On the equivalence between different canonical forms of F(R) theory of gravity

TL;DR

This paper demonstrates that different canonical forms of F(R) gravity, including those derived via Lagrange multipliers, are classically equivalent when the Hamiltonian constraint is properly incorporated, resolving previous discrepancies.

Contribution

It proves the classical equivalence of various canonical structures of F(R) gravity by including the Hamiltonian constraint, clarifying the relationship between different formulations.

Findings

All three canonical structures admit the same symmetries.

Inclusion of the Hamiltonian constraint ensures equivalence.

Discrepancies between frames are resolved at the classical level.

Abstract

Classical equivalence between Jordan's and Einstein's frame counterparts of F(R) theory of gravity has recently been questioned, since the two produce different Noether symmetries, which couldn't be translated back and forth using transformation relations. Here we add the Hamiltonian constraint equation, which is essentially the time-time component of Einstein's equation, through a Lagrange multiplier to the existence condition for Noether symmetry to show that all the three different canonical structures of F(R) theory of gravity, including the one which follows from Lagrange multiplier technique, admit each and every available symmetry independently. This establishes classical equivalence amongst all the three.