In three dimensions, one can give the graviton a mass using the analogue of a Chern-Simons topological term. This so-called `New Massive Gravity' can be made free of kinetic ghosts in the vielbein formalism. However, when generalizing this construction to four dimensions, giving `critical gravity', it is not known how to remove the kinetic ghosts. This paper suggests such a procedure, by introducing a second (auxilliary) spin-2 field (which integrates out to give critical gravity), and then promoting it to a dynamical field. The resulting theory resembles New Massive Gravity, but can now be defined in four-dimensions without kinetic ghosts thanks to more degrees of freedom.

This is an interesting alternative to the dRGT construction of mass terms in four dimensions, which are also ghost free. While dRGT terms are ghost-free even with a non-dynamical auxially spin-2 field, one can promote this to a dynamical second metric (yielding Hassan-Rosen bigravity). The authors provide an interesting comparison between their New Bigravity, and Hassan-Rosen bigravity.