Bondi–Sachs Coordinates and Expansion

Bondi–Sachs coordinates turn the vacuum Einstein equation near future null infinity into a radial hierarchy. The angular metric contains two freely varying radiative components; the remaining leading coefficients are obtained from hypersurface equations or propagated by asymptotic constraints. This division is more informative than a list of powers of $1/r$: it says what must be supplied, what the field equations determine, and which coordinate conditions make those statements true. We assume vacuum outside a compact source and a smooth power expansion in an asymptotic Bondi frame; logarithmic asymptotics and detector-specific reconstruction are not treated here.

Helpful background. Einstein Equations supplies the vacuum equations placed into characteristic form; Waves, Green Functions, and Geometric Optics supplies outgoing null characteristics; Causality and Global Structure supplies future null infinity; Gravitational Radiation and Waveforms supplies the physical interpretation of radiative coefficients; and Formulations of General Relativity supplies the distinction among gauge, constraints, and freely specified characteristic data.

Bondi–Sachs coordinate gauge

Use coordinates $(u,r,x^A)$ adapted to outgoing null hypersurfaces. The level sets $u=\text{constant}$ are null, the angles $x^A$ remain constant along their generators, and $r$ is an areal radius: a surface of constant $(u,r)$ has area $4\pi r^2$. These choices imply

$$g^{uu}=0, \qquad g^{uA}=0, \qquad \det h_{AB}=\det q_{AB},$$

where $q_{AB}$ is the unit-sphere metric and $g_{AB}=r^2h_{AB}$. The most general metric in this gauge is

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

The determinant condition removes the scalar area degree of freedom from $h_{AB}$, leaving precisely two independent angular components. It is a coordinate condition, not an extra field equation. The areal radius can fail where the outgoing null congruence has zero expansion, so this chart is not promised to cover the entire spacetime.

The asymptotic expansion

An inertial Bondi frame approaches outgoing Minkowski coordinates as $r\to\infty$:

$$\beta\to0, \qquad U^A\to0, \qquad \frac Vr\to1, \qquad h_{AB}\to q_{AB}.$$

Expand the conformal angular metric as

$$h_{AB} =q_{AB}+\frac{C_{AB}}r+\frac{E_{AB}}{r^2}+O(r^{-3}).$$

Taking the determinant order by order gives

$$q^{AB}C_{AB}=0, \qquad q^{AB}E_{AB}=\frac12 C_{AB}C^{AB}.$$

The second relation follows without using dynamics. For a small matrix perturbation, $\det(I+\epsilon A+\epsilon^2B)$ has second-order coefficient

$$\operatorname{tr}B +\frac12\left[(\operatorname{tr}A)^2-\operatorname{tr}(A^2)\right].$$

Apply this formula to $q^{-1}h$ with $\epsilon=1/r$. The first-order coefficient sets $\operatorname{tr}C=0$; the second then sets $\operatorname{tr}E=\tfrac12\operatorname{tr}(C^2)$. This is an independent algebraic check that no sign from the spacetime metric enters the two-dimensional determinant condition.

Thus $C_{AB}$ is trace-free and has two components. The determinant alone does not eliminate the trace-free part of $E_{AB}$. The additional vacuum no-logarithm condition makes that part divergence-free; smoothness on the complete two-sphere then forces it to vanish. Consequently $E_{AB}=\tfrac14q_{AB}C_{CD}C^{CD}$. Allowing logarithms or angular singularities changes this conclusion; their absence is part of the present regime Mädler and Winicour 2018, §3, eqs. (48)–(54), pp. 13–14, PDF.

Substitute the series into the radial Einstein equations in their hierarchical order. The $rr$ equation first determines $\beta$; the $rA$ equation then determines $U^A$; the angular trace determines $V$. At the first consequential orders,

$$\beta=-\frac{C_{AB}C^{AB}}{32r^2}+O(r^{-3}),$$ $$U^A=-\frac{D_BC^{AB}}{2r^2}+O(r^{-3}), \qquad V=r-2m+O(r^{-1}).$$

Here $D_A$ is the covariant derivative of $q_{AB}$ and $m(u,x^A)$ is the Bondi mass aspect, with dimensions of length because $G$ remains explicit. This chain, including the determinant restrictions and the next-order angular-momentum coefficient, is derived in Mädler and Winicour 2018, §3, pp. 10–16, PDF.

Notice the different radial orders in physical components: $g_{AB}=r^2q_{AB}+rC_{AB}+O(1)$, while $g_{uA}$ has a finite leading angular-gradient term and $g_{uu}=-1+2m/r+O(r^{-2})$. Radiation, longitudinal frame structure, and Coulombic mass therefore occupy distinct coefficients even though they belong to one metric.

Free radiative data and constrained coefficients

The hierarchy separates three roles:

Coefficient	How it enters	Status in the formal asymptotic solution
$C_{AB}(u,x^A)$	$1/r$ coefficient of $h_{AB}$	Freely prescribed radiative data over the retarded-time interval
$m(u,x^A)$	$O(1)$ correction in $V=r-2m+\cdots$	Initial value supplied; retarded-time evolution fixed by a supplementary constraint
$L_A(u,x^A)$	Next independent coefficient in $U^A$	Initial value supplied; evolution fixed by the angular supplementary constraint
$\beta$, leading $U^A$, and higher radial coefficients	Radial metric functions	Determined sequentially on each outgoing null hypersurface

Calling $C_{AB}$ “free” does not mean that arbitrary data automatically produce a complete global solution. In this formal expansion, $C_{AB}$ is supplied for all $u$ in the interval; it is not predicted from one initial cut. A well-posed characteristic evolution also needs suitable data on an initial null hypersurface or worldtube and compatibility at their intersection. That continuum formulation belongs to the characteristic-evolution treatment; the present result identifies the asymptotic coefficients.

The news convention used throughout this volume is $N_{AB}=\partial_uC_{AB}$. Mädler and Winicour define $N^{\rm MW}_{AB}=\tfrac12\partial_uC_{AB}$, so their constraint formulas must be translated using $N^{\rm MW}_{AB}=N_{AB}/2$ before comparison.

Stationary and dimensional checks

Minkowski space is recovered with

$$C_{AB}=0, \qquad m=0, \qquad L_A=0,$$

which sets $\beta=U^A=0$ and $V=r$. Outgoing Eddington–Finkelstein Schwarzschild is recovered with $C_{AB}=0$, $U^A=\beta=0$, and $m=GM$, giving $V=r-2GM$. These limits check the sign and dimension of the mass term. They also show that a nonzero mass aspect is Coulombic data, not by itself radiation: both stationary metrics have vanishing shear and news.

Exercise

Assume $E_{AB}=a\,q_{AB}C_{CD}C^{CD}$ after its trace-free part has been excluded. Use the determinant condition to determine $a$.

Solution

Contract the ansatz with $q^{AB}$. Because the unit sphere is two-dimensional, $q^{AB}q_{AB}=2$, so $q^{AB}E_{AB}=2a\,C_{CD}C^{CD}$. The determinant expansion requires this trace to equal $\tfrac12C_{CD}C^{CD}$. For nonzero shear the coefficients therefore obey $2a=1/2$, hence $a=1/4$; continuity gives the same result when the shear vanishes.

References

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