An action principle for the Einstein-Weyl equations

TL;DR

This paper introduces the first action principle for the three-dimensional Einstein-Weyl equations with a non-exact Weyl vector, involving metric affine f(R) gravity, Lagrange multipliers, and Chern-Simons terms.

Contribution

It provides a novel action principle for Einstein-Weyl equations in three dimensions with non-exact Weyl vectors, expanding the theoretical framework.

Findings

Weyl vector dynamics governed by a generalized monopole equation

The model includes metric affine f(R) gravity with additional terms

Equations reduce to Einstein-Weyl equations when traceless part is zero

Abstract

A longstanding open problem in mathematical physics has been that of finding an action principle for the Einstein-Weyl (EW) equations. In this paper, we present for the first time such an action principle in three dimensions in which the Weyl vector is not exact. More precisely, our model contains, in addition to the Weyl nonmetricity, a traceless part. If the latter is (consistently) set to zero, the equations of motion boil down to the EW equations. In particular, we consider a metric affine $f(R)$ gravity action plus additional terms involving Lagrange multipliers and gravitational Chern-Simons contributions. In our framework, the metric and the connection are considered as independent objects, and no a priori assumptions on the nonmetricity and the torsion of the connection are made. The dynamics of the Weyl vector turns out to be governed by a special case of the generalized monopole equation, which represents a conformal self-duality condition in three dimensions.