Angular-Momentum Flux

Radiative angular momentum is the Noether charge associated with asymptotic rotations, not a positive norm like radiated energy. Its flux therefore carries a sign, depends on the chosen spatial origin, and at null infinity requires a specified Bondi frame. Once those choices are fixed, a rotation generator acting on the complex strain gives an unambiguous flux, and each azimuthal waveform mode carries angular momentum in the ratio $m/\omega$ to its energy. This page constructs that result without importing binary equations of motion or detector conventions.

Helpful background. Einstein Equations supplies the asymptotic constraints; Waves, Green Functions, and Geometric Optics supplies retarded outgoing fields; Gravitational Radiation and Waveforms supplies strain modes; Causality and Global Structure supplies Bondi cuts and BMS transformations; and Black-Hole Geometry and Mechanics supplies horizon generators and rotational charges.

Rotations of the radiative field

Work in a future-directed Bondi frame at $\mathscr I^+$ with the outward unit-sphere measure $d\Omega$. Use

$$H=C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r(h_+-ih_\times), \qquad \mathcal N=\partial_uH,$$

where $m^A\bar m_A=1$. Choose a spatial origin and a right-handed Cartesian triad. The rotation about its $z$ axis is generated on the standard spin frame by $\partial_\phi$. The outward $z$-angular-momentum flux is

$$\mathcal F_{J_z}^{\infty} =-\frac{1}{16\pi G} \operatorname{Re}\int_{S^2} (\partial_\phi H)\,\overline{\mathcal N}\,d\Omega.$$

The minus sign is fixed by the outward orientation and the $h=h_+-ih_\times$ convention. For rotations about other axes, $\partial_\phi$ is replaced by the corresponding spin-weighted rotation operator; its action includes the dyad rotation, not just a scalar directional derivative. The STF and pure-spin-harmonic derivations of the full vector flux are tabulated in Thorne 1980, §§II–IV and Tables I–II, while the equivalent radiative-moment flux is displayed in Blanchet 2024, §2.4, eqs. (179)–(180), pp. 79–80, PDF.

Unlike energy flux, $\mathcal F_{J_z}^{\infty}$ need not be positive. A pattern rotating in the positive azimuthal direction carries positive $J_z$; a counterrotating pattern carries negative $J_z$. An axisymmetric time-dependent wave may carry energy, but because $\partial_\phi H=0$, it carries no $z$ angular momentum.

The azimuthal mode check

Expand in orthonormal spin-weight $-2$ spherical harmonics,

$$H=\sum_{\ell,m}H_{\ell m}(u)\,{}_{-2}Y_{\ell m}, \qquad \partial_\phi{}_{-2}Y_{\ell m}=im\,{}_{-2}Y_{\ell m}.$$

Sphere orthogonality then gives

$$\mathcal F_{J_z}^{\infty} =\frac{1}{16\pi G} \sum_{\ell,m}m\, \operatorname{Im}\!\left( H_{\ell m}\dot{\bar H}_{\ell m} \right).$$

For $H_{\ell m}=A_{\ell m}e^{-i\omega u}$ with real $\omega>0$, one has $H_{\ell m}\dot{\bar H}_{\ell m}=i\omega\lvert A_{\ell m}\rvert^2$. Hence a mode with $m>0$ carries positive $J_z$ and satisfies

$$\mathcal F_{J_z}^{\infty} =\frac{m}{\omega}\mathcal F_E^{\infty}.$$

This is an independent sign check, not a universal relation for a broadband superposition. Cross terms between distinct orthonormal $(\ell,m)$ modes vanish in the sphere integral, but a time-dependent change of frame can mix modes and contributes its own rotation rate.

Origin and supertranslation dependence

Ordinary angular momentum already depends on origin: under a translation by $a^i$, the spatial angular-momentum tensor changes by a term proportional to $a^{[i}P^{j]}$. Its flux changes correspondingly when linear momentum is radiated. A recoil calculation that shifts the origin during the waveform without transforming the charge and flux together therefore creates a spurious torque.

At null infinity there is an additional issue. A supertranslation changes the cut and adds a time-independent electric-parity term to the shear. The news transforms homogeneously after the fields are pulled back, but the shear appearing in the angular-momentum flux changes. Consequently an instantaneous angular-momentum charge or flux is not a frame-free scalar. Boyle gives the required BMS transformations of asymptotic waveform data in Boyle 2016, §§II–VI, pp. 2–17, PDF. Nichols explains how angular-momentum flux participates in the spin-memory balance law and why its magnetic-parity content must remain distinct from displacement memory Nichols 2017, §§II–III, PDF.

The practical rule is to state the Bondi rest frame, spatial origin, triad, polarization dyad, retarded-time origin, and supertranslation frame. If two descriptions use different cuts, transform both charge and flux before comparing them. A change of frame may move a total-derivative contribution between the endpoint charge and the integrated radiative flux; it does not create new physical radiation.

Source and horizon signs

Let $J_z^{\rm source}$ denote the angular momentum of an exterior mechanical subsystem, excluding the black holes’ evolving spins, in the same asymptotic frame. Let $\mathcal F_{J_z}^{\infty}$ be the signed outward flux through $\mathscr I^+$ and $\mathcal F_{J_z}^{H}$ the signed transfer into future horizons, when a consistent approximate rotational generator is available. A balance law has the orientation

$$\frac{dJ_z^{\rm source}}{du} =-\mathcal F_{J_z}^{\infty}-\mathcal F_{J_z}^{H} \quad\text{up to stored-field, Schott, and nonadiabatic terms}.$$

For an ordinarily absorbed corotating mode, $\mathcal F_{J_z}^{H}>0$ and the exterior subsystem loses additional angular momentum. In a superradiant channel the horizon can return rotational energy and angular momentum, so the horizon term may be negative. If the charge instead includes the holes’ spins, their increase cancels this internal transfer; it must not be counted again as loss of total angular momentum. Generic precessing or dynamical horizons require a specified quasi-local rotational prescription before their charges can be combined with asymptotic $J_z$.

Exercise

A single normalized mode has $H_{32}=A e^{-i\omega u}$ with $\omega>0$. Compute its energy and $z$-angular-momentum fluxes. Repeat for $m=0$ at the same amplitude and frequency, and explain the difference.

Solution

For the $m=2$ mode,

$$\mathcal F_E^\infty =\frac{\omega^2\lvert A\rvert^2}{16\pi G}, \qquad \mathcal F_{J_z}^\infty =\frac{2\omega\lvert A\rvert^2}{16\pi G} =\frac{2}{\omega}\mathcal F_E^\infty.$$

For $m=0$, the energy flux is unchanged because it depends on $\lvert\dot H\rvert^2$, but $\partial_\phi H=0$ and therefore $\mathcal F_{J_z}^\infty=0$. Time dependence alone carries energy; azimuthal phase winding is what carries $z$ angular momentum.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI. Open PDF.
• Boyle, Michael. “Transformations of Asymptotic Gravitational-Wave Data.” Physical Review D 93 (2016): 084031. DOI. Open PDF.
• Nichols, David A. “Spin Memory Effect for Compact Binaries in the Post-Newtonian Approximation.” Physical Review D 95 (2017): 084048. DOI. Open PDF.
• Thorne, Kip S. “Multipole Expansions of Gravitational Radiation.” Reviews of Modern Physics 52 (1980): 299–339. DOI.