Lessons from $f(R,R_c^2,R_m^2,L_m)$ gravity: Smooth Gauss-Bonnet limit, energy-momentum conservation and nonminimal coupling

Abstract

This paper studies a generic fourth-order theory of gravity with Lagrangian density $f(R,R_c^2,R_m^2,{\mathcal{L}}_m)$. By considering explicit $R^2$ dependence and imposing the "coherence condition" $f_{R^2}=f_{R_m^2}=-f_{R_c^2}/4$, the field equations of $f(R,R^2,R_c^2,R_m^2,{\mathcal{L}}_m)$ gravity can be smoothly reduced to that of $f(R,\mathcal{G},{\mathcal{L}}_m)$ generalized Gauss-Bonnet gravity. We use Noether's conservation law to study the $f({\mathcal{R}}_1,{\mathcal{R}}_2\dots ,{\mathcal{R}}_n,{\mathcal{L}}_m)$ model with nonminimal coupling between ${\mathcal{L}}_m$ and Riemannian invariants ${\mathcal{R}}_i$, and conjecture that the gradient of nonminimal gravitational coupling strength ${\nabla }^{\mu }{f}_{{\mathcal{L}}_m}$ is the only source for energy-momentum non-conservation. This conjecture is applied to the $f(R,R_c^2,R_m^2,{\mathcal{L}}_m)$ model, and the equations of continuity and non-geodesic motion of different matter contents are investigated. Finally, the field equation for Lagrangians including the traceless-Ricci square and traceless-Riemann (Weyl) square invariants is derived, the $f(R,R_c^2,R_m^2,{\mathcal{L}}_m)$ model is compared with the $f(R,R_c^2,R_m^2,T)+2\kappa {\mathcal{L}}_m$ model, and consequences of nonminimal coupling for black hole and wormhole physics are considered.

Final version: 29 pages, 1 table, modified RevTex format in one column. To appear in Physical Review D

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• To conserve, or not to conserve: A review of nonconservative theories of gravity
• Non-minimal geometry-matter couplings in Weyl-Cartan space-times: $f(R,T,Q,T_m)$ gravity
• Friedmann equations from nonequilibrium thermodynamics of the Universe: A unified formulation for modified gravity
• Constrained transit cosmological models in $f(R,L_m,T)$-gravity