Relativistic Lindblad Torques

• Relativistic Lindblad torques are angular momentum and energy exchanges in accretion discs defined by relativistic orbital frequencies and metric perturbations.
• They govern resonant coupling in EMRIs by modifying torque amplitude and sign compared to classical Newtonian predictions, especially near the ISCO.
• Strong-field effects, including pericenter precession and frame dragging, can amplify or reverse the torque, impacting gravitational waveform dephasing.

Relativistic Lindblad torques are the angular-momentum and energy exchanges produced when an orbiting perturber interacts resonantly with a disc in a strong-field spacetime, so that the classical Lindblad-resonance picture must be reformulated using relativistic orbital frequencies, metric perturbations, and Hamiltonian perturbation theory. In the context of accretion discs around black holes, they govern resonant coupling between disc material and a secondary body such as a compact object in an extreme-mass-ratio inspiral (EMRI), and they differ qualitatively from their Newtonian counterparts because pericenter precession, frame dragging, and the behavior of the radial epicyclic frequency near the innermost stable circular orbit (ISCO) modify both the location and the strength of resonances [(Hirata, 2010); (Hirata, 2010); (R. et al., 24 Sep 2025); (R. et al., 3 Oct 2025)].

1. Relativistic definition and physical setting

Relativistic Lindblad torques arise in thin discs whose fluid elements or test particles execute nearly circular equatorial motion in a stationary, axisymmetric spacetime, while being forced by an external perturbation or by the gravitational field of an embedded secondary. In the black-hole accretion context, the relevant backgrounds are Schwarzschild and Kerr, and the perturber may be a companion black hole in a binary-disc system or a stellar-mass compact object embedded in an accretion disc around a supermassive black hole [(Hirata, 2010); (Hirata, 2010)].

The Newtonian theory of disc-satellite interaction associates torque exchange with Lindblad resonances determined by orbital commensurabilities. The relativistic generalization preserves the resonant character of the interaction but replaces Keplerian orbital dynamics with geodesic motion and replaces the Newtonian potential by metric perturbations or, in EMRI applications, by a disturbing Hamiltonian built from self-force and Hamiltonian perturbation theory [(Hirata, 2010); (R. et al., 24 Sep 2025); (R. et al., 3 Oct 2025)].

Recent EMRI studies specialize this framework to a small compact object of mass $m \ll M$ on a nearly circular, equatorial orbit around a supermassive black hole, embedded in a thin equatorial accretion disc of surface density $\Sigma(r)$. In that setting, the goal is to compute the secular rates at which the compact object exchanges energy and angular momentum with the disc at Lindblad resonances, using relativistically accurate expressions for the orbital frequency $\Omega$ and radial epicyclic frequency $\kappa$ (R. et al., 24 Sep 2025, R. et al., 3 Oct 2025).

2. Resonance structure in strong gravity

The basic relativistic resonance condition generalizes the Newtonian statement that the forcing frequency matches the radial epicyclic frequency. In the general relativistic thin-disc treatment, Lindblad resonances occur where

$m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$

with $\Omega$ the orbital frequency, $\Omega_s$ the pattern speed of the perturber, and $\kappa$ the radial epicyclic frequency (Hirata, 2010).

In the EMRI Hamiltonian formulation, the resonance is written in action-angle language as

$$j\, \dot{\lambda}' - (j + k)\, \dot{\lambda} + k\, \dot{\varpi} = 0 \,, \qquad k = \pm1,$$

where $\dot{\lambda}$ is the mean-longitude precession rate and $\Sigma(r)$0 is the periapsis precession rate. In terms of $\Sigma(r)$1 and $\Sigma(r)$2, the resonance locations satisfy

$\Sigma(r)$3

with explicit spin dependence through the Kerr expressions for $\Sigma(r)$4 and $\Sigma(r)$5 (R. et al., 3 Oct 2025).

For a non-spinning black hole, the resonance location admits a large-$\Sigma(r)$6 expansion,

$\Sigma(r)$7

which recovers the Newtonian limit when $\Sigma(r)$8 (R. et al., 24 Sep 2025).

A central relativistic feature is the separation between $\Sigma(r)$9 and $\Omega$0, caused by strong-field precession. This produces resonances with no Newtonian Keplerian analogue. In particular, discs around a black hole possess an $\Omega$1 inner Lindblad resonance because pericenter precession splits the orbital and epicyclic frequencies; this resonance does not exist in Newtonian Keplerian discs where $\Omega$2 (Hirata, 2010). The existence of such resonances is one of the clearest indicators that “relativistic Lindblad torque” is not merely a correction to Newtonian migration formulae, but a structurally different regime.

