2.5 The Bondi–Metzner–Sachs group

The group of coordinate transformations at + ℑ 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 ℑ+ 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 [65, 42Jump To The Next Citation Point].

Specifically, the BMS group is given by the supertranslations, with -- α(ζ,ζ) an arbitrary regular differentiable function on S2:

^u = u + α (ζ, ζ) (2.63 ) B- B -- (^ζ,^ζ) = (ζ,ζ)
and the Lorentz transformations, with (a, b,c,d ) the complex parameters of SL (2,ℂ),
^u = Ku , (2.64 ) B B -- --------------1-+-ζζ-------------- K = (aζ + b)(aζ-+ b) + (cζ + d)(cζ-+ d), ^ζ = aζ-+-b, ad − bc = 1. cζ + d

If -- α(ζ,ζ) is expanded in spherical harmonics,

∑ ml α(ζ,ζ¯) = α Ylm (ζ, ¯ζ), (2.65 ) m,l
the l = 0,1 terms represent the Poincaré translations, i.e.,
√ -- α(P)(ζ, ¯ζ) = daˆla =--2d0Y00− 1diY10i. (2.66 ) 2 2
Details about the representation theory, with applications, are given later.
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