The trace of field equations for higher-derivative gravity and an equality associating the Lagrangian density with a divergence term

TL;DR

This paper derives the explicit form of the trace of field equations in higher-derivative gravity theories and establishes an equality linking the Lagrangian density to a divergence term, with applications to complex curvature-dependent Lagrangians.

Contribution

It provides a general explicit expression for the trace of field equations and relates the Lagrangian density to a divergence in higher-derivative gravity theories.

Findings

Derived the explicit trace of field equations for generic higher-derivative gravity.

Established an equality linking the Lagrangian density with a divergence term.

Expressed certain complex Lagrangians as covariant divergences of vector fields.

Abstract

We figure out the explicit expression for the trace of the field equations associated to generic higher derivative theories of gravity endowed with Lagrangians depending upon the metric and its Riemann tensor, together with arbitrary order covariant derivatives of the Riemann tensor. Then an equality linking the Lagrangian density with the covariant divergence of a vector field is put forward in terms of the trace of the field equations. As a significant application, we particularly concentrate on a broad range of higher derivative theories of gravity with the Lagrangian density constructed from the contraction of the product for metric tensors with the product of the Riemann tensors and the arbitrary order covariant derivatives of the Riemann tensor. By utilizing the trace for the equations of motion, such a type of Lagrangian density is expressed as the covariant divergence of a vector field.