On the structure and applications of the Bondi–Metzner–Sachs group

Abstract

This work is a pedagogical review dedicated to a modern description of the Bondi–Metzner–Sachs (BMS) group. Minkowski space-time has an interesting and useful group of isometries, but, for a generic space-time, the isometry group is simply the identity and hence provides no significant informations. Yet symmetry groups have important role to play in physics; in particular, the Poincaré group describing the isometries of Minkowski space-time plays a role in the standard definitions of energy-momentum and angular-momentum. For this reason alone it would seem to be important to look for a generalization of the concept of isometry group that can apply in a useful way to suitable curved space-times. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work, the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations between coordinate systems of a certain type in asymptotically flat space-times. In the third section the conformal method and the notion of “asymptotic simplicity” are introduced, following mainly the works of Penrose. This section prepares us for another derivation of the BMS group which will involve the conformal structure, and is thus more geometrical and fundamental. In the subsequent sections we discuss the properties of the BMS group, e.g. its algebra and the possibility to obtain as its subgroup the Poincaré group, as we may expect. The paper ends with a review of the BMS invariance properties of classical gravitational scattering discovered by Strominger, that are finding application to black hole physics and quantum gravity in the literature.

Carrollian conservation laws and Ricci-flat gravity

“…When the bulk metric is expressed in an appropriate gauge, usually given by imposing the radial coordinate be null, Einstein equations can reduce in some instances to equations defined on null infinity I + . 15 Its null nature makes it a natural host for a Carrollian geometry and the gravitational dynamics will be shown to match with Carrollian conservation laws. This section can be considered as a precursor of a full asymptotically flat holographic scheme.…”

A fuzzy bipolar celestial sphere

“…In order to describe the action of such transformations on the celestial sphere, we start by introducing complex stereographic coordinates (ζ,ζ) for each point of the sphere. Then, it turns out [19] that any Lorentz transformation on I + for the stereographic coordinates is given by a Möbius map,…”

“…This is the classical phenomenon of stellar aberration. The infinitesimal transformations of (3.2), (3.3) are described by the following vector fields on on I + , 15) and it is easy to prove [19] that they are a representation of the Lorentz algebra on I + and on the celestial sphere, having fixed the value of u:…”

Note on the symplectic structure of asymptotically flat gravity and BMS symmetries

“…In order to carry out an explicit calculation of the symplectic form in (29) we need to specify the boundary conditions on the field C zz . Such conditions play a key role because they account for the soft graviton zero modes [1,2].…”