A Unified Approach to Variational Derivatives of Modified Gravitational Actions

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 $n⩾3$. 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.

References in corpus (19)

• Wilkinson Microwave Anisotropy Probe (WMAP) Three Year Results: Implications for Cosmology
• Five-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Interpretation
• f(R) theories
• Cosmological Constraints from the SDSS Luminous Red Galaxies
• Massive Gravity in Three Dimensions
• Measuring the Baryon Acoustic Oscillation scale using the SDSS and 2dFGRS
• Chern-Simons Modified General Relativity
• More on Massive 3D Gravity
• Three-dimensional black holes, gravitational solitons, kinks and wormholes for BHT massive gravity
• Violation of the Equivalence Principle in Modified Theories of Gravity
• f(R) Gravity with Torsion: The Metric-Affine Approach
• The Cosmology of Ricci-Tensor-Squared Gravity in the Palatini Variational Approach
• Ghost Constraints on Modified Gravity
• Classification of solutions in topologically massive gravity
• Dynamical Aspects of Generalized Palatini Theories of Gravity
• Renormalizability of Massive Gravity in Three Dimensions
• Cosmological dynamics of exponential gravity
• Black Holes in Higher Dimensional Gravity Theory with Quadratic in Curvature Corrections
• Indistinguishable Macroscopic Behaviour of Palatini Gravities and General Relativity

Cited by in corpus (18)

• Hybrid metric-Palatini gravity
• Extended Gauss-Bonnet gravities in Weyl geometry
• Ghosts in pure and hybrid formalisms of gravity theories: a unified analysis
• Multi-Scalar-Tensor Equivalents for Modified Gravitational Actions
• Gauss-Bonnet lagrangian G ln G and cosmological exact solutions
• Swampland conjectures in hybrid metric-Palatini gravity
• An alternative derivation of the Minimal massive 3D gravity
• Covariant Differential Identities and Conservation Laws in Metric-Torsion Theories of Gravitation. I. General Consideration
• Maxwell extension of $f(R)$ gravity
• Covariant Differential Identities and Conservation Laws in Metric-Torsion Theories of Gravitation. II. Manifestly Generally Covariant Theories
• Black Holes in Multi-Metric Gravity
• Energy definition for quadratic curvature gravities
• Variational derivatives of gravitational actions
• Gravitational wave solutions of quadratic curvature gravity using a null coframe formulation
• Quadratic Curvature Gravity with Second Order Trace and Massive Gravity Models in Three Dimensions
• Modified Einstein-Gauss-Bonnet gravity: Riemann-Cartan and Pseudo-Riemannian cases
• Generalized Sparling-Thirring form in the Brans-Dicke theory
• Impulsive gravitational waves in general massive 3D gravity