Weighing the curvature invariants

TL;DR

This paper establishes inequalities between curvature invariants that constrain singularities, applicable to various spacetime families and solutions of Einstein's equations, with implications for understanding gravitational singularities.

Contribution

It introduces new inequalities between curvature invariants that apply broadly to different spacetime families and solutions of Einstein's equations, enhancing understanding of curvature singularities.

Findings

Certain inequalities hold for a wide class of spacetimes regardless of field equations.

Other inequalities are valid specifically for Einstein solutions with various matter fields.

Examples demonstrate the different behaviors of curvature invariants and the scope of the inequalities.

Abstract

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--Lema\^itre--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.