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The current paper is dedicated to developing a (3+1) decomposition for theminimal gravitational Standard-Model Extension. Our setting is explicitdiffeomorphism violation and we focus on the background fields known in theliterature as $u$ and $s^{\mu\nu}$. The Hamiltonian formalism is developed forthese contributions, which amounts to deriving modified Hamiltonian andmomentum constraints. We then study the connection between these modifiedconstraints and the modified Einstein equations. Implications are drawn on theform of the background fields to guarantee the internal consistency of thecorresponding modified-gravity theories. In the course of our analysis, weobtain a set of consistency requirements for $u$ and certain sectors of$s^{\mu\nu}$. We argue that the constraint structure remains untouched whenthese conditions are satisfied. Our results shed light on explicit violationsof diffeomorphism invariance and local Lorentz invariance in gravity. They mayturn out to be valuable for developing a better understanding of effectivemodified-gravity theories.

Hamiltonian formulation of an effective modified gravity with nondynamical background fields