Adiabatic Flux-Balance Evolution

At leading order in a small mass ratio, an EMRI may be evolved without evaluating the instantaneous self-force along every orbit. One computes gravitational-wave fluxes for a fixed Kerr geodesic, averages them over its fast phases, and lets the orbital constants drift slowly. This adiabatic construction is reliable only away from resonances and separatrices, and only for the leading accumulated phase.

Helpful background. The Einstein Equation fixes the background field equation, Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge control, and Presymplectic Potentials and Currents clarifies conserved fluxes. Black-Hole Perturbation Theory supplies horizon and infinity amplitudes, Relativistic Two-Body Dynamics supplies the mass-ratio expansion, and Gravitational Radiation and Waveforms supplies radiative normalization.

Two timescales and averaged actions

Let the Kerr black hole have mass $M_\bullet$, the secondary have mass $m_\star$, and $\epsilon=m_\star/M_\bullet\ll1$. A bound geodesic has fast phases $\vartheta_\alpha$ and slowly evolving actions $J_A$. Schematically,

$$\frac{d\vartheta_\alpha}{dt} =\Omega_\alpha(J)+\epsilon\,g_\alpha(\vartheta,J)+O(\epsilon^2),$$ $$\frac{dJ_A}{dt} =\epsilon\,G_A(\vartheta,J) +\epsilon^2G_A^{(2)}(\vartheta,J)+O(\epsilon^3).$$

At fixed dimensionless orbital parameters, the orbital time is $T_{\mathrm{orb}}\sim GM_\bullet$, while the radiation-reaction time is $T_{\mathrm{rr}}\sim GM_\bullet/\epsilon$. Away from a resonance, a near-identity transformation removes the oscillatory part of $G_A$ at leading order and leaves

$$\frac{dJ_A}{dt} =\epsilon\left\langle G_A\right\rangle(J)+O(\epsilon^2).$$

The brackets are a two-torus average over independent radial and polar phases, not necessarily a finite-time average along a resonant orbit. This distinction is essential: a nonresonant trajectory densely samples the torus, whereas a resonant trajectory does not.

The two-timescale expansion, including the nonresonance hypothesis and phase counting, is developed in Hinderer and Flanagan 2008, §§III–VI.

Energy and angular-momentum balance

For a stationary, axisymmetric background, the orbit-averaged losses of specific energy and axial angular momentum equal the negative radiative fluxes through future null infinity and the future horizon:

$$\left\langle\frac{dE}{dt}\right\rangle =-\left( \mathcal F_E^\infty+\mathcal F_E^{H} \right)/m_\star,$$ $$\left\langle\frac{dL_z}{dt}\right\rangle =-\left( \mathcal F_{L_z}^\infty+\mathcal F_{L_z}^{H} \right)/m_\star.$$

Here $E=-u_t$ and $L_z=u_\phi$ are specific quantities, while the $\mathcal F$ are physical total energy and angular-momentum fluxes. Multiplying the equations by the fixed $m_\star$ gives the balance laws for total orbital energy $E_{\mathrm{orb}}=m_\star E$ and angular momentum $L_{z,\mathrm{orb}}=m_\star L_z$. The fluxes are quadratic in the first-order field and scale as $m_\star^2$ at fixed background; division by $m_\star$ explains why specific orbital quantities drift at first order in $\epsilon$. Horizon flux can be negative for a superradiant mode; the sum still represents the correct change in orbital energy. Omitting the horizon term is therefore not merely a small numerical approximation—it changes the balance law and must be justified for the chosen spin and orbit.

Frequency-domain perturbation theory decomposes the field into modes with

$$\omega_{mkn} =m\Omega_\phi+k\Omega_\theta+n\Omega_r.$$

For a radiative mode with $\omega_{mkn}\gt0$, the infinity energy flux is nonnegative and the horizon energy flux has the sign of the horizon-frame frequency $\omega_{mkn}-m\Omega_H$. Negative-frequency conjugate modes describe the same real radiation and must follow the chosen mode-counting convention. Summing converged mode families and translating the normalizations yields the averaged drifts.

For a circular equatorial sequence, remaining circular imposes the consistency relation

$$\left\langle\dot E_{\mathrm{orb}}\right\rangle =\Omega_\phi \left\langle\dot L_{z,\mathrm{orb}}\right\rangle.$$

The same relation holds for the corresponding total fluxes. A numerical flux table that violates it beyond truncation error is not a valid circular inspiral input.

