Generically, non-linear bimetric theories of gravity suffer from the same Boulware-Deser ghost instability as non-linear theories of massive gravity. However, recently proposed theories of massive gravity have been shown to be ghost-free. These theories are formulated with respect to a flat, non-dynamical reference metric. In this work we show that it is possible to give dynamics to the reference metric in such a way that the consistency of the theory is maintained. The result is a non-linear bimetric theory of a massless spin-2 field interacting with a massive spin-2 field that is free of the BoulwareDeser ghost. To our knowledge, this is the first construction of such a ghost-free bimetric theory.

We analyze the ghost issue in the recently proposed models of non-linear massive gravity in the ADM formalism. We show that, in the entire two-parameter family of actions, the Hamiltonian constraint is maintained at the complete non-linear level and we argue for the existence of a nontrivial secondary constraint. This implies the absence of the pathological Boulware-Deser ghost to all orders. To our knowledge, this is the first demonstration of the existence of a consistent theory of massive gravity at the complete non-linear level, in four dimensions.Introduction and Summary: The search for a consistent theory of massive gravity is motivated by both theoretical and observational considerations. Since the construction of a linear theory of massive gravity by Fierz and Pauli in 1939 [1, 2], a proof of the existence of a consistent non-linear generalization has remained elusive, making it a theoretically intriguing problem. On the observational side, the recent discovery of dark energy and the associated cosmological constant problem has prompted investigations into long distance modifications of general relativity. An obvious such modification is massive gravity.

The grand challenges of contemporary fundamental physics—dark matter, dark energy, vacuum energy, inflation and early universe cosmology, singularities and the hierarchy problem—all involve gravity as a key component. And of all gravitational phenomena, black holes stand out in their elegant simplicity, while harbouring some of the most remarkable predictions of General Relativity: event horizons, singularities and ergoregions. The hitherto invisible landscape of the gravitational Universe is being unveiled before our eyes: the historical direct detection of gravitational waves by the LIGO-Virgo collaboration marks the dawn of a new era of scientific exploration. Gravitational-wave astronomy will allow us to test models of black hole formation, growth and evolution, as well as models of gravitational-wave generation and propagation. It will provide evidence for event horizons and ergoregions, test the theory of General Relativity itself, and may reveal the existence of new fundamental fields. The synthesis of these results has the potential to radically reshape our understanding of the cosmos and of the laws of Nature. The purpose of this work is to present a concise, yet comprehensive overview of the state of the art in the relevant fields of research, summarize important open problems, and lay out a roadmap for future progress. This write-up is an initiative taken within the framework of the European Action on ‘Black holes, Gravitational waves and Fundamental Physics’.