The universality of vacuum Einstein equations with cosmological constant

Abstract

It is shown that for a wide class of analytic Lagrangians which depend only\non the scalar curvature of a metric and a connection, the application of the\nso--called ``Palatini formalism'', i.e., treating the metric and the connection\nas independent variables, leads to ``universal'' equations. If the dimension\n$n$ of space--time is greater than two these universal equations are Einstein\nequations for a generic Lagrangian and are suitably replaced by other universal\nequations at bifurcation points. We show that bifurcations take place in\nparticular for conformally invariant Lagrangians $L=R^{n/2} \\sqrt g$ and prove\nthat their solutions are conformally equivalent to solutions of Einstein\nequations. For 2--dimensional space--time we find instead that the universal\nequation is always the equation of constant scalar curvature; the connection in\nthis case is a Weyl connection, containing the Levi--Civita connection of the\nmetric and an additional vectorfield ensuing from conformal invariance. As an\nexample, we investigate in detail some polynomial Lagrangians and discuss their\nbifurcations.\n