All Fierz-Paulian massive gravity theories have ghosts or superluminal modes

TL;DR

The paper analyzes nonlinear FP gravity by framing it as $S=S_{EH}+S_{FP}$ and examining the FP2 two-parameter Lagrangian, arguing that all ghost-free candidates still admit superluminal propagation in flat spacetime. Using a flat-background for the four scalars $\phi^A$ and a perturbation $\delta\phi^A$, it derives a quadratic Lagrangian for the scalar perturbation with phase velocity $v=\pm 1+\frac{2c_1+1}{4}\epsilon$, revealing superluminality except at $c_1=-\tfrac{1}{2}$, and showing that the special case still yields superluminal behavior via a higher-order term $-{1\over 16}\epsilon^3\dot{\chi}\chi'$. This result suggests that no nonlinear FP theory can be both ghost-free and causal, effectively ruling out FP theories as viable massive gravity models. Consequently, the work constrains the landscape of Lorentz-invariant massive gravity theories by highlighting inherent tension between ghost freedom and subluminal propagation.

Abstract

We show that all non-linear completions of the Fierz-Pauli massive gravity are ruled out, because all theories which might be ghost-free have superluminal modes.