[Paper Review] Nonpropagating ghost in covariant $f(Q)$ gravity

Kun Hu, Makishi Yamakoshi+3 more|arXiv (Cornell University)|Oct 24, 2023

Cosmology and Gravitation Theories4 citations

TL;DR

This paper investigates the ghost scalar mode in covariant $f(Q)$ gravity, demonstrating that although the scalar mode exhibits negative kinetic energy in the scalar-nonmetricity formulation, it is non-propagating and eliminated via second-class constraints in the ADM formalism. By reformulating $f(Q)$ gravity as a higher-order scalar-tensor theory using vector St"uckelberg fields, the authors show the theory remains healthy and ghost-free, resolving concerns about instability in the absence of gauge fixing.

ABSTRACT

$f(Q)$ gravity is an extension of the symmetric teleparallel equivalent to general relativity (STEGR). This work shows that based on the scalar-nonmetricity formulation, a scalar mode in $f(Q)$ gravity has a negative kinetic energy. This conclusion holds regardless of the coincident gauge frequently used in STEGR and $f(Q)$ gravity. To study the scalar mode, we further consider the covariant $f(Q)$ gravity as a special class in higher-order scalar tensor (HOST) theory and rewrite the four scalar fields, which play a role of the Stüeckelberg fields associated with the diffeomorphism, by vector fields. Applying the standard Arnowitt-Deser-Misner (ADM) formulation to the new formulation of the $f(Q)$ gravity, we demonstrate that the ghost scalar mode can be eliminated by the second-class constraints, thus ensuring that $f(Q)$ gravity is a healthy theory.

Motivation & Objective

• To address the controversy over the presence of a ghost scalar mode in $f(Q)$ gravity, particularly in the absence of gauge fixing.
• To confirm whether the scalar mode with negative kinetic energy is physically observable or removable through constraints.
• To establish a gauge-invariant formulation of $f(Q)$ gravity by embedding it within higher-order scalar-tensor (HOST) theory.
• To demonstrate that the theory remains unitary and free of physical ghost modes despite the negative kinetic energy in the scalar sector.
• To provide a Hamiltonian analysis in arbitrary gauge, avoiding reliance on the coincident gauge commonly used in prior works.

Proposed method

• Reformulate $f(Q)$ gravity as a higher-order scalar-tensor (HOST) theory by introducing vector fields as St"uckelberg fields to replace the four scalar fields associated with diffeomorphism symmetry.
• Apply the standard Arnowitt-Deser-Misner (ADM) decomposition to the new formulation, separating spatial and temporal components of the metric and vector fields.
• Identify second-class constraints in the ADM formalism that eliminate the ghost scalar mode despite its negative kinetic energy.
• Use conformal rescaling of the action to analyze the scalar-nonmetricity formulation and isolate the scalar degree of freedom.
• Perform a 3+1 decomposition of the covariant derivative $\nabla_\mu A_\nu^a$ and the tensor $C_a^{\alpha\mu\nu}$ to express the action in terms of ADM variables.
• Derive the full ADM action and analyze the constraint structure to show that the ghost mode is unphysical and does not propagate.

Experimental results

Research questions

• RQ1 Does the scalar mode in $f(Q)$ gravity with negative kinetic energy lead to physical instabilities?
• RQ2 Can the ghost mode be removed through constraints in a gauge-invariant formulation?
• RQ3 Is $f(Q)$ gravity unitary and free of physical degrees of freedom with negative energy?
• RQ4 How does the ghost mode behavior change when moving from the coincident gauge to a general gauge?
• RQ5 What is the role of second-class constraints in eliminating non-propagating ghost modes in the ADM formalism?

Key findings

• The scalar mode in $f(Q)$ gravity has negative kinetic energy in the scalar-nonmetricity formulation, regardless of gauge choice.
• Despite the negative kinetic energy, the ghost mode is non-propagating and does not contribute to physical observables.
• The ghost mode is eliminated by second-class constraints in the ADM formulation, ensuring the theory's unitarity.
• The covariant $f(Q)$ gravity can be consistently embedded into the higher-order scalar-tensor (HOST) theory framework using vector St"uckelberg fields.
• The Hamiltonian analysis in an arbitrary gauge confirms that the theory remains healthy and ghost-free, resolving prior controversies.
• The action in the ADM formalism shows that the scalar degree of freedom is constrained and does not propagate, even though its kinetic term is negative.