[Paper Review] A note on stability and Cauchy time functions

Miguel Sánchez|arXiv (Cornell University)|Apr 21, 2013

Advanced Differential Geometry Research16 references3 citations

TL;DR

This paper investigates the stability of global hyperbolicity in spacetimes by analyzing Cauchy temporal functions under metric perturbations. It proves that any Cauchy temporal function for a globally hyperbolic spacetime remains Cauchy temporal under sufficiently small metric variations, thereby establishing the stability of global hyperbolicity under such deformations.

ABSTRACT

Since the solution of the so-called folk problems of smoothability, there has been a special interest in the properties of classical time and volume functions of spacetimes. Here we supply some information that complements the one provided in arXiv:1108.5120v3 and arXiv:1301.2909v1, and discuss some results. As an illustration, we show that any Cauchy temporal function for a globally hyperbolic spacetime remains Cauchy temporal for close metrics ---which, in particular, implies stability for global hyperbolicity.

Motivation & Objective

• To address gaps in the understanding of time and volume functions in spacetimes, particularly in relation to global hyperbolicity.
• To clarify the behavior of Cauchy temporal functions under metric variations, especially in the context of stability.
• To provide complementary results to prior works on smoothability and global hyperbolicity in Lorentzian geometry.
• To establish that global hyperbolicity is stable under small metric perturbations via the persistence of Cauchy temporal functions.

Proposed method

• Analyzing the properties of Cauchy temporal functions in globally hyperbolic spacetimes using differential geometry and Lorentzian metric theory.
• Employing perturbation techniques to examine how small changes in the spacetime metric affect the existence and behavior of Cauchy temporal functions.
• Applying results from prior works (arXiv:1108.5120v3 and arXiv:1301.2909v1) to derive stability conclusions.
• Using topological and analytical arguments to show that the causal structure remains intact under small metric deformations.
• Establishing that the level sets of a Cauchy temporal function remain spacelike and acausal under small metric variations.
• Leveraging the fact that a Cauchy temporal function's level sets foliate the spacetime and are compact in the appropriate topology.

Experimental results

Research questions

• RQ1 Under what conditions does a Cauchy temporal function remain Cauchy temporal when the spacetime metric is perturbed?
• RQ2 How does the causal structure of a globally hyperbolic spacetime behave under small metric variations?
• RQ3 What is the relationship between the stability of Cauchy temporal functions and the stability of global hyperbolicity?
• RQ4 Can the persistence of Cauchy temporal functions under metric perturbations be used to infer broader stability results for spacetime structure?

Key findings

• Any Cauchy temporal function for a globally hyperbolic spacetime remains Cauchy temporal under sufficiently small metric perturbations.
• The stability of Cauchy temporal functions under metric variation implies the stability of global hyperbolicity for the spacetime.
• The result confirms that global hyperbolicity is robust under small changes in the Lorentzian metric.
• The persistence of Cauchy temporal functions under perturbations is a direct consequence of the topological and causal structure being preserved under small metric deformations.
• The analysis provides a geometric explanation for the stability of global hyperbolicity based on the behavior of time functions.