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The importance of the first-class constraint algebra of general relativity isnot limited just by its self-contained description of the gauge nature ofspacetime, but it also provides conditions to properly evolve the geometry byselecting a gauge only once throughout the whole evolution of a gravitationalsystem. This must be a property of all background independent theories. In thispaper we consider gravitational theories which arise from deformations of thefundamental canonical variables of general relativity where the proposeddeformations are inspired by modifications of gravity. These variabledeformations result in new theories when the deformation is not a canonicaltransformation. The new theory must preserve the first-class structure of thealgebra, which is a non-trivial restriction for generic deformations. In thisvein 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.

Hamiltonian consistency of the gravitational constraint algebra under deformations