Adiabatic, Post-Adiabatic, and Resonant Timescales

An inspiral contains fast orbital phases and slowly changing orbital elements. Adiabatic evolution averages over the fast motion; post-adiabatic evolution restores controlled corrections; transient resonances require a separate local scaling because ordinary averaging loses the phase that becomes dynamically important. These are statements about timescale separation, not synonyms for weak field.

Required background. The Einstein Equation supplies the underlying nonlinear dynamics, and Weak Metric Perturbations and Gauge Freedom supplies the perturbative organization and its gauge limitations.

Helpful background. Presymplectic Potentials and Currents clarifies actions and slowly varying invariants, while Black-Hole Perturbation Theory supplies fluxes and local forces in the extreme-mass-ratio limit.

Fast phases and slow actions

For an integrable conservative system, introduce angle variables $q_\alpha$ and actions $J_a$:

$$\dot q_\alpha=\Omega_\alpha(\mathbf J), \qquad \dot J_a=0.$$

Weak dissipation or another small perturbation of size $\varepsilon$ changes the equations to

$$\dot q_\alpha =\Omega_\alpha(\mathbf J) +\varepsilon g_\alpha(\mathbf q,\mathbf J) +O(\varepsilon^2),$$ $$\dot J_a =\varepsilon G_a(\mathbf q,\mathbf J) +\varepsilon^2G_a^{(2)}(\mathbf q,\mathbf J) +O(\varepsilon^3).$$

The orbital time is $T_{\mathrm{orb}}\sim\Omega^{-1}$, while order-unity evolution of the actions takes $T_{\mathrm{rr}}\sim T_{\mathrm{orb}}/\varepsilon$. For a small body of mass ratio $\epsilon=m_2/m_1$, one usually has $\varepsilon\sim\epsilon$ after quantities are nondimensionalized. For a PN quasi-circular inspiral, the leading estimate is instead $\varepsilon\sim\nu x^{5/2}$.

Introduce slow time

$$\widetilde t=\varepsilon t.$$

The phases accumulate as $q_\alpha=O(\varepsilon^{-1})$ over a radiation-reaction time. Consequently, an $O(\varepsilon)$ error in an instantaneous frequency can produce an $O(1)$ phase error. Local force accuracy and long-time phase accuracy are different requirements.

Adiabatic averaging

Away from resonance, expand the forcing in harmonics,

$$G_a(\mathbf q,\mathbf J) =\sum_{\mathbf k\in\mathbb Z^n} G_{a,\mathbf k}(\mathbf J)e^{i\mathbf k\cdot\mathbf q}.$$

If $\mathbf k\cdot\boldsymbol\Omega\neq0$ for every nonzero mode that matters, the oscillatory terms average to zero over many orbits. The leading slow evolution is

$$\frac{dJ_a^{(0)}}{d\widetilde t} =\langle G_a\rangle =G_{a,\mathbf0}(J^{(0)}).$$

For stationary black-hole inspirals, averaged energy and angular-momentum fluxes at infinity and horizons can determine this leading evolution. For generic Kerr orbits, the Carter constant requires either an appropriate flux relation or local self-force information. Adiabatic EMRI dynamics and the distinction between flux balance and local evolution are reviewed in Barack and Pound 2019, §§5.1–5.2.

The averaging theorem has hypotheses: smooth dependence, bounded nonresonant denominators, a sufficiently regular torus, and evolution that stays away from a separatrix. It does not say that the instantaneous trajectory follows an exact conservative geodesic, only that the slowly evolving actions are approximated over the stated time.

Post-adiabatic information

A two-timescale expansion takes the form

$$J_a(t,\varepsilon) =J_a^{(0)}(\widetilde t) +\varepsilon J_a^{(1)}(\mathbf q,\widetilde t) +O(\varepsilon^2),$$ $$q_\alpha(t,\varepsilon) =\varepsilon^{-1}\psi_\alpha^{(0)}(\widetilde t) +\psi_\alpha^{(1)}(\widetilde t) +O(\varepsilon).$$

The leading phase is $O(\varepsilon^{-1})$. The first post-adiabatic phase is $O(1)$ and generally depends on:

• oscillatory pieces of the first-order dissipative force;
• the first-order conservative force;
• averaged second-order dissipation;
• the map between orbital elements and observables;
• initial phase and near-identity transformation conventions.

This accounting is derived systematically for extreme-mass-ratio inspirals in Hinderer and Flanagan 2008, §§II–V. It explains why a leading flux model can predict the dominant chirp yet fail a phase-accurate requirement. It also shows that “post-adiabatic” names an order in a multiscale expansion, not one universal correction formula.

Near-identity transformations remove nonresonant oscillatory forcing order by order. Their intermediate variables are not unique; observable phases and invariant evolution at matched order are the appropriate comparisons. Initial-data layers can also contribute at the same formal order and must not be discarded without a preparation argument.

