Conformal collineations of the Ricci and energy–momentum tensors in static plane symmetric space–times

We provide a complete classification of static plane symmetric space–times according to conformal Ricci collineations {(}CRCs{\rm)} and conformal matter collineations (CMCs) in both the degenerate and nondegenerate cases. In the case of a nondegenerate Ricci tensor, we find a general form of the vector field generating CRCs in terms of unknown functions of $t$ and $x$ subject to some integrability conditions. We then solve the integrability conditions in different cases depending upon the nature of the Ricci tensor and conclude that the static plane symmetric space–times have a $7$-, $10$- or $15$-dimensional Lie algebra of CRCs. Moreover, we find that these space–times admit an infinite number of CRCs if the Ricci tensor is degenerate. We use a similar procedure to study CMCs in the case of a degenerate or nondegenerate matter tensor. We obtain the exact form of some static plane symmetric space–time metrics that admit nontrivial CRCs and CMCs. Finally, we present some physical applications of our obtained results by considering a perfect fluid as a source of the energy–momentum tensor.