Causality and its violation in $$f(R,\mathcal {L}_m,\phi ,g^{\mu u } abla _\mu \phi abla _ u \phi )$$ gravity

Abstract A modified gravitational model whose action is given by an arbitrary function of the Ricci scalar, the matter Lagrangian density, a scalar field, and its kinetic term is investigated as an extension of the gravitational sector including an additional dynamical degree of freedom. Within this framework, the causal structure of rotating cosmological solutions is analyzed by considering Gödel and Gödel-type spacetimes as background geometries used as theoretical probes of the model consistency. Different matter sources are examined, including a perfect fluid and scalar-field configurations. It is found that the standard Gödel metric is not compatible with the scalar sector of the theory unless the model reduces to the General Relativity limit. In contrast, Gödel-type geometries admit a wider class of solutions whose causal properties depend on the model parameters and on the matter content. In particular, perfect-fluid sources may lead to either causal or noncausal configurations, whereas scalar-field configurations constrain the geometry to the causal limit, preventing the formation of closed timelike curves, highlighting that the scalar field plays a nontrivial dynamical role distinct from that of a cosmological constant.