Post-Adiabatic and Resonant Corrections

Leading adiabatic evolution retains only the averaged first-order dissipative drift. First-post-adiabatic accuracy also needs oscillatory and conservative first-order effects, averaged second-order dissipation, and consistent phase reconstruction. At a resonance, the nonresonant transformation develops a small denominator; an inner phase-sensitive solution must replace it and be matched to adiabatic outer solutions on both sides.

Helpful background. The Einstein Equation fixes the dynamics, Weak Metric Perturbations and Gauge Freedom supplies perturbative variables, Presymplectic Potentials and Currents supplies actions, Partial Differential Equations and Green Functions supplies retarded forcing, Gravitational Radiation and Waveforms supplies phase diagnostics, and Relativistic Two-Body Dynamics supplies approximation boundaries.

First-post-adiabatic structure

Measure time in a fixed orbital scale $t_0$ and normalize the actions, so the variables $t$ and $J_A$ below are dimensionless. For an EMRI one may take $t_0=GM_\bullet$ and $\epsilon=m_\star/M_\bullet$. Write

$$\dot J_A =\epsilon G_A^{(1)}(q,J) +\epsilon^2G_A^{(2)}(q,J)+\cdots,$$ $$\dot q^\alpha =\Omega^\alpha(J) +\epsilon g_{(1)}^\alpha(q,J)+\cdots.$$

After a nonresonant near-identity transformation, the slow action equation through first-post-adiabatic order is

$$\dot{\widetilde J}_A =\epsilon\langle G_A^{(1)}\rangle +\epsilon^2 \left[ \langle G_A^{(2)}\rangle +\mathcal Q_A(G^{(1)},g_{(1)}) \right] +O(\epsilon^3),$$

where $\mathcal Q_A$ contains derivatives and products generated by removing first-order oscillations. The phase equation receives corresponding averaged frequency and reconstruction terms.

The accumulated phase is

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

$\Phi_{-1}$ comes from leading averaged dissipation. $\Phi_0$ requires all terms displayed at the next order; including only a conservative frequency correction or only an improved flux is incomplete.

The resonant small denominator

A nonzero Fourier harmonic is removed using a generator proportional to

$$\frac{G_{A,\mathbf k}^{(1)}} {i\mathbf k\cdot\boldsymbol\Omega}.$$

Suppose $\mathbf k_*\cdot\boldsymbol\Omega(J_*)=0$ at $t_*$. Define the resonant phase $\psi=\mathbf k_*\cdot\mathbf q$. Near a simple crossing,

$$\dot\psi \simeq\alpha(t-t_*), \qquad \alpha=O(\epsilon).$$

The inner scale is

$$T=\sqrt{\epsilon}\,(t-t_*),$$

Define $\delta J_A=J_A-J_A^{\mathrm{ad}}$, where the smooth adiabatic solution is continued through the crossing. In variables with nonresonant oscillations removed, retaining the resonant harmonics gives

$$\frac{d\delta J_A}{dT} =\sqrt{\epsilon} \sum_{s\ne0} G_{A,s\mathbf k_*}^{(1)}(J_*) e^{is\psi(T)} +O(\epsilon).$$

The full $J_A$ equation also contains $\sqrt\epsilon\langle G_A^{(1)}\rangle$ at this same order. It has canceled here only because $\delta J_A$ measures the excess over smooth drift. In the weak-resonant-forcing limit, evaluate $\psi(T)=\psi_*+(\alpha/\epsilon)T^2/2$ on the unperturbed crossing. Stationary-phase integration then produces an $O(\sqrt\epsilon)$ action jump depending on the entry phase $\psi_*$. This is larger than the ordinary $O(\epsilon)$ oscillatory correction and cannot be recovered from a phase-averaged flux.

The resonance mechanism and jump scaling are derived in Flanagan and Hinderer 2012, p. 4, “Analytic Derivation”. For stronger resonant forcing, solve the coupled inner action and phase equations: feedback on the detuning is not suppressed merely by sending $\epsilon$ to zero. The transient-resonance page derives the real jump and its entry-phase dependence explicitly.

Matched inner and outer solutions

The overlap must be outside the resonance width but shorter than the inspiral scale. In the dimensionless time used here it requires

$$\epsilon^{-1/2}\ll\lvert t-t_*\rvert\ll\epsilon^{-1}, \qquad 1\ll\lvert T\rvert\ll\epsilon^{-1/2}.$$

The inner solution is expanded for large negative $T$ and matched term by term to that outer expansion. Its large positive-$T$ behavior supplies shifted actions and phases for the post-resonance outer solution. The matching fixes otherwise arbitrary integration constants and prevents double counting the smooth adiabatic drift already present inside the layer.

A practical implementation chooses matching surfaces within the common domain and varies them. The resulting post-resonance invariant frequencies should change only at the first omitted order. An instantaneous jump formula is a compressed representation of this matched solution, not a literal discontinuous worldline.

A separatrix requires a different inner balance because an orbital period diverges or a restoring frequency vanishes. The familiar circular inspiral-to-plunge transition uses radial width $O(\epsilon^{2/5})$ and time $O(\epsilon^{-1/5})$, not the resonance’s square-root scaling. If resonance and separatrix layers overlap, neither isolated formula is uniform.

Checks and limitations

An independent calculation should:

1. recover the nonresonant first-post-adiabatic equations far from the layer;
2. reproduce the stationary-phase jump for a single harmonic;
3. compare matched and direct phase-sensitive evolution;
4. vary the matching interval and harmonic cutoff;
5. set the resonant coefficient to zero and recover smooth adiabatic passage.

The analysis assumes a simple isolated resonance, slowly varying amplitudes, and a controlled perturbative force. Degenerate crossings, overlapping resonances, abrupt environmental forcing, and merger require a new inner problem.

Initial data deserve the same ordering discipline. The post-adiabatic outer solution and resonant inner solution must represent the same physical point, including phase offsets generated by the near-identity map. Matching only the actions while resetting the fast phases can alter the predicted jump at leading resonant order and defeat an otherwise correct force calculation.

Exercises

Recover the square-root jump scaling from the resonant integral.

Solution

The leading contribution is

$$\epsilon \int e^{i\alpha(t-t_*)^2/2}\,dt =\epsilon \sqrt{\frac{2\pi}{\lvert\alpha\rvert}} e^{i\pi\operatorname{sgn}\alpha/4}.$$

Since $\alpha=O(\epsilon)$, the result is $O(\sqrt\epsilon)$. Its complex phase combines with the entry phase and the Fourier coefficient to give a real action jump.

Why must smooth adiabatic drift be subtracted when defining a jump?

Solution

The actions change by ordinary averaged radiation reaction throughout the resonance interval. The jump is the additional phase-sensitive change relative to the matched continuation of that smooth outer solution. Failing to subtract it makes the answer depend on the arbitrary width chosen for the layer.

References

• Flanagan, Éanna É., and Tanja Hinderer. “Transient Resonances in the Inspirals of Point Particles into Black Holes.” Physical Review Letters 109 (2012): 071102. DOI.

Further reading

• 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.