Bondi–Sachs Gauge in General Relativity

• Bondi–Sachs gauge is a null-coordinate formulation in general relativity that adapts coordinates to outgoing null hypersurfaces and fixes the areal radial coordinate.
• It structures the Einstein equations in a hierarchical (characteristic) system, enabling clear extraction of radiative degrees of freedom and gravitational mass loss.
• Modern generalizations extend its framework to include cosmological constant effects, partial gauge fixings, and a broader asymptotic symmetry analysis via the BMS group.

Bondi–Sachs gauge is the null-coordinate gauge underlying the Bondi–Sachs formalism of general relativity: it fixes much of the coordinate freedom by adapting coordinates to outgoing null hypersurfaces, choosing an areal radial coordinate, and imposing a determinant condition on the angular metric. In this gauge, the Einstein equations acquire a characteristic hierarchical structure, the radiative degrees of freedom are encoded directly in the angular metric, and the Bondi mass, Bondi news, and Bondi–Metzner–Sachs group emerge in a transparent way (Mädler et al., 2016). Later work extends the same geometric strategy to asymptotically de Sitter settings, generalized asymptotic symmetry analyses, and partial gauge fixings that include Bondi–Sachs and Newman–Unti gauges as special cases (He et al., 2018, Flanagan et al., 2023, Geiller et al., 2024).

1. Definition and metric form

In the standard construction, one introduces coordinates

$x^a=(u,r,x^A),\qquad A=2,3,$

where $u$ labels outgoing null hypersurfaces, $r$ is an areal radial coordinate along the null rays, and $x^A$ are angular coordinates on the transverse 2-surfaces (Mädler et al., 2016). The defining geometric conditions are that $u=\mathrm{const}$ are null, the angular coordinates are constant along the null generators, and the angular metric has fixed determinant.

The corresponding Bondi–Sachs metric is

$$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$

with metric functions $\beta(u,r,x^A)$, $V(u,r,x^A)$, $U^A(u,r,x^B)$, and $h_{AB}(u,r,x^C)$, together with

$u$0

where $u$1 is the determinant of a fixed unit-sphere metric $u$2 (Mädler et al., 2016). In the common $u$3 parameterization, $u$4 can be written in terms of two functions $u$5 and $u$6, leaving two independent radiative degrees of freedom (Mädler et al., 2016).

For the general non-axisymmetric parameterization used in asymptotically de Sitter analyses,

$u$7

with $u$8 and $u$9 (He et al., 2018). In the axisymmetric special case, $r$0 and $r$1, and the metric simplifies accordingly (He et al., 2018).

2. Gauge conditions and residual coordinate freedom

The gauge conditions may be summarized as follows. First, null slicing: $r$2 Second, the angular coordinates are constant along the null generators: $r$3 Third, the areal-radius condition is imposed through

$r$4

so that the surfaces of constant $r$5 have area $r$6 (Mädler et al., 2016). From these conditions one obtains

$r$7

which are the standard Bondi–Sachs gauge equations (Mädler et al., 2016).

The determinant condition implies

$r$8

so the radial and retarded-time derivatives of $r$9 are trace-free with respect to $x^A$0 (Mädler et al., 2016). This is the mechanism by which the angular metric carries only the two gravitational-wave polarizations.

These conditions do not exhaust the diffeomorphism freedom. At future null infinity, the residual diffeomorphisms preserving Bondi–Sachs gauge and its asymptotic structure form the BMS group. In inertial coordinates at $x^A$1, the generators take the form

$x^A$2

where $x^A$3 is a conformal Killing vector of the unit 2-sphere and $x^A$4 generates supertranslations (Mädler et al., 2016). This suggests that Bondi–Sachs gauge is simultaneously a coordinate choice and the natural stage on which asymptotic symmetry acts.

A weaker framework, the partial Bondi gauge, keeps only

$x^A$5

while leaving the traces in the angular expansion free. In that setting, Bondi–Sachs and Newman–Unti gauges arise as distinct complete gauge fixings, the former by imposing the determinant condition and the latter by setting $x^A$6 (Geiller et al., 2024).

3. Characteristic Einstein equations

A central feature of Bondi–Sachs gauge is that the Einstein equations decompose into a hierarchical characteristic system. In the vacuum metric formulation, the main equations are taken to be

$x^A$7

while the remaining components become supplementary conditions propagated by the contracted Bianchi identities (Mädler et al., 2016).

The hypersurface equations are radial equations solved sequentially along each null cone. For $x^A$8,

$x^A$9

For $u=\mathrm{const}$0,

$u=\mathrm{const}$1

For $u=\mathrm{const}$2,

$u=\mathrm{const}$3

with $u=\mathrm{const}$4 the Ricci scalar of $u=\mathrm{const}$5 (Mädler et al., 2016).

