The ADM Formulation of the SME Gravity

The Hamiltonian formulation of the gravitational sector of the Standard-Model Extension (SME) with nondynamical fields $u$ and $s^{μν}$ is studied. We provide the relevant Hamiltonians that describe the constrained phase space and the dynamics of the induced metric on the ADM hypersurface. The generalization of the Gibbons-Hawking-York boundary term has been crucial to preventing second time-derivatives of the metric tensor in the Hamiltonians. By extracting the dynamics and constraints from the Einstein equations we have proved the equivalence between the Lagrangian and Hamiltonian formulations.