Tetrads in Yang-Mills geometrodynamics

TL;DR

This paper introduces a new set of tetrads in SU(2) x U(1) Yang-Mills theories within curved spacetimes, enabling explicit diagonalization of the stress-energy tensor and establishing isomorphisms between gauge and spacetime transformation groups.

Contribution

The paper develops gauge-dependent and gauge-invariant tetrads that diagonalize the Yang-Mills stress-energy tensor and reveals isomorphisms between internal gauge groups and spacetime transformation groups.

Findings

New tetrads diagonalize the Yang-Mills stress-energy tensor.

Established isomorphisms between gauge groups and Lorentz transformation groups.

Presented gauge-invariant objects for tensor diagonalization.

Abstract

A new set of tetrads is introduced within the framework of SU(2) X U(1) Yang-Mills field theories in four dimensional Lorentz curved spacetimes. Each one of these tetrads diagonalizes separately and explicitly each term of the Yang-Mills stress-energy tensor. Therefore, three pairs of planes also known as blades, can be defined, and make up the underlying geometrical structure, at each point. These tetrad vectors are gauge dependent on one hand, and also in their definition, there is an additional inherent freedom in the choice of two vector fields. In order to get rid of the gauge dependence, another set of tetrads is defined, such that the only choice we have to make is for the two vector fields. A particular choice is made for these two vector fields such that they are gauge dependent, but the transformation properties of these tetrads are analogous to those already known for curved spacetimes where only electromagnetic fields are present. This analogy allows to establish group isomorphisms between the local gauge group SU(2), and the tensor product of the groups of local Lorentz tetrad transformations, either on blade one or blade two. These theorems show explicitly that the local internal groups of transformations are isomorphic to local spacetime groups of transformations. As an example of application of these new tetrads, we exhibit three new gauge invariant objects, and using these objects we show how to diagonalize the Yang-Mills stress-energy tensor in a gauge invariant way.