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 well-motivated matter source. 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 Einstein's field equations with a cosmological constant. Employing
this far-reaching result, we determine a general G\"odel-type whose matter
source is a combination of a scalar with an electromagnetic field plus a
perfect fluid. This general G\"odel-type solution contains special solutions in
which the essential parameter $m^2$ can be $m^{2} > 0$, $m=0$, and $m^{2} < 0$,
covering thus all classes of homogeneous G\"odel-type spacetimes. This general
solution also contains all previously known solution as special cases. The bare
existence of these G\"odel-type solutions makes apparent that hybrid
metric-Palatini gravity does not remedy causal anomaly in the form of closed
timelike curves that are permitted in general relativity.
