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

We examine the question as to whether the Palatini $f(R)$ gravity theories permit space-times in which the causality is violated. We show that every perfect-fluid Gödel-type solution of Palatini $f(R)$ gravity with density $\rho$ and pressure $p$ that satisfy the weak energy condition $\rho +p\ge 0$ is necessarily isometric to the Gödel geometry, demonstrating therefore that these theories present causal anomalies in the form of closed timelike curves. This result extends a theorem on Gödel-type models to the framework of Palatini $f(R)$ gravity theory. We concretely examine the Gödel-type perfect-fluid solutions in specific $f(R)=R-\alpha /R^n$ Palatini gravity theory, where the free parameters $\alpha$ and $n$ have been recently constrained by observational data. We show that for positive matter density and for $\alpha$ and $n$ within the interval permitted by the observations, this theory does not admit the Gödel geometry as a perfect-fluid solution of its field equations. In this sense, this theory remedies the causal pathology in the form of closed timelike curves which is allowed in general relativity. We derive an expression for a critical radius $r_c$ (beyond which the causality is violated) for an arbitrary Palatini $f(R)$ theory. The expression makes apparent that the violation of causality depends on the form of $f(R)$ and on the matter content components. We also examine the violation of causality of Gödel-type by considering a single scalar field as the matter content. For this source we show that Palatini $f(R)$ gravity gives rise to a unique Gödel-type solution with no violation of causality.