Boulware-Deser ghost in extended quasidilaton massive gravity

TL;DR

This paper investigates the extended quasidilaton massive gravity model and demonstrates that it does not eliminate the Boulware-Deser ghost through a nonlinear Hamiltonian analysis.

Contribution

It provides a detailed Hamiltonian analysis showing the absence of a primary constraint to remove the Boulware-Deser ghost in this gravity theory.

Findings

The Hessian matrix is invertible, indicating no primary constraint.

The equations of motion determine the lapse and shift variables without constraints.

The Boulware-Deser ghost persists in the extended quasidilaton model.

Abstract

In the extended quasidilaton massive gravity we perform a nonlinear transformation of the shift vector and then calculate the second derivatives of the Hamiltonian density with respect to the lapse function and the (nonlinearly transformed) shift vector. It is then shown that the $4\times 4$ Hessian matrix is invertible, meaning that the equations of motion for the lapse function and the shift vector simply determine themselves. Therefore, there is no primary constraint that removes the Boulware-Deser ghost.