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

TL;DR

The paper derives an explicit expression for the trace of the field equations in generic higher-derivative gravity theories whose Lagrangian depends on the metric, the Riemann tensor, and arbitrary covariant derivatives of the Riemiemann tensor, using a surface-term-based approach in a diffeomorphism-invariant setting. It proves an identity linking the Lagrangian density to the covariant divergence of a vector field through the trace of the equations of motion, enabling the Lagrangian to be written as a divergence under a concrete constraint and without appealing to $rac{ abla L}{ abla g_{ ext{...}}}$ terms. As a significant application, the authors consider Lagrangians built from contractions of the metric with products of Riemann tensors and their derivatives, derive the on-shell trace relation $rac{1}{2}ig( ext{sum}(i+2)k_i - Dig) ilde L + abla_ u ilde V^ u = 0$ and express $ ilde L$ as a covariant divergence $ ilde L=rac{2}{D- ext{sum}(i+2)k_i} abla_ u ilde V^ u$, with explicit vector forms in examples $L_1$ and $L_2$. These results have implications for simplifying Euclidean gravitational actions and for studying higher-derivative gravity solutions, with future work proposed on including matter fields and extending to non-vacuum spacetimes.

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.