Field equations and Noether potentials for higher-order theories of gravity with Lagrangians involving ${\square }^{i}R$, ${\square }^i{R}_{\mu \nu }$ and ${\square }^i{R}_{\mu \nu \rho \sigma }$

Abstract

In this paper, we aim to perform a systematical investigation on the field equations and Noether potentials for the higher-order gravity theories endowed with Lagrangians depending on the metric and the Riemann curvature tensor, together with $i$th ($i=1,2,\cdot \cdot \cdot$) powers of the Beltrami-d'Alembertian operator $\square$ acting on the latter. We start with a detailed derivation of the field equations and the Noether potential corresponding to the Lagrangian $\sqrt{-g}{L}_R(R,\square R,\cdot \cdot \cdot ,{\square }^{m}R)$ through the direct variation of the Lagrangian and a method based upon the conserved current. Next the parallel analysis is extended to a more generic Lagrangian $\sqrt{-g}{L}_{\text{Ric}}(g^{\mu \nu },R_{\mu \nu },\square R_{\mu \nu },\cdot \cdot \cdot ,{\square }^m{R}_{\mu \nu })$, as well as to the generalization of the Lagrangian $\sqrt{-g}{L}_{\text{Ric}}$, which depends on the metric $g^{\mu \nu }$, the Riemann tensor $R_{\mu \nu \rho \sigma }$ and ${\square }^i{R}_{\mu \nu \rho \sigma }$s. Finally, all the results associated to the three types of Lagrangians are extended to the Lagrangian relying on an arbitrary tensor and the variables via ${\square }^i$ acting on such a tensor. In particular, we take into consideration of equations of motion and Noether potentials for nonlocal gravity models. For Lagrangians involving the variables ${\square }^{i}R$, ${\square }^i{R}_{\mu \nu }$ and ${\square }^i{R}_{\mu \nu \rho \sigma }$, our investigation provides their concrete Noether potentials and the field equations without the derivative of the Lagrangian density with respect to the metric. Besides, the Iyer-Wald potentials associated to these Lagrangians are also presented.

arXiv comment74 pages, No figures,several typos are corrected