Abstract
We discuss the no-ghost theorem in the massive gravity in a covariant manner. Using the BRST formalism and St\"{u}ckelberg fields, we first clarify how the Boulware-Deser ghost decouples in the massive gravity theory with Fierz-Pauli mass term. Here we find that the crucial point in the proof is that there is no higher (time) derivative for the St\"{u}ckelberg `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 St\"{u}ckelberg scalar field, but show that this does not introduce the ghost into the theory.
Highlights
There has been renewed interest in the search for the modification of gravity at large distances by adding the mass terms for graviton
It is interesting to explore the possibility of formulating theory of massive spin-2 field
There remain six degrees of freedom in general. Five out of these are the modes of massive spin-2 graviton, but it turns out that the sixth scalar mode is a ghost with a negative metric
Summary
There has been renewed interest in the search for the modification of gravity at large distances by adding the mass terms for graviton. Using the noncovariant Arnowitt-Deser-Misner (ADM) decomposition, it is shown that the mass term introduces nonlinear terms for the shifts so that these do not produce any constraint, but the lapse function remains linear and we are left with one constraint instead of four in general relativity. In this noncovariant approach, we have six degrees of freedom for the propagating modes from the spatial metric gij, but one of them is removed by the above constraint from the lapse, leaving correct five degrees of freedom for a massive spin without ghost. We show that naively it looks that there appears higher time derivatives on the Stuckelberg scalar field, but our definition of the Stuckelberg fields avoids the trouble, so that there is no BD ghost in the theory
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