Energy, Momentum, and Angular-Momentum Fluxes

Gravitational radiation transports energy, linear momentum, and angular momentum across null boundaries. At future null infinity, the news determines those fluxes directly. At a black-hole horizon, regular ingoing fields and horizon shear determine absorption or superradiant extraction. Conservation then relates the boundary fluxes to changes of a source charge, provided stored near-field terms and nonadiabatic limitations are kept visible. This chapter builds that common sign and normalization system; it does not derive orbital dynamics, numerical extraction algorithms, or detector inference.

Helpful background. Einstein Equations supplies asymptotic and horizon constraints; Waves, Green Functions, and Geometric Optics supplies outgoing and ingoing characteristic fields; Gravitational Radiation and Waveforms supplies strain and news; Causality and Global Structure supplies future null infinity; and Black-Hole Geometry and Mechanics supplies horizon generators and charges.

Enter the flux system

Read the six pages in order when first learning the subject:

1. Energy Flux at Null Infinity fixes the positive news-squared normalization and converts it to strain modes.
2. Angular-Momentum Flux applies asymptotic rotation generators and exposes origin and supertranslation dependence.
3. Linear-Momentum Flux and Recoil extracts the dipolar part of the radiative intensity and fixes the recoil direction.
4. Bondi Mass Loss integrates the asymptotic mass constraint and separates gravitational from matter flux.
5. Flux Through Black-Hole Horizons relates regular horizon fields, shear, area growth, absorption, and superradiant signs.
6. Balance Laws and Secular Evolution joins infinity and horizon boundaries to source changes and states when averaging is justified.

Readers who already know Bondi–Sachs theory may begin with the energy page. Any use of balance in a rotating black-hole system should include the horizon page, because “absorption” is not sign definite with respect to asymptotic Killing energy.

One convention dictionary

Use retarded Bondi coordinates $(u,r,x^A)$ with $u$ future increasing, $r$ outward increasing, and $d\Omega$ the positively oriented unit-sphere measure. The angular metric, shear, and news are

$$g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad N_{AB}=\partial_uC_{AB}.$$

For a dyad normalized by $m^A\bar m_A=1$, define

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

Then $N_{AB}N^{AB}=2\lvert\mathcal N\rvert^2$. Mädler and Winicour use $N^{\rm MW}_{AB}=N_{AB}/2$; their asymptotic constraints must be translated before combining them with these formulas Mädler and Winicour 2018, §3, eqs. (58)–(61), pp. 15–16, PDF.

The boundary orientations are deliberately asymmetric:

Boundary	Positive direction	Energy-flux interpretation
Future null infinity	Outward from the system	$\mathcal F_E^\infty\ge0$ for gravitational news
Future horizon	Into the horizon along future generators	$\mathcal F_E^H>0$ for ordinary absorption; it may be negative for superradiance
Exterior mechanical charge	Subsystem excluding the holes’ changing masses or spins	Its loss includes both boundary fluxes and stored-field changes

This convention makes the schematic source balance

$$\frac{dQ_{\rm source}}{dt} =-\mathcal F_Q^\infty-\mathcal F_Q^H -\frac{dQ_{\rm near}}{dt}.$$

Here $Q$ is energy or angular momentum with the necessary approximate symmetry and $Q_{\rm near}$ records reversible field storage or a Schott-type total derivative. Dropping it requires an exact endpoint argument or a controlled averaging limit. Including the holes’ changing charges in the total Bondi system cancels the horizon transfer; only flux to infinity leaves that total system. Horizon linear momentum needs a separate prescription because there is no preferred horizon translation symmetry.

The core flux relations

The fluxes at $\mathscr I^+$ use the same radiative data, but angular momentum also depends on strain and the chosen rotation generator:

$$\mathcal F_E^\infty =\frac{1}{32\pi G}\int N_{AB}N^{AB}d\Omega,$$ $$\mathcal F_{P^i}^\infty =\frac{1}{32\pi G}\int n^iN_{AB}N^{AB}d\Omega,$$ $$\mathcal F_{J_z}^\infty =-\frac{1}{16\pi G} \operatorname{Re}\int(\partial_\phi H)\overline{\mathcal N}\,d\Omega.$$

Energy is the monopolar integral of a nonnegative density. Linear momentum is its dipolar, direction-weighted integral. Angular momentum instead measures the action of a rotation on the strain and is sensitive to the origin and BMS frame. Thorne’s STF and pure-spin-harmonic treatment supplies a common multipolar derivation Thorne 1980, §§II–IV and Tables I–II; Blanchet gives the modern radiative-moment flux and balance formulas Blanchet 2024, §2.4, eqs. (179)–(182), pp. 79–81, PDF.

