New Tetrads for General Relativity

TL;DR

This paper introduces a one-parameter family of q-gauge theories based on quantum groups, demonstrating that classical General Relativity emerges as an invariant sector despite underlying non-commutative gauge fields.

Contribution

It constructs a novel family of deformed gauge theories using q-differential calculus, linking quantum group structures to classical Einstein gravity.

Findings

Classical Einstein gravity is recovered as an invariant sector.

Gauge fields are non-commutative but lead to ordinary geometric objects.

The theory maintains invariance under a deformation parameter q.

Abstract

Using the tools of q--differential calculus and quantum Lie algebras associated to quantum groups, we find a one--parameter family of q-gauge theories associated to the quantum group $ISO_q(3,1)$. Although the gauge fields, that is the spin--connection and the vierbeins are non--commuting objects depending on a deformation parameter, $q$, it is possible to construct out of them a metric theory which is insensitive to the deformation. The Christoffel symbols and the Riemann tensor are ordinary commuting objects. Hence it is argued that torsionless Einstein's General Relativity is the common invariant sector of a one--parameter family of deformed gauge theories.