Classification of ruled surfaces in Lorentz-Minkowski space that are stationary for the moment of inertia

In this paper, we study hypersurfaces in Lorentz-Minkowski space ${\mathbb{L}}^{n+1}$ that are stationary for the moment of inertia with respect to the origin. After giving examples and applications of the maximum principle, we classify, in ${\mathbb{L}}^3$, all stationary surfaces that are ruled surfaces. We prove that planes are the only cylindrical stationary surfaces. If the surface is not cylindrical, the classification depends on the causal character of the rulings. In addition, we provide explicit parametrizations of all ruled stationary surfaces.