Bondi–Metzner–Sachs Frame Overview

• BMS frame is a specific choice of Bondi coordinates at future null infinity that fixes a unique cross-section and angular mapping.
• It employs gauge-fixing conditions on the sphere metric, shear, and mass aspect to reduce the supertranslation ambiguity.
• BMS transformations relate different cuts, influencing gravitational charge definitions and memory effects in asymptotically flat spacetimes.

A Bondi–Metzner–Sachs (BMS) frame is the choice of Bondi coordinates and boundary data at future null infinity $\mathscr I^+$ that singles out a particular representative of the asymptotic-symmetry orbit of an asymptotically flat spacetime. Physically, a BMS frame consists of a choice of cross-section (cut) $u=0$ of future null infinity, and an identification of its angular coordinates $(\theta,\phi)$ with the unit sphere. Equivalently, it is a choice of Bondi gauge together with additional conditions on the sphere metric, shear, mass aspect, or asymptotic charges that reduce the infinite-dimensional BMS freedom to a preferred Poincaré subgroup. The notion is central because supertranslations shift the origin cut and mix asymptotic data, while more general extensions also act on the conformal structure of the celestial sphere (Helfer, 2021, Flanagan et al., 2023).

1. Null infinity, Bondi gauge, and the Bondi frame

For asymptotically simple spacetimes, $\scri$ is a null hypersurface whose components $\scri^+$ and $\scri^-$ each have topology $S^2\times\mathbb R$ and carry a degenerate universal conformal metric of signature $(0,+,+)$. Near $\scri^+$, one introduces Bondi–Sachs coordinates $(u,r,\theta^A)$, where $u=0$0 is retarded time, $u=0$1 is an areal or affine radial parameter along null generators, and $u=0$2 label the generators. In Bondi gauge,

$u=0$3

and the metric is written as

$u=0$4

Asymptotic flatness implies expansions in $u=0$5. A convenient form is

$u=0$6

The corresponding Bondi-gauge fields are the shear $u=0$7, the Bondi news $u=0$8, the mass aspect $u=0$9, the angular-momentum aspect $(\theta,\phi)$0, and the sphere metric $(\theta,\phi)$1. The Bondi frame at future null infinity is then the manifold chart $(\theta,\phi)$2 obtained as $(\theta,\phi)$3 (Flanagan et al., 2023, Prinz et al., 2021).

2. BMS transformations and the meaning of “frame”

The BMS group is the group of diffeomorphisms of $(\theta,\phi)$4 that preserve the Bondi gauge and asymptotic fall-offs. Abstractly,

$(\theta,\phi)$5

where $(\theta,\phi)$6 is the additive abelian group of supertranslations and $(\theta,\phi)$7 is the proper orthochronous Lorentz group. The $(\theta,\phi)$8 spherical harmonics of the supertranslation parameter are the ordinary translations; the $(\theta,\phi)$9 modes are genuine supertranslations.

In stereographic coordinate $\scri$0, the most general diffeomorphism preserving Bondi gauge and the standard fall-offs can be written as

$\scri$1

$\scri$2

Here $\scri$3 is the supertranslation and $\scri$4 is the sphere-conformal factor.

Mirzaiyan–Esposito analyze the four Möbius classes of the angular map $\scri$5: parabolic, elliptic, hyperbolic, and loxodromic. In that classification, elliptic transformations have $\scri$6, while hyperbolic and loxodromic transformations have conformal factors

$\scri$7

whose monotonicity as functions of $\scri$8 distinguishes the cases. These formulas describe how a BMS frame changes under Lorentz redefinitions of the celestial sphere together with angle-dependent shifts of $\scri$9 (Mirzaiyan et al., 2023, Prinz et al., 2021).

3. Gauge fixing and canonical Bondi frames

A “frame” is a choice of representative in an orbit of the asymptotic symmetry action on Bondi data. In the nonlinear BMSW setting, the group acts transitively on the space of Bondi data $\scri^+$0, so choosing a frame amounts to choosing a slice through each group orbit. A standard procedure is to fix the unit-sphere metric $\scri^+$1 to be the round metric, thereby using up all sphere diffeomorphisms and Weyl rescalings, and then supertranslate so that the asymptotic shear on a reference cut vanishes: $\scri^+$2 The resulting coordinates are the canonical Bondi frame, in which

$\scri^+$3

and the remaining freedom is the finite Poincaré group (Flanagan et al., 2023).

A related and older strategy is the good-cut or rest-frame construction. Under a supertranslation, the shear transforms as

$\scri^+$4

so a shear-free cut $\scri^+$5 can be obtained by solving

$\scri^+$6

An alternative is the vanishing mass-dipole condition: expanding

$\scri^+$7

one imposes $\scri^+$8. Either choice eliminates all but the constant and $\scri^+$9 modes of the supertranslation freedom, leaving an ordinary Poincaré subgroup and thus a canonical rest-frame cut of $\scri^-$0 (Mädler et al., 2016).

4. Center-of-mass cuts and the supertranslation ambiguity

In the Dray–Streubel charge construction, angular momentum is encoded in a two-cut dependent charge $\scri^-$1, where $\scri^-$2 is the active cut at which one evaluates angular momentum and $\scri^-$3 is the passive cut serving as the choice of origin. The natural center-of-mass prescription is to define the center-of-mass cut $\scri^-$4 by requiring the time-space components to vanish,

$\scri^-$5

in a Bondi frame in which the four-momentum $\scri^-$6 is aligned along the $\scri^-$7-axis.

