Bondi Mass Aspect and the Asymptotic Constraint

The Bondi mass aspect is the angle-dependent coefficient through which the asymptotic metric records energy on a cut of future null infinity. It is not freely specifiable at every retarded time. The null Einstein equation propagates it: gravitational news and matter flux decrease its sphere average, while an angular-divergence term redistributes the local aspect without changing the total mass.

Required background. Variational and Symplectic Geometry supplies charge and flux language. Helpful background. Lorentzian Metrics and Causal Cones and Geodesics and Jacobi Fields supply null infinity; Causality and Global Structure supplies its completion; Formulations of General Relativity derives the constraints; Waves, Green Functions, and Geometric Optics supplies radiation propagation.

The mass aspect in Bondi gauge

Use retarded Bondi coordinates and the round-sphere conformal frame. Fix the convention

$$g_{uu}=-1+\frac{2m(u,x^A)}{r}+O(r^{-2}), \qquad g_{AB}=r^2q_{AB}+rC_{AB}+O(1),$$

with $q^{AB}C_{AB}=0$ and news

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

Here $m$ has dimensions of length because $G$ is explicit. The Bondi mass on a cut $S_u$ is

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

For Schwarzschild in outgoing coordinates, $m=GM$ and $C_{AB}=N_{AB}=0$, giving $M_B=M$. For Minkowski space all three vanish in the standard Bondi frame. These checks fix the normalization.

Here the customary name “Bondi mass” denotes the energy component in the selected Bondi frame. More generally the translation charges are

$$P_B^\mu=\frac{1}{4\pi G}\int m\,(1,\widehat x^i)\,d\Omega.$$

When $P_B^\mu$ is future timelike, its invariant rest mass is $\sqrt{(P_B^0)^2-\delta_{ij}P_B^iP_B^j}$; it equals $M_B=P_B^0$ only in a rest frame. This distinction prevents a coordinate energy from being mistaken for a Lorentz scalar.

The angle dependence contains supermomentum information beyond total energy. Its transformation under BMS symmetries involves the shear and the choice of cut, so $m$ is not an invariant scalar function detached from its Bondi frame. The charge paired with a supertranslation $f(x^A)$ is schematically an integral of $fm/G$ plus prescription-dependent shear terms when radiation and integrability are included.

The null constraint

Expand the Einstein equation $G_{uu}=8\pi G T_{uu}$ at the first nontrivial inverse-radius order. In the stated convention it yields

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

where

$$T_{uu}^{(2)}=\lim_{r\to\infty}r^2T_{uu}$$

is the outgoing matter energy flux density. Coefficients can move under alternate definitions of $C_{AB}$ or $m$; the metric expansion, news definition, and constraint must always be quoted as one convention set. The Bondi expansion and its constraint hierarchy are derived in Compère and Fiorucci 2019, §§3.1–3.3, pp. 69–84.

The three terms play different roles. The double divergence changes the angle-dependent aspect but integrates to zero on a smooth compact sphere. The news square is nonnegative with the positive sphere metric, so its minus sign describes gravitational-wave energy escaping through $\mathscr I^+$. The matter term has the same sign when the asymptotic null energy flux is nonnegative.

Integrating over $S^2$ gives

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

Thus the total Bondi mass is nonincreasing under nonnegative outgoing matter flux. The detailed waveform normalization, detector observables, and radiative flux derivation belong to Bondi Mass Loss; this page establishes the global asymptotic constraint that makes that balance possible.

Local redistribution and supermomentum

It is incorrect to read $\partial_um$ pointwise as a negative energy flux. The divergence term can have either sign at a given angle. It represents angular redistribution and contributes to higher-harmonic supermomentum balance. It disappears from the four translation charges, but not from a general angle-weighted charge.

Multiply the constraint by a smooth supertranslation parameter $f(x^A)$ and integrate. Two integrations by parts give

$$\int fD_AD_BN^{AB}\,d\Omega =\int N^{AB}D_AD_Bf\,d\Omega.$$

For $\ell=0$, the Hessian vanishes and one recovers total mass loss. For $\ell=1$, the trace-free news kills the pure-trace Hessian, yielding momentum balance. For $\ell\geq2$, the term is generally nonzero and couples the shear/news to supermomentum. This hierarchy is the charge-language counterpart of angle-dependent energy flow.

When news vanishes over an interval and no matter reaches infinity, the constraint gives $\partial_um=0$. The shear can nevertheless be a nonzero, time-independent vacuum shear related to another frame by a supertranslation. Stationarity of $m$ and zero news do not uniquely select $C_{AB}=0$.

Charges, flux, and integrability

In a radiative phase space the variation of a BMS generator contains an integrable cut charge and a nonintegrable piece involving news times $\delta C_{AB}$. The latter is the symplectic flux through null infinity. One should not hide it by calling every cut integral conserved. Instead, state a balance law between two cuts:

$$Q_f[S_{u_2}]-Q_f[S_{u_1}] =-\mathcal F_f[\Delta\mathscr I^+],$$

with both charge and flux defined in one prescription.

Potential shifts of the presymplectic form and boundary counterterms can redistribute terms between charge and flux while preserving the total balance. A reference vacuum fixes additive constants. General integrability and ambiguity analysis is developed in Charge Integrability and Ambiguities.

For the time-translation mode, the mass-loss equation supplies a robust normalization because it agrees with Schwarzschild mass and positive outgoing flux. For more general supertranslations or Lorentz transformations, corner terms, vacuum frames, and angular-momentum aspects require additional care.

Assumptions and limitations

The derivation assumes a four-dimensional asymptotically flat Bondi expansion with adequate differentiability, a compact smooth sphere of generators, and matter falloff making $T_{uu}^{(2)}$ finite. Logarithmic or polyhomogeneous terms can modify intermediate expansions. Massive matter does not reach null infinity in the same way as massless flux and contributes through timelike/spatial matching rather than a naive $T_{uu}^{(2)}$ term alone.

The monotonicity statement also assumes a nonnegative total outgoing matter flux. Quantum fields can violate pointwise energy conditions, and alternative boundary conditions can inject energy. Finally, the Bondi mass is a cut-dependent asymptotic charge, not a local mass density and not automatically equal to ADM mass at spatial infinity. Their relation needs matching at $i^0$ and an accounting of radiation.

Exercise

Let the only nonzero news during an interval be $N_{AB}=a(u)Y_{AB}$, where $Y_{AB}$ is a smooth trace-free tensor harmonic normalized by $\int Y_{AB}Y^{AB}d\Omega=1$, and let matter flux vanish. Find the change in Bondi mass.

Solution

The sphere-integrated divergence term vanishes. The mass-loss law gives

$$\frac{dM_B}{du} =-\frac{a(u)^2}{32\pi G}.$$

Hence between $u_1$ and $u_2$,

$$M_B(u_2)-M_B(u_1) =-\frac{1}{32\pi G}\int_{u_1}^{u_2}a(u)^2\,du\leq0.$$

The result is independent of the angular sign changes of $Y_{AB}$ because the flux is quadratic. Equality holds only if the news vanishes almost everywhere in the interval.

Supertranslations and Angular-Momentum Ambiguity next explains why the analogous Lorentz charges require a choice of cuts and vacuum frame.

References

• Compère, Geoffrey, and Adrien Fiorucci, Advanced Lectures on General Relativity, §§3.1–3.3, pp. 69–84, Springer, 2019.
• Frauendiener, Jörg, “Conformal Infinity”, Living Reviews in Relativity 7, article 1, §§1–2, 2004.