EOB Radiation Reaction and Wave Generation

EOB inspiral is produced by coupling a resummed conservative Hamiltonian to a dissipative force constructed from the same multipolar waveform that carries energy and angular momentum to infinity. This common origin is essential: the orbital phase used in the modes must agree with the phase entering the torque, and the mechanical losses must satisfy balance up to explicitly identified near-zone, or Schott, terms. The discussion here assumes retarded, no-incoming-radiation boundary conditions and focuses on quasicircular black-hole binaries before merger.

Required background. Presymplectic Potentials and Currents supplies Hamiltonian evolution and the distinction between conservative flow and non-Hamiltonian forcing.

Helpful background. The Einstein Equation supplies the radiative field equations; Weak Metric Perturbations and Gauge Freedom supplies the leading radiation limit; Black-Hole Perturbation Theory supplies strong-field test-body fluxes and ringdown; and Relativistic Two-Body Dynamics supplies the orbital invariants and approximation regimes.

Forced EOB dynamics and balance

For planar motion use physical canonical variables $(r,p_r,\phi,p_\phi)$ and the real binding Hamiltonian $H_{\rm EOB}=H_{\rm real}-M$. Here $p_r$ has momentum units and $p_\phi$ is an angular momentum; neither is the dimensionless reduced momentum used on the potentials page. A general local radiation-reaction force gives

$$\dot r=\frac{\partial H_{\rm EOB}}{\partial p_r}, \qquad \dot\phi=\frac{\partial H_{\rm EOB}}{\partial p_\phi} \equiv\Omega,$$ $$\dot p_r=-\frac{\partial H_{\rm EOB}}{\partial r} +\mathcal F_r, \qquad \dot p_\phi=\mathcal F_\phi.$$

Along the forced trajectory,

$$\frac{dH_{\rm EOB}}{dt} =\dot r\,\mathcal F_r+\Omega\mathcal F_\phi.$$

The fluxes $\Phi_E$ and $\Phi_J$ carried out of the near zone obey

$$\frac{dH_{\rm EOB}}{dt} +\Phi_E+\frac{dE_{\rm Schott}}{dt}=0,$$ $$\dot p_\phi+\Phi_J +\frac{dJ_{\rm Schott}}{dt}=0.$$

Schott terms are reversible energy and angular momentum stored in the near-zone field. They depend on the radiation-reaction gauge and generally cannot be set to zero for arbitrary eccentric motion. In a standard quasicircular gauge, $\mathcal F_r$ is set to zero at leading adiabatic order, the Schott contribution is negligible at that order, and $\Phi_E=\Omega\Phi_J$. Then

$$\mathcal F_\phi =-\Phi_J=-\frac{\Phi_E}{\Omega}.$$

This is a prescription with stated assumptions, not an identity for generic motion. The original EOB treatment introduces a resummed azimuthal damping force and follows the loss of adiabaticity into plunge Buonanno and Damour 1999, § VII, PDF.

Multipolar waveform and flux

At a large extraction radius $R$, decompose the complex strain as

$$h_+-ih_\times =\frac{1}{R} \sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} H_{\ell m}(t)\,{}_{-2}Y_{\ell m}(\iota,\varphi).$$

Here $t$ denotes asymptotic retarded time and $H_{\ell m}=Rh_{\ell m}$ has dimensions of length. Normalize the spin-weighted harmonics by $\int|{}_{-2}Y_{\ell m}|^2d\Omega=1$. The energy flux to infinity is

$$\Phi_E =\frac{1}{16\pi G} \sum_{\ell,m}\left\lvert\dot H_{\ell m}\right\rvert^2.$$

With the same polarization and azimuthal convention, the outward angular-momentum flux along the orbital axis is

$$\Phi_J =-\frac{1}{16\pi G} \sum_{\ell,m}m\,\operatorname{Im} \left(H_{\ell m}^*\dot H_{\ell m}\right).$$

For $\dot H_{\ell m}=-im\Omega H_{\ell m}$, every nonzero-$m$ mode gives nonnegative angular-momentum loss when $\Omega\gt0$, and the two formulas directly yield $\Phi_E=\Omega\Phi_J$. This derivation also makes clear why the equality is restricted to circular harmonics. Horizon fluxes, when included, are separate boundary contributions.

