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Some inequalities among curvature invariants

Sebastian J. Szybka, Yaroslava Kravetska, Kornelia Nikiel

TL;DR

The paper addresses coordinate-free inequalities among curvature invariants of symmetric rank-2 tensors in Lorentzian spacetimes, focusing on Segre types $A1$, $A3$, and $B$. Using the Segre canonical forms and invariant traces $P_n$, it proves an infinite family of inequalities $P_s^{2m} \leq D^{2m-s} P_{2m}^s$ and connects these to known relations among Ricci invariants. A key result relates the second Ricci invariant to the Kretschmann scalar via $K = I_1 + \frac{4}{D-2} R_2 - \frac{2}{(D-1)(D-2)} R_1^2$, yielding $2 R_2 \le (D-1) K$ when $I_1 \ge 0$ and highlighting the special case $R_1^2 \le D R_2$ for $s=m=1$. The findings extend prior work (WCI) to broader Segre classes, illustrate the role of algebraic type in inequality validity, and provide practical tools and illustrative examples (e.g., Vaidya, Schmidt) for spacetime analysis.

Abstract

We prove an infinite sequence of inequalities among scalar polynomial invariants of symmetric rank-2 tensors of Segre type A1, A3, and B. In particular, these inequalities apply to the Ricci tensor and the energy-momentum tensor. We use one of them to generalize the known relation between the second Ricci invariant and the Kretschmann scalar.

Some inequalities among curvature invariants

TL;DR

The paper addresses coordinate-free inequalities among curvature invariants of symmetric rank-2 tensors in Lorentzian spacetimes, focusing on Segre types , , and . Using the Segre canonical forms and invariant traces , it proves an infinite family of inequalities and connects these to known relations among Ricci invariants. A key result relates the second Ricci invariant to the Kretschmann scalar via , yielding when and highlighting the special case for . The findings extend prior work (WCI) to broader Segre classes, illustrate the role of algebraic type in inequality validity, and provide practical tools and illustrative examples (e.g., Vaidya, Schmidt) for spacetime analysis.

Abstract

We prove an infinite sequence of inequalities among scalar polynomial invariants of symmetric rank-2 tensors of Segre type A1, A3, and B. In particular, these inequalities apply to the Ricci tensor and the energy-momentum tensor. We use one of them to generalize the known relation between the second Ricci invariant and the Kretschmann scalar.

Paper Structure

This paper contains 5 sections, 3 theorems, 20 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Segre classification
  3. Invariants
  4. Examples and discussion
  5. Conclusions

Key Result

Lemma 1

Let $\bm{ P'}$ be related to a symmetric rank-2 tensor $\bm P$ at $p\in M$ by with $\alpha,\beta\in \mathbb{R}$ and $\alpha\neq 0$. Then $\bm{P'}$ is of the same Segre type as $\bm P$.

Figures (1)

  • Figure 1: Regions where the inequalities $R_s^{2m}\leq D^{2m-s} R_{2m}^s$ and $2 R_2\leq (D-1)K$ are violated for the Schmidt spacetime. The bold black lines correspond to $y\,z=0$, where the Ricci tensor is of Segre type $A1$ and Theorem \ref{['thm1']} holds.

Theorems & Definitions (6)

  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof