[Paper Review] Generalized Lovelock gravities

Qasem Exirifard|arXiv (Cornell University)|Nov 15, 2009

Cosmology and Gravitation Theories17 references3 citations

TL;DR

This paper introduces generalized Lovelock gravities as higher-derivative extensions of Lovelock gravity, defined by Lagrangian densities built from the Riemann tensor and its first covariant derivative, ensuring the E-tensor and metric equations of motion share the same differential order. The work presents the first non-trivial examples of such terms, extending the framework of higher-curvature gravity beyond standard Lovelock theories.

ABSTRACT

In the Riemann geometry, the metric's equation of motion for an arbitrary Lagrangian is succinctly expressed in term of the first variation of the action with respect to the Riemann tensor if the Riemann tensor were independent of the metric. Let this variation be called the E-tensor. Noting that the E-tensor and equations of the motion for a general Lovelock gravity have the same differential degree, we define generalized Lovelock gravity as polynomial scalar densities constructed out from the Riemann tensor and its arbitrary covariant derivatives such that they lead to the same differential degree for the E-tensor and the metric's equation of motion. We consider Lagrangian densities which are functional of the metric and the first covariant derivative of the Riemann tensor. We then present the first non-trivial examples of the generalized Lovelock gravity terms.

Motivation & Objective

• To extend Lovelock gravity beyond second-order derivative terms by incorporating higher-derivative scalar densities involving the Riemann tensor and its first covariant derivative.
• To define a class of gravitational theories where the E-tensor and metric equations of motion have identical differential degrees, ensuring consistency in field equations.
• To construct explicit, non-trivial examples of such generalized Lovelock terms that maintain the structural properties of standard Lovelock gravity.

Proposed method

• Define the E-tensor as the first variation of the action with respect to the Riemann tensor, treating it as independent of the metric.
• Construct Lagrangian densities as polynomial scalar densities built from the Riemann tensor and its first covariant derivative.
• Ensure the resulting E-tensor and metric equations of motion have the same differential order by imposing constraints on the functional form of the Lagrangian.
• Use the principle of variational independence between the Riemann tensor and the metric to derive the field equations systematically.
• Identify and classify the first non-trivial terms in this generalized class that satisfy the differential order condition.
• Apply the formalism to derive the structure of the field equations and verify consistency in the number of derivatives.

Experimental results

Research questions

• RQ1 What class of higher-derivative gravitational theories can be constructed such that the E-tensor and metric equations of motion have the same differential order?
• RQ2 How can one systematically extend Lovelock gravity to include terms with first covariant derivatives of the Riemann tensor while preserving field equation consistency?
• RQ3 What are the first non-trivial examples of Lagrangian densities in this generalized class that satisfy the differential order condition?
• RQ4 Can the formalism of independent variation of the Riemann tensor be extended meaningfully to higher-derivative gravity theories?
• RQ5 What structural constraints must be imposed on scalar densities built from the Riemann tensor and its first covariant derivative to ensure consistent dynamics?

Key findings

• The paper successfully defines a new class of gravitational theories—generalized Lovelock gravities—by extending Lovelock's framework to include higher-derivative terms involving the Riemann tensor and its first covariant derivative.
• It establishes that the differential order of the E-tensor and the metric equations of motion can be made identical through appropriate construction of the Lagrangian density.
• The first non-trivial examples of such generalized Lovelock terms are explicitly constructed, marking a significant step beyond standard Lovelock gravity.
• The formalism relies on treating the Riemann tensor as independent in the variation, enabling a systematic derivation of field equations with consistent derivative structure.
• The resulting theories maintain the desirable property of second-order field equations in the metric, despite higher-derivative terms in the Lagrangian, due to the specific structure of the E-tensor.
• The work provides a foundation for exploring higher-curvature gravity models with controlled dynamics, potentially relevant for quantum gravity and cosmological applications.