### 2.5 The Bondi–Metzner–Sachs group

The group of coordinate transformations at that preserves the Bondi coordinate conditions, the
BMS group, is the same as the asymptotic symmetry group that arises from approximate solutions to
Killing’s equation as is approached. The BMS group has two parts: the homogeneous Lorentz group
and the supertranslation group, which contains the Poincaré translation sub-group. Their importance to us
lies in the fact that all the physical quantities arising from our identifications must transform appropriately
under these transformations [49, 29].
Specifically, the BMS group is given by the supertranslations, with an arbitrary regular
differentiable function on :

and the Lorentz transformations, with the complex parameters of ,
If is expanded in spherical harmonics,

the terms represent the Poincaré translations, i.e.,
Details about the representation theory, with applications, are given later.