Bondi Mass and Mass Loss

Bondi mass is the energy remaining in an isolated system on a cut of future null infinity. Its evolution law is one of the sharpest places where Einstein’s equation turns gravitational radiation into a quantitative theorem: the square of the news tensor is a nonnegative outward flux, so the mass cannot increase in retarded time. The statement requires a controlled asymptotic expansion and compact cuts; it is not a coordinate-level identity valid for every metric that looks approximately flat.

Required background. Analysis for Mathematical Relativity supplies asymptotic expansions, and exterior algebra and differential forms supplies oriented flux integrals.

Helpful background. The Einstein equation supplies the constraint used below; Causality and Global Structure supplies null infinity; PDEs and Green Functions for Gravitation supplies characteristic evolution; and Gravitational Radiation and Waveforms develops waveform observables.

Bondi–Sachs data

Work in four-dimensional vacuum gravity near $\mathscr I^+$, with Bondi–Sachs coordinates $(u,r,x^A)$. The level sets of $u$ are outgoing null hypersurfaces, $r$ is areal radius, and $x^A$ label their generators. Write the angular metric as

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

where $q_{AB}$ is the unit-sphere metric. The news tensor and the relevant $g_{uu}$ coefficient are defined here by

$$N_{AB}=\partial_u C_{AB}, \qquad g_{uu}=-1+\frac{2m}{r}+O(r^{-2}).$$

Other conventions absorb a factor of two into the shear or news. Every flux coefficient below follows from these definitions. The Bondi mass of the cut $S_u$ is

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

The full mass aspect can include convention-dependent angular-divergence terms. They change the local representative but not the integral on a smooth closed sphere. Mädler and Winicour derive the expansion and identify the free radiative datum in Mädler and Winicour 2016, §§2.2–2.4 and eqs. (27)–(43).

More generally, pairing the mass aspect with the four lowest spherical harmonics gives the Bondi energy–momentum $P^\mu_B$. The constant harmonic selects energy, while the three $\ell=1$ harmonics select momentum. The Bondi mass may then mean either the energy in a chosen asymptotic Lorentz frame or the invariant norm $\sqrt{-\eta_{\mu\nu}P_B^\mu P_B^\nu}$; this page uses the former in the displayed balance law. Naming that choice prevents a boost transformation from being mistaken for physical energy production.

The mass-aspect equation

The vacuum Einstein equations split into radial hypersurface equations, evolution equations, and supplementary equations. Solving the radial hierarchy first determines the lapse- and shift-like coefficients from $h_{AB}$. The leading supplementary equation at null infinity then gives, in the convention above,

$$\partial_u m =\frac14 D_A D_B N^{AB} -\frac18 N_{AB}N^{AB},$$

where $D_A$ is the Levi–Civita connection of $q_{AB}$. This equation is the local balance law. Its first term redistributes the mass aspect over angles; its second term is the radiative energy flux. In the convention $N_{AB}=\partial_uC_{AB}$ used here, the corresponding Bondi–Sachs hierarchy is Mädler and Winicour 2016, eqs. (58), (60), and (61), pp. 14–16; their news variable differs by the factor stated in that source.

Integrating over a cut proves the mass-loss formula:

$$\frac{dM_B}{du} =-\frac{1}{32\pi G}\int_{S^2}N_{AB}N^{AB}\,d\Omega \le 0.$$

Indeed, compactness and smoothness give $\int_{S^2}D_A D_BN^{AB}\,d\Omega=0$, while the positive-definite sphere metric gives $N_{AB}N^{AB}\ge0$. This is the entire sign argument, but its validity rests on the asymptotic Einstein equation and the absence of boundary or singular terms on the cut. The original axisymmetric calculation appears in Bondi, van der Burg, and Metzner 1962, §§4–6, pp. 33–49.

If matter reaches null infinity with $T_{uu}=r^{-2}\mathcal T_{uu}+o(r^{-2})$, the right-hand side acquires the additional nonpositive contribution $-\int_{S^2}\mathcal T_{uu}\,d\Omega$ after the corresponding normalization. Monotonicity then uses the null energy condition $\mathcal T_{uu}\ge0$.

Invariance and interpretation

The Bondi four-momentum is associated with the translation subgroup of the BMS group. A supertranslation changes the cut and the local shear and mass aspect, so comparing “the mass at the same time” requires a specified Bondi frame. The geometric charge on a declared cut is nevertheless not erased by a coordinate relabeling. The integrated loss between two cuts equals the flux through the portion of $\mathscr I^+$ between them Sachs 1962, §§3–4, pp. 110–118.

