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Bigravity (for bimetric gravity) refers to a GR-like gravity model based upon a pair of metric formulas rather than a single formula, presumably with some criteria regarding under what circumstances each apply. The concept has been used to construct a GR-like gravity consistent with MOND.
Massive bigravity is a bigravity variant of massive gravity, i.e., a gravity model for which the corresponding gravitons have mass. Bigravity would have two different graviton types corresponding to the two metrics, and in massive bigravity, one of the two graviton types would have mass.