Modified-gravity theories with nondynamical background fields

We study the dynamics of a modified-gravity theory, which is supplemented by an extended Gibbons-Hawking-York boundary term and incorporates diffeomorphism violation through nondynamical background fields denoted as $u$ $s^{\mu\nu}$ in literature. An ADM decomposition allows us to project modified Einstein equations into purely spacelike hypersurfaces, implies field for induced dynamical three-metric. also obtain Hamilton-Jacobi motion canonical variables theory based on its Hamiltonian, was derived previous work. The computations show that obtained from Lagrangian Hamiltonian approaches are consistent with each other. Connections Brans-Dicke ghost-free massive gravity established.