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Local regularity of spacetimes under Ricci curvature and Lie derivative conditions

Bing-Long Chen

TL;DR

The paper establishes a local regularity theory for 4D spacetimes under bounds on the Ricci curvature and the Lie derivative of the Lorentzian metric along a timelike field. By introducing an associated Riemannian metric $\hat{g}$ and developing a hybrid analytic-geometric framework (including harmonic coordinates, a groupoid approach, and a smoothing argument), it proves a $W^{1,p}$-type regularity result under scale-invariant bounds, with consequences for the timelike field $X$ and higher-regularity outcomes. A key contribution is the combination of Gromov-Hausdorff compactness, a local-to-global patching via groupoids, and a smoothing procedure that yields global curvature control, enabling intrinsic intrinsic curvature estimates and an intrinsic coordinate-free regularity theory. The work also generalizes the main theorem to normalized $L^{p}$-bounds, broadening applicability to relativistic fluids and other settings where natural timelike frames arise. Overall, it provides a robust regularity mechanism for Lorentzian manifolds under geometric-analytic constraints with potential impact on stability, limits, and the Cauchy problem in general relativity.

Abstract

In this paper, we derive a general regularity estimate for any 4-d spacetime, in terms of a priori bounds of the Ricci curvature and Lie derivative of the Lorentzian metric with respect to an arbitrary timelike vector field.

Local regularity of spacetimes under Ricci curvature and Lie derivative conditions

TL;DR

The paper establishes a local regularity theory for 4D spacetimes under bounds on the Ricci curvature and the Lie derivative of the Lorentzian metric along a timelike field. By introducing an associated Riemannian metric and developing a hybrid analytic-geometric framework (including harmonic coordinates, a groupoid approach, and a smoothing argument), it proves a -type regularity result under scale-invariant bounds, with consequences for the timelike field and higher-regularity outcomes. A key contribution is the combination of Gromov-Hausdorff compactness, a local-to-global patching via groupoids, and a smoothing procedure that yields global curvature control, enabling intrinsic intrinsic curvature estimates and an intrinsic coordinate-free regularity theory. The work also generalizes the main theorem to normalized -bounds, broadening applicability to relativistic fluids and other settings where natural timelike frames arise. Overall, it provides a robust regularity mechanism for Lorentzian manifolds under geometric-analytic constraints with potential impact on stability, limits, and the Cauchy problem in general relativity.

Abstract

In this paper, we derive a general regularity estimate for any 4-d spacetime, in terms of a priori bounds of the Ricci curvature and Lie derivative of the Lorentzian metric with respect to an arbitrary timelike vector field.

Paper Structure

This paper contains 9 sections, 13 theorems, 214 equations.

Table of Contents

  1. Introduction
  2. Gromov-Hausdorff convergence
  3. Einstein equations
  4. Groupoid approach
  5. $C^{\alpha}$-convergence
  6. Strong maximum principle
  7. A smoothing argument
  8. Harmonic radius
  9. A generalization of Theorem \ref{['t1.1']}

Key Result

Theorem 1.1

Let $(\bar{M}, \bar{g})$ be a four dimensional spacetime with a timelike vector field $X$. Suppose the closure of the $\hat{g}$-geodesic ball ${B}_{\hat{g}}(x_0, a)$ of radius $a>0$ is compact and where $\bar{R}ic$ is the Ricci curvature of $\bar{g}$, $\mathcal{L}_{X}\bar{g}$ is the Lie derivative, $|\cdot|_{\hat{g}}$ is the norm defined by the Riemannian metric $\hat{g}$. Then for any $\epsilon>

Theorems & Definitions (18)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Proposition 1.4
  • Definition 2.1
  • Remark 2.1
  • Claim 1
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 3.3
  • ...and 8 more