[Paper Review] Comment on `Modified F(R) Ho\v{r}ava-Lifshitz gravity: a way to accelerating FRW cosmology' by M. Chaichian, S. Nojiri, S. D. Odintsov, M. Oksanen, A. Tureanu

N. Kiriushcheva, S. V. Kuzmin+1 more|arXiv (Cornell University)|Dec 26, 2011

Black Holes and Theoretical Physics9 references3 citations

TL;DR

This paper critiques the Hamiltonian analysis in a 2011 paper on modified F(R) Hoľrava-Lifshitz gravity, demonstrating that the choice of Nk (covariant shift) as an independent variable—instead of the standard Nk (contravariant shift)—leads to a fundamentally different algebra of constraints in the canonical formulation. The authors show that the Poisson brackets of constraints, particularly involving the momentum constraint Hi and Hamiltonian constraint H0, are non-zero and structurally more complex than claimed, invalidating the assumption of equivalence to the standard GR constraint algebra.

ABSTRACT

The partial Hamiltonian analysis of the Ho\v{r}ava-type action presented in the paper by M. Chaichian, S. Nojiri, S. D. Odintsov, M. Oksanen, A. Tureanu (Class. Quant. Grav. 27 (2011) 185021) is incorrect; for the authors' choice of variables, a covariant shift, instead of a contravariant shift which is the one usually used in General Relativity (GR) in ADM variables, the true algebra of constraints differs from what they presented. The algebra of constraints for their choice of variables is explicitly given for GR and compared with the standard algebra.

Motivation & Objective

• To identify and correct errors in the Hamiltonian constraint algebra of a modified F(R) Hoľrava-Lifshitz gravity model as presented in a 2011 paper.
• To demonstrate that the choice of Nk as an independent variable (covariant shift) instead of Nk (contravariant shift) fundamentally alters the Poisson bracket structure of constraints.
• To show that the constraint algebra derived in the original paper is inconsistent with standard Dirac procedure and GR canonical structure.
• To prove that the claimed closure of the constraint algebra—especially the vanishing of {Hi, ∫H0dy}—is incorrect under the chosen parametrization.
• To emphasize that gauge structure and constraint algebra depend critically on field parametrization, especially in non-GR theories with anisotropic scaling.

Proposed method

• Performs a re-analysis of the Hamiltonian formulation using the same action and variables as in the original paper (arXiv:1001.4102), focusing on the choice of Nk as an independent variable.
• Derives the fundamental Poisson brackets (PBs) for the non-standard parametrization, showing that {Ni(x), πkl(y)} = 0, while {Np(x), πkl(y)} ≠ 0 due to metric dependence.
• Calculates specific Poisson brackets involving the momentum constraint Hi and the Hamiltonian constraint H0, particularly the term ∫dy {Hi(x), (−1/(3μ)) (πB/√g gpq) (y)}.
• Compares the resulting constraint algebra with the standard GR algebra in ADM variables (using Nk), showing that the non-standard choice leads to field-dependent structure functions and non-trivial PBs.
• Uses the standard form of Dirac's procedure for singular systems, emphasizing that PBs depend on the choice of independent variables and that raising/lowering indices via the spatial metric affects PB structure.
• Demonstrates that the gauge generator and physical symmetries depend on the correct constraint algebra, which is distorted by the non-standard parametrization.

Experimental results

Research questions

• RQ1 Does the choice of Nk as an independent variable instead of Nk alter the canonical structure of the Hoľrava-Lifshitz gravity model?
• RQ2 Is the claimed closure of the constraint algebra—specifically {Hi, ∫H0dy} = 0—valid under the non-standard parametrization?
• RQ3 How does the Poisson bracket structure of constraints differ between the standard ADM choice (Nk) and the non-standard choice (Nk) in this model?
• RQ4 Can the constraint algebra derived in the original paper be reconciled with the known algebra of GR in ADM variables?
• RQ5 What are the implications of field-dependent structure functions in the constraint algebra for gauge symmetry and physical consistency?

Key findings

• The Poisson bracket {Hi(x), ∫dy (−1/(3μ)) (πB/√g gpq)(y)} produces a non-zero contribution of 1/(3μ) (1/√g) gij ∂jB (πB)²(x), contradicting the claim in the original paper that this bracket vanishes.
• The constraint algebra for the non-standard choice (Nk as independent variable) is significantly more complex than for the standard ADM choice (Nk), with all PBs (19)–(21) exhibiting field-dependent structure constants.
• The standard GR constraint algebra (e.g., {H0, ∫NH0dy} = N,kHk + (NHk),k) is not recovered under the non-standard parametrization; instead, new terms involving πpq and spatial derivatives of the shift appear.
• The assumption that the constraint algebra is isomorphic to the standard Hoľrava algebra is incorrect, as the non-standard parametrization breaks the expected symmetry and structure.
• The gauge generator and physical symmetries derived from the Hamiltonian formulation depend on the correct constraint algebra, which is distorted by the non-standard choice of shift variable.
• The paper concludes that the original claim of a consistent accelerating FRW cosmology via this modified F(R) Hoľrava-Lifshitz model is undermined by the incorrect Hamiltonian analysis.