Embeddings of the “New Massive Gravity”

Abstract

Here we apply different types of embeddings of the equations of motion of the linearized “New Massive Gravity” in order to generate alternative and even higher-order (in derivatives) massive gravity theories in D= 2 + 1. In the first part of the work we use the Weyl symmetry as a guiding principle for the embeddings. First we show that a Noether gauge embedding of the Weyl symmetry leads to a sixth-order model in derivatives with either a massive or a massless ghost, according to the chosen overall sign of the theory. On the other hand, if the Weyl symmetry is implemented by means of a Stueckelberg field we obtain a new scalar–tensor model for massive gravitons. It is ghost-free and Weyl invariant at the linearized level around Minkowski space. The model can be nonlinearly completed into a scalar field coupled to the NMG theory. The elimination of the scalar field leads to a nonlocal modification of the NMG. In the second part of the work we prove to all orders in derivatives that there is no local, ghost-free embedding of the linearized NMG equations of motion around Minkowski space when written in terms of one symmetric tensor. Regarding that point, NMG differs from the Fierz–Pauli theory, since in the latter case we can replace the Einstein–Hilbert action by specific f(R,□R) generalizations and still keep the theory ghost-free at the linearized level.