Bondi Mass Loss

Bondi energy is the energy remaining in an isolated system as seen on a cut of future null infinity. The asymptotic Einstein constraint evolves its angle-dependent mass aspect. A sphere divergence redistributes that aspect without changing the total, while gravitational news and outward matter flux reduce the sphere average. The result is an exact asymptotic balance law under Bondi falloffs, not merely the high-frequency effective-stress approximation. Its monotonicity requires positive gravitational news norm and, when matter reaches null infinity, an appropriate positive-energy condition.

Helpful background. Einstein Equations supplies the supplementary field equation; Waves, Green Functions, and Geometric Optics supplies outgoing retarded propagation; Gravitational Radiation and Waveforms supplies the news convention; Causality and Global Structure supplies cuts of future null infinity; and Black-Hole Geometry and Mechanics distinguishes Bondi energy from horizon charges.

The mass-aspect constraint

Adopt retarded Bondi coordinates $(u,r,x^A)$ with $u$ future increasing, $r$ outward increasing, and $d\Omega$ the measure of the unit-sphere metric $q_{AB}$. Write

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

and let $m(u,x^A)$ be the Bondi mass aspect in the radial convention for which Schwarzschild has $m=GM$. In vacuum, the leading supplementary Einstein equation is

$$\partial_um =\frac14D_AD_BN^{AB} -\frac18N_{AB}N^{AB}.$$

Mädler and Winicour write their tensor news as $N^{\rm MW}_{AB}=N_{AB}/2$. Translating that factor and their mass-aspect normalization produces the displayed equation Mädler and Winicour 2018, §3, eqs. (58)–(61), pp. 15–16, PDF. The first term is a double divergence on the unit sphere and can have either sign at a point. The second term is nonpositive because $q_{AB}$ is positive definite.

If matter has a finite outward null flux, define

$$T_{uu}^{(2)}(u,x^A)=\lim_{r\to\infty}r^2T_{uu}.$$

In the same convention the constraint becomes

$$\partial_um =\frac14D_AD_BN^{AB} -\frac18N_{AB}N^{AB} -4\pi G\,T_{uu}^{(2)}.$$

This matter term is distinct from gravitational radiation: it is the leading physical stress–energy flux through $\mathscr I^+$. Under the null energy condition, $T_{uu}^{(2)}\ge0$. Without that condition, a monotonicity claim for the combined Bondi energy needs qualification.

Sphere integration and mass loss

Define Bondi energy by

$$E_B(u)=\frac{1}{4\pi G}\int_{S^2}m(u,x^A)d\Omega.$$

This is the time component of Bondi four-momentum in the chosen asymptotic frame. When that four-momentum is timelike, its invariant mass is $M_B=\sqrt{E_B^2-\lvert\boldsymbol P_B\rvert^2}$. Energy and mass coincide in a rest frame, which explains the conventional name “Bondi mass-loss formula”; they should not be identified after an arbitrary boost.

On a closed unit sphere, integration by parts gives

$$\int_{S^2}D_AD_BN^{AB}d\Omega=0.$$

Therefore

$$\frac{dE_B}{du} =-\frac{1}{32\pi G}\int_{S^2}N_{AB}N^{AB}d\Omega -\int_{S^2}T_{uu}^{(2)}d\Omega.$$

The two terms on the right are precisely minus the positive-outward gravitational and matter luminosities. With $\mathcal N=N_{AB}\bar m^A\bar m^B=\partial_uH$ and $m^A\bar m_A=1$, the identity $N_{AB}N^{AB}=2\lvert\mathcal N\rvert^2$ recovers

$$\mathcal F_{E,\rm grav}^{\infty} =\frac{1}{16\pi G}\int_{S^2}\lvert\mathcal N\rvert^2d\Omega.$$

This coefficient agrees with the large-radius strain formula because $H=\lim r(h_+-ih_\times)$. It also explains why inserting the Mädler–Winicour news into the present formula without translation would fail.

What is and is not monotone

The total Bondi energy is nonincreasing in vacuum, and remains so with positive outward matter flux. The local mass aspect need not decrease at every angle because the double divergence can move Coulombic information across the sphere. A negative value of $\partial_um$ at one angle is not a local gravitational energy density, and a positive value at another angle does not contradict total mass loss.

Several limiting cases provide sharp checks:

• Minkowski data have $m=C_{AB}=N_{AB}=0$, so $E_B=0$.
• Schwarzschild data in a rest Bondi frame have $m=GM$ and $N_{AB}=0$, so $E_B=M$ is constant.
• On any news-free vacuum interval, $E_B$ is constant even if the time-independent shear is nonzero in the chosen supertranslation frame.
• A boost changes the components of Bondi four-momentum. A translation changes the cuts on which data are compared; it does not Lorentz-transform the four-vector on a fixed geometric cut. Neither operation invalidates a consistently transformed balance law.

Bondi energy should not be silently identified with near-zone mechanical energy, a finite-radius quasi-local mass, or a horizon mass. Those quantities can be joined in a larger conservation budget only after common orientations, time parameters, and stored-field terms have been specified. Rigorous positivity and the relation between Bondi and ADM mass require additional global hypotheses beyond this asymptotic derivation.

Cuts, matter, and assumptions

The derivative $dE_B/du$ compares neighboring cuts in one Bondi foliation. Under a supertranslation, the cuts and mass aspect both change; comparing two numerical answers at the same coordinate label without pulling the data into a common frame is not a comparison of the same physical cut. The integrated energy–momentum balance is BMS covariant, but the split into angle-dependent aspect, divergence term, and radiative term is tied to the chosen Bondi description.

The formula also assumes that all matter reaching the boundary has the stated $r^{-2}$ flux limit. Massive particles end at timelike infinity rather than crossing $\mathscr I^+$, so their energy is encoded in the changing Coulombic field and endpoint Bondi momentum, not in $T_{uu}^{(2)}$ as though they were null radiation. Long-range fields can require additional falloffs and boundary contributions.

Finally, the quadratic news term measures flux during the burst, whereas the shear difference $\Delta C_{AB}=\int N_{AB}du$ measures memory. Two bursts can have the same memory but different radiated energies because the energy depends on the integral of $N_{AB}N^{AB}$, not merely on the endpoint shear. The smooth-pulse exercise gives an explicit example whose energy decreases when the same memory is accumulated over a longer time.

Exercise

Let $N_{AB}(u,x^A)=f(u)Q_{AB}(x^A)$, where $Q_{AB}$ is smooth, trace free, and normalized by $\int Q_{AB}Q^{AB}d\Omega=K>0$. Suppose no matter reaches $\mathscr I^+$ and $f$ has support in $u_0<u<u_1$. Find the Bondi-energy change.

Solution

The sphere divergence drops out after integration, while

$$\int_{S^2}N_{AB}N^{AB}d\Omega=Kf(u)^2.$$

Integrating the mass-loss equation gives

$$E_B(u_1)-E_B(u_0) =-\frac{K}{32\pi G}\int_{u_0}^{u_1}f(u)^2du\le0.$$

Equality holds only if $f=0$ almost everywhere. The conclusion concerns the sphere average; the local mass aspect may still be redistributed by $D_AD_BN^{AB}/4$ during the burst.

References

• Mädler, Thomas, and Jeffrey Winicour. “Bondi–Sachs Formalism.” Scholarpedia 11, no. 12 (2016): 33528; expanded arXiv version, revised 2018. Open PDF.