Deformations of spacelike hypersurfaces in space-time play an important role in discussions of general covariance and slicing independence in gravitational theories. In a canonical formulation, they provide the geometrical meaning of gauge transformations generated by the diffeomorphism and Hamiltonian constraints. However, it has been known for some time that the relationship between hypersurface deformations and general covariance is not a kinematical equivalence but holds only on the solution space of the constraints and requires their gauge equations and equations of motion to be used. The off-shell behavior of hypersurface deformations on their own, without imposing constraint and gauge equations, is therefore different from space-time diffeomorphisms. Its complete understanding is important for potential quantizations or modifications of general relativity in canonical form and of compatible space-time geometries that may be implied by them. Here, a geometrical analysis of hypersurface deformations is performed, allowing for a dependence of hypersurface deformation generators (the lapse function and the shift vector) on the phase-space degrees of freedom given by the geometry of an embedded spacelike hypersurface. The result is compared in detail with Poisson brackets of the gravitational constraints. As a new implication of physical relevance, covariance conditions are obtained for theories of emergent modified gravity without symmetry restrictions.

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• Alternative gravity theories
• General relativity