The coupling to matter in massive, bi- and multi-gravity

In this paper we construct a family of ways in Inch matter can couple to one or more metrics'/spin-2 fields in the vielbein formulation. We do so subject to requiring the weak equivalence principle and the absence of ghosts from pure spin-2 interactions generated by the matter action. Results are presented for Massive, and Multi-Gravity theories and we give explicit expressions for the effective matter metric in all of these cases.