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

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.

arXiv comment6 pages. Matches version to be published in Physical Review D; section reviewing Cartan geometry removed, references added

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• Scale-invariant cosmology in de Sitter gauge theory
• Dark energy as a kinematic effect
• A first-order approach to conformal gravity
• de Sitter invariant special relativity and galaxy rotation curves
• de Sitter-invariant special relativity and the dark energy problem
• An introduction to the physics of Cartan gravity
• Entropy in locally-de Sitter spacetimes
• Basic automorphism of Cartan foliations covered by fibrations
• From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras