* There is already an article named Gravitational self-force in the mainspace. This is a draft article. It is a work in progress open to editing by anyone. Please ensure core content policies are met before publishing it as a live Wikipedia article. Find sources: Google (books · news · scholar · free images · WP refs) · FENS · JSTOR · TWL Easy tools: Citation bot (help) | Advanced: Fix bare URLs · Article logs · Draft logs. Last edited by Fortek67 (talk | contribs) 15 hours ago. (Update) Finished drafting? Submit for review or Publish now

This article may incorporate text from a large language model, which is prohibited in Wikipedia articles. It may include hallucinated information, copyright violations, claims not verified in cited sources, original research, or fictitious references. Any such material should be removed. (June 2026) (Learn how and when to remove this message)

In gravitational wave physics, the gravitational self-force (also known as gravitational radiation reaction) is the correction to the motion of a small body in general relativity arising from the back-reaction of its own gravitational field. A compact body moving through the curved spacetime generated by a much larger primary mass perturbs that spacetime; those metric perturbations in turn exert a force on the body, causing its trajectory to deviate from a geodesic of the background spacetime. The gravitational self-force formalism provides a systematic method for solving the relativistic two-body problem in the extreme-mass-ratio regime by expanding the dynamics in powers of the mass ratio ${\displaystyle q=m_{2}/m_{1}\ll 1}$. It is currently regarded as the primary theoretical tool for computing accurate gravitational wave templates for extreme mass-ratio inspirals (EMRIs), which are a key science target of the space-based detector LISA.^[1][2]

The concept of radiation reaction—the back-reaction of a body's own emitted field on its motion—was first made quantitatively precise in classical electrodynamics. Paul Dirac (1938) derived the relativistic equation of motion for a radiating point electron in flat spacetime, yielding the Abraham–Lorentz–Dirac equation.^[3] Although Dirac's derivation produced a finite self-force, the resulting equation is third-order in time and admits unphysical "runaway" self-accelerating solutions.

The generalization to curved spacetime was accomplished by Bryce DeWitt and Robert Brehme (1960), who derived the equation of motion for an electrically charged point particle in an arbitrary curved background.^[4] Their result introduced a feature absent from flat spacetime: a tail term, representing radiation that backscatters off the background spacetime curvature and returns to the particle at later times, making the self-force sensitive to the entire past history of the particle's worldline. A correction to the DeWitt–Brehme result was later supplied by Hobbs (1968), who clarified the contribution of the Ricci curvature to the equations of motion.^[5]

The gravitational analogue—a first-order equation of motion for a small body including the leading correction from its own gravitational field—was derived independently by two groups in 1997. Mino, Sasaki, and Tanaka used a method extending the DeWitt–Brehme formalism to the gravitational case, constructing a conserved rank-two symmetric tensor and integrating over a world-tube surrounding the particle's worldline to extract the finite self-force.^[6] Quinn and Wald independently derived the same result using an axiomatic approach: the comparison axiom states that particles with equal acceleration magnitude but different charge-to-mass ratios experience self-force differences determined by ordinary force laws acting on their field differences.^[7] Both derivations showed that the deviation from geodesic motion arises entirely from a tail term. The combined result is known as the MiSaTaQuWa equation, after the initials of the five authors, and forms the foundation of modern gravitational self-force theory.^[8]

A rigorous derivation of the MiSaTaQuWa equation using the method of matched asymptotic expansions was later provided by Gralla and Wald (2008), using a one-parameter family of metrics in which the secondary body is scaled to zero size.^[9]

The primary computational difficulty is that the retarded metric perturbation produced by a point particle diverges at the particle's own location, making a direct evaluation of the self-force impossible. Barack and Ori (2000) introduced the mode-sum regularization method to overcome this obstacle: the retarded field is decomposed into spherical-harmonic modes, each of which is individually finite at the particle; analytically computed regularization parameters are subtracted from each mode before the sum is taken.^[10] Barack (2001) extended the procedure to the gravitational case.^[11] Barack and Ori (2003) subsequently derived the regularization parameters required for generic bound geodesics in Kerr spacetime, making practical self-force calculations in Kerr possible for the first time.^[12]

