Hamiltonian formulation of an effective modified gravity with nondynamical background fields

TL;DR

This paper develops a Hamiltonian formulation for a modified gravity theory with explicit diffeomorphism violation, analyzing the constraints and consistency conditions related to background fields in the Standard-Model Extension.

Contribution

It provides a Hamiltonian formalism for the minimal gravitational Standard-Model Extension with explicit diffeomorphism violation, deriving modified constraints and analyzing their implications.

Findings

Derived modified Hamiltonian and momentum constraints.

Identified conditions for the background fields to ensure consistency.

Clarified the impact of explicit symmetry violations on the constraint structure.

Abstract

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