Transient resonances

Ordinary averaging fails when an integer combination of frequencies becomes small:

$$\mathbf k_\ast\cdot\boldsymbol\Omega =k_r\Omega_r+k_\theta\Omega_\theta+k_\phi\Omega_\phi \approx0.$$

For stationary axisymmetric perturbations, the relevant generic Kerr orbital resonance is often $k_r\Omega_r+k_\theta\Omega_\theta=0$. The resonant phase

$$\Phi_\ast=\mathbf k_\ast\cdot\mathbf q$$

then varies slowly, so its Fourier term no longer averages away.

Let the resonance be crossed at $t=t_\ast$, with a nonzero sweep rate $\alpha$ of order $T_{\mathrm{orb}}^{-2}$. If

$$\mathbf k_\ast\cdot\boldsymbol\Omega \simeq \alpha\varepsilon(t-t_\ast),$$

then

$$\Phi_\ast(t) \simeq \Phi_\ast(t_\ast) +\frac12\alpha\varepsilon(t-t_\ast)^2.$$

The phase changes by order unity across

$$\lvert t-t_\ast\rvert\sim\varepsilon^{-1/2}T_{\mathrm{orb}}.$$

During this layer, the resonant contribution to $\dot J$ is $O(\varepsilon)$ and acts for $O(\varepsilon^{-1/2})$ orbital times. After subtracting the smooth averaged drift, it produces a change in normalized actions

$$\Delta J_a=O(\varepsilon^{1/2}).$$

The jump depends on the phase $\Phi_\ast(t_\ast)$, information deliberately erased by ordinary averaging. This square-root scaling and phase dependence are derived in Flanagan and Hinderer 2012, pp. 1–4, Eqs. (1)–(12).

Not every commensurability produces a large effect. The relevant Fourier coefficient may vanish by symmetry, the crossing may occur too close to plunge for the asymptotics to separate, or the system may be captured into resonance rather than cross it. Each case requires a local model and matching to the outer adiabatic solution.

Separatrices and other breakdowns

A separatrix causes a different failure. As an orbit approaches an unstable limiting orbit, one period can diverge logarithmically, invalidating the assumed uniform bound on $T_{\mathrm{orb}}$. The transition from inspiral to plunge uses a local expansion of the radial potential rather than resonant averaging.

Other failure modes include:

• an internal stellar mode with $\omega_n\approx m\Omega$, requiring dynamical tides;
• rapid spin precession comparable to orbital evolution;
• eccentric encounters with no long sequence of orbital periods;
• chaotic or nonintegrable conservative motion, where global action-angle variables may not exist;
• abrupt changes in external environment.

The remedy is not to call every such case “nonadiabatic.” One identifies the small denominator or degenerating timescale, chooses an inner scaling, solves the local problem, and matches it to the valid outer expansions.

Exercises

Suppose $\dot J=\varepsilon[A+B\cos\vartheta]$ and $\dot\vartheta=\Omega(J)$ with $\Omega$ bounded away from zero. What is the leading adiabatic equation, and why does the oscillatory term still matter at post-adiabatic order?

Solution

The average over $\vartheta$ is $\langle\cos\vartheta\rangle=0$, so

$$\frac{dJ^{(0)}}{d\widetilde t}=A.$$

At the next order, the $B\cos\vartheta$ term produces an $O(\varepsilon)$ oscillation in $J$. To leading local order, $J=\bar J+\varepsilon B\sin\vartheta/\Omega(\bar J)+\cdots$. The corresponding instantaneous frequency correction is oscillatory, so integrating its magnitude times the full inspiral time would overestimate the secular phase. It can matter through the initial-data map and its coupling to the other terms of the consistently averaged expansion. For constant $A,B,\Omega$, it produces only a bounded action oscillation and no secular frequency correction. An $O(1)$ phase shift is therefore a possible post-adiabatic contribution, not an automatic consequence of every oscillatory force.

Derive the width and action-jump scaling of a simple resonance from $\dot\Phi=\alpha\varepsilon(t-t_\ast)$ and $\dot J=\varepsilon C\cos\Phi$.

Solution

Integration gives $\Phi-\Phi_\ast=\alpha\varepsilon(t-t_\ast)^2/2$. The resonant term remains coherent while this phase is $O(1)$, hence $\Delta t=O(\varepsilon^{-1/2})$. Integrating $\dot J=O(\varepsilon)$ over that interval gives $\Delta J=O(\varepsilon^{1/2})$. The coefficient depends on the Fresnel-type integral and on $\Phi_\ast$.

References

• Barack, Leor, and Adam Pound. “Self-Force and Radiation Reaction in General Relativity.” Reports on Progress in Physics 82 (2019): 016904. DOI.
• Flanagan, Éanna É., and Tanja Hinderer. “Transient Resonances in the Inspirals of Point Particles into Black Holes.” Physical Review Letters 109 (2012): 071102. DOI.
• 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.