An introduction to the physics of Cartan gravity

Abstract

A distance can be measured by monitoring how much a wheel has rotated when rolled without slipping. This simple idea underlies the mathematics of Cartan geometry. The Cartan-geometric description of gravity consists of a SO(1,4) gauge connection AAB(x) and a gravitational Higgs field VA(x) which breaks the gauge symmetry. The clear similarity with symmetry-broken Yang–Mills theory suggests strongly the existence of a new field VA in nature: the gravitational Higgs field. By treating VA as a genuine dynamical field we arrive at a natural generalization of General Relativity with a wealth of new phenomenology. Importantly, General Relativity is reproduced exactly in the limit that the SO(1,4) norm V2(x) tends to a positive constant. We show that in regions wherein V2 varies–but has a definite sign–the Cartan-geometric formulation is a particular version of a scalar–tensor theory (in the sense of gravity being described by a scalar field ϕ, metric tensor gμν, and possibly a torsion tensor Tμνρ). A specific choice of action yields the Peebles–Ratra quintessence model whilst more general actions are shown to exhibit propagation of torsion. Regions where the sign of V2 changes correspond to a change in signature of the geometry. Specifically, a simple choice of action with FRW symmetry imposed yields, without any additional ad hoc assumptions, a classical analogue of the Hartle–Hawking no-boundary proposal with the big bang singularity replaced by signature change. Cosmological solutions from more general actions are described, none of which have a big bang singularity, with most solutions reproducing General Relativity, or its Euclidean version, for late cosmological times. Requiring that gravity couples to matter fields through the gauge prescription forces a fundamental change in the description of bosonic matter fields: the equations of motion of all matter fields become first-order partial differential equations with the scalar and Dirac actions taking on structurally similar first-order forms. All matter actions reduce to the standard ones in the limit V2→const. We argue that Cartan geometry may function as a novel platform for inspiring and exploring modified theories of gravity with applications to dark energy, black holes, and early-universe cosmology. We end by listing a set of open problems.