Universality of Einstein Equations for the Ricci Squared Lagrangians

TL;DR

This paper demonstrates that Einstein equations and Komar energy-momentum complex are universally valid for a broad class of nonlinear Ricci squared Lagrangians in the first order formalism, extending previous results for scalar curvature.

Contribution

It extends the universality of Einstein equations and Komar energy-momentum complex to nonlinear Lagrangians depending on Ricci squared invariants in the first order formalism.

Findings

Einstein equations hold for Ricci squared Lagrangians after a conformal transformation.

Komar energy-momentum complex remains valid in this extended framework.

Universality applies to a wider class of gravitational Lagrangians.

Abstract

It has been recently shown that, in the first order (Palatini) formalism, there is universality of Einstein equations and Komar energy-momentum complex, in the sense that for a generic nonlinear Lagrangian depending only on the scalar curvature of a metric and a torsionless connection one always gets Einstein equations and Komar's expression for the energy-momentum complex. In this paper a similar analysis (also in the framework of the first order formalism) is performed for all nonlinear Lagrangians depending on the (symmetrized) Ricci square invariant. The main result is that the universality of Einstein equations and Komar energy-momentum complex also extends to this case (modulo a conformal transformation of the metric).