[Paper Review] Comment on the Hojman conservation quantities in Cosmology

Andronikos Paliathanasis, P. G. L. Leach|arXiv (Cornell University)|Mar 29, 2015

Cosmology and Gravitation Theories3 citations

TL;DR

This paper critiques the application of Hojman's method for deriving conservation laws in cosmology, particularly in scalar field models. By applying the ansatz ϕ(t) = ϕ(a(t)), it demonstrates that the resulting Hojman conservation quantity reduces to the Noether momentum conservation law of a free particle, revealing that the method does not yield new physical constraints and is equivalent to Noether's theorem under generalized transformations in Hamiltonian systems.

ABSTRACT

We comment upon the application of Hojman's method for the determination of conservation laws in Cosmology, which has been introduced by Capozziello \& Roshan (Phys. Lett. B 726 (2013) 471 (arXiv:1308.3910)), and has been applied recently in the cosmological scenario of a nonminimally coupled scalar field by Paolella \& Capozziello (Phys. Lett. A (2015), in press (arXiv:1503.00098)). We apply the Ansatz, $ϕ\left( t ight) =ϕ\left( a\left( t ight) ight) $, which was introduced by the cited authors for a minimally-coupled scalar field, and we study the Lie and Noether point symmetries for the reduced equation. We show that under this Ansatz the unknown function of the model cannot be constrained by the requirement of the existence of a conservation law and that the Hojman conservation quantity which arises for the reduced equation is nothing more than the functional form of the Noether conservation law of momentum for the free particle. Finally we show that Hojman's method for Hamiltonian systems, in which the Hamiltonian function is one of the involved equations of the system, is equivalent with the application of Noether's Theorem for generalized transformations.

Motivation & Objective

• To examine the validity and physical significance of Hojman's method in deriving conservation laws within cosmological models involving scalar fields.
• To assess whether the ansatz ϕ(t) = ϕ(a(t)) used in prior work genuinely constrains the model's functional form through conservation laws.
• To investigate the relationship between Hojman's method and Noether's theorem in Hamiltonian systems.
• To determine whether the conservation quantities derived via Hojman's method represent novel physical insights or are mathematically equivalent to known Noether symmetries.

Proposed method

• The authors apply the ansatz ϕ(t) = ϕ(a(t)) to reduce the dynamical equations of a nonminimally coupled scalar field in cosmology.
• They analyze the Lie and Noether point symmetries of the reduced second-order differential equation.
• The study compares the Hojman conservation quantity derived from the reduced equation with the Noether conservation law for a free particle.
• The authors examine the Hamiltonian formulation of the system, identifying the Hamiltonian as one of the system's equations.
• They establish equivalence between Hojman's method and Noether's theorem under generalized transformations in Hamiltonian systems.
• The analysis relies on symmetry reduction and functional form comparison to determine the physical content of the derived conservation laws.

Experimental results

Research questions

• RQ1 Does the ansatz ϕ(t) = ϕ(a(t)) lead to physically meaningful constraints on the scalar field model through Hojman's conservation law?
• RQ2 Is the Hojman conservation quantity derived from the reduced equation genuinely new, or is it equivalent to a known conservation law?
• RQ3 Can Hojman's method in Hamiltonian systems be interpreted as a special case of Noether's theorem with generalized transformations?
• RQ4 What is the functional form of the Hojman conserved quantity in the reduced cosmological model?
• RQ5 Does the existence of a conservation law under this ansatz uniquely determine the scalar field potential or coupling function?

Key findings

• The Hojman conservation quantity derived from the reduced equation is mathematically equivalent to the Noether conservation law for momentum in a free particle system.
• The ansatz ϕ(t) = ϕ(a(t)) does not constrain the unknown function of the model through the existence of a conservation law.
• Hojman's method for Hamiltonian systems, where the Hamiltonian is part of the system's equations, is shown to be equivalent to applying Noether's theorem with generalized transformations.
• The conservation law obtained via Hojman's method does not provide new physical insight beyond standard Noether symmetries in the reduced system.
• The functional form of the conserved quantity is not tied to the specific dynamics of the scalar field but reflects the underlying symmetry of a free particle.
• The study concludes that Hojman's method in this context does not yield independent or novel conservation laws beyond those already implied by Noether's theorem.