3. Formal torque expressions and invariant formulation

A foundational result of the general relativistic theory is that the Lindblad torque can be written explicitly in terms of metric perturbations for an equatorial thin disc in a general axisymmetric, time-stationary spacetime with a plane of symmetry, and that the resulting torque formula is gauge-invariant (Hirata, 2010). The torque density is

$\Omega$3

where $\Omega$4 is the azimuthal mode number, $\Omega$5 contains the disc surface density and orbital normalization, $\Omega$6 is the resonant amplitude, and $\Omega$7 (Hirata, 2010).

The resonant amplitude can be expressed directly in terms of the $\Omega$8-th Fourier components of the metric perturbation and their derivatives, evaluated on unperturbed circular orbits. A covariant form is also available, written through contractions of $\Omega$9 with the circular-orbit 4-velocity $\kappa$0. The formalism further admits an epicyclic-geodesic representation,

$\kappa$1

which makes the resonance strength an overlap integral over a small neighborhood of the resonance (Hirata, 2010).

The companion computation paper relates the torque to gravitational wave amplitudes. There, the normalized torque is

$\kappa$2

and the resonant amplitude is connected to the power delivered to a test particle at the resonance by the $\kappa$3-th Fourier component of the perturbation. This establishes a direct relation between Lindblad torques and the gravitational waveforms emitted by the perturber and by a test particle in a slightly eccentric orbit at the resonance radius (Hirata, 2010).

In EMRI applications, the formalism is recast in self-force and Hamiltonian perturbation theory. The secular rates of exchange at each resonance obey

$\kappa$4

and, for large $\kappa$5,

$\kappa$6

with fully relativistic coefficients that depend on spin, radius, and surface density (R. et al., 3 Oct 2025). In the non-spinning case, an explicit leading-order per-resonance expression involves modified Bessel functions $\kappa$7 and $\kappa$8, and the Newtonian limit is recovered for $\kappa$9 (R. et al., 24 Sep 2025).

4. Strong-field effects: enhancement, asymmetry, and torque reversal

The most important physical result of the recent EMRI literature is that strong-field effects can substantially amplify Lindblad torques relative to standard Newtonian expressions. For non-spinning black holes, relativistic corrections can enhance the magnitude of the torque by $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$0–$m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$1 orders of magnitude compared to purely Newtonian expressions when the compact-object orbit is smaller than $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$2 Schwarzschild radii, and the relativistic torque formula is described as crucial for reliable estimates when the orbit is closer than $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$3 Schwarzschild radii to the supermassive black hole (R. et al., 24 Sep 2025). For spinning black holes, the torque can likewise be $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$4–$m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$5 orders of magnitude larger than the Newtonian torque routinely used in the literature, and in some comparisons the enhancement reaches factors of $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$6–$m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$7, sometimes up to $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$8, near the ISCO (R. et al., 3 Oct 2025).

This enhancement is tied to strong radial variation in $m[\Omega(R)-\Omega_s] \simeq \pm \kappa(R),$9, especially its approach to zero near the ISCO, together with relativistic precession and the clustering of resonances in the strong-field regime (R. et al., 24 Sep 2025, R. et al., 3 Oct 2025). In Newtonian discs, outer Lindblad resonances are typically closer to the perturber and dominate the net torque, producing inward migration. The relativistic analyses show that this hierarchy can invert. For a Schwarzschild black hole, strong relativity shifts the inner Lindblad resonances closer to the compact object than the outer Lindblad resonances when the compact object is closer than $\Omega$0 Schwarzschild radii to the supermassive black hole, and for $\Omega$1 the inner resonances become closer, while for $\Omega$2 their contribution can dominate (R. et al., 24 Sep 2025).

This rearrangement can reverse the sign of the net torque. The 2025 spinning-Kerr treatment states that strong relativistic effects can potentially cause a reversal in the direction of the torque on the small compact object if the surface density gradient is not too large, so that the disc can transfer angular momentum to the compact object rather than remove it (R. et al., 3 Oct 2025). In the same framework, if terms involving the surface-density gradient are ignored, the sign of the net torque is set by

$\Omega$3

and the torque reversal radius is determined by solving

$\Omega$4

(R. et al., 3 Oct 2025)

Several papers note that torque reversal suggests the possibility of transiently stalled or “floating” behavior if the disc is sufficiently dense, although such discs are described as not astrophysically favored in the non-spinning EMRI study (R. et al., 24 Sep 2025). A plausible implication is that relativistic Lindblad torques can qualitatively reshape orbital migration near the ISCO even when they remain subdominant to gravitational-wave losses.

5. Spin dependence and current points of tension

Black-hole spin modifies both resonance strengths and the radial structure of the torque. The early Kerr calculations found that the $\Omega$5 inner Lindblad resonance is enhanced for retrograde spins and suppressed for prograde spins, with the difference growing toward the ISCO and at higher $\Omega$6 (Hirata, 2010). For $\Omega$7 inner Lindblad resonances, the torque is enhanced relative to the nonrelativistic case, and in Schwarzschild the $\Omega$8 inner Lindblad resonance is stronger than the Newtonian prediction by a factor of $\Omega$9 even when the perturber is at $\Omega_s$0 (Hirata, 2010).

