Hamiltonian Structure of a Friedmann-Robertson-Walker Universe with Torsion

TL;DR

This paper formulates a Hamiltonian approach to a torsion-inclusive $R^{2}$ gravity model in a closed FRW universe, deriving constraints and field equations, and discusses future directions like solving these equations and quantization.

Contribution

It introduces a Hamiltonian formulation for an $R^{2}$ gravity model with torsion in a cosmological setting, highlighting new constraints and detailed field equations.

Findings

Torsion induces primary and secondary constraints in the Hamiltonian formulation.

Full field equations are explicitly derived for the model.

Lays groundwork for quantum cosmology with torsion.

Abstract

We study a $R^{2}$ model of gravity with torsion in a closed Friedmann-Robertson-Walker universe. The model is cast in Hamiltonian form subtracting from the original Lagrangian the total time derivative of $f_{K}f_{R}$, where $f_{K}$ is proportional to the trace of the extrinsic curvature tensor, and $f_{R}$ is obtained differentiating the Lagrangian with respect to the highest derivative. Torsion is found to lead to a primary constraint linear in the momenta and a secondary constraint quadratic in the momenta, and the full field equations are finally worked out in detail. Problems to be studied for further research are the solution of these equations and the quantization of the model. One could then try to study a new class of quantum cosmological models with torsion.