In this article, firstly, the isomorphism between the subset of the tangent bundle of Lorentzian unit sphere, T (M) over tilde, and Lorentzian unit sphere, S-1(2) is represented. Secondly, the isomorphism between the subset of hyperbolic unit sphere, T (M) over cap, and hyperbolic unit sphere, H-2 is given. According to E. Study mapping, any curve on S-1(2) or H-2 corresponds to a ruled surface in R-1(3). By constructing these isomorphisms, we correspond to any natural lift curve on T (M) over tilde or T (M) over cap a unique ruled surface in R-1(3). Then we calculate striction curve, shape operator, Gaussian curvature and mean curvature of these ruled surfaces. We give developability condition of these ruled surfaces. Finally, we give examples to support the main results.