Balance Laws and Dissipative Forces

Radiative flux is measured at infinity and horizons, whereas radiation reaction is a local force on orbital variables. They agree only after the near-zone field energy stored around the source—the Schott contribution—is included. Instantaneously, mechanical losses need not equal outgoing flux. For periodic or adiabatic motion the average Schott derivative vanishes, recovering the familiar balance law.

Helpful background. The Einstein Equation fixes stress-energy conservation, Weak Metric Perturbations and Gauge Freedom supplies radiative perturbations, Presymplectic Potentials and Currents supplies charges and fluxes, Partial Differential Equations and Green Functions supplies retarded propagation, Gravitational Radiation and Waveforms defines asymptotic fluxes, and Relativistic Two-Body Dynamics supplies the orbital variables.

Local work and asymptotic flux

Let a reduced orbital description have mechanical energy $E_{\mathrm{mech}}$ and dissipative acceleration $\mathbf a_{\mathrm{RR}}$. At leading radiation-reaction order its local power, using Newtonian momenta, is

$$\dot E_{\mathrm{mech}} =\sum_A m_A\mathbf v_A\cdot\mathbf a_{A,\mathrm{RR}}$$

At higher PN orders the velocity–momentum relation and conservative energy also change. In a canonical description the corresponding expression is $\dot E_{\mathrm{mech}}=\sum_A\dot{\mathbf x}_A\cdot\mathbf F_{A,\mathrm{RR}}$ with a consistently defined canonical force. Conservation of the complete gravitational system instead gives

$$\frac{d}{dt} \left( E_{\mathrm{mech}}+E_{\mathrm{Schott}} \right) =-\mathcal F_E^\infty-\mathcal F_E^H.$$

Our sign convention makes each positive $\mathcal F_E$ an outward or horizon-directed loss from the orbital system. The Schott energy is a near-zone field contribution. It is gauge and variable dependent, just as the split into mechanical and field energy is, while the complete balance is not.

Angular momentum obeys the analogous relation

$$\frac{d}{dt} \left( \mathbf J_{\mathrm{mech}}+\mathbf J_{\mathrm{Schott}} \right) =-\boldsymbol{\mathcal F}_J^\infty -\boldsymbol{\mathcal F}_J^H.$$

Linear momentum and center-of-mass balance require their own Schott terms and are essential for recoil. One cannot obtain them by multiplying the energy flux by a velocity.

For a bound periodic source,

$$\left\langle\dot E_{\mathrm{Schott}}\right\rangle=0, \qquad \left\langle\dot{\mathbf J}_{\mathrm{Schott}}\right\rangle=0,$$

so the averaged mechanical losses equal minus the total fluxes. In an adiabatic inspiral, the averages vanish through the stated leading slow order but leave post-adiabatic corrections. In a scattering event, integrate the Schott derivative between asymptotic times and verify that its endpoints vanish in the chosen variables.

The PN balance construction, including hereditary effects and compact-binary fluxes, is reviewed in Blanchet 2024, §§2–3.

Circular and eccentric checks

For a quasicircular nonprecessing conservative sequence at fixed intrinsic masses and spins,

$$\dot E_{\mathrm{mech}} =\Omega\,\dot J_{\mathrm{mech}}$$

along the circular sequence. After orbit averaging, consistency requires

$$\mathcal F_E^\infty+\mathcal F_E^H =\Omega \left( \mathcal F_J^\infty+\mathcal F_J^H \right).$$

This relation checks flux normalization, harmonic signs, and interpolation. If horizon absorption is also used to evolve the bodies’ masses or spins, the associated parameter derivatives of the conservative energy must be included at their retained order; the fixed-parameter relation is not the complete evolution law in that case. It does not hold instantaneously for a generic eccentric orbit, where energy and angular-momentum losses contain distinct harmonics and Schott oscillations.

At leading quadrupole order for a Newtonian circular binary,

with $M=m_1+m_2$ and $\mu=m_1m_2/M$,

$$E_{\mathrm N} =-\frac{G\mu M}{2r}, \qquad \mathcal F_E =\frac{32}{5}G\mu^2r^4\Omega^6, \qquad \Omega^2=\frac{GM}{r^3}.$$

Balance gives

$$\dot r =-\frac{64}{5} \frac{G^3\mu M^2}{r^3},$$

with $c=1$. The negative sign is an immediate test of the flux convention. Substitution back into $\dot E_{\mathrm N}=(dE_{\mathrm N}/dr)\dot r$ recovers $-\mathcal F_E$.

Force ambiguity and physical equivalence

A dissipative force is not unique. A near-identity coordinate transformation or addition of a total time derivative can move terms between $E_{\mathrm{mech}}$, $E_{\mathrm{Schott}}$, and $\mathbf a_{\mathrm{RR}}$. Two forces are physically equivalent when they give the same asymptotic fluxes and invariant orbital evolution after the variables are transformed consistently.

This ambiguity is why “derive the force from flux balance” is underdetermined without a gauge choice. Balance constrains the force; a retarded field, near-zone potential, or in-in effective action supplies a definite representative. The potential/radiation split and the distinction between local reaction and asymptotic flux are treated in Porto 2016, §§7.3–7.7, pp. 52–68.

The regime here assumes weakly nonstationary orbital dynamics with a clean radiation zone and a perturbative dissipative force. Near merger, during a nonadiabatic resonance, or with incoming radiation, the instantaneous and averaged balance statements need appropriate boundary and matching terms.

A practical implementation should report the instantaneous balance residual and its accumulated integral. Refining timestep, multipole order, and waveform extraction should reduce that residual at the expected rate. A small orbit-averaged residual alone can hide large oscillatory errors that later couple to eccentricity or a resonance.

Exercises

Derive the leading circular inspiral rate from the displayed energy and flux.

Solution

Since

$$\frac{dE_{\mathrm N}}{dr} =\frac{G\mu M}{2r^2},$$

balance gives

$$\dot r =-\frac{\mathcal F_E}{dE_{\mathrm N}/dr}.$$

Using Kepler’s law in the flux, $\mathcal F_E=(32/5)G^4\mu^2M^3/r^5$, yields $\dot r=-(64/5)G^3\mu M^2/r^3$.

Why may the instantaneous mechanical power be positive during part of an orbit even though the averaged system loses energy?

Solution

Near-zone field energy can flow back into the mechanical variables, giving an oscillatory Schott derivative. The complete instantaneous relation includes that derivative. Over a periodic orbit its average vanishes, leaving negative mechanical loss equal to positive total radiative flux.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI.
• Porto, Rafael A. “The Effective Field Theorist’s Approach to Gravitational Dynamics.” Physics Reports 633 (2016): 1–104. DOI.