On the Nature of Bondi–Metzner–Sachs Transformations

Abstract

This paper investigates, as a first step, the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable ζ used in the metric for cuts of null infinity, there is a corresponding BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of ζ. In the loxodromic and hyperbolic cases, this conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of ζ. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold.

Homogeneous Projective Coordinates for the Bondi-Metzner-Sachs Group

“…As was pointed out in Ref. [19], the complex homogeneous coordinates associated to the Bondi-Metzner-Sachs transformation (2) have modulus ≤ 1, which is the equation of a unit circle, and are…”

“…The next step of the program initiated in Ref. [19] consists in realizing that, much in the same way as the affine transformations in the Euclidean plane…”

“…The work in Ref. [19] has outlined the resulting geometric picture, where (w 0 , w 1 , w 2 ) are viewed as homogeneous coordinates in a complex projective plane. In our paper we have instead a less abstract and more concrete task: since the Bondi-Metzner-Sachs transformation (2) becomes linear when expressed in terms of z 0 and z 1 , we are aiming to develop the Bondi-Metzner-Sachs formalism with the associated Killing vector fields by using the pair of variables (z 0 , z 1 ) instead of (ζ, ζ).…”

“…In particular, we have in mind the concept of superrotations [21,22,26] on the one hand, and the physical applications of the Segre manifold advocated in Ref. [19]. In other words, since our Eq.…”

“…The Bondi-Metzner-Sachs [1-3] asymptotic symmetry group of asymptotically flat spacetime has received again much attention over the last decade by virtue of its relevance for black-hole physics [4][5][6], the group-theoretical structure of general relativity [7][8][9][10][11][12][13][14][15][16][17][18][19][20] and the infrared structure of fundamental interactions [21][22][23][24]. The appropriate geometric framework can be summarized as follows.…”