Fully Covariant Van Dam-Veltman-Zakharov Discontinuity, and Absence Thereof

TL;DR

The paper addresses whether the vDVZ discontinuity persists beyond linearized massive gravity by constructing a fully covariant nonlinear extension of the Pauli-Fierz theory, showing that a scalar sector can sustain the discontinuity through covariant nonlocal constraints such as $G_{mu nu} - M^2 (Box^{-1} G_{mu nu})^T + \tfrac{1}{2} M^2 g_{mu nu} Box^{-1} R = 8\pi G T_{mu nu}$ and $(Box - M^2)\psi = -\tfrac{8\pi}{3} G T$. It then analyzes the DGP model, a covariant brane-world construction, where the linear theory exhibits the vDVZ sign, but nonlinear brane bending eliminates the discontinuity. Key insights include the role of nonlocal covariant terms and the scale at which nonlinearities become important, such as $r \ll (G M L^2)^{1/3}$ with $L = \hat G / G$, leading to recovery of GR on the brane. Together, these results clarify how covariant massive gravity can be compatible with standard gravity at observable scales and highlight the importance of nonlinear dynamics in resolving the vDVZ issue.

Abstract

In both old and recent literature, it has been argued that the celebrated van Dam-Veltman-Zakharov (vDVZ) discontinuity of massive gravity is an artifact due to linearization of the true equations of motion. In this letter, we investigate that claim. First, we exhibit an explicit -albeit somewhat arbitrary- fully covariant set of equations of motion that, upon linearization, reduce to the standard Pauli-Fierz equations. We show that the vDVZ discontinuity still persists in that non-linear, covariant theory. Then, we restrict our attention to a particular system that consistently incorporates massive gravity: the Dvali-Gabadadze-Porrati (DGP) model. DGP is fully covariant and does not share the arbitrariness and imperfections of our previous covariantization, and its linearization exhibits a vDVZ discontinuity. Nevertheless, we explicitly show that the discontinuity does disappear in the fully covariant theory, and we explain the reason for this phenomenon.