Attractors in spacetimes and time functions

TL;DR

This paper introduces a novel method for establishing the existence of time functions in Lorentzian manifolds by leveraging dynamical systems theory, specifically Conley's work on Lyapunov functions, to generalize Hawking's result.

Contribution

It presents a new approach using dynamical systems concepts to prove the existence of time functions, extending previous results to a broader class of spacetimes.

Findings

Recovered Hawking's result for stably causal spacetimes

Established a general method for constructing time functions

Linked attractor concepts with spacetime causality properties

Abstract

We develop a new approach to the existence of time functions on Lorentzian manifolds, based on Conley's work regarding Lyapunov functions for dynamical systems. We recover Hawking's result that a stably causal admits a time function through a more general result giving the existence of a continuous function that is non decreasing along all future directed causal curves, and increasing along such curves that lie outside a special region of the spacetime, called the chain recurrent set, which is empty for stably causal spacetimes. The construction is based on a notion of attractive sets in spacetimes.