A complementary approach was introduced by Steven Detweiler and Whiting (2003), who showed that the full retarded perturbation can be split into a singular part—which has the same local divergence as the retarded field but exerts no force on the particle—and a smooth regular part that is entirely responsible for the physical self-force.^[13] This Detweiler–Whiting decomposition provided both a cleaner conceptual framework and an alternative numerical strategy—the puncture (or effective source) method.

The first full numerical implementation of the gravitational self-force in Lorenz gauge was achieved by Barack and Lousto (2005), who formulated a set of wavelike equations and a time-domain characteristic numerical scheme for a particle in circular orbit around a Schwarzschild black hole.^[14] Barack and Sago (2007) carried out the first calculation of both the dissipative and conservative components of the gravitational self-force for circular Schwarzschild orbits, validating the energy-dissipation component against the gravitational-wave energy flux computed independently, and determining the conservative correction to the orbital frequency for the first time.^[15]

Detweiler (2008) introduced the gauge-invariant redshift observable ${\displaystyle u^{t}}$—the contravariant time-component of the four-velocity of a particle on a circular Schwarzschild orbit—and showed that this conservative self-force quantity agrees with post-Newtonian predictions at sixth order in velocity.^[16] Barack and Sago (2010) extended the Lorenz-gauge time-domain calculation to eccentric orbits around a Schwarzschild black hole.^[17] Van de Meent (2018) completed the program for Schwarzschild with the first calculation of the gravitational self-force on generic bound geodesics—inclined and eccentric—in Kerr spacetime, using a frequency-domain method based on reconstruction of the metric perturbation from the Weyl scalar.^[18]

Template-quality waveforms for LISA require the gravitational self-force to be known through second order in the mass ratio. Pound (2012) derived the second-order gravitational self-force equation using matched asymptotic expansions, demonstrating that a compact secondary still moves on a geodesic of a locally defined smooth effective metric satisfying the Einstein equations at second order in ${\displaystyle q}$.^[19] An extension to gauge choices suited to numerical computation was given in a subsequent paper.^[20] Wardell, Pound, Warburton, Miller, Durkan, and Le Tiec (2023) produced the first gravitational waveforms from a complete second-order self-force calculation, using a two-timescale expansion of the Einstein equations. The resulting waveforms agree remarkably well with those from numerical relativity even for comparable-mass binaries such as ${\displaystyle q=1/10}$.^[21]

The gravitational self-force formalism addresses the regime of the relativistic two-body problem where one body (called the primary) is much more massive than the other body (the secondary). It aims to solve the dynamics as a systematic expansion in powers of the mass ratio ${\displaystyle q=m_{2}/m_{1}}$. In typical situations both bodies are assumed to be black holes, but this assumption is not strictly necessary and most results hold for general bodies, as long as the secondary is compact, meaning that the length scales associated with the secondary are proportional to ${\displaystyle Gm_{2}/c^{2}}$.^[8] The gravitational self-force formalism exploits the hierarchy of length scales in these systems: the typical length scale of the secondary is much smaller than the curvature length scale associated with the primary. As a consequence the metric "far" away from the secondary can be described as a perturbation of the metric that would have been produced by the primary alone:^[8][22][23]

${\displaystyle g_{\mu \nu }^{\rm {far}}=g_{\mu \nu }^{\rm {primary}}+qh_{\mu \nu }^{(1)}+q^{2}h_{\mu \nu }^{(2)}+{\mathcal {O}}(q^{3}).}$

At the same time, the equivalence principle implies that in some sufficiently small neighborhood "near" the secondary the metric is described as

