Bigravity in Hamiltonian formalism

Theory of bigravity is one of approaches proposed to solve the dark energy problem of the Universe. It deals with two metric tensors, each one is minimally coupled to the corresponding set of matter fields. The bigravity Lagrangian equals to a sum of two General Relativity Lagrangians with the different gravitational coupling constants and different fields of matter accompanied by the ultralocal potential. As a rule, such a theory has 8 gravitational degrees of freedom: the massless graviton, the massive graviton and the ghost. A special choice of the potential, suggested by de Rham, Gabadadze and Toley (dRGT), allows to avoid of the ghost. But the dRGT potential is constructed by means of the matrix square root, and so it is not an explicit function of the metrics components. One way to do with this difficulty is to apply tetrads. Here we consider an alternative approach. The potential as a differentiable function of metrics components is supposed to exist, but we never appeal to the explicit form of this function. Only properties of this function necessary and sufficient to exclude the ghost are studied. The final results are obtained from the constraint analysis and the Poisson brackets calculations. The gravitational variables are the two induced metrics and their conjugated momenta. Also lapse and shift variables for both metrics are involved. After the exclusion of 3 auxiliary variables we stay with 4 first class constraints and 2 second class ones responsible for the ghost exclusion. The requirements for the potential are as follows:

• the potential should satisfy a system of the first order linear differential equations;
• the potential should satisfy the homogeneous Monge-Ampere equation in 4 auxiliary variables;
• the Hessian of the potential in 3 auxiliary variables is non-degenerate.