Homogeneous G\"odel-type solutions in hybrid metric-Palatini gravity.

[Abridged] If gravitation is to be described by a hybrid metric-Palatini $f(\mathcal{R})$ gravity theory there are a number of issues that ought to be examined in its context, including the question as to whether its equations allow homogeneous G\"odel-type solutions, which necessarily leads to violation of causality. Here, to look further into the potentialities and difficulties of $f(\mathcal{R})$ theories, we examine whether they admit G\"odel-type solutions for some physically well-motivated matter sources. We first show that under certain conditions on the matter sources the problem of finding out space-time homogeneous solutions in $f(\mathcal{R})$ theories reduces to the problem of determining solutions of this type in $f(R)$ gravity in the metric formalism. Employing this result, we determine a perfect-fluid G\"odel-type solution in $f(\mathcal{R})$ gravity, and show that it is isometric to the G\"odel geometry, and therefore exhibits violation of causality. This extends a theorem on G\"odel-type models, which was established in the framework of general relativity. We also show that a single massless scalar field gives rise to the only ST-homogeneous G\"odel-type solution with no violation of causality. Since the perfect-fluid and scalar field solutions are in the hyperbolic family, i.e. the essential parameter is positive $m^{2} > 0$, we further determine a general G\"odel-type solution with a combination of a scalar with an electromagnetic field plus a perfect fluid as matter source, which contains G\"odel-type solutions with $m=0$ and with $m^{2} < 0$, as well as the previous solutions as special cases. The bare existence of these G\"odel-type solutions makes apparent that hybrid metric-Palatini $f(\mathcal{R})$ gravity does not remedy causal anomaly in the form of closed timelike curves that are permitted in general relativity.