The trace-free part of $u=\mathrm{const}$6 is the evolution equation for the angular metric, equivalently for the radiative variables. Operationally, the characteristic algorithm is

$u=\mathrm{const}$7

followed by radial determination of $u=\mathrm{const}$8, which advances the solution in retarded time (Mädler et al., 2016).

The same hierarchical structure persists in Einstein–scalar systems. In the zero-cosmological-constant Einstein–massless-scalar case, the equations again split into six main equations, one trivial equation, and three supplementary equations, yielding seven independent equations arranged as hypersurface equations for $u=\mathrm{const}$9 and evolution equations for $$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$0 (Li et al., 2024). In spherical symmetry with cosmological constant, a Bondi–Sachs-type null slicing also yields a first-order strongly hyperbolic formulation after passage to $$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$1 variables, with lapse and shift as characteristic fields and a constraint-preserving initial boundary value problem constructed from the Bianchi identity (Cao et al., 2023).

4. Radiation, news, and mass loss

In an asymptotically flat Bondi frame,

$$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$2

and the angular metric admits the expansion

$$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$3

with

$$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$4

(Mädler et al., 2016).

The hypersurface equations then determine the leading asymptotics: $$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$5

$$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$6

$$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$7

where $$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$8 is the angular momentum aspect and $$g_{ab}dx^a dx^b = -\frac{V}{r}e^{2\beta}\,du^2 -2e^{2\beta}\,du\,dr +r^2 h_{AB}(dx^A-U^Adu)(dx^B-U^Bdu),$$9 is the mass aspect (Mädler et al., 2016).

The Bondi news tensor is

$\beta(u,r,x^A)$0

and in dyad notation the news function is

$\beta(u,r,x^A)$1

so the real and imaginary parts of $\beta(u,r,x^A)$2 correspond to the $\beta(u,r,x^A)$3 and $\beta(u,r,x^A)$4 polarizations (Mädler et al., 2016). The leading supplementary equation gives

$\beta(u,r,x^A)$5

which integrates to the Bondi mass-loss formula

$\beta(u,r,x^A)$6

(Mädler et al., 2016).

In the Einstein–massless-scalar extension, the scalar field contributes an additional radiative channel. With

$\beta(u,r,x^A)$7

the Bondi 4-momentum is defined using a modified mass aspect $\beta(u,r,x^A)$8, and the mass-loss formula becomes

$\beta(u,r,x^A)$9

so $V(u,r,x^A)$0 is the scalar radiation flux term (Li et al., 2024).

A conformally invariant reformulation of Bondi–Sachs energy-momentum avoids dependence on a Bondi system altogether. In that formulation, the mass aspect $V(u,r,x^A)$1, the Bondi news $V(u,r,x^A)$2, and the space of asymptotic translations are defined directly on arbitrary cuts of $V(u,r,x^A)$3, with the Bondi–Sachs energy-momentum computed by integrating $V(u,r,x^A)$4 against conformally normalized asymptotic translations (Frauendiener et al., 2021). This suggests that the standard Bondi gauge is a particularly convenient presentation of invariant structures rather than their only definition.

5. Cosmological constant and asymptotically de Sitter variants

With a nonzero cosmological constant, the Bondi–Sachs metric keeps the same structural gauge conditions—null hypersurfaces, luminosity radius, unit-determinant angular metric, and $V(u,r,x^A)$5—but the asymptotic behavior changes (Xie et al., 2017, Ge et al., 2011). In the $V(u,r,x^A)$6 four-dimensional metric used by Xie and Zhang,

$V(u,r,x^A)$7

the leading term in $V(u,r,x^A)$8 is

$V(u,r,x^A)$9

and additional asymptotic functions $U^A(u,r,x^B)$0 enter nontrivially (Xie et al., 2017). Under their natural boundary condition—Sommerfeld radiation condition together with $U^A(u,r,x^B)$1-independence of $U^A(u,r,x^B)$2—the Weyl scalars satisfy the usual peeling hierarchy

$U^A(u,r,x^B)$3

even though the asymptotics are not the asymptotically flat ones (Xie et al., 2017).