For a slowly evolving circular orbit choose the phase convention

$$H_{\ell m}=A_{\ell m}e^{-im\phi},$$

so that $\dot H_{\ell m}\simeq-im\Omega H_{\ell m}$. The same convention gives

$$\Phi_E \simeq\frac{1}{16\pi G} \sum_{\ell,m}(m\Omega)^2 \left\lvert H_{\ell m}\right\rvert^2.$$

Using $e^{+im\phi}$ in one part of a code and $e^{-im\phi}$ in another reverses signs in phase derivatives and can produce an incorrect torque even if waveform amplitudes look reasonable. A robust implementation computes the flux from the actual complex modes and verifies $-\Omega\mathcal F_\phi=\Phi_E$ on a circular sequence.

The leading quadrupole result is recovered with

$$x=(GM\Omega)^{2/3}, \qquad \Phi_E^{\rm N} =\frac{32}{5G}\nu^2x^5.$$

Together with the Newtonian circular binding energy $E=-\mu x/2$, this gives

$$\dot x =\frac{64}{5GM}\nu x^5.$$

These expressions check the powers of $G$, the mass-ratio normalization, and the sign of the dissipative force. The PN derivation and higher-order hereditary contributions to flux and phase are reviewed in Blanchet 2024, § 3.3, especially Eqs. (392)–(401).

Factorized and resummed modes

A Taylor-expanded PN mode deteriorates as the orbital velocity increases. EOB wave generation instead reorganizes each inspiral–plunge mode into physically motivated factors. Define the parity index by $\pi_{\ell m}=(\ell+m)\bmod2$, with value zero for even-parity mass-type modes and one for odd-parity current-type modes:

$$H_{\ell m}^{\rm IP} =H_{\ell m}^{\rm N} \widehat S_{\rm eff}^{(\pi_{\ell m})} T_{\ell m} e^{i\delta_{\ell m}} \left(\rho_{\ell m}\right)^\ell N_{\ell m}.$$

The factors have distinct roles:

• $H_{\ell m}^{\rm N}$ contains the leading multipolar amplitude and the phase $e^{-im\phi}$.
• $\widehat S_{\rm eff}^{(\pi_{\ell m})}$ uses an effective energy or angular momentum and captures a known strong-field source behavior.
• $T_{\ell m}$ resums leading logarithmic tails generated by propagation in the monopolar background.
• $\delta_{\ell m}$ is a residual phase correction.
• $\rho_{\ell m}$ is the $\ell$-th root of the residual amplitude, introduced to avoid coefficients that grow artificially with $\ell$.
• $N_{\ell m}$ denotes nonquasicircular corrections, which approach unity during adiabatic circular inspiral and become important near plunge.

The factorization and the $\rho_{\ell m}$ prescription are developed and tested against test-mass numerical waveforms in Damour, Iyer, and Nagar 2009, § II, Eqs. (4)–(22), and §§ III–IV. Re-expanding the product to the known PN order must reproduce the original Taylor mode. Agreement after re-expansion is a necessary algebraic check; improvement in the strong field remains an empirical claim that must be tested independently.

The test-mass limit supplies more than a leading coefficient. As $\nu\to0$, the conservative trajectory becomes a geodesic of the background, and the factorized amplitudes and flux can be compared with solutions of the Regge–Wheeler–Zerilli or Teukolsky equations with ingoing horizon and outgoing infinity boundary conditions. This constrains tail phases, horizon absorption, and strong-field amplitude behavior. At finite $\nu$, PN, self-force, and numerical-relativity information may be added, but its domain must be stated.

From inspiral through plunge and ringdown

During early inspiral, the radiation-reaction timescale greatly exceeds the orbital period and the circular balance prescription is controlled. Near the last stable orbit, radial momentum grows and the hierarchy weakens. EOB evolves the forced Hamilton equations through this transition rather than terminating the waveform at a PN stability estimate. Nonquasicircular factors correct the mode amplitude and frequency using quantities such as $p_{r_*}$, $\dot r$, and $\ddot r$.

