Covariant approach to the no-ghost theorem in massive gravity

We discuss the no-ghost theorem in massive gravity in a covariant manner. Using the Becchi-Rouet-Stora-Tyutin (BRST) formalism and Stückelberg fields, we first clarify how the Boulware–Deser ghost decouples in massive gravity theory with the Fierz–Pauli mass term. Here we find that the crucial point in the proof is that there is no higher (time) derivative for the Stückelberg “scalar” field. We then analyze the nonlinear massive gravity proposed by de Rham, Gabadadze, and Tolley, and show that there is no ghost for general admissible backgrounds. In this process, we find a very nontrivial decoupling limit for general backgrounds. We end the paper by demonstrating the general results explicitly in a nontrivial example where there apparently appear higher time derivatives for the Stückelberg scalar field, but show that this does not introduce the ghost into the theory.