A Unified Approach to Variational Derivatives of Modified Gravitational Actions

TL;DR

This paper introduces a coframe variational method as a unified framework for deriving field equations from various modified gravitational actions involving curvature scalars, including new models with Chern-Simons and three-dimensional gravity.

Contribution

It develops a master equation for variational derivatives of generalized gravitational actions and applies it to a broad class of theories, including novel models with Chern-Simons terms.

Findings

Derived a master variational equation for curvature-based actions.

Applied the method to well-known and new modified gravity models.

Presented new formulations for three-dimensional gravity and Chern-Simons models.

Abstract

Our main aim in this paper is to promote the coframe variational method as a unified approach to derive field equations for any given gravitational action containing the algebraic functions of the scalars constructed from the Riemann curvature tensor and its contractions. We are able to derive a master equation which expresses the variational derivatives of the generalized gravitational actions in terms of the variational derivatives of its constituent curvature scalars. Using the Lagrange multiplier method relative to an orthonormal coframe, we investigate the variational procedures for modified gravitational Lagrangian densities in spacetime dimensions $n\geqslant 3$. We study well-known gravitational actions such as those involving the Gauss-Bonnet and Ricci-squared, Kretchmann scalar, Weyl-squared terms and their algebraic generalizations similar to generic $f(R)$ theories and the algebraic generalization of sixth order gravitational Lagrangians. We put forth a new model involving the gravitational Chern-Simons term and also give three dimensional New massive gravity equations in a new form in terms of the Cotton 2-form.