Note About Hamiltonian Structure of Non-Linear Massive Gravity

TL;DR

This paper analyzes the Hamiltonian structure of a non-linear massive gravity theory built along the Hassan–Rosen framework. By performing a Dirac constraint analysis with a shift redefinition that renders the action linear in the lapse $N$, the authors show that the Hamiltonian constraint $\tilde{\mathcal{H}}_T$ is second class at generic points in phase space, while the momentum constraints acquire a mixed first/second-class character. Consequently, the theory contains seven second-class constraints alongside one first-class constraint, yielding $11$ physical degrees of freedom and an additional problematic $\tfrac{1}{2}$ degree of freedom that cannot be readily interpreted. The results suggest that ghost elimination is not automatically guaranteed in this non-linear construction and point to further work, potentially using Stückelberg fields, to clarify the physical content.

Abstract

We perform the Hamiltonian analysis of non-linear massive gravity action studied recently in arXiv:1106.3344 [hep-th]. We show that the Hamiltonian constraint is the second class constraint. As a result the theory possesses an odd number of the second class constraints and hence all non physical degrees of freedom cannot be eliminated.