${\displaystyle g_{\mu \nu }^{\rm {near}}=g_{\mu \nu }^{\rm {secondary}}+qH_{\mu \nu }^{(1)}+q^{2}H_{\mu \nu }^{(2)}+{\mathcal {O}}(q^{3}).}$

Taking general solutions in each regime and matching them in an intermediate region using the method of matched asymptotic expansions leads to an effective description of the binary in terms of an effective spacetime

${\displaystyle g_{\mu \nu }^{\rm {eff}}=g_{\mu \nu }^{\rm {primary}}+qh_{\mu \nu }^{R1}+q^{2}h_{\mu \nu }^{R2}+{\mathcal {O}}(q^{3}),}$

and the motion of the secondary is represented as a worldline in this spacetime. If the secondary is spherically symmetric (e.g. a Schwarzschild black hole) then its worldline will be a geodesic in the effective spacetime. Or equivalently, expressed relative to the background spacetime ${\displaystyle g_{\mu \nu }^{\rm {primary}}}$, the 4-velocity of the worldline ${\displaystyle u^{\mu }}$ satisfies

${\displaystyle u^{\nu }\nabla _{\nu }u^{\mu }=0+qF_{(1)}^{\mu }[h_{\mu \nu }^{R1}]+q^{2}F_{(2)}^{\mu }[h_{\mu \nu }^{R1},h_{\mu \nu }^{R2}]+{\mathcal {O}}(q^{3}),}$

where the left-hand side is the geodesic equation in the background spacetime, and the right-hand side represents an effective force term correcting the motion—the gravitational self-force. Specifically, ${\displaystyle F_{(1)}^{\mu }}$ is known as the first-order gravitational self-force and ${\displaystyle F_{(2)}^{\mu }}$ as the second-order gravitational self-force.

If the secondary is not spherically symmetric, its compactness implies that the above result is modified by the multipole moments of the secondary's gravitational field, with higher-order multipole moments entering at higher order in the mass ratio. At zeroth order in ${\displaystyle q}$, the worldline is still a geodesic of the background spacetime. At linear order, the motion is corrected by a force term from the current-dipole moment (the spin) of the secondary coupling to the background curvature—the linear Mathisson–Papapetrou–Dixon force—while the first-order effective metric perturbation ${\displaystyle h_{\mu \nu }^{R1}}$ depends only on the monopole moment (the mass) of the secondary. At second order, further corrections appear from the quadratic Mathisson–Papapetrou–Dixon force, a coupling of the secondary's quadrupole moment to the background curvature, and a spin-sourced contribution to ${\displaystyle h_{\mu \nu }^{R2}}$.^[8]

The retarded metric perturbation ${\displaystyle h_{\mu \nu }^{\rm {ret}}}$ produced by a point particle diverges as ${\displaystyle 1/\epsilon }$, where ${\displaystyle \epsilon }$ is the geodesic distance from the field point to the particle, and therefore cannot be directly evaluated at the particle's location. In the mode-sum method, the retarded field is expanded in scalar (or tensor) spherical harmonics labelled by the multipole index ${\displaystyle \ell }$. Each individual ${\displaystyle \ell }$-mode contribution to the self-force is finite at the particle, even though the sum over all modes diverges. The divergence is cancelled by subtracting from each mode analytically derived regularization parameters ${\displaystyle A_{\alpha }}$, ${\displaystyle B_{\alpha }}$, ${\displaystyle C_{\alpha }}$, ... that encode the local singular structure of the Detweiler–Whiting S-field in the mode basis.^[10][11] After mode-by-mode subtraction, the sum converges and yields the physical self-force. The regularization parameters depend on the geodesic and the background geometry but are independent of the specific perturbation and can be computed analytically for Schwarzschild^[11] and Kerr spacetimes.^[12][8]

The Detweiler–Whiting decomposition provides an alternative regularization framework by splitting the full retarded perturbation as^[13]

${\displaystyle h_{\mu \nu }^{\rm {ret}}=h_{\mu \nu }^{\rm {S}}+h_{\mu \nu }^{\rm {R}}.}$

