Linear-Momentum Flux and Recoil

Radiation carries net linear momentum when its angular energy distribution has a nonzero dipole. The corresponding source momentum changes in the opposite direction, producing recoil. Anisotropy alone is insufficient: an intensity that is equal in opposite directions has zero net momentum. At future null infinity the construction is especially direct: weight the nonnegative news intensity by the outward unit vector $n^i$ and integrate over the sphere. The result defines the asymptotic momentum flux; converting its time integral into the motion of a finite remnant belongs to black-hole dynamics, while detector inference lies still farther downstream.

Helpful background. Einstein Equations supplies the Bondi momentum constraint; Waves, Green Functions, and Geometric Optics supplies outgoing retarded propagation; Gravitational Radiation and Waveforms supplies the news and strain conventions; Causality and Global Structure supplies future null infinity and its cuts; and Black-Hole Geometry and Mechanics distinguishes momentum lost to infinity from flux into horizons.

Momentum in the outgoing radiation

Fix one asymptotic Bondi frame, initially at rest with the source, an origin, and a right-handed Cartesian triad. Keep that frame fixed while recoil develops. On each unit-sphere direction let $n^i(x^A)$ be the outward radial unit vector. With

$$N_{AB}=\partial_uC_{AB}, \qquad \mathcal N=N_{AB}\bar m^A\bar m^B=\partial_uH,$$

and $d\Omega$ the unit-sphere measure, the positive-outward energy and signed linear-momentum fluxes are

$$\mathcal F_E^\infty =\frac{1}{32\pi G}\int_{S^2}N_{AB}N^{AB}d\Omega,$$ $$\mathcal F_{P^i}^\infty =\frac{1}{32\pi G}\int_{S^2}n^iN_{AB}N^{AB}d\Omega =\frac{1}{16\pi G}\int_{S^2}n^i\lvert\mathcal N\rvert^2d\Omega.$$

Every angular element transports future-directed null four-momentum proportional to $k^\mu=(1,n^i)$. Therefore $\lvert\boldsymbol{\mathcal F}_P^\infty\rvert\le\mathcal F_E^\infty$: the magnitude of the net momentum cannot exceed the emitted energy in $c=1$ units. The formula follows as the $\ell=1$ part of the Bondi four-momentum constraint after translating Mädler and Winicour’s news normalization $N^{\rm MW}_{AB}=N_{AB}/2$ Mädler and Winicour 2018, §3, eqs. (58)–(61), pp. 15–16, PDF.

The sign is geometric. If more radiation leaves toward $+z$, then $\mathcal F_{P^z}^\infty>0$ and the isolated source acquires negative $z$ momentum. The flux vanishes whenever the news intensity obeys antipodal symmetry,

$$\lvert\mathcal N(u,n^i)\rvert^2 =\lvert\mathcal N(u,-n^i)\rvert^2,$$

because $n^i$ is odd under the antipodal map.

How waveform modes carry net momentum

Writing $H=\sum_{\ell m}H_{\ell m}\,{}_{-2}Y_{\ell m}$ makes the energy flux diagonal only after an unweighted sphere integral. Momentum inserts the $\ell=1$ function $n^i$, so the relevant integrals are triple products of harmonics. The recoil is therefore controlled by the dipolar part of the quadratic intensity: interference among allowed radiative modes, and unequal power in symmetry-related modes, can survive the angular integral. A single complex spin-weighted mode can already have asymmetric intensity; interference between distinct $(\ell,m)$ labels is not mandatory. The exercise below makes this distinction explicit.

In a source-multipole expansion, the leading nonzero terms couple the mass quadrupole to the mass octupole and the mass quadrupole to the current quadrupole. A conserved mass monopole or center-of-mass dipole cannot radiate by itself. The explicit leading flux and its source balance are derived in Blanchet 2024, §1.1.4, eqs. (35)–(39), pp. 21–23, PDF; the general radiative-moment hierarchy appears in Blanchet 2024, §2.4, eqs. (181)–(182), pp. 80–81, PDF. Thorne gives the corresponding STF angular integrals and multipole normalizations Thorne 1980, §§II–IV and Tables I–II.

These statements identify the flux mechanism without predicting a particular merger kick. Strong-field waveform production, remnant formation, finite-radius corrections, and the choice of a remnant center-of-mass worldline are separate inputs.

