An introduction to the physics of Cartan gravity

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 A(AB) (x) and a gravitational Higgs field V-A (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 V-A 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 V-2 (x) tends to a positive constant. We show that in regions wherein V-2 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 phi, metric tensor g(mu nu), and possibly a torsion tensor T-mu nu(rho)). 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 V-2 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 V-2 -> 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. (C) 2015 Elsevier Inc. All rights reserved.