Proper Time Foliations of Lorentz Manifolds

TL;DR

This paper explores the mathematical structure of proper time foliations in Lorentz manifolds, comparing them with standard coordinate-based foliations, and discusses their extension and properties in the context of Minkowski space and general Lorentz manifolds.

Contribution

It introduces formal definitions of proper time foliations, contrasts them with coordinate-based foliations, and analyzes their extension in Lorentz manifolds, advancing the mathematical understanding of spacetime foliation structures.

Findings

Proper time foliation of Minkowski space is defined and characterized.

Comparison between proper time foliation and coordinate-based foliation.

Discussion on extending proper time foliations to general Lorentz manifolds.

Abstract

Some standard definitions and results concerning foliations of dimension one and codimension one are introduced. A proper time foliation of Minkowski space is defined and contrasted with the foliation that is defined by the time coordinate. The extent to which a Lorentz structure on a manifold defines foliations, and the issues concerning the extension of the proper time foliation of Minkowski space to a Lorentz manifold are discussed, such as proper time sections of a geodesic flow.