Recoil as a balance statement

Let $P^i_B$ be the Bondi momentum of the complete isolated system in the chosen frame. In vacuum at $\mathscr I^+$,

$$\frac{dP^i_B}{du}=-\mathcal F_{P^i}^\infty.$$

Absorption by a black hole transfers momentum within this system; it must not be subtracted a second time from the Bondi budget. If one follows only an exterior mechanical subsystem, its momentum balance also needs stored-field terms and a specified prescription for exchanges with each hole. Unlike stationary Kerr energy and axial angular momentum, a generic horizon has no preferred spatial-translation Killing field and hence no universal horizon linear-momentum flux obtained merely by choosing an asymptotic triad.

If the initial system is at rest and the final state is one isolated remnant, integration gives $P_f^i=-P_{\rm rad}^i$. Include any escaping matter or additional surviving bodies in that endpoint budget. In the chosen inertial frame the exact remnant velocity is

$$v_f^i=\frac{P_f^i}{E_f} =-\frac{P_{\rm rad}^i}{E_i-E_{\rm rad}}, \qquad M_f^2=E_f^2-\lvert\boldsymbol P_f\rvert^2.$$

Only for nonrelativistic recoil does $E_f\simeq M_f$ justify $v_f^i\simeq-P_{\rm rad}^i/M_f$. A boost changes four-momentum components, while a translation changes the waveform origin and retarded cuts. Data in different asymptotic frames must be transformed consistently before comparison.

Symmetry and frame checks

Three tests should precede any recoil interpretation. First, reflect the waveform through every exact source symmetry: if one reflection reverses a candidate momentum component while leaving the radiation invariant, that component must vanish. Second, compute the four-vector inequality $\lvert\boldsymbol{\mathcal F}_P^\infty\rvert\le\mathcal F_E^\infty$ at each retarded time; a violation exposes an inconsistent normalization or angular quadrature. A consistently truncated waveform still obeys this inequality, so passing it does not prove convergence in mode number. Third, repeat the integral after adding successively higher modes: a mode set adequate for total luminosity can still miss the small asymmetry responsible for recoil.

The instantaneous flux can oscillate rapidly and even reverse direction. The physically relevant accumulated momentum is its retarded-time integral over a declared interval, including early-time transients and late-time radiation. A quoted “kick” without the integration bounds, asymptotic frame, mode content, and horizon treatment is not a reproducible flux result. Conversely, a zero time-averaged flux does not imply that the instantaneous momentum flux vanished.

Exercise

Suppose the news has only one normalized spin-weighted mode in the stated sphere frame,

$$\mathcal N=a(u)\,{}_{-2}Y_{22}, \qquad {}_{-2}Y_{22} =\sqrt{\frac{5}{64\pi}}(1+\cos\theta)^2e^{2i\phi}.$$

Compute $\mathcal F_{P^z}^\infty/\mathcal F_E^\infty$ and the direction of recoil. What changes if an equal-power $m=-2$ mode, with intensity proportional to $(1-\cos\theta)^4$, is included?

Solution

Put $x=\cos\theta$. The azimuthal integral and normalization cancel in the ratio, leaving

$$\frac{\mathcal F_{P^z}^\infty}{\mathcal F_E^\infty} =\frac{\int_{-1}^1x(1+x)^4dx}{\int_{-1}^1(1+x)^4dx} =\frac{64/15}{32/5}=\frac23.$$

Thus $\mathcal F_E^\infty=\lvert a\rvert^2/(16\pi G)$ and the source recoils toward $-z$. A lone complex strain mode is allowed: its real and imaginary parts specify two real polarizations, so metric reality alone does not require equal $m$ and $-m$ amplitudes. The added equal-power $m=-2$ mode has the opposite momentum flux; the cross term integrates to zero in $\phi$, and the net $z$ flux vanishes. Equal-power pairing can follow from additional source symmetries, but it is not the reality condition for a spin-weight $-2$ field.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI. Open PDF.
• Mädler, Thomas, and Jeffrey Winicour. “Bondi–Sachs Formalism.” Scholarpedia 11, no. 12 (2016): 33528; expanded arXiv version, revised 2018. Open PDF.
• Thorne, Kip S. “Multipole Expansions of Gravitational Radiation.” Reviews of Modern Physics 52 (1980): 299–339. DOI.