Two Approaches to Hamiltonian Bigravity

Abstract

Bigravity is one of the most natural modifications of General Relativity (GR), as it is based on the equivalence principle. However, its canonical structure appears rather complicated because of the unusual form of the interaction between two metrics. As a consequence, there are different approaches that are difficult to compare in detail. This work is a first attempt to obtain a synthetic picture of the Hamiltonian formalism for bigravity. Here, we are trying to combine two rather different approaches to gain a binocular view of the theory. The first publications on the subject were based on metric formalism. It was proved that both massive gravity and bigravity with de Rham–Gabadadze–Tolley (dRGT) potential were free of Boulware–Deser (BD) ghosts. This proof was based on the transformation of variables involving a 3×3-matrix which could be treated as the root of a quadratic equation involving two spatial metrics and a new 3-vector introduced instead of the standard shift variable. Therefore, this matrix occurred as an implicit function of the abovementioned variables. After a substantial amount of time, it became possible to calculate the algebra of constraints in full using this method. However, in another approach also based on metric variables and implicit functions, similar calculations were completed earlier. It is not a new matrix, but the potential itself has been taken as an implicit function of two spatial metrics and four functions constructed of two pairs of lapses and shifts. Finally, a straightforward route to canonical bigravity is to apply tetrad (or vierbein) variables. The matrix square root involved in the dRGT potential can be explicitly extracted if tetrads fulfill the symmetry condition. A full treatment has been developed in first-order formalism by treating tetrads and connections as independent variables. In that case, the theory contains many more variables and constraints than in metric formalism. An essential simplification occurs in second-order vierbein formalism. The potential is given explicitly as a polynomial of bilinear combinations of the two tetrads. The 3×3-matrix introduced in the pioneer papers can be expressed explicitly through canonical coordinates, and the celebrated transformation of variables arises in the Dirac constraint analysis.