The singular field ${\displaystyle h^{\rm {S}}}$ is a particular solution of the inhomogeneous perturbation equations constructed to reproduce the local divergent behavior of the retarded field near the particle; by construction, it exerts no force on the particle. The regular field ${\displaystyle h^{\rm {R}}=h^{\rm {ret}}-h^{\rm {S}}}$ is a smooth, homogeneous (source-free) solution of the linearized Einstein equations throughout a neighborhood of the particle and drives the entire self-force:

${\displaystyle F_{\rm {self}}^{\mu }=F^{\mu }[h^{\rm {R}}].}$

Because ${\displaystyle h^{\rm {R}}}$ is smooth at the particle, the self-force can be evaluated from it without further regularization. In numerical practice this leads to the puncture method (or effective source method): the particle source is analytically split into the known singular piece and a smooth effective source, and only the smooth part is evolved numerically, avoiding the singularity altogether.^[8]

Self-force calculations employ two principal families of numerical methods:

Time-domain methods

The perturbation equations—typically cast in the Lorenz gauge^[14]—are evolved as hyperbolic partial differential equations in time and radius on a characteristic or hyperboloidal grid. Time-domain codes handle eccentric and inclined orbits in a unified framework and are naturally adapted to the puncture method. They are computationally expensive, however, particularly at small orbital radii.

Frequency-domain methods

The orbital motion is decomposed into a discrete set of Fourier frequencies, reducing the perturbation equations to systems of ordinary differential equations for each frequency mode. The solutions can be obtained with high numerical precision. Frequency-domain codes achieve roughly three orders of magnitude improvement in speed over time-domain codes for circular orbits.^[24] Van de Meent (2018) used a frequency-domain approach based on the Teukolsky formalism with metric reconstruction to compute the first gravitational self-force on generic Kerr geodesics.^[18]

For long-duration inspiral waveforms, both approaches are coupled with a two-timescale (multiscale) expansion that formally separates the short orbital timescale ${\displaystyle M}$ from the long radiation-reaction timescale ${\displaystyle M/q}$. At leading (adiabatic) order the inspiral is driven by the orbit-averaged dissipative self-force; post-adiabatic corrections at order ${\displaystyle q}$ require the full first-order conservative self-force and the leading second-order self-force.^[25][26]

An extreme mass-ratio inspiral consists of a stellar-mass compact object—a black hole, neutron star, or white dwarf—with mass ${\displaystyle m\sim 1}$–${\displaystyle 100\,M_{\odot }}$ spiraling into a massive black hole of mass ${\displaystyle M\sim 10^{4}}$–${\displaystyle 10^{7}\,M_{\odot }}$, yielding a mass ratio ${\displaystyle q\sim 10^{-4}}$–${\displaystyle 10^{-7}}$. The inspiral proceeds through roughly ${\displaystyle 10^{4}}$–${\displaystyle 10^{5}}$ cycles in the strong-gravity regime close to the massive black hole, before the final plunge. The emitted gravitational waves fall in the millihertz frequency band, making EMRIs a primary science target for LISA.^[2][1]

Accurate EMRI waveform templates must track the orbital phase to within a fraction of a radian over the full inspiral, since accumulated phase errors of order unity render the templates useless for matched filtering. Meeting this requirement demands the self-force to be known at second order in ${\displaystyle q}$.^[8] With templates of this quality, LISA is expected to measure EMRI parameters with fractional statistical errors of ${\displaystyle 10^{-4}}$–${\displaystyle 10^{-6}}$ on the redshifted masses, determine the spin magnitude of the central massive black hole to ${\displaystyle \sim 10^{-4}}$, locate the source in the sky to within several square degrees, and test the multipolar structure of the central Kerr spacetime at the percent level.^[2]

In addition, the slowly evolving EMRI signal encodes a detailed map of the central spacetime's multipolar structure. Deviations from the Kerr metric would imprint systematic phase drifts on the gravitational waveform, enabling precision tests of general relativity in the strong-field regime inaccessible to current ground-based detectors.^[2]

