Conformal relations and Hamiltonian formulation of fourth-order gravity

TL;DR

This paper explores new conformal equivalences in fourth-order gravity theories and introduces an alternative Hamiltonian formulation that avoids constraints, with implications for the stability of cosmological models.

Contribution

It presents a novel conformal equivalence of fourth-order gravity theories and proposes a Hamiltonian formulation without constraints, differing from Ostrogradski's approach.

Findings

New conformal equivalence between fourth-order gravity and other theories.

A Hamiltonian formulation without constraints for second-order derivative Lagrangians.

Discussion on stability properties of cosmological models in this framework.

Abstract

The conformal equivalence of fourth-order gravity following from a non-linear Lagrangian L(R) to theories of other types is widely known, here we report on a new conformal equivalence of these theories to theories of the same type but with different Lagrangian. For a quantization of fourth-order theories one needs a Hamiltonian formulation of them. One of the possibilities to do so goes back to Ostrogradski in 1850. Here we present another possibility: A Hamiltonian H different from Ostrogradski's one is discussed for the Lagrangian L depending on first and second order drivatives of the position variable q. We add a suitable divergence to L. Contrary to other approaches no constraint is needed. One of the canonical equations becomes equivalent to the fourth-order Euler-Lagrange equation of L. Finally, we discuss the stability properties of cosmological models within fourth-order gravity.