On dual formulations of gravity

Abstract

In this paper we consider a possibility to construct dual formulation of gravity where the main dynamical field is the connection and not that of tetrad eμa or metric gμν. Our approach is based on the usual dualization procedure which uses first order parent Lagrangians but in (Anti) de Sitter space and not in the flat Minkowski one. At first we consider dual formulation based on the usual terad formalism with Lorentz connection ωμab as new dynamical field. It turns out that in d = 3 dimensions such dual formulation is related with the so called exotic parity-violating interactions for massless spin-2 particles. Then we construct a dual formulation of gravity where the main dynamical object is affine connection starting with the well known first order Palatini formulation but in (Anti) de Sitter background. The final result obtained by solving equations for the metric is the Lagrangian written by Eddington in his book in 1924. Also there is an interesting connection with attempts to construct gravitational analog of Born-Infeld electrodynamics. © SISSA 2006.