In the weak-field, slow-motion limit, gravitational self-force predictions can be compared directly against post-Newtonian expansions of the two-body dynamics. The redshift invariant ${\displaystyle u^{t}}$ first computed numerically by Barack and Sago (2007) was found by Detweiler (2008) to agree with sixth-order post-Newtonian predictions.^[16] Such comparisons have enabled the extraction of previously unknown post-Newtonian coefficients from self-force data, and have confirmed the first law of binary black-hole mechanics relating the binding energy to gauge-invariant self-force observables.^[8]

The effective one-body formalism (EOB) resums the post-Newtonian two-body dynamics into a Hamiltonian describing the relative motion in an effective spacetime. Gravitational self-force calculations provide exact-in-mass-ratio results for conservative gauge-invariant quantities—such as the binding energy at fixed frequency, the periastron advance, and the redshift invariant—which are used to calibrate the EOB potentials and improve their accuracy in regimes beyond the reach of post-Newtonian theory.^[8]

Numerical relativity solves the full Einstein equations and is complementary to self-force methods: it works best for comparable-mass binaries but becomes computationally prohibitive at extreme mass ratios. Cross-validation at intermediate mass ratios is nonetheless possible. The second-order self-force waveforms of Wardell et al. (2023) were found to agree with numerical-relativity waveforms even for mass ratios as large as ${\displaystyle q=1/10}$, demonstrating an unexpectedly wide domain of validity of the post-adiabatic approximation and suggesting potential for self-force methods to contribute to the broader gravitational-wave modeling effort.^[21]

The "Capra Meeting on Radiation Reaction in General Relativity" is the annual meeting of the international community of scientists working on gravitational self-force (colloquially known as the Capra community). The first Capra meeting was organized in 1998 by Patrick Brady to discuss "Radiation Reaction in General Relativity",^[27] including the then-recent papers by Mino, Sasaki, and Tanaka^[6] and Quinn and Wald.^[7] The meeting was held at the Capra ranch, which had been bequeathed to Caltech by the film director Frank Capra. Subsequent meetings have been held annually at various locations worldwide; while never returning to the original ranch, they have retained the Capra name. The first meeting was attended by only ten people, including Eric Poisson and Kip Thorne.^[27] The most recent editions have attracted over 100 participants.^[28]

• Extreme mass-ratio inspiral
• Laser Interferometer Space Antenna
• Abraham–Lorentz force
• Gravitational wave
• Post-Newtonian expansion
• Effective one-body formalism
• Numerical relativity

• Barack, Leor; Pound, Adam (2019). "Self-force and radiation reaction in general relativity". Reports on Progress in Physics. 82 (1): 016904. arXiv:1805.10385. Bibcode:2019RPPh...82a6904B. doi:10.1088/1361-6633/aae552. PMID 30270849. — Comprehensive review of the formal foundations, computational methods, and astrophysical applications.
• Poisson, Eric; Pound, Adam; Vega, Ian (2011). "The Motion of Point Particles in Curved Spacetime". Living Reviews in Relativity. 14 (1): 7. arXiv:1102.0529. Bibcode:2011LRR....14....7P. doi:10.12942/lrr-2011-7. PMID 28179832. — Definitive reference on the mathematical foundations of radiation reaction in curved spacetime.
• Pound, Adam; Wardell, Barry (2021). "Black Hole Perturbation Theory and Gravitational Self-Force". Handbook of Gravitational Wave Astronomy. pp. 1–119. arXiv:2101.04592. doi:10.1007/978-981-15-4702-7_38-1. ISBN 978-981-15-4702-7. — Overview of perturbation-theory methods including the two-timescale waveform framework.

Category:General relativity Category:Gravitational-wave astronomy Category:Theoretical physics Category:Mathematical physics Category:Astrophysics Category:Equations of physics