Matter collineations of Spacetime Homogeneous G\"odel-type Metrics

The spacetime homogeneous G"odel-type spacetimes which have four classes of metrics are studied according to their matter collineations. The obtained results are compared with Killing vectors and Ricci collineations. It is found that these spacetimes have infinite number of matter collineations in degenerate case, i.e. det$(T_{ab})=0$, and do not admit proper matter collineations in non-degenerate case, i.e. det$(T_{ab})\mathrm{\ne }0$. The degenerate case has the new constraints on the parameters $m$ and $w$ which characterize the causality features of the G"odel-type spacetimes.