A study on causality in $f(R,\varphi ,X)$ theory

The $k$-essence modified $f(R)$ gravity model, i.e., $f(R,\varphi ,X)$ theory is studied. The question of violation of causality, in the framework of G"{o}del-type universes, is investigated in this gravitational model. Causal and non-causal solutions are allowed. A critical radius for non-causal solution is calculated. It is shown that the violation of causality depends on the content of matter.