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 in a potential, which is a mechanism describing a spontaneous violation of spacetime symmetries. A broad class of spacetime-dependent potentials can be chosen allowing for a spacetime-dependent vacuum expectation value of the scalar field. The spacetime dependence of the potential can be parametrized by either another dynamical field or a nondynamical one. The second possibility introduces an aspect of explicit symmetry violation into the model, which is then interpreted as a hybrid. Based on the ($3+1$) decomposition, we demonstrate how to develop the Hamiltonian formulation for the hybrid theory. Having done so, our primary interest is to understand how spontaneous spacetime symmetry violation caused by the dynamical scalar via a mechanism 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 field. This is the case even when the potential involves a nondynamical field, i.e., the latter is unessential for the constraint structure of the model. These results emphasize the beauty of spontaneous spacetime symmetry violation in gravity from the viewpoint of the Hamiltonian formulation. They may pose as 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.

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• Alternative gravity theories