The Group Law for the New Internal-Spacetime Mapping between the Group of Internal Electromagnetic Gauge Transformations and the Groups LB1 and LB2 of Spacetime Tetrad Transformations
Alcides Garat
Abstract
In this note we will demonstrate the nature of the group law for the internal-spacetime mapping between the local group of internal electromagnetic gauge transformations and the local groups of tetrad transformations LB1 and LB2 introduced previously. This proof is not trivial and that is why we present the manifest deductions step by step. New tetrads have been introduced in four-dimensional Lorentz spacetimes with remarkable properties. These new tetrads are gauge dependent and enable the proof of results in group theory. It has been already proved using these new tetrads that the local Abelian group of electromagnetic gauge transformations can be mapped into the local groups of tetrad transformations LB1 and LB2. In Einstein–Maxwell four-dimensional spacetimes at every point we can define two orthogonal planes. The timelike-spacelike local plane one. The orthogonal spacelike local plane two. All vectors on these local orthogonal planes have been proven to be eigenvectors of the Einstein–Maxwell stress-energy tensor. LB1 is the local group of tetrad transformations on plane one. It is made up of the boosts $$SO(1,1)$$ plus two discrete transformations. LB2 is the local group of tetrad transformations on the spacelike plane two, that is $$SO(2)$$ . The local electromagnetic gauge transformations of the tetrads spanning both local orthogonal planes will result in these tetrads not leaving these local planes after the transformation thus ensuring the invariance of the metric tensor. We will display in this note the explicit nature of the group law for these internal-spacetime mappings through the use of the new tetrads. We will show the details for LB1, the proper sector and also the special improper sector. We will do the same for LB2. This map is at the core of grand field unification, since it explicitly displays the nontrivial mechanism of tetrad unification between the electromagnetic field and the gravitational field. We will show in a theorem that these mappings between the local internal group of electromagnetic gauge transformations and separately both LB1 and LB2 satisfy the group law.