Balance Laws and Secular Evolution

Flux balance replaces a difficult local radiation-reaction calculation by accounting for conserved quantities crossing distant boundaries. The replacement is exact only for a fully defined total charge and all its boundaries. For a reduced mechanical source, near-zone field storage, gauge-dependent total derivatives, averaging, and nonadiabatic evolution must be controlled. This page states that interface: it uses waveform fluxes to infer secular changes of source invariants, but does not derive binary equations of motion or detector phasing.

Helpful background. Einstein Equations supplies covariant conservation; Waves, Green Functions, and Geometric Optics supplies boundary fluxes; Gravitational Radiation and Waveforms supplies strain and news; Causality and Global Structure supplies future null infinity; and Black-Hole Geometry and Mechanics supplies future-horizon charges and orientations.

A closed charge budget

Fix a common asymptotic time, Bondi frame, spatial origin, and boundary orientation. First take $Q$ to be an exterior mechanical energy or angular momentum, excluding the evolving black-hole masses or spins. Define $\mathcal F_Q^\infty$ as the signed flux outward through $\mathscr I^+$ and $\mathcal F_Q^H$ as the signed flux into future horizons, using the appropriate approximate stationary Killing fields. If $Q_{\rm near}$ denotes the associated exterior field contribution, a useful balance is

$$\frac{dQ_{\rm mech}}{dt} =-\mathcal F_Q^\infty -\mathcal F_Q^H -\frac{dQ_{\rm near}}{dt}.$$

For energy, $\mathcal F_E^\infty\ge0$. Ordinary horizon absorption has $\mathcal F_E^H>0$, whereas a superradiant rotating-horizon channel can have $\mathcal F_E^H<0$. Angular-momentum fluxes are signed and need not be positive. Adding the changing horizon charges into the total system cancels the horizon term: in particular, total Bondi energy loses only the flux reaching null infinity. The two budgets must not be combined in a way that counts absorption twice.

Linear momentum has its own exact Bondi balance at infinity. Extending the mechanical formula above to a separate horizon momentum requires an additional, stated quasi-local prescription; a generic horizon has no preferred translation generator. For whole-system recoil, use the Bondi endpoint momentum budget.

The last term includes reversible field storage and “Schott” terms—total derivatives that depend on how mechanical and field contributions are split. Over a complete periodic cycle or an adiabatic averaging interval, a bounded $Q_{\rm near}$ often has zero average derivative. One then obtains the secular relation

$$\left\langle\frac{dQ_{\rm mech}}{dt}\right\rangle =-\left\langle\mathcal F_Q^\infty+\mathcal F_Q^H\right\rangle.$$

Blanchet derives this structure at leading order by comparing radiation-reaction work with the fluxes and displaying the total-time-derivative terms Blanchet 2024, §1.1.2, eqs. (9)–(10), pp. 11–13, PDF. At high post-Newtonian order, the same review explicitly records the domain and evidence for the flux-balance assumption rather than presenting it as an unrestricted identity Blanchet 2024, §3.4, eq. (447), p. 194, PDF.

When secular balance is justified

Flux balance is most powerful when three conditions hold:

1. Closed boundaries. Every relevant escape channel—null infinity, each horizon, and any matter boundary—is included with one orientation.
2. Separated timescales. The radiation-reaction time is long compared with the orbital or oscillation time, so averaging defines slowly evolving invariants.
3. A matched charge. The source energy or angular momentum is expressed in the same asymptotic time, rotational frame, approximation order, and parameter convention as the flux.

For a slowly evolving, nonprecessing circular system with fixed masses and spins at the conservative order used, the conservative sequence supplies $E(\Omega)$ and $J(\Omega)$. The leading adiabatic waveform is approximately helically symmetric: each oscillatory mode has $\omega=m\Omega$. This gives

$$\mathcal F_E=\Omega\mathcal F_J, \qquad \frac{dE}{d\Omega}=\Omega\frac{dJ}{d\Omega},$$

at that adiabatic order, so energy and angular-momentum balances describe the same secular drift. An exactly helically symmetric, eternally radiating binary is not an isolated asymptotically flat system with finite total radiated energy; helical symmetry here is the instantaneous circular approximation. Changes in masses or spins, slow amplitude drift, and nonoscillatory memory require their corresponding higher-order terms. The general multipolar fluxes and their time-averaged balance equations are given in Blanchet 2024, §2.4, eqs. (179)–(182), pp. 79–81, PDF, with independent STF normalizations in Thorne 1980, §§II–IV and Tables I–II.

Where flux balance is insufficient

A boundary flux does not determine the instantaneous local self-force. Several situations require more information:

• During a short burst, plunge, merger, or other nonadiabatic episode, the near-field derivative need not average away.
• At a transient resonance, phase-dependent changes can survive an otherwise adiabatic averaging scheme.
• For eccentric or precessing motion, energy and one angular-momentum component do not determine every evolving invariant; additional actions or local information may be required.
• A finite-radius waveform can contain gauge and stored-field contributions that are absent from the Bondi flux.
• An incomplete mode sum can bias momentum and angular-momentum flux more strongly than total energy.
• A changing BMS frame, time coordinate, or spatial origin changes the charge components and can manufacture an apparent imbalance.

Balance can still be used across a nonadiabatic interval if one compares well-defined initial and final charges and includes every boundary and residual field term. What fails is the inference that an instantaneous mechanical evolution law follows from the asymptotic luminosity alone.

Circular quadrupole check

For two masses with total physical mass $M$, reduced mass $\mu=\eta M$, and a Newtonian circular frequency $\Omega$, the leading conservative binding energy and outward quadrupole luminosity are

$$E(\Omega) =-\frac12\mu(GM\Omega)^{2/3}, \qquad \mathcal F_E^\infty =\frac{32}{5G}\eta^2(GM\Omega)^{10/3}.$$

The flux normalization follows from the Einstein quadrupole formula Blanchet 2024, §1.1.2, eq. (5), pp. 10–11, PDF. With no horizon at this order, $\dot E=-\mathcal F_E^\infty$. Since $dE/d\Omega<0$, the frequency must increase: radiative energy loss drives an inspiral along the circular sequence. This is a consistency check on signs, not a derivation of the underlying radiation-reaction force.

Exercise

Use the two formulas above and $\dot E=(dE/d\Omega)\dot\Omega$ to derive the leading secular $\dot\Omega$. Show explicitly that it is positive.

Solution

Differentiating the binding energy gives

$$\frac{dE}{d\Omega} =-\frac13\eta M(GM)^{2/3}\Omega^{-1/3}.$$

Substituting $\dot E=-\mathcal F_E^\infty$ therefore yields

$$\dot\Omega =\frac{96}{5}\eta(GM)^{5/3}\Omega^{11/3}>0.$$

Both $dE/d\Omega$ and $\dot E$ are negative, so their ratio is positive. A positive ordinary horizon-absorption flux would add to the total loss and increase this drift; a negative superradiant horizon contribution would partially offset it within the regime where adiabatic balance remains valid.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI. Open PDF.
• Thorne, Kip S. “Multipole Expansions of Gravitational Radiation.” Reviews of Modern Physics 52 (1980): 299–339. DOI.