Lovelock Tensor as Generalized Einstein Tensor

Abstract

We show that the splitting feature of the Einstein tensor, as the first term of the Lovelock tensor, into two parts, namely the Ricci tensor and the term proportional to the curvature scalar, with the trace relation between them is a common feature of any other homogeneous terms in the Lovelock tensor. Motivated by the principle of general invariance, we find that this property can be generalized, with the aid of a generalized trace operator which we define, for any inhomogeneous Euler-Lagrange expression that can be spanned linearly in terms of homogeneous tensors. Then, through an application of this generalized trace operator, we demonstrate that the Lovelock tensor analogizes the mathematical form of the Einstein tensor, hence, it represents a generalized Einstein tensor. Finally, we apply this technique to the scalar Gauss-Bonnet gravity as an another version of string-inspired gravity.

16 pages, added refs., corrected a mistyped eq. (the last one in p12), TeX file (using phyzzx.tex)

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• Introduction to Modified Gravity and Gravitational Alternative for Dark Energy
• Gauss-Bonnet dark energy
• Dark energy cosmology from higher-order, string-inspired gravity and its reconstruction
• String-inspired Gauss-Bonnet gravity reconstructed from the universe expansion history and yielding the transition from matter dominance to dark energy
• Cosmology and Astrophysical Constraints of Gauss-Bonnet Dark Energy
• The fate of (phantom) dark energy universe with string curvature corrections
• Dark energy and cosmological solutions in second-order string gravity
• String-inspired cosmology: Late time transition from scaling matter era to dark energy universe caused by a Gauss-Bonnet coupling
• Non-Trivial Vacua in Higher-Derivative Gravitation
• On Higher Order Gravities, Their Analogy to GR, and Dimensional Dependent Version of Duff's Trace Anomaly Relation
• Classical Trace Anomaly
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• The relation between F(R) gravity and Einstein-conformally invariant Maxwell source
• Birkhoff's Theorem in Higher Derivative Theories of Gravity
• On Higher Order Gravities, Their Analogy to GR, and Dimensional Dependent Version of Duff's Trace Anomaly Relation
• General features of Bianchi-I cosmological models in Lovelock gravity
• Exact exponential solutions in Einstein-Gauss-Bonnet flat anisotropic cosmology
• Gauss-Bonnet lagrangian G ln G and cosmological exact solutions
• Dynamical compactification in Einstein-Gauss-Bonnet gravity from geometric frustration
• Gravitoelectromagnetism and other decompositions of the Riemann tensor
• Big Rip and Little Rip solutions in scalar model with kinetic and Gauss Bonnet couplings
• Autonomous Dynamical System of Einstein-Gauss-Bonnet Cosmologies
• Constant volume exponential solutions in Einstein-Gauss-Bonnet flat anisotropic cosmology with a perfect fluid
• The dynamics of the flat anisotropic models in the Lovelock gravity. I: The even-dimensional case
• Late time cosmological scenarios from scalar field with Gauss Bonnet and non-minimal kinetic couplings
• Quintessential and phantom power-law solutions in scalar tensor model of dark energy
• Phase space analysis for a scalar-tensor model with kinetic and Gauss-Bonnet couplings
• Perturbation Spectra of Warm Inflation in $f(Q,T)$ Gravity
• Noncommutative Universe and Chameleon Field Dynamics
• Cosmological Solutions of Chameleon Scalar Field Model
• Euler-Lagrange formulas for pseudo-Kaehler manifolds
• Third Order Lagrangians, Weyl Invariants & Classical Trace Anomaly in Six Dimensions
• Constraints on a scalar-tensor model with Gauss-Bonnet coupling from SN Ia and BAO observations
• Metric Field as Emergence of Hilbert Space
• Entire Four-Graviton EFT from the Duality Between Color and Kinematics