Note: An Unusual Route to General Relativity Equation in Presence of Dust in Irrotational Motion

Abstract

We show that the general relativity theory equation, in presence of pressureless matter (dust) in irrotational motion, can be recovered from a scalar-tensor like variational approach. In this approach, the kinetic energy, ∂_σ ϕ∂^σϕ, of a dynamical scalar field ϕ, couples directly to gravity. The lagrangian, exempt of explicit matter term, is varied in the framework of the first order formalism, and a conformal transformation, restoring riemannian geometry, is made. In this approach, it turns out that a non-empty spacetime is necessarily four-dimensional.