Weighing the curvature invariants

We prove several inequalities between the curvature invariants, which impose constraints on curvature singularities. Some of the inequalities hold for a family of spacetimes which include static, Friedmann–Lemaître–Robertson–Walker, and Bianchi type I metrics, independently of whether they are solutions of some particular field equations. In contrast, others hold for solutions of Einstein’s gravitational field equation and a family of energy–momentum tensors (featuring ideal fluids, scalar fields and nonlinear electromagnetic fields), independently of the specific form of the spacetime metric. We illustrate different behaviour of the basic curvature invariants with numerous examples and discuss the consequences and limitations of the proven results.