Bigravity in Kuchar's Hamiltonian formalism. 2. The special case

TL;DR

This paper establishes four specific conditions on the interaction potential in bigravity theory that are necessary and sufficient to avoid ghost modes, ensuring a consistent Hamiltonian formulation.

Contribution

It provides a rigorous proof that four conditions on the potential eliminate ghost modes in bigravity and massive gravity within Kuchar's Hamiltonian formalism.

Findings

Four conditions on the potential prevent ghost modes.

Conditions include differential equations, Monge-Ampere equation, and Hessian nondegeneracy.

Proof applies to both bigravity and massive gravity cases.

Abstract

It is proved, that, in order to avoid the ghost mode in bigravity theory, it is sufficient to impose four conditions on the potential of interaction of the two metrics. First, the potential should allow its expression as a function of components of the two metrics' 3+1 decomposition. Second, the potential must satisfy the first order linear differential equations which are necessary for the presence of four first class constraints in bigravity. Third, the potential should be a solution of the Monge-Ampere equation, where the lapse and shift are considered as variables. Fourth, the potential must have a nondegenerate Hessian in the shift variables. The proof is based on the explicit derivation of the Hamiltonian constraints, the construction of Dirac brackets on the base of a part of these constraints, and calculation of other constraints' algebra in these Dirac brackets. As a byproduct, we prove that these conditions are also sufficient in the massive gravity case.