Isomorphism between the local Poincare generalized translations group and the group of spacetime transformations (x LB1)4

TL;DR

This paper demonstrates an isomorphism between the Poincare translation subgroup and a group of tetrad transformations called LB1, extending the concept to local translations and connecting it with the BMS supertranslations.

Contribution

It establishes a direct isomorphism between the Poincare translation subgroup and the LB1 tetrad transformation group, generalizing to local translations including supertranslations.

Findings

Proves the isomorphism between the translation subgroup and (x LB1)4.

Introduces differential equations involving various fields to facilitate the proof.

Shows that generalized translations are tensor products of LB1 groups.

Abstract

We will prove that there is a direct relationship between the Poincare subgroup of translations, and the group of tetrad transformations LB1 introduced in a previous manuscript. LB1 is the group composed by SO(1; 1) plus two kinds of discrete transformations. Translations have been extensively studied under the scope of gauge theories. By using the geometric structures built to prove this elementary result we will generalize it to the case of what we might call local translations. A special case of the latter is the Bondi-Metzner-Sachs subgroup of supertranslations. In order to accomplish this goal and since the group of translations is four-dimensional we will prove first that it is isomorphic to (x LB1)4. In order to prove this claim we will introduce a system of differential equations involving several kinds of fields. Abelian, non-Abelian, spinor, gravitational. These fields will constitute the structure needed to build local tetrads of a new kind that allow for the proof to be carried out with simplicity. Results already obtained involving similar but not equal tetrads will be useful in our constructions and demonstrations. Translations and generalized translations isomorphic to tensor products of LB1 groups are not trivial results. Because the LB1 group is composed by SO(1; 1) and two discrete transformations.