Energy Flux at Null Infinity

Gravitational waves carry a positive outward energy flux through future null infinity. In a Bondi frame that flux is the squared norm of the news; in a large-radius wave zone it is the squared retarded-time derivative of the complex strain. The two descriptions agree only after their radius, dyad, and normalization conventions are matched. This page performs that match with $G$ explicit and with outward flux defined as positive. It treats asymptotically flat four-dimensional general relativity; finite-radius extraction, cosmological redshift, detector response, and source dynamics lie outside the calculation.

Helpful background. Einstein Equations supplies the asymptotic field equation behind the mass constraint; Waves, Green Functions, and Geometric Optics supplies outgoing retarded propagation; Gravitational Radiation and Waveforms supplies the complex-strain convention; Causality and Global Structure supplies future null infinity; and Black-Hole Geometry and Mechanics supplies the distinction between null infinity and a horizon boundary.

News as an energy-carrying field

Use retarded Bondi coordinates $(u,r,x^A)$, with $u$ increasing toward the future, $r$ an areal radius, and $d\Omega$ the positively oriented measure of the unit-sphere metric $q_{AB}$. The outward radial direction points toward increasing $r$. The angular metric has the expansion

$$g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad q^{AB}C_{AB}=0,$$

and this volume defines

$$N_{AB}=\partial_uC_{AB}.$$

Choose a complex unit-sphere dyad with $m^A\bar m_A=1$, and set

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

The rescaled complex strain is $H=\lim_{r\to\infty}r(h_+-ih_\times)$. Thus $h=h_+-ih_\times=H/r+O(r^{-2})$ in this Bondi frame. The dyad identity gives

$$N_{AB}N^{AB}=2\lvert\mathcal N\rvert^2.$$

Mädler and Winicour instead write $N^{\rm MW}_{AB}=N_{AB}/2$. Translating that factor before importing their mass constraint is essential; otherwise the luminosity is wrong by a factor of four. Their asymptotic hierarchy and mass-loss relation are given in Mädler and Winicour 2018, §3, eqs. (58)–(61), pp. 15–16, PDF.

The positive outward gravitational energy flux is

$$\frac{dE_{\rm out}}{du\,d\Omega} =\frac{1}{32\pi G}N_{AB}N^{AB} =\frac{1}{16\pi G}\lvert\mathcal N\rvert^2.$$

This is an asymptotic statement, not a local gravitational stress tensor. In a short-wavelength wave zone, the Isaacson effective stress–energy gives the same leading result after averaging over several wavelengths Flanagan and Hughes 2005, §5.3, eqs. (5.35)–(5.40), pp. 28–29, PDF. The Bondi expression itself does not require the radiation to be monochromatic or weak near the source; it requires an asymptotically flat region, an admissible Bondi conformal frame, and the stated smoothness and falloffs at $\mathscr I^+$.

Strain and mode formulas

At a large areal radius, the same luminosity can be written

$$\frac{dE_{\rm out}}{du} =\lim_{r\to\infty}\frac{r^2}{16\pi G} \int_{S^2} \left(\dot h_+^2+\dot h_\times^2\right)d\Omega =\frac{1}{16\pi G}\int_{S^2}\lvert\dot H\rvert^2d\Omega,$$

where a dot denotes $\partial_u$. The formula presumes that $r$ is the Bondi areal radius and that $h$ has been placed in the same asymptotic frame as $H$. Replacing $r$ by an arbitrary coordinate radius or differentiating a finite-radius gauge waveform does not reproduce a Bondi flux automatically.

Expand the spin-weight $-2$ field in orthonormal harmonics,

$$H(u,x^A)=\sum_{\ell=2}^{\infty}\sum_{m=-\ell}^{\ell} H_{\ell m}(u)\,{}_{-2}Y_{\ell m}(x^A), \qquad \int_{S^2}{}_{-2}Y_{\ell m}\,{}_{-2}\bar Y_{\ell' m'}\,d\Omega =\delta_{\ell\ell'}\delta_{mm'}.$$

Orthogonality removes cross terms only after the full sphere integral, giving

$$\frac{dE_{\rm out}}{du} =\frac{1}{16\pi G} \sum_{\ell,m}\lvert\dot H_{\ell m}\rvert^2.$$

This expression is positive term by term and invariant under a time-independent rotation of the polarization dyad. A time-dependent dyad rotation adds a frame-rotation term to $\dot H$ and therefore must be removed or included explicitly before interpreting the result as physical luminosity. Thorne’s STF and pure-spin-harmonic flux formulas give the equivalent multipolar description Thorne 1980, §§II–IV and Tables I–II; Blanchet supplies the modern radiative-moment dictionary and its flux checks Blanchet 2024, §2.4, pp. 79–103, PDF.

Energy loss versus outward luminosity

The quantity above is the energy crossing $\mathscr I^+$ outward. It is not, by definition, the instantaneous mechanical energy lost by a chosen near-zone subsystem. Integrating the Bondi constraint over a cut gives

$$\frac{dE_B}{du}=-\frac{dE_{\rm out}}{du}$$

in vacuum at null infinity, including when the isolated system contains black holes. Horizon absorption transfers energy within that total system and is not an additional loss from Bondi energy. For a different budget that follows only an exterior mechanical subsystem, such as an orbit excluding the holes’ changing masses, define $\mathcal F_E^H$ as flux directed into the future horizons. The corresponding balance has the sign convention

$$\frac{dE_{\rm source}}{du} =-\mathcal F_E^\infty-\mathcal F_E^H \quad\text{up to stored-field and nonadiabatic boundary terms}.$$

Here $\mathcal F_E^\infty=dE_{\rm out}/du\ge0$, and the mechanical energy and horizon rates must use one consistently matched time and approximation. The horizon term is positive for ordinary absorption but can be negative for a superradiant rotating-horizon mode. Adding the evolving horizon masses back into the energy budget cancels that internal transfer and recovers the Bondi loss to infinity.

Three quick checks catch most normalization errors. First, $N_{AB}=0$ makes the luminosity vanish, even if a time-independent shear remains. Second, $N_{AB}N^{AB}\ge0$ because $q_{AB}$ is positive definite. Third, $C_{AB}$ and $H$ have dimensions of length when $u$ does, so $\mathcal N$ is dimensionless and $G^{-1}\lvert\mathcal N\rvert^2$ has the dimensions of power in $c=1$ units.

Exercise

Let one normalized waveform mode be $H_{\ell m}(u)=A e^{-i\omega u}$ with real $\omega>0$, and let every other mode vanish. Compute its outward luminosity and verify directly that a constant phase rotation of $A$ changes no energy flux.

Solution

The derivative is $\dot H_{\ell m}=-i\omega A e^{-i\omega u}$, so orthonormality gives

$$\mathcal F_E^\infty =\frac{1}{16\pi G}\lvert\dot H_{\ell m}\rvert^2 =\frac{\omega^2\lvert A\rvert^2}{16\pi G}.$$

Replacing $A$ by $e^{i\alpha}A$ leaves $\lvert A\rvert^2$ unchanged. The result is positive, has the expected quadratic dependence on amplitude and frequency, and agrees with the news-squared expression because $\mathcal N=-i\omega H$.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024): 4. DOI. Open PDF.
• Flanagan, Éanna É., and Scott A. Hughes. “The Basics of Gravitational Wave Theory.” New Journal of Physics 7 (2005): 204. 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.