The Carter constant is different

The Carter constant $Q$ is associated with a rank-two Killing tensor, not with a spacetime translation or rotation whose current can simply be measured at infinity. Its averaged evolution cannot be obtained by inventing a standalone “Carter flux.” Instead one evaluates the radiative self-force or an equivalent mode-sum formula derived from the hidden symmetry. Define $K=K_{\mu\nu}u^\mu u^\nu$ and $Q=K-(L_z-aE)^2$, as on the Kerr geodesics page. For fixed background $a=J_\bullet/M_\bullet$, the coordinate-time rate is

$$\begin{aligned} \left\langle\frac{dQ}{dt}\right\rangle_t &=\left\langle\frac{2K_{\mu\nu}u^\mu a_{\mathrm{rad}}^\nu}{u^t} \right\rangle_t\\ &\quad-2(L_z-aE)\left( \left\langle\frac{dL_z}{dt}\right\rangle_t -a\left\langle\frac{dE}{dt}\right\rangle_t\right). \end{aligned}$$

The constants in the second line are held fixed on the averaging geodesic. The factor $1/u^t$ converts a proper-time force into a coordinate-time drift; it cannot be dropped. These distinctions follow directly by differentiating $K$ and $Q$, as shown on Osculating Orbits and Local Self-Consistent Evolution. In action variables the three drifts can be treated uniformly, but that does not turn $Q$ into an asymptotic Noether charge.

For circular equatorial motion $Q=0$ remains zero by reflection symmetry. For generic inclination, a separate $\dot Q$ computation is indispensable. The adiabatic Kerr construction and the relation between radiative fields, fluxes, and orbital elements are reviewed in Pound and Wardell 2021, §§6.2 and 7.1, pp. 65–100.

What phase accuracy adiabatic evolution provides

Over $T_{\mathrm{rr}}\sim GM_\bullet/\epsilon$, the orbital phase has the asymptotic form

$$\Phi =\frac{1}{\epsilon}\Phi_{-1} +\Phi_0 +O(\epsilon).$$

The averaged dissipative first-order self-force determines $\Phi_{-1}$. The oscillatory first-order dissipative force, the conservative first-order force, and the averaged dissipative second-order force contribute to $\Phi_0$. Consequently a leading adiabatic inspiral can be an excellent qualitative trajectory and still miss an order-unity phase over a long observation.

Adiabatic flux balance requires:

• $\epsilon\ll1$ and a stable bound geodesic at each slow time;
• orbital periods much shorter than the drift time;
• no low-order radial–polar resonance in the averaging interval;
• adequate distance from the separatrix, where the radial period grows;
• consistently normalized and converged horizon and infinity mode sums.

It does not provide the local self-force, conservative frequency shifts, post-adiabatic phase, transient-resonance jump, or transition-to-plunge dynamics. Those are separate controlled corrections, not tunable pieces of the leading flux.

Exercises

Derive the circular consistency relation from a sequence of circular equatorial geodesics.

Solution

Along a one-parameter circular sequence, the geodesic first law gives $dE=\Omega_\phi\,dL_z$. Differentiating along the slow inspiral yields

$$\dot E=\Omega_\phi\dot L_z.$$

Balance then demands

$$\mathcal F_E^\infty+\mathcal F_E^H =\Omega_\phi(\mathcal F_{L_z}^\infty+\mathcal F_{L_z}^H).$$

The relation is a check on the summed fluxes and on the interpolation used to evolve the circular sequence.

Why does an $O(\epsilon^2)$ averaged force affect the phase at order unity?

Solution

The force changes a frequency drift by $O(\epsilon^2)$, but the inspiral lasts $O(\epsilon^{-1})$ in units of $GM_\bullet$. Its accumulated frequency correction is $O(\epsilon)$, and integrating that frequency correction over another $O(\epsilon^{-1})$ interval produces an $O(1)$ phase. This secular enhancement is why second-order dissipative information enters first-post-adiabatic phasing.

References

• Hinderer, Tanja, and Éanna É. Flanagan. “Two Timescale Analysis of Extreme Mass Ratio Inspirals in Kerr. I. Orbital Motion.” Physical Review D 78 (2008): 064028. DOI.
• Pound, Adam, and Barry Wardell. “Black Hole Perturbation Theory and Gravitational Self-Force.” In Handbook of Gravitational Wave Astronomy, edited by Cosimo Bambi, Stavros Katsanevas, and Konstantinos D. Kokkotas, 1–119. Singapore: Springer (2021). DOI.