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 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
$V^2\rightarrow const$. 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.