Under an actual translation by an asymptotically constant vector $\scri^-$8, the change-of-origin law is

$\scri^-$9

One may therefore solve for a complementary translation $S^2\times\mathbb R$0, unique up to constants parallel to $S^2\times\mathbb R$1, and define

$S^2\times\mathbb R$2

The ambiguity enters because the BMS group is $S^2\times\mathbb R$3, its translations are precisely the $S^2\times\mathbb R$4 modes of the supertranslation generators, and there is no invariant notion of “$S^2\times\mathbb R$5-free” supertranslation. The condition $S^2\times\mathbb R$6 eliminates only the $S^2\times\mathbb R$7 part of the shift. The $S^2\times\mathbb R$8 parts remain unfixed, so the quotient

$S^2\times\mathbb R$9

parameterizes the distinct center-of-mass cuts. Cuts which differ by an $(0,+,+)$0 supertranslation are physically incompatible choices of center-of-mass frame.

The same issue appears in spin. In Helfer’s formulation, the charge associated with a Lorentz generator $(0,+,+)$1 is

$(0,+,+)$2

For pure translations one has $(0,+,+)$3, so the space-space components identified with spin are translation-invariant. For a general supertranslation with $(0,+,+)$4, however, $(0,+,+)$5 unless the mass aspect is constant. Thus the natural definition of spin is translation-, but not supertranslation-, invariant. To recover uniqueness one must restrict the class of admissible cuts, for example to stationary good cuts or by adding a further convention such as requiring the shear-potential to have vanishing $(0,+,+)$6 part (Helfer, 2021).

5. Generalized BMS frames, BMSW frames, and vacuum structure

The notion of frame broadens when one enlarges the asymptotic symmetry group. In the generalized BMS group, $(0,+,+)$7 is an arbitrary sphere diffeomorphism and the Weyl factor is constrained by preservation of the volume form. In the Weyl BMS group, one allows arbitrary $(0,+,+)$8, arbitrary supertranslation $(0,+,+)$9, and arbitrary Weyl rescaling $\scri^+$0. A Bondi-gauge-preserving diffeomorphism then acts by

$\scri^+$1

For the induced metric and shear, a compact expression is

$\scri^+$2

In this setting, a BMS or BMSW frame is again a representative of an orbit, but now the orbit includes transformations of the boundary metric as well as of the shear and mass aspect (Flanagan et al., 2023).

Allowing the boundary metric to fluctuate introduces additional vacuum structure. One description writes

$\scri^+$3

with $\scri^+$4 and a boundary Liouville field $\scri^+$5. If $\scri^+$6 is fixed to the round metric but $\scri^+$7 is allowed, vacua are labeled by a $\scri^+$8-independent shift $\scri^+$9, interpreted as the supertranslation field; transitions $(u,r,\theta^A)$0 are displacement-memory effects. If $(u,r,\theta^A)$1 is allowed to fluctuate, one finds superboost and superrotation fields, and finite superboost transformations act inhomogeneously on $(u,r,\theta^A)$2. The impulsive limit of such a superboost transition is an impulsive gravitational wave whose memory is a velocity-kick/refraction effect at $(u,r,\theta^A)$3. This suggests that different generalized BMS frames are not merely coordinate choices but distinct asymptotic vacua related by memory transitions (Ruzziconi, 2020).

6. Charges, memory, and contemporary frame fixing

The modern charge-based viewpoint makes the role of the frame explicit. In nonradiative vacuum regions, BMS charges include the Bondi four-momentum, intrinsic angular momentum, center-of-mass location, and infinitely many supermomentum charges. In the extended setting with superrotations, one also obtains electric-parity super center-of-mass charges and magnetic-parity superspin charges. A canonical Bondi frame can then be fixed in stages: boost to the rest frame so that $(u,r,\theta^A)$4, impose $(u,r,\theta^A)$5, eliminate supertranslations by demanding $(u,r,\theta^A)$6, and eliminate superrotations by imposing $(u,r,\theta^A)$7. In that frame,

$(u,r,\theta^A)$8

The same formalism ties supermomentum to ordinary gravitational-wave memory, super center-of-mass charges to total memory, and superspin charges to the ordinary piece of spin memory (Flanagan et al., 2015).

Recent black-hole perturbation theory has made BMS-frame control a practical issue. On a Kerr background, one may iteratively transform an arbitrary asymptotically flat perturbation to Bondi–Sachs gauge and then fix the residual BMS freedom at a reference $(u,r,\theta^A)$9 by imposing

$u=0$00

$u=0$01

$u=0$02

Time translations and $u=0$03-rotations remain free because they are background Killing symmetries. In multiscale self-force expansions, “forgetful gauges” forget memory because the $u=0$04 modes of the first-order shear vanish in the near zone; passing to Bondi–Sachs gauge introduces a slowly evolving supertranslation $u=0$05 and corresponding soft hair,

$u=0$06

In that framework, memory and “memory distortion” appear naturally, and the second-order source becomes integrable toward $u=0$07, curing the infrared divergence encountered in forgetful gauges. The same framework could also be used for ringdown analysis and is expected to be vital for comparisons with numerical relativity and post-Newtonian theory (Spiers et al., 23 Jun 2026).