Bimetric interactions based on metric congruences

In massive gravity and bigravity, spin-2 interactions are defined in terms of a square root matrix that involves two metrics. In this work, the interactions are constructed using a congruence matrix between the metrics. It is established that the primary square root matrix function is the only power series solution to the equations of motion for the congruence. Moreover, the shift vector redefinition that is used in the bimetric ghost-free proofs follows from the $N+1$ form of the equations of motion. The analysis also gives an insight into the vielbein formulation of spin-2 interactions since the bimetric formulation in terms of a congruence is algebraically equivalent to the unconstrained vielbein formulation.