  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.