The sphere-integrated mass constraint yields

$$\frac{dE_B}{du}=-\mathcal F_E^\infty$$

in vacuum. With outward matter at null infinity, its positive flux is added to the right-hand loss. Locally, however, the Bondi mass aspect can increase at some angles because a sphere divergence redistributes it.

Massive matter ending at timelike infinity is not an outward null flux and must enter through endpoint charges or another boundary. This distinction prevents a convenient Bondi formula from being applied outside its causal domain.

For averaged fluxes into a weakly perturbed nonextremal rotating horizon, with background generator $\chi^a=t^a+\Omega_H\phi^a$,

$$\mathcal F_E^H-\Omega_H\mathcal F_J^H =\frac{\kappa}{8\pi G}\left\langle\frac{dA}{dv}\right\rangle\ge0.$$

This rate form describes the averaged change between nearby stationary states. Before averaging, Raychaudhuri retains an area-acceleration term: $\mathcal P_\chi=(\kappa\dot A-\ddot A)/(8\pi G)$ at the retained perturbative order. The horizon page derives why an event horizon’s instantaneous area growth cannot simply be read as the current absorbed power Poisson 2004, §IV, pp. 10–12, PDF.

For $e^{-i\omega t+im\phi}$, the horizon frequency is $\omega-m\Omega_H$. A positive-frequency mode is superradiant when $0<\omega<m\Omega_H$; then the hole loses Killing energy and angular momentum even though its area grows. The ingoing horizon condition and Kerr critical frequency are reviewed in Berti, Cardoso, and Starinets 2009, §§3.1 and 5.3, pp. 17–18 and 38–39, PDF.

Checks before using a balance law

Every calculation should pass four tests:

• Translate news, strain, dyad, Fourier, and harmonic normalizations before comparing formulas.
• Confirm that energy at infinity is nonnegative and that $\lvert\boldsymbol{\mathcal F}_P^\infty\rvert\le\mathcal F_E^\infty$.
• State the Bondi frame, spatial origin, rotation axis, horizon generator, and all integration intervals.
• Include every boundary and verify that any discarded near-field derivative vanishes by averaging or endpoint conditions.

These checks do not replace a local radiation-reaction calculation. They determine when asymptotic fluxes can reliably supply secular changes and when the missing instantaneous information matters.

Review the chapter

A burst has gravitational luminosity $\mathcal F_E^\infty=4\epsilon$, ordinary horizon absorption $\mathcal F_E^H=\epsilon$, and a near-field storage term $dE_{\rm near}/dt=-2\epsilon$ at one instant, where $\epsilon>0$. Find the instantaneous mechanical-energy change. What is the averaged change if the near-field term integrates to zero over a complete cycle while the two boundary fluxes retain the stated cycle averages?

Solution

The closed budget gives

$$\frac{dE_{\rm mech}}{dt} =-4\epsilon-\epsilon-(-2\epsilon) =-3\epsilon.$$

Instantaneously, release of stored near-field energy offsets part of the boundary loss. Over a complete cycle with $\langle dE_{\rm near}/dt\rangle=0$,

$$\left\langle\frac{dE_{\rm mech}}{dt}\right\rangle =-4\epsilon-\epsilon=-5\epsilon.$$

This difference is precisely why an asymptotic luminosity does not determine an instantaneous mechanical self-force unless the stored-field term is controlled.

References

• Berti, Emanuele, Vitor Cardoso, and Andrei O. Starinets. “Quasinormal Modes of Black Holes and Black Branes.” Classical and Quantum Gravity 26 (2009): 163001. DOI. Open PDF.
• 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.
• Poisson, Eric. “Absorption of Mass and Angular Momentum by a Black Hole: Time-Domain Formalisms for Gravitational Perturbations, and the Small-Hole/Slow-Motion Approximation.” Physical Review D 70 (2004): 084044. DOI. Open PDF.
• Thorne, Kip S. “Multipole Expansions of Gravitational Radiation.” Reviews of Modern Physics 52 (1980): 299–339. DOI.