Partial Orderings of Curvature Invariants

TL;DR

This paper develops a hierarchy of curvature invariants in spacetimes by establishing inequalities among them, clarifying how higher-order invariants relate to lower-order ones across various spacetime types.

Contribution

It introduces a systematic set of inequalities among curvature invariants, including conditions bounding Zakhary--McIntosh invariants by the Kretschmann scalar in 4D spacetimes.

Findings

Established pointwise inequalities among curvature invariants.

Identified conditions bounding invariants by the Kretschmann scalar.

Clarified the algebraic control of higher-order invariants by lower-order ones.

Abstract

We establish a new set of pointwise inequalities that order curvature invariants across various Petrov and Segre types of spacetimes. In arbitrary spacetime dimension, we systematically analyze inequalities among contractions of the Ricci tensor. We further explore the conditions under which all Zakhary--McIntosh invariants in $(1+3)$-dimensional spacetimes are bounded above (up to appropriate powers) by the Kretschmann scalar. These results establish a practical hierarchy among curvature scalars and clarify the extent to which higher-order invariants are algebraically controlled by lower-order ones or vice versa.