# Regularity for Lorentz metrics under curvature bounds

@article{Anderson2002RegularityFL,
title={Regularity for Lorentz metrics under curvature bounds},
author={Michael T. Anderson},
journal={Journal of Mathematical Physics},
year={2002},
volume={44},
pages={2994-3012}
}
Let (M, g) be an (n+1)-dimensional space–time, with bounded curvature, with respect to a bounded framing. If (M, g) is vacuum, or satisfies a weak condition on the stress-energy tensor, then it is shown that (M, g) locally admits coordinate systems in which the Lorentz metric g is well-controlled in the (space–time) Sobolev space L2,p, for any p<∞. This result is essentially optimal. The result allows one to control the regularity of limits of sequences of space–times, with uniformly bounded…
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## References

SHOWING 1-10 OF 24 REFERENCES

This paper surveys aspects of the convergence and degeneration of Riemannian metrics on a given manifold M, and some recent applications of this theory to general relativity. The basic point of view
• Mathematics
• 1982
In this paper, we prove a-priori estimates for harmonic mappings between Riemannian manifolds which solve a Dirichlet problem. These estimates employ geometrical methods and depend only on geometric
• Mathematics
• 1981
Introduction - Riemannian themes in Lorentzian geometry connections and curvature Lorentzian manifolds and causality Lorentzian distance examples of space-times completness and extendibility
The local extendibility of causal geodesically incomplete space‐times is examined. It is shown that for a space‐time including an incomplete inextendible causal geodesic curve γ there exists a
This book develops three related tools that are useful in the analysis of partial differential equations, arising from the classical study of singular integral operators: pseudodifferential
• Physics
• 1973
The theory of the causal structure of a general space-time is developed, and is used to study black holes and to prove a number of theorems establishing the inevitability of singualarities under certain conditions.
A singularity reached on a timelike curve in a globally hyperbolic space-time must be a point at which the Riemann tensor becomes infinite (as a curvature or intermediate singularity) or is of typeD
• Mathematics
• 1997
We study in this chapter a class of partial differential equations that generalize and are to a large extent represented by Laplace’s equation. These are the elliptic partial differential equations
1. Introductory 2. The Riemann tensor 3. Boundary constructions 4. Existence theory and differentiability 5. The analytic extension problem 6. Attributes of singularities 7. Extension theories.
• Mathematics
• 1972
7 Scalar and Vector Ultra-Distributions.- 1. Scalar-Valued Functions of Class Mk.- 1.1 The Sequences {Mk}.- 1.2 The Space $${D_{{M_k}}}\left( H \right)$$.- 1.3 The Spaces {D_{{M_k}}}\left( H