The Bondi-Sachs Formulation

Bondi–Sachs coordinates adapt one coordinate to outgoing null hypersurfaces and another to their areal radius. The determinant gauge removes the screen-area degree of freedom from the radiative two-metric. Vacuum Einstein equations then form a radial hierarchy for auxiliary fields, followed by one retarded-time equation for the two radiative polarizations.

Required background. Timelike Congruences and Spatial Projection supplies the congruence and deformation ideas applied to outgoing null generators.

Helpful background. We use The Einstein Equation, Hypersurface Normals and Tangential Projectors, Formulations of General Relativity, Causality and Global Structure, and PDEs and Green Functions for Gravitation.

Areal null coordinates and fields

Let $u=\mathrm{constant}$ label outgoing null hypersurfaces, $r$ be areal radius, and $x^A$ label generators. Let $q_{AB}$ be the unit round-sphere metric in those angular coordinates. On a region where $r\gt0$ increases strictly along the outgoing generators, the Bondi–Sachs metric is

$$\begin{aligned} ds^2={}&-\left(e^{2\beta}\frac Vr-r^2h_{AB}U^AU^B\right)du^2 -2e^{2\beta}du\,dr \\ &-2r^2h_{AB}U^Bdu\,dx^A +r^2h_{AB}dx^A dx^B . \end{aligned}$$

The determinant condition

$$\det h_{AB}=\det q_{AB}$$

sets $4\pi r^2$ as the area of a spherical cross-section and implies $h^{AB}\partial_rh_{AB}=0$. Thus $h_{AB}$ has two independent components. They carry the radiative conformal screen geometry; $\beta$, $U^A$, and $V$ are determined radially once boundary or vertex values are supplied.

Choose the future outgoing and ingoing null vectors

$$\ell=e^{-2\beta}\partial_r, \qquad n=\partial_u+U^A\partial_A-\frac{V}{2r}\partial_r, \qquad \ell\cdot n=-1.$$

This fixes their orientation relative to increasing $r$ and $u$. A reparameterization of $u$ induces a compensating boost of $\ell$ and $n$; quantities such as expansion or news must be compared only after that normalization is fixed.

In fact $\ell_\mu=-\nabla_\mu u$, so $\ell$ is affinely parameterized: a null gradient obeys $\ell^\nu\nabla_\nu\ell_\mu=\tfrac12\nabla_\mu(\ell^2)=0$. The coordinate vector $\partial_r=e^{2\beta}\ell$ is generally nonaffine. Its distinction from $\ell$ matters when applying Raychaudhuri.

The radial hierarchy

The first hypersurface equation, $R_{rr}=0$ in vacuum, is

$$\partial_r\beta =\frac r{16} h^{AC}h^{BD} (\partial_rh_{AB})(\partial_rh_{CD}).$$

It determines $\beta$ along each generator from $h_{AB}$. Next, $R_{rA}=0$ gives a second-order radial equation for $U^A$, or a pair of first-order equations after introducing a radial momentum. The trace $h^{AB}R_{AB}=0$ then determines $V$. Finally, the tracefree part of $R_{AB}=0$ evolves the two independent components of $h_{AB}$ from one null hypersurface to the next.

An independent check of the first coefficient uses the physical screen metric $r^2h_{AB}$. Its expansion is $\theta=2e^{-2\beta}/r$, while

$$\sigma_{AB}\sigma^{AB} =\frac14e^{-4\beta}h^{AC}h^{BD} (\partial_rh_{AB})(\partial_rh_{CD}).$$

Substituting these into the affine vacuum Raychaudhuri equation cancels the $-2e^{-4\beta}/r^2$ terms and gives $\partial_r\beta=r\lVert\partial_rh\rVert_h^2/16$. The factor $1/16$ therefore follows from the determinant gauge and affine normalization together.

The order matters:

$$h_{AB} \longrightarrow \beta \longrightarrow U^A \longrightarrow V \longrightarrow \partial_u h_{AB}.$$

The remaining $u\mu$ equations are supplementary conditions. Contracted Bianchi identities propagate them once they hold on the inner worldtube or at a regular vertex. Treating all ten Einstein components as unrelated radial equations would overdetermine the metric.

The determinant condition and this main/supplementary hierarchy are developed explicitly in Winicour 2012, §§ 3.2–3.3. At a finite inner worldtube, the hierarchy needs compatible boundary values. At a regular cone vertex, smoothness imposes stronger parity and leading-order conditions; one cannot freely transplant worldtube data to a vertex.

Radiation near null infinity

For an asymptotically flat solution in Bondi gauge,

$$h_{AB} =q_{AB}+\frac{c_{AB}(u,x^C)}r+O(r^{-2}), \qquad q^{AB}c_{AB}=0.$$

In an asymptotically inertial Bondi frame with a smooth inverse-radius expansion, define the news tensor by

$$N_{AB}=\frac12\partial_u c_{AB}.$$

The factor $1/2$ is a convention; some sources call $\partial_uc_{AB}$ the news instead. This normalization agrees with Mädler and Winicour 2016, arXiv v3, § 3, equation (42), p. 12, PDF. Indices on $c_{AB}$ and $N_{AB}$ are raised with $q^{AB}$. A finite-worldtube coordinate frame need not already be this inertial asymptotic frame.

The expansion $V=r-2Gm+O(r^{-1})$ identifies a mass aspect once the residual asymptotic gauge is fixed. Vanishing news means no radiative change in the asymptotic shear in that Bondi frame; it does not require the spacetime to be flat or the mass aspect to vanish.

This page stops at the continuum hierarchy. Asymptotic charges, BMS transformations, detector observables, and production numerical algorithms require additional structures in their respective volumes. The areal chart stops at generator caustics, where $(u,r,x^A)$ ceases to be one-to-one. It can also fail earlier if the outgoing expansion reaches zero: $r$ then ceases to be a valid generator parameter even when the screen area is nonzero. A double-null chart does not require areal radius to be monotone and can pass through such a marginal screen if its other coordinates remain regular.

Exercises

Set

$$\beta=0,\qquad U^A=0,\qquad V=r,\qquad h_{AB}=q_{AB}.$$

Show that the metric is Minkowski spacetime in outgoing null coordinates and check the determinant, first radial equation, news, and mass aspect.

Solution

The line element becomes

$$ds^2=-du^2-2du\,dr+r^2d\Omega_2^2.$$

With $u=t-r$, this is $-dt^2+dr^2+r^2d\Omega_2^2$. The determinant condition holds identically because $h_{AB}=q_{AB}$. Since $\partial_rh_{AB}=0$, the right-hand side of the $\beta$ equation vanishes, consistent with $\beta=0$.

There is no $1/r$ coefficient $c_{AB}$, so $N_{AB}=0$. Also $V=r$ has no constant term $-2Gm$, so the Bondi mass aspect is zero. These four checks recover the flat member of the hierarchy rather than merely a metric with vanishing news.

References

• Mädler, Thomas, and Jeffrey Winicour. “Bondi–Sachs Formalism.” Scholarpedia 11, no. 12 (2016): 33528. DOI. Open PDF.
• Winicour, Jeffrey. “Characteristic Evolution and Matching.” Living Reviews in Relativity 15 (2012): article 2. arXiv:0810.1903. doi:10.12942/lrr-2012-2.