Self-force and radiation reaction in general relativity

TL;DR

The paper surveys gravitational self-force (GSF) theory in curved spacetime, focusing on EMRIs and the perturbative framework built from matched asymptotic expansions that yield the MiSaTaQuWa equation and its Detweiler–Whiting reinterpretation. It reviews computational methods (mode-sum and puncture) and evolution formalisms (two-timescale and osculating geodesics) for incorporating the GSF into long-term orbital dynamics, including dissipative and conservative effects, resonances, and finite-size corrections. The review further explores synergies with PN theory, NR, and EOB modeling, highlighting how GSF data calibrate and inform universal two-body descriptions across all mass ratios. It concludes with open foundational and computational questions, emphasizing second-order GSF, gauge issues, and the practical infrastructure required to deliver accurate EMRI waveforms for LISA.

Abstract

[Abridged] This review surveys the theory of gravitational self-force in curved spacetime and its application to the gravitational two-body problem in the extreme-mass-ratio regime. We first lay the relevant formal foundation, describing the rigorous derivation of the equation of self-forced motion using matched asymptotic expansions and other ideas. We then review the progress that has been achieved in numerically calculating the self-force and its physical effects in the astrophysical scenario of a compact object inspiralling into a (rotating) massive black hole. We highlight the way in which, nowadays, self-force calculations make a fruitful contact with other approaches to the two-body problem and help inform an accurate universal model of binary black hole inspirals, valid across all mass ratios. We conclude with a summary of the state of the art, open problems and prospects. Our review is aimed at non-specialist readers and is for the most part self-contained and non-technical; only elementary-level acquaintance with General Relativity is assumed. Where useful, we draw on analogies with familiar concepts from Newtonian gravity or classical electrodynamics.

Figures (11)

• Figure 1: Relevant points for the retarded field $A^\mu_+$ (left), advanced field $A^\mu_-$ (middle), and singular and regular fields $A^\mu_{\rm S}$ and $A^\mu_{\rm R}$ (right) in flat spacetime. $A^\mu_+$ at the point $x^\mu$ depends on the state of the particle at the retarded point $z^\mu_{\rm ret}=z^\mu(\tau_{\rm ret})$, where the particle's worldline intersects $x^\mu$'s past light cone. $A^\mu_-$ at $x^\mu$ depends on the state of the particle at the advanced point $z^\mu_{\rm adv}=z^\mu(\tau_{\rm adv})$, where the particle's worldline intersects $x^\mu$'s future light cone. $A^\mu_{\rm S}$ and $A^\mu_{\rm R}$ each depend on the state of the particle at both $z^\mu_{\rm ret}$ and $z^\mu_{\rm adv}$.
• Figure 2: Relevant points for (left to right) the retarded field $A^\mu_+$, advanced field $A^\mu_-$, singular field $A^\mu_{\rm S}$, and regular field $A^\mu_{\rm R}$ in curved spacetime. $A^\mu_+$ at the point $x^\mu$ depends not just on the state of the particle at the retarded point on $x^\mu$'s past light cone, but also on the particle's state at all points within the past light cone. Analogously, $A^\mu_-$ depends on the state of the particle at all points on and within $x^\mu$'s future light cone. $A^\mu_{\rm S}$ depends on the state of the particle at all points on and outside$x^\mu$'s past and future light cones. $A^\mu_{\rm R}$ depends on the state of the particle at the advanced point $z^\mu_{\rm adv}=z^\mu(\tau_{\rm adv})$ and at all prior points $z^\mu(\tau<\tau_{\rm adv})$.
• Figure 3: Regions involved in matched asymptotic expansions, specialized to the case of an EMRI. The body zone corresponds to distances $r\sim m$ from the small object; the inner expansion, performed in the limit $\epsilon\to0$ at fixed $r/\epsilon$, is presumed to be accurate there. The external universe corresponds to distances $r\sim M$; the outer expansion, performed in the limit $\epsilon\to0$ at fixed $r$, is presumed to be accurate there. The buffer region corresponds to $m\ll r\ll M$, lying between the other two; the double expansions in the limits $\epsilon\to0$ and $r\to0$ are expected to be accurate there.
• Figure 4: Left: Typical geodesic orbit around a Kerr black hole. The orbit is 3-periodic, and it ergodically fills the interior of the outlined torus-shaped region. Right: The special case of a resonant orbit. Here the radial and longitudinal periods are in a $3\, {:}\, 2$ ratio, and the motion is no longer ergodic. Note how the orbit is instead confined to a certain 2-dimensional surface (topologically, a self-intersecting cylinder). The rightmost figure expands the central region of the resonant orbit, for clarity.
• Figure 5: The shape---and radiative dynamics---of a resonant orbit depend sensitively on the resonant phase $\psi^{\rm res}$, here defined as the value of the longitudinal angle $\theta$ at a periapsis of the resonant orbit. The two 3:2 resonant orbits viewed here, and the orbit shown on the right in Fig. \ref{['fig:orbit']}, all have the same parameters $\{E,\,L_z,\,Q\}$ but different resonant phases ($\psi^{\rm res}=20^\circ$ for the orbit in Fig. \ref{['fig:orbit']}, $\psi^{\rm res}=90^\circ$ for the orbit on the left here, and $\psi^{\rm res}=45^\circ$ for the orbit on the right here). The two orbits are viewed from just above the equatorial plane. Next to each orbit we display the corresponding Lissajous curve, showing the orbit in the plane of radius $r$ (horizontal axis, in units of $M$) and inclination angle (vertical axis, measured from the equator in degrees).