Homogeneous Gödel-type solutions in hybrid metric-Palatini gravity

The hybrid metric-Palatini $$f({\mathscr {R}})$$ gravity is a recently devised approach to modified gravity in which it is added to the metric Ricci scalar R, in the Einstein–Hilbert Lagrangian, a function $$f({\mathscr {R}})$$ of Palatini curvature scalar $${\mathscr {R}}$$ , which is constructed from an independent connection. These hybrid metric-Palatini gravity theories provide an alternative way to explain the current accelerating expansion without a dark energy matter component. If gravitation is to be described by a hybrid metric-Palatini $$f({\mathscr {R}})$$ 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 Godel-type solutions, which necessarily leads to violation of causality. Here, to look further into the potentialities and difficulties of $$f({\mathscr {R}})$$ theories, we examine whether they admit Godel-type solutions for physically well-motivated matter source. We first show that under certain conditions on the matter sources the problem of finding out space-time homogeneous (ST-homogeneous) solutions in $$f({\mathscr {R}})$$ 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 ST-homogeneous Godel-type solution whose matter source is a combination of a scalar with an electromagnetic fields plus a perfect fluid. This general Godel-type solution contains special solutions in which the essential parameter $$m^2$$ can be $$m^{2} > 0$$ hyperbolic family, $$m=0$$ linear class, and $$m^{2} < 0$$ trigonometric family, covering thus all classes of homogeneous Godel-type spacetimes. This general solution also contains all previously known solutions as special cases. The bare existence of these Godel-type solutions makes apparent that hybrid metric-Palatini $$f({\mathscr {R}})$$ gravity does not remedy causal anomaly in the form of closed timelike curves that are permitted in general relativity.