The Universality of Einstein Equations

TL;DR

This paper demonstrates that for a broad class of Lagrangians depending on scalar curvature, the Palatini formalism yields universal equations that are essentially Einstein equations in higher dimensions, with special cases in two dimensions.

Contribution

It establishes the universality of Einstein equations from a wide class of Lagrangians using the Palatini formalism, including bifurcation analysis and conformal invariance considerations.

Findings

Universal equations are Einstein equations for generic Lagrangians in dimensions greater than two.

Bifurcations occur at specific conformally invariant Lagrangians like R^{n/2} √g.

In two dimensions, the universal equation reduces to constant scalar curvature with a Weyl connection.

Abstract

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