Confirmation of the Secondary Constraint and Absence of Ghost in Massive Gravity and Bimetric Gravity

TL;DR

The paper provides a rigorous Hamiltonian-constraint analysis of nonlinear massive gravity and its bimetric extension, explicitly proving the existence of a secondary constraint ${\cal C}_{(2)}$ in addition to the Hamiltonian constraint ${\cal C}=0$. By introducing shift-like variables and carefully evaluating the Poisson brackets, the authors show ${\{ {\cal C}(x),{\cal C}(y) \}} \approx 0$, enabling a secondary constraint to arise as ${\cal C}_{(2)} = {\{ {\cal C}, {\cal H}_0 \}} = 0$; this eliminates the momentum conjugate to the would-be ghost and leaves five propagating degrees of freedom for the massive graviton. The analysis extends to bimetric gravity, where the same constraints persist and additional linear and coordinate-invariance constraints reduce the total to 14 phase-space degrees of freedom, corresponding to seven physical modes (massive and massless gravitons). This work resolves previous concerns about the presence of a tertiary constraint and solidifies the ghost-free nature of these nonlinear theories, with significant implications for consistent infrared modifications of gravity.

Abstract

In massive gravity and in bimetric theories of gravity, two constraints are needed to eliminate the two phase-space degrees of freedom of the Boulware-Deser ghost. For recently proposed non-linear theories, a Hamiltonian constraint has been shown to exist and an associated secondary constraint was argued to arise as well. In this paper we explicitly demonstrate the existence of the secondary constraint. Thus the Boulware-Deser ghost is completely absent from these non-linear massive gravity theories and from the corresponding bimetric theories.