An inspiral–plunge mode is then joined near its amplitude peak to a merger–ringdown representation,

$$H_{\ell m}^{\rm RD}(t) =\sum_n C_{\ell mn} e^{-i\omega_{\ell mn}(t-t_{\rm match})},$$

where the complex Kerr quasinormal frequencies depend on the modeled remnant mass and spin. Continuity conditions determine the coefficients over a chosen matching interval. This attachment is not derived from the balance law. Its time, mode content, remnant fits, and any calibration are additional modeling inputs. The early EOB transition analysis and quasinormal-mode matching logic are presented in Buonanno and Damour 2000, §§ II–IV.

The displayed sum is a simplified attachment retaining one angular family. Kerr quasinormal modes use spin-weighted spheroidal harmonics, whereas $H_{\ell m}$ above uses spherical harmonics. At nonzero remnant spin, one spherical mode can therefore contain several spheroidal angular indices at the same $m$, each with its own frequencies and mixing coefficients. This effect must be included when it matters at the requested accuracy Berti and Klein 2014, § I, Eqs. (1)–(5).

Independent checks and limitations

Four checks should be performed together. First, re-expand every factorized mode and recover the known PN amplitude and phase. Second, take $\nu\to0$ and compare fluxes and modes with perturbation theory on the same orbit. Third, on a quasicircular sequence verify both $\Phi_E=\Omega\Phi_J$ and $dH_{\rm EOB}/dt=-\Phi_E$ within the retained order. Fourth, compute the Newtonian chirp above from the implemented torque.

The simple balance prescription breaks down for highly eccentric motion, hyperbolic encounters, strong precession, and during merger. Radial radiation reaction and Schott terms then matter; the instantaneous relation $\Phi_E=\Omega\Phi_J$ is no longer valid. Horizon absorption adds a second physical boundary flux. Tidal or matter systems require additional internal-energy channels. Finally, a small mismatch between two completed waveforms does not identify whether the error came from conservative dynamics, flux, nonquasicircular factors, ringdown attachment, or calibration. Those sectors should be compared separately whenever possible.

Common pitfalls

Equating flux with force without a gauge choice. Balance constrains combinations of the force and Schott terms. Setting $\mathcal F_r=0$ is a quasicircular radiation-reaction gauge, not a general theorem.

Using inconsistent mode phases. The sign in $e^{-im\phi}$, the definition of $h_+-ih_\times$, and the torque must form one convention set.

Treating ringdown attachment as flux balance. Quasinormal-mode matching is a separate strong-field completion with its own parameters and validation.

Exercise

Starting from the Newtonian circular energy and quadrupole flux, derive $\dot x$ and the accumulated orbital phase as functions of $x$. State which sign corresponds to inspiral.

Solution

Balance gives

$$\frac{dE}{dx}\dot x=-\Phi_E, \qquad \frac{dE}{dx}=-\frac{\mu}{2}.$$

Therefore

$$\dot x =\frac{2\Phi_E}{\mu} =\frac{64}{5GM}\nu x^5\gt 0.$$

The positive sign means the frequency parameter grows as the binary loses energy. Since $\dot\phi=\Omega=x^{3/2}/(GM)$,

$$\frac{d\phi}{dx} =\frac{\Omega}{\dot x} =\frac{5}{64\nu}x^{-7/2}.$$

Integrating toward a formal coalescence value gives

$$\phi_c-\phi =\frac{1}{32\nu}x^{-5/2}.$$

The approximation is valid only while the adiabatic, quasicircular Newtonian assumptions hold; the divergence as $x\to0$ represents the large number of early-inspiral cycles, not a finite-time strong-field prediction.

References

• Berti, Emanuele, and Antoine Klein. “Mixing of Spherical and Spheroidal Modes in Perturbed Kerr Black Holes.” Physical Review D 90, 064012 (2014). DOI. Open PDF.
• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27, 4 (2024). DOI.
• Buonanno, Alessandra, and Thibault Damour. “Effective One-Body Approach to General Relativistic Two-Body Dynamics.” Physical Review D 59, 084006 (1999). DOI. Open PDF.
• Buonanno, Alessandra, and Thibault Damour. “Transition from Inspiral to Plunge in Binary Black Hole Coalescences.” Physical Review D 62, 064015 (2000). DOI. Open PDF.
• Damour, Thibault, Bala R. Iyer, and Alessandro Nagar. “Improved Resummation of Post-Newtonian Multipolar Waveforms from Circularized Compact Binaries.” Physical Review D 79, 064004 (2009). DOI. Open PDF.