Hamiltonian formulation for scalar model of spontaneous spacetime symmetry violation in gravity

The focus of this article is on a modification of General Relativity (GR) governed by a dynamical scalar field. The latter is able to acquire a nonzero spacetime-dependent vacuum expectation value, which gives rise to a spontaneous violation of spacetime symmetries. Based on the (3+1) decomposition, we demonstrate how to develop the Hamiltonian formulation for this model. Having done so, our primary interest is to understand how spontaneous spacetime symmetry violation manifests itself in such a setting. In particular, we find that the constraint structure of GR is preserved, although the constraints are clearly modified by the scalar background field. These results emphasize the beauty of spontaneous spacetime symmetry violation in gravity from the viewpoint of the Hamiltonian formulation. They may pose the base for further studies of more sophisticated models of vector and higher-rank tensor fields. Moreover, the description developed can bear fruits when applied within phenomenological quests for spacetime symmetry violation in gravity, in particular, at cosmological scales.