In asymptotically de Sitter spacetime, He, Jing, and Cao formulate Bondi–Sachs coordinates $U^A(u,r,x^B)$4 by solving the eikonal equation

$U^A(u,r,x^B)$5

on the perturbed de Sitter background, with

$U^A(u,r,x^B)$6

(He et al., 2018). The outgoing boundary condition is modified to

$U^A(u,r,x^B)$7

with

$U^A(u,r,x^B)$8

so the radiative $U^A(u,r,x^B)$9 coefficients are derivatives of the leading $h_{AB}(u,r,x^C)$0-dependent pieces (He et al., 2018). Matching to transverse-traceless perturbations yields

$h_{AB}(u,r,x^C)$1

where $h_{AB}(u,r,x^C)$2 and $h_{AB}(u,r,x^C)$3 are explicit combinations of derivatives of the mass and pressure quadrupoles (He et al., 2018). The same paper estimates that $h_{AB}(u,r,x^C)$4-effects on gravitational-wave detection become important only when

$h_{AB}(u,r,x^C)$5

which for observationally relevant frequencies is far beyond realistic distance scales (He et al., 2018).

A separate $h_{AB}(u,r,x^C)$6 application is the Bondi–Sachs rocket family, where accelerated pure-radiation solutions retain Bondi–Sachs gauge and exhibit Bondi mass loss balanced by emitted pure radiation in special cases, with no gravitational radiation because the shear vanishes (Ge et al., 2011).

6. Modern generalizations and limitations

Recent work generalizes Bondi–Sachs gauge by relaxing how fully the radial coordinate and boundary metric are fixed. In the partial Bondi gauge, one retains only

$h_{AB}(u,r,x^C)$7

with metric

$h_{AB}(u,r,x^C)$8

and

$h_{AB}(u,r,x^C)$9

leaving the traces $u$00 free (Geiller et al., 2024). Bondi–Sachs gauge is then recovered by the determinant condition, which fixes

$u$01

whereas Newman–Unti gauge is obtained by setting $u$02, leaving $u$03 free (Geiller et al., 2024).

In asymptotically flat vacuum near $u$04, a different generalization allows arbitrary induced 2-metric $u$05 on the celestial sphere and studies the fully nonlinear action of the Weyl–BMS group on the leading Bondi-gauge metric functions

$u$06

with Bondi news

$u$07

(Flanagan et al., 2023). In that framework, supertranslations act by

$u$08

while Weyl-like transformations rescale $u$09 and shift $u$10 inhomogeneously (Flanagan et al., 2023). This suggests that Bondi–Sachs gauge is not a single rigid choice but a family of null gauges supporting progressively larger asymptotic symmetry groups.

At the same time, PDE analyses show a limitation of Bondi-like gauges. In linearized studies of the Einstein equations in Bondi-like coordinates, the principal symbol decomposes into gauge, constraint, and physical blocks, and the gauge block is only weakly hyperbolic. Giannakopoulos et al. argue that Bondi-like gauges therefore lead, under quite general conditions, to weakly hyperbolic free-evolution systems, rendering the characteristic initial boundary value problem ill-posed in the simplest norms one would like to employ (Giannakopoulos et al., 2021). This does not negate the geometric utility of Bondi–Sachs gauge, but it sharply distinguishes geometric transparency from PDE well-posedness.

A complementary development is the Special Double Null gauge, designed to treat $u$11 and $u$12 democratically with null holographic directions rather than a spacelike areal radius. That framework is presented explicitly as complementary to Bondi and Ashtekar–Hansen gauges rather than a replacement for Bondi–Sachs gauge (Krishnan et al., 2021).

7. Broader significance

Bondi–Sachs gauge provided the first convincing evidence that gravitational radiation is a nonlinear effect of general relativity and that the emission of gravitational waves from an isolated system is accompanied by mass loss from the system (Mädler et al., 2016). It also revealed the asymptotic symmetry group at null infinity to be larger than the Poincaré group, namely the BMS group (Mädler et al., 2016).

The same gauge continues to organize several distinct research programs. In characteristic numerical relativity, it underlies worldtube–null-cone evolution and clean wave extraction at $u$13 (Mädler et al., 2016). In asymptotic symmetry, it is the natural arena for BMS, generalized BMS, and Weyl–BMS analyses (Flanagan et al., 2023). In asymptotically de Sitter and anti–de Sitter studies, it remains the reference null gauge from which modified boundary conditions, source multipole identifications, and constraint-preserving boundary conditions are formulated (He et al., 2018, Cao et al., 2023). In conformal treatments, it serves as the benchmark gauge whose standard notions—mass aspect, news, and translations—can be reconstructed in a gauge-independent form on arbitrary cuts of null infinity (Frauendiener et al., 2021).

Taken together, these developments show that Bondi–Sachs gauge is both a classical coordinate gauge and a durable geometric framework: it isolates radiative and Coulombic data, organizes the Einstein equations into a characteristic hierarchy, and anchors the modern understanding of null infinity, asymptotic symmetries, and gravitational radiation (Mädler et al., 2016, Geiller et al., 2024).