Bondi-Sachs formalism

The Bondi-Sachs formalism uses outgoing null hypersurfaces to describe gravitational radiation. A null coordinate $u$ labels wavefronts, areal radius $r$ labels cross-sections, and angular coordinates label null generators.

The Einstein field equations split into radial relations and evolution equations for the angular geometry. At future null infinity, changing Bondi shear gives the Bondi news function, linking radiation to loss of Bondi mass in nonlinear general relativity.^[ref]

Bondi-Sachs coordinates $(u,r,x^A)$ follow outgoing null hypersurfaces. The retarded time $u$ labels the hypersurfaces, the angles $x^A$ label their null generators, and the areal radius $r$ fixes the angular area element. The coordinate conditions include $g_{rr}=g_{rA}=0$ and $\det g_{AB}=r^4\det q_{AB}$, where $q_{AB}$ is the unit-sphere metric.

Flat spacetime already illustrates the construction: setting $u=t-r$ in units $c=1$ gives $d{s}^2=-d{u}^2-2dudr+r^{2}d{\Omega }^2$. Constant-$u$ surfaces are outgoing light cones, although $g_{uu}$ is nonzero. The null condition concerns the hypersurface normal and gives $g^{uu}=0$.^[ref]

The Bondi shear is the leading trace-free angular distortion in an asymptotically flat spacetime described by Bondi-Sachs coordinates. In a common expansion, the angular metric is $h_{AB}=q_{AB}+C_{AB}/r+O(r^{-2})$, with $q^{AB}{C}_{AB}=0$.

Its retarded-time derivative determines the Bondi news function, with normalization depending on convention. The distinction matters: a nonzero but time-independent shear need not represent ongoing radiation, while changing shear describes radiative degrees of freedom. Changes between early and late configurations are also connected with gravitational memory.^[ref]

The Bondi mass measures the total mass-energy remaining in an isolated system at a retarded time, as defined at future null infinity. It is obtained by integrating the Bondi mass aspect over a sphere, with normalization set by the metric convention.

Outgoing gravitational radiation carries energy away, so the Bondi mass decreases by a nonnegative news-squared flux in the vacuum mass-loss formula. Additional outgoing matter contributes its own flux. This is distinct from the ADM mass defined at spatial infinity: the two quantities use different asymptotic cuts and account differently for radiation that has escaped.^[ref]

The Bondi news function is the outgoing-null-time derivative of Bondi shear at future null infinity. Write the angular metric as $g_{AB}=r^2{q}_{AB}+r{C}_{AB}+O(1)$, with $q^{AB}{C}_{AB}=0$. One convention defines $N_{AB}=\frac{1}{2}{\partial }_u{C}_{AB}$; its two components encode changing gravitational-wave polarizations.

The outgoing gravitational-wave energy flux is quadratic in the news, decreasing total Bondi mass in an isolated vacuum system. Time-independent shear supplies no such flux. The shear also sets the leading detector strain at large distance.^[ref]