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Equivalence of the null energy condition to variable lower bounds on the timelike Ricci curvature for $C^2$-Lorentzian metrics

Melanie Graf, Yaver Gulusoy

Abstract

The null energy or null convergence condition (NEC) is one of the fundamental assumptions necessary for many celebrated results from Lorentzian Geometry and Mathematical General Relativity. As such there have been several recent efforts to find a good generalization of this condition to the new setting of Lorentzian length spaces or metric measure spacetimes. One important property any such generalization should fulfill is consistency with the classical formulation for a class of spacetimes as large as possible. The purpose of this note is to show that the recent reformulation of the NEC by McCann as variable lower timelike Ricci curvature bounds (arXiv:2304.14341) remains equivalent to the classical NEC not just for smooth but even for $C^2$-metrics, where McCann's original proof needs to be modified.

Equivalence of the null energy condition to variable lower bounds on the timelike Ricci curvature for $C^2$-Lorentzian metrics

Abstract

The null energy or null convergence condition (NEC) is one of the fundamental assumptions necessary for many celebrated results from Lorentzian Geometry and Mathematical General Relativity. As such there have been several recent efforts to find a good generalization of this condition to the new setting of Lorentzian length spaces or metric measure spacetimes. One important property any such generalization should fulfill is consistency with the classical formulation for a class of spacetimes as large as possible. The purpose of this note is to show that the recent reformulation of the NEC by McCann as variable lower timelike Ricci curvature bounds (arXiv:2304.14341) remains equivalent to the classical NEC not just for smooth but even for -metrics, where McCann's original proof needs to be modified.
Paper Structure (4 sections, 5 theorems, 41 equations, 1 figure)

This paper contains 4 sections, 5 theorems, 41 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Equivalence of Definitions \ref{['def:classicNEC']} and \ref{['def:RobertNEC']} for C2-Lorentzian metrics
  3. Discussion of implications for the equivalence to the synthetic NEC of mccann2024synthetic
  4. Properties of the flow map Phi

Key Result

Theorem 2.1

Let $M$ be a smooth manifold with a $C^0$ semi-Riemannian metric $g$. Let $F$ be a $(0,2)$-tensor field of regularity $C^{0}$ satisfying the null energy condition Then, for each compact subset $Z \subseteq M$ there is a constant $C_Z \in \mathbb{R}$ such that for all $\mathbf{v} \in TZ:=TM|_Z$.

Figures (1)

  • Figure 1: Approaching $\left(p_{\infty},v_{\infty}\right)$ along the null projections $(p_{m},\hat{v}_{m})$ at $S_0$ (for illustrative purposes $p_{m} = p_{\infty}$).

Theorems & Definitions (14)

  • Definition 1: Classical null energy condition
  • Definition 2: Variable timelike curvature condition
  • Theorem 2.1
  • proof : Proof of Theorem \ref{['thm:null-bounds']}.
  • Corollary 2.2
  • Remark 2.3: Weighted null energy vs weighted Ricci bounds
  • Definition 3.1: Definition 29 in mccann2024synthetic
  • Theorem 3.2: Consistency with the null energy condition
  • proof
  • Remark 3.3
  • ...and 4 more