Recent EMRI studies agree that spin changes the radial dependence of the torque, but they do not present a single identical picture for the reversal radius. One relativistic-disc study states that the torque reversal is highly spin-dependent, shifting progressively closer to the ISCO as the spin of the central black hole increases, and reports that for $\Omega_s$1 the strong-field torque can be fit with a slope parameter $\Omega_s$2, compared to the Newtonian value $\Omega_s$3, within the parametrization

$\Omega_s$4

It also lists fitted strong-field values $\Omega_s$5 and $\Omega_s$6 for $\Omega_s$7 and $\Omega_s$8, respectively (Duque et al., 2 Oct 2025).

By contrast, the analytic Kerr EMRI treatment states that the location $\Omega_s$9 increases with decreasing supermassive-black-hole spin for prograde orbits, that for retrograde discs and orbits the reversal can occur at much larger radii, and that for nearly maximal retrograde spin the reversal can occur at up to $\kappa$0 ISCO radii. The same work further states that the ratio $\kappa$1 is approximately insensitive to spin, except for very high spins (R. et al., 3 Oct 2025).

These statements are not identical. The shared conclusion is that spin matters, but the detailed mapping between spin and reversal location remains model-dependent in the currently cited literature. This suggests that the outcome depends sensitively on how the disc model, pressure regularization, surface-density gradient, and resonance summation are implemented, rather than on geodesic spin effects alone.

6. Astrophysical role, observational implications, and limitations

Relativistic Lindblad torques are primarily important as environmental effects in EMRIs and as mechanisms for angular-momentum transport and heating in relativistic discs. For EMRIs, the main consequence is waveform dephasing. The non-spinning EMRI study states that disk torques, even when subdominant to gravitational-wave emission, can accumulate measurable dephasing over long inspirals of more than $\kappa$2 cycles, and that the difference between Newtonian and relativistic evaluations can reach tens of percent in dephasing estimates (R. et al., 24 Sep 2025). The spinning-Kerr EMRI study similarly emphasizes that relativistic torques can be much larger than Newtonian estimates precisely where the gravitational-wave signal is generated and observed with LISA (R. et al., 3 Oct 2025).

The observational argument has been extended to phenomenological parameterizations. One recent study investigates whether Lindblad torques can be approximated by power laws and argues that the spin- and disc-dependent slope parameter $\kappa$3 may be measurable in “golden” EMRIs. It cites current forecasts of parameter recovery with $\kappa$4 and therefore predicts that LISA could distinguish between different disc configurations through their relativistic Lindblad torque signatures (Duque et al., 2 Oct 2025). This suggests that relativistic Lindblad torques may serve not only as nuisance environmental corrections but also as probes of disc structure in the inner regions that are otherwise inaccessible to electromagnetic observations.

The present analytic models also have explicit limitations. The EMRI calculations cited here neglect detailed disc microphysics such as MHD turbulence, disc self-gravity, radiative transport, and hydrodynamic backreaction, and they do not treat more generic eccentric or inclined orbits or corotation and saturation effects (R. et al., 3 Oct 2025). Pressure regularization and mode cutoffs are model ingredients in relativistic-disc calculations, with the number of contributing modes cut off at $\kappa$5, where the aspect ratio $\kappa$6 is set by disc microphysics (Duque et al., 2 Oct 2025). Accordingly, the current results define the leading-order gravitational component of the torque in thin-disc, nearly circular, equatorial settings rather than a complete environmental theory.

7. Relation to Newtonian theory and conceptual significance

The relativistic theory reduces to the classical Lindblad-resonance formalism in the nonrelativistic limit. In the Newtonian case, the resonant amplitude takes the simpler form

$\kappa$7

with $\kappa$8 the $\kappa$9-th azimuthal Fourier component of the perturbing potential (Hirata, 2010). The relativistic construction generalizes this by promoting the relevant orbital quantities to their strong-field forms and by coupling to the full metric perturbation rather than to a scalar potential alone (Hirata, 2010).

The conceptual significance of relativistic Lindblad torques lies in three linked results. First, the torque formula can be cast in a gauge-invariant form in general relativity (Hirata, 2010). Second, the resonance strengths can be computed through gravitational-wave amplitudes, linking disc torque physics directly to black-hole perturbation theory (Hirata, 2010). Third, in EMRI applications, strong-field corrections are not perturbatively small near the ISCO: they can change the magnitude of the torque by orders of magnitude and can reverse its sign (R. et al., 24 Sep 2025, R. et al., 3 Oct 2025, Duque et al., 2 Oct 2025).

In that sense, relativistic Lindblad torques occupy the intersection of disc dynamics, Hamiltonian resonance theory, self-force methods, and black-hole perturbation theory. Their modern role is to provide a relativistically accurate description of disc-mediated angular-momentum exchange in regimes where Newtonian migration recipes cease to be reliable, especially near spinning supermassive black holes.