[Paper Review] A Trouble with Hoÿrava-Lifshitz Gravity

Miao Li, Yi Pang|arXiv (Cornell University)|May 19, 2009

Black Holes and Theoretical Physics8 references71 citations

TL;DR

This paper investigates the phase space structure in Hoïrava-Lifshitz gravity and finds that the theory's constraints lead to an inconsistent canonical framework: the Poisson brackets of the Hamiltonian density do not close, generating additional constraints that eliminate all physical degrees of freedom or reduce the phase space to an odd number of fields, implying a fundamental inconsistency in the theory's dynamics.

ABSTRACT

We study the structure of the phase space in Hoyrava-Lifshitz theory. With the constraints derived from the action, the phase space is described by five fields, thus there is a lack of canonical structure. The Poisson brackets of the Hamiltonian density do not form a closed structure, resulting in many new constraints. Taking these new constraints into account, it appears that there is no degree of freedom left, or the phase space is reduced to one with an odd number of fields.

Motivation & Objective

• To analyze the canonical structure of Hoïrava-Lifshitz gravity using constraints derived from its action.
• To investigate whether the theory supports a consistent phase space with well-defined dynamics.
• To determine whether the constraints lead to a reduction or elimination of physical degrees of freedom.
• To assess whether the Poisson brackets of the Hamiltonian density form a closed algebra, as required for a consistent Hamiltonian formulation.

Proposed method

• Derive the primary and secondary constraints from the action of Hoïrava-Lifshitz gravity.
• Analyze the Poisson brackets of the Hamiltonian density to test closure of the algebra.
• Identify new constraints arising from the non-closure of Poisson brackets.
• Examine the resulting phase space structure to determine the number of independent fields and degrees of freedom.
• Assess whether the final phase space has an odd number of fields, which would violate canonical quantization rules.

Experimental results

Research questions

• RQ1 Does the Hamiltonian density in Hoïrava-Lifshitz gravity generate a closed algebra under Poisson brackets?
• RQ2 What is the resulting phase space structure after including all constraints from the action?
• RQ3 Are there physical degrees of freedom remaining after imposing all derived constraints?
• RQ4 Does the final phase space contain an odd number of fields, indicating a breakdown of canonical structure?

Key findings

• The Poisson brackets of the Hamiltonian density in Hoïrava-Lifshitz gravity do not form a closed algebra, leading to additional constraints.
• The derived constraints reduce the phase space to one with only five fields, indicating a lack of canonical structure.
• The phase space is inconsistent with standard Hamiltonian mechanics due to the presence of an odd number of fields.
• The theory appears to have no physical degrees of freedom left after imposing all constraints.