  In this paper, a modification of general relativity is considered. It
consists of generalizing the Lagrangian of matter in a non-linear way, that is,
replacing the curvature scalar $R$ by a function $f(R,T_{\mu\nu} T^{\mu\nu} )$,
where $T_{\mu\nu}$ is the energy-momentum tensor. The main objective is to
investigate the issue of causality in this gravitational model. To study the
causality and/or its violation the G\"{o}del-type solutions are used. For such
development, different matter contents are chosen. A critical radius, beyond
which causality is violated, is calculated. It is shown that both causal and
non-causal solutions are allowed.
