Section VI, cartan geometrodynamics, of the following paper. It talks a bit of what Garrett is talking about, but with a simpler group structure. This and the other papers also they also talk about an observer space.
http://arxiv.org/abs/1111.7195v4
Spontaneously broken Lorentz symmetry for Hamiltonian gravity
Steffen Gielen, Derek K. Wise
(Submitted on 30 Nov 2011 (v1), last revised 10 May 2012 (this version, v4))
In Ashtekar's Hamiltonian formulation of general relativity, and in loop quantum gravity, Lorentz covariance is a subtle issue that has been strongly debated. Maintaining manifest Lorentz covariance seems to require introducing either complex-valued fields, presenting a significant obstacle to quantization, or additional (usually second class) constraints whose solution renders the resulting phase space variables harder to interpret in a spacetime picture. After reviewing the sources of difficulty, we present a Lorentz covariant, real formulation in which second class constraints never arise. Rather than a foliation of spacetime, we use a gauge field y, interpreted as a field of observers, to break the SO(3,1) symmetry down to a subgroup SO(3)_y. This symmetry breaking plays a role analogous to that in MacDowell-Mansouri gravity, which is based on Cartan geometry, leading us to a picture of gravity as 'Cartan geometrodynamics.' We study both Lorentz gauge transformations and transformations of the observer field to show that the apparent breaking of SO(3,1) to SO(3) is not in conflict with Lorentz covariance.
http://arxiv.org/abs/1210.0019
Lifting General Relativity to Observer Space
Steffen Gielen, Derek K. Wise
(Submitted on 28 Sep 2012 (v1), last revised 4 May 2013 (this version, v3))
The `observer space' of a Lorentzian spacetime is the space of future-timelike unit tangent vectors. Using Cartan geometry, we first study the structure a given spacetime induces on its observer space, then use this to define abstract observer space geometries for which no underlying spacetime is assumed. We propose taking observer space as fundamental in general relativity, and prove integrability conditions under which spacetime can be reconstructed as a quotient of observer space. Additional field equations on observer space then descend to Einstein's equations on the reconstructed spacetime. We also consider the case where no such reconstruction is possible, and spacetime becomes an observer-dependent, relative concept. Finally, we discuss applications of observer space, including a geometric link between covariant and canonical approaches to gravity.
http://arxiv.org/abs/1304.5430
Extensions of Lorentzian spacetime geometry: From Finsler to Cartan and vice versa
Manuel Hohmann
(Submitted on 19 Apr 2013 (v1), last revised 27 Jun 2013 (this version, v2))
We briefly review two recently developed extensions of the Lorentzian geometry of spacetime and prove that they are in fact closely related. The first is the concept of observer space, which generalizes the space of Lorentzian observers, i.e., future unit timelike vectors, using Cartan geometry. The second is the concept of Finsler spacetimes, which generalizes the Lorentzian metric of general relativity to an observer-dependent Finsler metric. We show that every Finsler spacetime possesses a well-defined observer space that can naturally be equipped with a Cartan geometry. Conversely, we derive conditions under which a Cartan geometry on observer space gives rise to a Finsler spacetime. We further show that these two constructions complement each other. We finally apply our constructions to two gravity theories, MacDowell-Mansouri gravity on observer space and Finsler gravity, and translate their actions from one geometry to the other.The fist paper of the list cites the case of SO(4,1)/(SO(3,1)) as being studied by West and Stelle and another by mansouri. Those are behind paywalls, but I checked them, and they are not quite as explicit at that.
I also found this, about discrete representations of deformations of SO(3,1):
http://www.math.utah.edu/~kapovich/EPR/def.pdf
Curiously, it uses Dehn surgery on knots to find such classifications.
That is not very different with what Torsten uses, except that he does it directly on a generic 4 manifold, through the analog of Dehn surgery in 4 dimensions, Casson Handles, and then he identifies the gauge fields with the different types of torus on tangent space:
http://arxiv.org/pdf/1006.2230.pdf
The propagating solitons are identified with propagating exotic structures here:
http://arxiv.org/pdf/1303.1632.pdf
