We consider some aspects of the geometry of surfaces of revolution in three-dimensional Minkowski space. First, we show that Clairaut's theorem, which gives a well-known characterization of geodesics on a surface of revolution in Euclidean space, has an analogous result in three-dimensional Minkowski space. We then illustrate the significant differences between the two cases which arise in spite of their formal similarity.
Anis Saad. Robert J. Low. "Clairaut's Theorem in Minkowski Space." J. Geom. Symmetry Phys. 28 105 - 112, 2012. https://doi.org/10.7546/jgsp-28-2012-105-112