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 $A^{AB}(x)$ and a symmetry-breaking field $V^A(x)$. The clear similarity with symmetry-broken Yang-Mills theory suggests strongly the existence of a new field in nature: the gravitational Higgs field $V^A$. By treating $V^A$ as a genuine dynamical field we arrive at a natural generalization of General Relativity with a wealth of new phenomenology and with General Relativity reproduced exactly in the limit where $V^2$ tends to a positive constant. We show that in regions wherein $V^2(x)$ varies, but has a definite sign, the Cartan-geometric formulation is a form of a scalar-tensor theory. 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; a simple choice of action with FRW symmetry yields, without any additional ad hoc assumptions, a classical analogue of the Hartle-Hawking no-boundary proposal. Solutions from more general actions are discussed. A gauge prescription for coupling matter to gravity is described and matter actions are presented which reduce to standard ones in the limit $V^2\to const$. It becomes clear 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.

Updated to reflect content of published version

References in corpus (16)

• The Barbero-Immirzi Parameter as a Scalar Field: K-Inflation from Loop Quantum Gravity?
• New Action Principle for General Relativity
• Barbero-Immirzi field in canonical formalism of pure gravity
• Peccei--Quinn mechanism in gravity and the nature of the Barbero--Immirzi parameter
• Cosmological dynamics with propagating Lorentz connection modes of spin zero
• Chern-Simons Modified Gravity as a Torsion Theory and its Interaction with Fermions
• Einstein-Cartan Theory
• Bimetric structure formation: non-Gaussian predictions
• Remarks on Barbero-Immirzi parameter as a field
• Disformal quintessence
• Gauge Gravity: a forward-looking introduction
• Weak equivalence principle for self-gravitating bodies: A sieve for purely metric theories of gravity
• Cartan geometry of spacetimes with a nonconstant cosmological function $\Lambda$
• Cosmological signature change in Cartan Gravity with dynamical symmetry breaking
• Gauge theory of gravity and matter
• A First-Principles Implementation of Scale Invariance Using Best Matching

Cited by in corpus (4)

• Dynamical system analysis of dark energy models in scalar coupled metric-torsion theories
• Clifford algebra Cl(0,6) approach to beyond the standard model and naturalness problems
• Lie Group Cosmology
• From Special Relativity to embedded generators in Cartan subalgebras of rank-4 spin algebras