The Cauchy problem on spacetimes that are not globally hyperbolic

TL;DR

This paper explores the initial value problem on non-globally hyperbolic spacetimes, including those with closed timelike curves, and discusses conditions for existence and uniqueness of solutions.

Contribution

It broadens the class of spacetimes where the initial value problem is well-posed, including non-orientable and certain causally complex spacetimes, and suggests criteria for uniqueness.

Findings

Initial value problem well-defined on broader spacetimes

Examples with closed timelike curves admitting solutions

Uniqueness depends on the causal structure and confinement of closed timelike curves

Abstract

The initial value problem is well-defined on a class of spacetimes broader than the globally hyperbolic geometries for which existence and uniqueness theorems are traditionally proved. Simple examples are the time-nonorientable spacetimes whose orientable double cover is globally hyperbolic. These spacetimes have generalized Cauchy surfaces on which smooth initial data sets yield unique solutions. A more difficult problem is to characterize the class of spacetimes with closed timelike curves that admit a well-posed initial value problem. Examples of spacetimes with closed timelike curves are given for which smooth initial data at past null infinity has been recently shown to yield solutions. These solutions appear to be unique, and uniquesness has been proved in particular cases. Other examples, however, show that confining closed timelike curves to compact regions is not sufficient to guarantee uniqueness. An approach to the characterization problem is suggested by the behavior of congruences of null rays. Interacting fields have not yet been studied, but particle models suggest that uniqueness (and possibly existence) is likely to be lost as the strength of the interaction increases.