Pure Lorentz spin connection theories and uniqueness of General Relativity

Abstract: General Relativity can be reformulated as a diffeomorphism invariant gauge theory of the Lorentz group, with Lagrangian of the type $f(F\wedge F)$, where $F$ is the curvature 2-form of the spin connection. A theory from this class with a generic $f$ is known to propagate eight degrees of freedom: a massless graviton, a massive graviton and a scalar. General Relativity in this formalism avoids extra degrees of freedom because the function $f$ is special and leads to the appearance of six extra primary constraints on the phase space variables. Our main new result is that there are other theories of the type $f(F\wedge F)$ that lead to six extra primary constraints. However, only in the case of GR the dynamics is such that these six primary constraints get supplemented by six secondary constraints, which gives the end result of two propagating degrees of freedom. This is how uniqueness of GR manifests itself in this ``pure spin connection" formalism. The other theories we discover are shown to give examples of irregular dynamical systems. At the linear level around (anti-)de Sitter space they have two degrees of freedom, as General Relativity, with the extra ones manifesting themselves only non-linearly.