It was originally formulated in 1962 by Hermann Bondi, M. G. Van der Burg, A. W. Metzner[1] and Rainer K. Sachs[2] in order to investigate the flow of energy at infinity due to propagating gravitational waves.
Instead of the expected ordinary four spacetime translations of special relativity associated with the well-known conservation of momentum and energy, they found, much to their puzzling surprise, a novel infinite superset of direction-dependent time translations, which were named supertranslations.
Half a century later, this work of Bondi, Van der Burg, Metzner, and Sachs is considered pioneering and seminal.
[4]: 79 The group of supertranslations is key to understanding the connections to quantum fields and gravitational wave memories.
To give some context for the general reader, the naive expectation for asymptotically flat spacetime symmetries, i.e., symmetries of spacetime seen by observers located far away from all sources of the gravitational field, would be to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group, also called the inhomogeneous Lorentz group,[2] which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations.
Expectations aside, the first step in the work of Bondi, Van der Burg, Metzner, and Sachs was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, with no a priori assumptions made about the nature of the asymptotic symmetry group — not even the assumption that such a group exists.
Then after artfully designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields.
This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity.
These additional non-Lorentz asymptotic symmetries, which constitute an infinite superset of the four spacetime translations, are named supertranslations.
[3]: 35 The coordinates used in the 1962 formulation were those introduced by Bondi[6] and generalized by Sachs,[7] which focused on null (i.e., light-like) geodesics, called null rays, along which the gravitational waves traveled.
can be expanded as an infinite series of spherical harmonics, it was shown that the first four terms (ℓ = 0, 1) reproduce the four ordinary spacetime translations, which form a subgroup of the supertranslations.
[2] Loosely speaking, "neighboring" points on the future null infinity with slightly different
Light rays from one point can't reach another, clocks can not be synchronized, and thus an arbitrary time offset
[11] Abstractly, the BMS group is an infinite-dimensional extension, or a superset, of the Poincaré group, in which four of the ten conserved quantities or charges of the Poincaré group (namely, the total energy and momentum associated with spacetime translations) are extended to include an infinite number of conserved supermomentum charges associated with spacetime supertranslations, while the six conserved Lorentz charges remain unchanged.
[12] After half a century lull, interest in the study of this asymptotic symmetry group of General Relativity (GR) surged, in part due to the advent of gravitational-wave astronomy (the hope of which prompted the pioneering 1962 studies).
Interestingly, the extension of ordinary four spacetime translations to infinite-dimensional supertranslations, viewed in 1962 with consternation, is interpreted, half a century later, to be a key feature of the original BMS symmetry.
For example, by imposing supertranslation invariance (using a smaller BMS group acting only on the future or past null infinity) on S-matrix elements involving gravitons, the resulting Ward identities turn out to be equivalent to Weinberg's 1965 soft graviton theorem.
In fact, such a relation between asymptotic symmetries and soft Quantum field theory theorems is not specific to gravitation alone, but is rather a general property of gauge theories including electromagnetism.
When a gravitating radiation pulse transit past arrays of detectors stationed near future null infinity in the vacuum, the relative positions and clock times of the detectors before and after the radiation transit differ by a BMS supertranslation.
The relative spatial displacement found for a pair of nearby detectors reproduces the well-known and potentially measurable gravitational memory effect.
Hence the displacement memory effect both physically manifests and directly measures the action of a BMS supertranslation.
[13] BMS supertranslations, the leading soft graviton theorem, and displacement memory effect form the three vertices of an IR triangle describing the leading infrared structure of asymptotically flat spacetimes at null infinity.
[3] In addition, BMS supertranslations have been utilized to motivate the microscopic origin of black hole entropy,[14] and that black hole formed by different initial star configurations would have different supertranslation hair.
[15] To sort out which GR asymptotic symmetry might represent the Universe, recent simulations suggest that determining which gravitational-wave (GW) memory terms, displacement and spin, would give the best fit to the GW data to be collected in next generation detectors might constrain the three model symmetry scenarios: (a) Poincaré group (no memory); original BMS group (only displacement memory); and (c) extended BMS group (both displacement and spin memories).