Bondi–Metzner–Sachs group

Description

The Bondi–Metzner–Sachs group, or BMS group, is the group of asymptotic symmetries of spacetimes that are asymptotically flat at future null infinity. Use coordinates $(u, z)$ on null infinity $\R \times S^2$, where $u$ is retarded time and $z \in \C \cup \set{\infty}$ is a stereographic coordinate on the sphere. The group acts by $$(u, z) \mapsto \Bigl( K(z) \bigl( u + \alpha(z) \bigr), \ \frac{az + b}{cz + d} \Bigr), \qquad K(z) = \frac{1 + \abs{z}^2}{\abs{az + b}^2 + \abs{cz + d}^2}.$$ Here $a, b, c, d \in \C$ with $ad - bc = 1$, and $\alpha$ is a smooth real function on $S^2$. The maps with $a = d = 1$, $b = c = 0$ are the supertranslations $u \mapsto u + \alpha(z)$; they form an infinite-dimensional abelian normal subgroup $S$. The maps with $\alpha = 0$ form a subgroup isomorphic to the proper orthochronous Lorentz group $SL(2, \C)/\set{\pm 1}$. So the BMS group is the semidirect product $S \rtimes SO^+(1, 3)$. Check of one case: the identity matrix gives $K(z) = (1 + \abs{z}^2)/(\abs{z}^2 + 1) = 1$, so the map is the supertranslation by $\alpha$.