Minimally modified theories of gravity: a playground for testing the uniqueness of general relativity

In a recent paper [1], it was introduced a new class of gravitational theories with two local degrees of freedom. The existence of these theories apparently challenges the distinctive role of general relativity as the unique non-linear theory of massless spin-2 particles. Here we perform a comprehensive analysis of these theories with the aim of (i) understanding whether or not these are actually equivalent to general relativity, and (ii) finding the root of the variance in case these are not. We have found that a broad set of seemingly different theories actually pass all the possible tests of equivalence to general relativity (in vacuum) that we were able to devise, including the analysis of scattering amplitudes using on-shell techniques. These results are complemented with the observation that the only examples which are manifestly not equivalent to general relativity either do not contain gravitons in their spectrum, or are not guaranteed to include only two local degrees of freedom once radiative corrections are taken into account. Coupling to matter is also considered: we show that coupling these theories to matter in a consistent way is not as straightforward as one could expect. Minimal coupling, as well as the most straightforward non-minimal couplings, cannot be used. Therefore, before being able to address any issues in the presence of matter, it would be necessary to find a consistent (and in any case rather peculiar) coupling scheme.