In gravitational theory,–Metzner– the Bondi–Metzner–Sachs (BMS) group, or the Bondi–van der Burg–Metzner–Sachs group, is an asymptotic symmetry group of asymptotically flat, Lorentzian spacetimes at null (i.e., light-like) infinity. It was originally formulated in 1962 by Hermann Bondi, M. G. van der Burg, A. W. Metzner and Rainer K. Sachs in order to investigate the flow of energy at infinity due to propagating gravitational waves. Half a century later, this work of Bondi, van der Burg, Metzner, and Sachs is considered pioneering and seminal. In his autobiography, Bondi considered the 1962 work as his "best scientific work".
1962 work of Bondi, van der Burg, Metzner, and Sachs
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, might be to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group, 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. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances.
The coordinates used in the 1962 formulation were those introduced by Bondi and generalized by Sachs, which focused on null (i.e., light-like) geodesics, called null rays, along which the gravitational waves traveled. The null rays form a null hypersurface, defined by the retarded time for outgoing waves and advanced time for incoming waves. The basic idea, which was novel then, was to use the family of outgoing (or incoming) null hypersurfaces to build spacetime coordinates that would describe outgoing (or incoming) gravitational waves. In addition to the retarded (or advanced) time are the space-like distance and the null-ray direction to complete the local spacetime coordinates . As is large and approaches infinity, the set of null hypersurfaces form the future null infinity, where the outgoing gravitational waves "exit". Similar considerations of null hypersurfaces as goes to infinity yield the past null infinity, where the incoming gravitational waves "enter". These two null (i.e., light-like) infinities, found using the non-inertial Bondi-Sachs coordinates, are not obvious in the inertial Cartesian coordinates of flat spacetime, where the two time-like infinities and the space-like infinity are obvious. All five infinities are revealed in the asymptotic conformal treatment of infinity by Penrose, where the future (or past) null infinity is denoted by script (or script ) and pronounced "scri plus" (or "scri minus").