Generalized Gauge Theories with Nonunitary Parallel Transport: General Relativity with Cosmological Constant as an Example

TL;DR

This paper introduces a generalized gauge theory framework for gravity, removing the unitarity constraint on parallel transporters, leading to a unified description of the vierbein and spin connection within a de Sitter gauge field, and deriving Einstein's equations with a cosmological constant.

Contribution

It proposes a novel gauge theory approach to general relativity that abandons unitarity of parallel transporters and introduces a unique *-operation, unifying the vierbein and spin connection.

Findings

Derives Einstein's equations with a nonzero cosmological constant from the new gauge action.

Shows the classical equations of motion are de Sitter covariant.

Unifies the vierbein and spin connection as parts of a single gauge field.

Abstract

In gauge theories parallel transporters (PTs) U(C) along paths C play an important role. Traditionally they are unitary or pseudoorthogonal maps between vector spaces. We propose to abandon unitarity of parallel transporters and with it the a priori assumption of metricity in general relativity. A *-operation on parallel transporters serves as a substitute for it, and this *-operation is proven to be unique on group theoretical grounds. The vierbein and the spin connection appear as distinguishable parts of a single de Sitter gauge field with field strength F. The action takes the form $\frac{3}{16\pi G\Lambda}\int tr(F \wedge F i \gamma_5)$ and both the Einstein field equations with arbitrarily small but nonvanishing cosmological constant $\Lambda$ and the condition of vanishing torsion are obtained from it. The equation of motion for classical massive bodies turns out to be de Sitter covariant.