Effective Space-time Geometry for Black Holes and Cosmology

Effective space-time geometries can be derived by evolving initial data sets through a modified Hamiltonian obtained within a canonical approach to quantum gravity. This problem can be formulated precisely by first selecting a \textit{reduced} Hilbert space of the full theory Kinematical Hilbert space and then preforming a symmetry reduction in order to derive a symmetric sector of the theory. This framework has been successfully applied in the cosmological case. Recently, this setting has been extended by implementing a choice of gauge relevant for a black hole geometry, which led to the derivation of an effective Hamiltonian by means of coherent states, which are the best candidates to describe classical geometries. The classical data entering the coherent states can be seen as the initial data set to be evolved with the effective Hamiltonian. We study the algebra that the effective Hamiltonian constraint of the theory generates, as well as the equations of motion that the reduced phase space variables satisfy. With the goal to achieve closure of the algebra of the effective constraints, we extract a candidate expression for the effective diffeomorphism constraint, which is also compatible with the effective evolution equations. This is required in order to restore (a deformed version of) general covariance and to guarantee the consistency of the effective dynamics to be solved for.