Gödel-type solutions within the $f(R,Q)$ gravity

In this paper, we deal with the $f(R,Q)$ gravity whose action depends, besides of the scalar curvature $R$, on the higher-derivative invariant $Q=R_{\mu \nu }{R}^{\mu \nu }$. In order to compare this theory with the usual general relativity (GR), we verify the consistency of Gödel-type solutions within the $f(R,Q)$ gravity and discuss the related causality issues are discussed. Explicitly, we show that in the $f(R,Q)$ gravity there are new Gödel-type completely causal solutions having no analogue in the general relativity. In particular, a remarkable Gödel-type solution corresponding to the conformally flat space and maximally symmetric for physically well-motivated matter sources, with no necessity of cosmological constant, has been considered. We demonstrate that, in contrast to GR framework, $f(R,Q)$ gravity supports new vacuum solutions with the requirement for the cosmological constant to be nonzero. Finally, causal solutions are obtained for a particular choice $f(R,Q)=R+\alpha R^2+\beta Q$.