation is a powerful tool for modelling EMRIs. Gravitational perturbations of a Schwarzschild black hole were first studied by Regge and Wheeler about 60 year ago, and then 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 be also found in the classic The Mathematical Theory of Black Holes by Chandrasekhar. 

In the language of general relativistic 2-body problem, the EMRI is dynamics of relatively small objects in a strongly curved spacetimes of the large rotating (Kerr-type) black hole. As such, we can apply the black hole perturbation to it, based on an expansion in the small mass ratio. At the "zeroth-order", it is geodesic motion on a background spacetime of the large black hole (no gravitational-wave emission!).  At the next "first-order", a small body creates a perturbation on that background. The perturbation back-reacts on the body via a "gravitational self force" that drives the slow inspiral. In turn, the perturbation generated by the accelerated orbit propagates to observatories as a gravitational wave signal. This expansion can be continued until the desired precision goal is achieved, and it is now known that one must carry the expansion to at least "second-order" for the waveform modeling to accurately extract the EMRI parameters.

The gravitational self-force theory is  presented by some of B.H.P.C. members: Mino, Sasaki and Tanaka for the first time in 1996. (Quinn and Wald also derived it independently nearly the same time. This is why the first-order, gravitational-self-force equation of motion is refered as MiSaTaQuWa eq.).  Since then, there has been a remarkable progress in our understanding of the self-force. The self-force theory, including its state of the art, is described much more thoroughly in reviews. As an entry point to this field, the reader is invited to consult the recent review by, e.g., Barack and Pound as well as to visit (so named) CAPRA community.