Hamiltonian consistency of the gravitational constraint algebra under deformations

The importance of the first-class constraint algebra of general relativity is not limited just by its self-contained description of the gauge nature of spacetime, but it also provides conditions to properly evolve the geometry by selecting a gauge only once throughout the whole evolution of a gravitational system. This must be a property of all background independent theories. In this paper we consider gravitational theories which arise from deformations of the fundamental canonical variables of general relativity where the proposed deformations are inspired by modifications of gravity. These variable deformations result in new theories when the deformation is not a canonical transformation. The new theory must preserve the first-class structure of the algebra, which is a non-trivial restriction for generic deformations. In this vein we present a general deformation scheme along with consistency conditions, so that the algebra of constraints is still satisfied in the resulting theory. This is illustrated both in metric theory as well as in tetrad theory.