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Partial Orderings of Curvature Invariants

Ivica Smolić

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.

Partial Orderings of Curvature Invariants

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 -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.
Paper Structure (10 sections, 6 theorems, 59 equations, 3 tables)

This paper contains 10 sections, 6 theorems, 59 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Elementary curvature invariants
  3. Inequalities between Ricci invariants
  4. Ideal fluid
  5. Electromagnetic field
  6. Spinors, ZM invariants and Petrov classification
  7. Spherically symmetric Petrov type D
  8. Discussion
  9. Ricci and Weyl components
  10. ZM invariants in spinor formalism

Key Result

Lemma 3.1

For any finite nonempty set of real numbers $\{x_1,\dots,x_D\}$ and $m,n,p,s \in \mathds{N}$, such that $1 \le 2n < p$ and $1 \le s < 2m$, the following inequalities hold

Theorems & Definitions (6)

  • Lemma 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Lemma 5.1
  • Theorem 5.2
  • Theorem 5.3