[Paper Review] Proper Time Foliations of Lorentz Manifolds

D. H. Delphenich|ArXiv.org|Nov 20, 2002

Geometric Analysis and Curvature Flows20 references3 citations

TL;DR

This paper introduces and analyzes proper time foliations in Lorentzian manifolds, defining them as spatial hypersurfaces orthogonal to timelike geodesic congruences. It contrasts these with standard coordinate-based time foliations in Minkowski space and investigates the conditions under which such foliations can be globally extended to general Lorentz manifolds, particularly focusing on geodesic flow sections and integrability conditions.

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.

Motivation & Objective

• To define and formalize the concept of a proper time foliation in Lorentzian geometry, where leaves are spacelike hypersurfaces orthogonal to timelike geodesics.
• To contrast proper time foliations with standard coordinate-based time foliations in Minkowski spacetime, highlighting differences in geometric and physical interpretation.
• To investigate the conditions under which the proper time foliation of Minkowski space can be extended to more general Lorentz manifolds.
• To analyze the role of geodesic flows and their sections in constructing consistent proper time foliations on curved spacetimes.
• To examine the integrability and global existence of such foliations, particularly in relation to the geometry of the underlying Lorentz manifold.

Proposed method

• Uses differential geometry to define one-dimensional foliations and codimension-one spacelike hypersurfaces in Lorentzian manifolds.
• Applies the concept of a timelike congruence generated by geodesics to define proper time sections as level sets of a scalar time function.
• Compares the proper time foliation with the standard coordinate time foliation in Minkowski space, emphasizing the orthogonality of leaves to integral curves.
• Analyzes the integrability conditions for the existence of such foliations, particularly focusing on the vanishing of the twist of the geodesic congruence.
• Examines the extension of the foliation structure from flat Minkowski space to curved Lorentz manifolds via geometric and topological constraints.
• Employs geometric arguments and tensorial analysis to assess the consistency and global validity of proper time foliations under curvature and non-trivial topology.

Experimental results

Research questions

• RQ1 How can a proper time foliation be rigorously defined on a Lorentzian manifold, and what geometric conditions must be satisfied?
• RQ2 What distinguishes a proper time foliation from a coordinate time foliation in Minkowski spacetime?
• RQ3 Under what conditions can the proper time foliation of Minkowski space be extended to a general Lorentz manifold?
• RQ4 How do the properties of geodesic flows influence the construction and regularity of proper time foliations?
• RQ5 What are the topological and geometric obstructions to the global existence of proper time foliations on curved spacetimes?

Key findings

• Proper time foliations are defined as codimension-one spacelike hypersurfaces that are orthogonal to a timelike geodesic congruence, ensuring that the normal vector field is tangent to the integral curves.
• In Minkowski space, the proper time foliation differs from the standard coordinate time foliation due to the orthogonality condition, which ensures that time intervals are measured along geodesics.
• The existence of a proper time foliation on a general Lorentz manifold depends on the integrability of the timelike geodesic congruence, particularly the vanishing of the twist (vorticity) of the congruence.
• The paper establishes that the proper time foliation can be globally extended to a Lorentz manifold only if the geodesic congruence is hypersurface-orthogonal and the manifold admits a global timelike vector field with vanishing twist.
• The analysis shows that the foliation structure is preserved under certain curvature and topological conditions, but global existence is not guaranteed in arbitrary Lorentz manifolds.
• The results indicate that proper time foliations are most naturally defined in spacetimes with sufficient symmetry or trivial topology, such as Minkowski space or globally hyperbolic spacetimes.