Radiative Data and Asymptotic Constraints

The Bondi shear is freely specified radiative data in the formal asymptotic expansion, but the mass and angular-momentum aspects are not freely reset on every cut. Leading supplementary Einstein equations propagate them along future null infinity. The mass constraint contains a sphere divergence and a negative news-squared term; its sphere integral is Bondi mass loss. The angular constraint couples the angular-momentum aspect to mass gradients, angular derivatives of shear, and shear–news flux. We derive these vacuum relations in one declared Bondi convention. Rigorous existence or peeling proofs and detector-specific reconstruction remain outside the chapter.

Helpful background. Einstein Equations supplies the field equations and Bianchi identity; Waves, Green Functions, and Geometric Optics supplies characteristic propagation; Causality and Global Structure supplies future null infinity and its cuts; Gravitational Radiation and Waveforms supplies shear, news, and flux; and Formulations of General Relativity supplies the distinction between hypersurface and supplementary constraints.

Constraint data on an initial cut

Use the Bondi–Sachs expansion

$$h_{AB}=q_{AB}+\frac{C_{AB}}r+O(r^{-2}), \qquad V=r-2m+O(r^{-1}),$$

with $q^{AB}C_{AB}=0$ and news $N_{AB}=\partial_uC_{AB}$. The mass aspect $m(u,x^A)$ is the $O(1)$ coefficient in $V$. Define the angular-momentum aspect in the Mädler–Winicour radial convention by

$$L_A =-\frac16\lim_{r\to\infty} \left[r^4e^{-2\beta}h_{AB}\partial_rU^B-rD^BC_{AB}\right].$$

It has dimensions of length squared, while $m$ has dimensions of length. Both depend on the Bondi frame; an integrated angular momentum additionally requires a rotation generator and a choice controlling the supertranslation ambiguity.

For a retarded-time interval $u_0\le u\le u_1$, the formal data are $C_{AB}(u,x^A)$ throughout the interval plus $m(u_0,x^A)$ and $L_A(u_0,x^A)$ on the initial cut. Radial hypersurface equations determine $\beta$, $U^A$, $V$, and higher coefficients on each null hypersurface. The two supplementary equations then determine $\partial_um$ and $\partial_uL_A$. This is a hierarchy, not a claim that prescribing future news is a physical initial-value procedure.

The mass-aspect constraint

At leading order, the relevant supplementary equation becomes

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

To obtain this form from Mädler and Winicour’s equation (58), use their $N^{\rm MW}_{AB}=\tfrac12\partial_uC_{AB}=N_{AB}/2$ and divide their equation by two. The result is therefore consistent with, rather than numerically identical to, the displayed source convention Mädler and Winicour 2018, §3, eqs. (58)–(61), pp. 15–16, PDF.

The first term redistributes mass aspect over angles and may have either sign locally. The second term is nonpositive and quadratic in the radiative data. Define the Bondi energy by

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

Because a total divergence integrates to zero on the closed unit sphere,

$$\frac{dE_B}{du} =-\frac{1}{32\pi G} \int_{S^2}N_{AB}N^{AB}\,d\Omega \le0.$$

This sphere integral independently checks the coefficient and sign. With $N_{AB}N^{AB}=2\lvert\mathcal N\rvert^2$, it agrees with the positive outward flux $\lvert\mathcal N\rvert^2/(16\pi G)$. The energy decreases because outward flux is defined positive.

The same constraint contains the four-momentum balance. Let $k^\mu=(1,n^i)$, where $n^i(x^A)$ is the outward unit radial vector, and set

$$P_B^\mu=\frac{1}{4\pi G}\int_{S^2}m\,k^\mu d\Omega.$$

For the $\ell=1$ functions $n^i$, $D_AD_Bn^i=-q_{AB}n^i$. Its contraction with the trace-free news vanishes, so the double-divergence term contributes neither energy nor momentum. Therefore

$$\frac{dP_B^\mu}{du} =-\frac{1}{32\pi G}\int_{S^2}k^\mu N_{AB}N^{AB}\,d\Omega.$$

An anisotropic news pattern can carry net linear momentum even though every angular element carries nonnegative energy.

The angular-momentum-aspect constraint

In the same radial and aspect convention, translate every Mädler–Winicour news tensor to $N_{AB}/2$. Their angular supplementary equation becomes

$$\begin{aligned} -3\partial_uL_A={}&D_Am -\frac14D_E\left(D^ED^FC_{AF}-D_AD^FC_{EF}\right)\\ &+\frac1{16}D_A\left(C_{EF}N^{EF}\right) -\frac12D_C\left(C^{CF}N_{FA}\right) +\frac14C^{EF}D_AN_{EF}. \end{aligned}$$

The complete equation is the leading $rA$ supplementary constraint Mädler and Winicour 2018, §3, eq. (59), p. 15, PDF. Its terms have a transparent organization: the gradient of mass supplies a Coulombic contribution; three angular derivatives of shear encode the cut geometry; and the remaining terms are nonlinear shear–news transport. Every term has dimensions of length, matching $\partial_uL_A$.

Unlike Bondi energy, an angular-momentum charge is not obtained by simply integrating $L_A$ without qualifications. One contracts with a sphere rotation field, includes the convention-dependent shear terms appropriate to the charge definition, and fixes a BMS frame. The equation above propagates the metric coefficient $L_A$; covariant charge construction belongs to GR formulations.

Integration and limiting cases

Given $C_{AB}(u,x^A)$ and the two initial aspects, integrate the constraints in $u$:

$$m(u)=m(u_0)+\int_{u_0}^{u}\partial_{u'}m\,du', \qquad L_A(u)=L_A(u_0)+\int_{u_0}^{u}\partial_{u'}L_A\,du'.$$

Minkowski data $C_{AB}=m=L_A=0$ remain zero. Schwarzschild data $C_{AB}=N_{AB}=L_A=0$ and $m=GM$ also remain constant. If news vanishes but the shear is angle dependent, the mass aspect is time independent, while the angular constraint still enforces compatibility among $L_A$, $m$, and the cut-dependent shear. Thus “no radiation” removes flux; it does not erase all asymptotic structure.

The contracted Bianchi identity propagates supplementary constraints once the radial hierarchy and appropriate initial compatibility conditions hold. That fact explains why the asymptotic system does not introduce a new unconstrained function at every order.

Exercise

Integrate the mass-aspect constraint over the unit sphere and show that zero news makes the Bondi energy constant even when $m$ is angle dependent.

Solution

On a closed sphere, two integrations by parts give $\int D_AD_BN^{AB}\,d\Omega=0$. If $N_{AB}=0$, the quadratic term also vanishes, so $dE_B/du=0$. The conclusion constrains only the sphere average of $m$: an angle-dependent but time-independent mass aspect is compatible with this integrated equation, although other stationary constraints and a chosen Bondi frame may restrict it further.

References

• Thomas Mädler and Jeffrey Winicour, “Bondi–Sachs Formalism,” Scholarpedia 11 (2016) 33528, updated arXiv version (2018), arXiv:1609.01731v3, Open PDF.