The ADM Formulation of the SME Gravity

TL;DR

This paper develops a Hamiltonian formulation for the gravitational sector of the Standard-Model Extension, clarifying the phase space and dynamics using ADM variables and boundary terms, and proves equivalence with the Lagrangian approach.

Contribution

It introduces a Hamiltonian formulation for SME gravity with nondynamical fields, including boundary term generalization and proof of equivalence with the Lagrangian formulation.

Findings

Derived Hamiltonians describing constrained phase space.

Established the role of boundary terms in the Hamiltonian.

Proved the equivalence between Lagrangian and Hamiltonian formulations.

Abstract

The Hamiltonian formulation of the gravitational sector of the Standard-Model Extension (SME) with nondynamical fields $u$ and $s^{\mu \nu}$ 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.