Cartan geometry of spacetimes with a nonconstant cosmological function $\Lambda$

TL;DR

This paper develops a Cartan geometric framework for spacetimes modeled locally on de Sitter spaces with a varying cosmological function, extending the symmetry group and analyzing geometric properties like torsion.

Contribution

It introduces a novel Cartan geometry with a nonconstant cosmological function and extends symmetry considerations to the de Sitter group, enriching the geometric description of such spacetimes.

Findings

Torsion receives additional contributions due to the variable cosmological function.

The structure group remains the Lorentz group, but nonlinear realizations extend symmetries to the de Sitter group.

Explicit expressions for spin connection, vierbein, curvature, and torsion are derived.

Abstract

We present the geometry of spacetimes that are tangentially approximated by de Sitter spaces whose cosmological constants vary over spacetime. Cartan geometry provides one with the tools to describe manifolds that reduce to a homogeneous Klein space at the infinitesimal level. We consider a Cartan geometry in which the underlying Klein space is at each point a de Sitter space, for which the combined set of pseudo-radii forms a nonconstant function on spacetime. We show that the torsion of such a geometry receives a contribution that is not present for a cosmological constant. The structure group of the obtained de Sitter-Cartan geometry is by construction the Lorentz group $SO(1,3)$. Invoking the theory of nonlinear realizations, we extend the class of symmetries to the enclosing de Sitter group $SO(1,4)$, and compute the corresponding spin connection, vierbein, curvature, and torsion.