Gödel-type universes in $f(R)$ gravity

The $f(R)$ gravity theories provide an alternative way to explain the current cosmic acceleration without a dark energy matter component. If gravity is governed by a $f(R)$ theory, a number of issues should be reexamined in this framework, including the violation of causality problem on nonlocal scale. We examine the question as to whether the $f(R)$ gravity theories permit space-times in which the causality is violated. We show that the field equations of these $f(R)$ gravity theories do not exclude solutions with breakdown of causality for a physically well-motivated perfect-fluid matter content. We demonstrate that every perfect-fluid Gödel-type solution of a generic $f(R)$ gravity satisfying the condition $df/dR>0$ is necessarily isometric to the Gödel geometry, and therefore presents violation of causality. This result extends a theorem on Gödel-type models, which has been established in the context of general relativity. We also derive an expression for the critical radius $r_c$ (beyond which the causality is violated) for an arbitrary $f(R)$ theory, making apparent that the violation of causality depends on both the $f(R)$ gravity theory and the matter content. As an illustration, we concretely take a recent $f(R)$ gravity theory that is free from singularities of the Ricci scalar and is cosmologically viable, and show that this theory accommodates noncausal as well as causal Gödel-type solutions.