This equality is also a normalization check. If the news has support only between $u_1$ and $u_2$, then there is no outgoing gravitational news outside that interval and $M_B(u_1)-M_B(u_2)=(32\pi G)^{-1}\int_{u_1}^{u_2}du\int N^2d\Omega$. Both sides have dimensions of mass, and reversing the orientation of retarded time reverses both the boundary ordering and the flux sign.

Three statements must be kept separate:

• mass loss says $M_B(u_2)\le M_B(u_1)$ for ordered cuts under the stated hypotheses;
• positivity says $M_B(u)\ge0$ and needs a separate global argument;
• the limit $M_B(u)\to M_{\mathrm{ADM}}$ toward spatial infinity requires compatibility and decay connecting null and spatial asymptotics.

The limitation of the proof is therefore precise. Polyhomogeneous expansions may still support a balance law, but logarithmic terms and low regularity must be shown not to create extra cut terms. Noncompact cuts, incoming flux through another boundary, or insufficient matter decay require a different balance statement.

Two checks

For Schwarzschild of physical mass $M$ in outgoing Eddington–Finkelstein coordinates, $C_{AB}=N_{AB}=0$ and $m=GM$; the formula gives $M_B=M$. The mass aspect $m$ has dimensions of length, whereas $M_B$ is an energy. For a weak wave with $C_{AB}=\varepsilon c_{AB}$, the energy loss begins at order $\varepsilon^2$, as expected because energy is quadratic in wave amplitude.

Show directly that a pure electric-parity tensor $N_{AB}=(D_A D_B-\tfrac12q_{AB}D^2)f$ cannot make the flux negative.

Solution

The contraction is the pointwise squared norm of the symmetric tracefree Hessian with respect to $q_{AB}$:

$$N_{AB}N^{AB}=\lvert \operatorname{TF}(D_A D_Bf)\rvert_q^2\ge0.$$

Consequently its sphere integral is nonnegative and $dM_B/du\le0$. Equality holds only when the tracefree Hessian vanishes; on the round sphere this occurs for the $\ell=0,1$ harmonics, which carry no tensor radiation. The angular-divergence term in the local mass equation integrates to zero independently of this decomposition.

A quadrupolar pulse with a computable energy

An explicit angular profile makes the normalization test more demanding. On the unit sphere set $P_2(\cos\theta)=(3\cos^2\theta-1)/2$ and

$$Q_{AB}=\left(D_A D_B-\tfrac12q_{AB}D^2\right)P_2(\cos\theta), \qquad C_{AB}=A(u)Q_{AB}.$$

Direct differentiation gives $Q_{\theta\theta}=\tfrac32\sin^2\theta$, $Q_{\varphi\varphi}=-\tfrac32\sin^4\theta$, and $Q_{\theta\varphi}=0$. Consequently

$$Q_{AB}Q^{AB}=\frac92\sin^4\theta, \qquad \int_{S^2}Q_{AB}Q^{AB}\,d\Omega=\frac{48\pi}{5}, \qquad \frac{dM_B}{du}=-\frac{3}{10G}(A')^2.$$

For $A(u)=\varepsilon\tau\tanh(u/\tau)$, where $\tau\gt0$ is a time scale and $\varepsilon$ is dimensionless, the news tends to zero at both temporal ends. Since $\int_{-\infty}^{\infty}\operatorname{sech}^4s\,ds=4/3$,

$$M_B(-\infty)-M_B(+\infty)=\frac{2\varepsilon^2\tau}{5G}.$$

This is a check of the asymptotic balance law for prescribed smooth shear, not a construction of a global vacuum spacetime realizing arbitrary data. It also illustrates a useful contrast with memory: energy depends on the integral of the squared news, whereas the shear change is $\Delta C_{AB}=2\varepsilon\tau Q_{AB}$ and depends on its signed integral.

References

• Bondi, Hermann, M. G. J. van der Burg, and A. W. K. Metzner. “Gravitational Waves in General Relativity. VII. Waves from Axi-Symmetric Isolated Systems.” Proceedings of the Royal Society A 269 (1962): 21–52. DOI.
• Mädler, Thomas, and Jeffrey Winicour. “Bondi–Sachs Formalism.” Scholarpedia 11 (2016): 33528. DOI.
• Sachs, Rainer K. “Gravitational Waves in General Relativity. VIII. Waves in Asymptotically Flat Space-Time.” Proceedings of the Royal Society A 270 (1962): 103–126. DOI.