THE UNIVERSALITY OF VACUUM EINSTEIN EQUATIONS WITH COSMOLOGICAL CONSTANT

It is shown that for a wide class of analytic Lagrangians, which depen d only on the scalar curvature of a metric and a connection, the appli cation of the so called 'Palatini formalism', i.e. treating the metric and the connection as independent variables, leads to 'universal' equ ations. If the dimension n of spacetime is greater than two these univ ersal equations are vacuum Einstein equations with cosmological consta nt for a generic Lagrangian and are suitably replaced by other univers al equations at degenerate points. We show that degeneracy takes place in particular for conformally invariant Lagrangians L = R(n/2) square -root g and we prove that their solutions are conformally equivalent t o solutions of Einstein's equations. For two-dimensional spacetimes we find instead that the universal equation is always the equation of co nstant scalar curvature; in this case the connection is a Weyl connect ion, containing the Levi-Civita connection of the metric and an additi onal vector field ensuing from conformal invariance. As an example, we investigate in detail some polynomial Lagrangians and discuss their d egenerate points.