Bigravity in the Kuchař Hamiltonian formalism: The general case

Abstract

We develop the Hamiltonian formalism of bigravity and bimetric theories for the general form of the interaction potential of two metrics. When studying the role of lapse and shift functions in theories with two metrics, we naturally use the Kuchař formalism in which these functions are independent of the choice of the space-time coordinate system. We find conditions on the potential necessary and sufficient for the existence of four first-class constraints. These constraints realize a well-known hypersurface deformation algebra in the framework of the formalism of Dirac brackets constructed on the base of all second-class constraints. Fixing one of the metrics, we obtain a bimetric theory not containing first-class constraints. Conserved quantities corresponding to symmetries of the background metric can then be expressed ultralocally in terms of the metric interaction potential.