Dynamics in Stationary, Non-Globally Hyperbolic Spacetimes

TL;DR

This paper extends a method for defining dynamics in static spacetimes to stationary spacetimes, ensuring well-posed evolution and smooth solutions, with implications for field quantization in non-globally hyperbolic settings.

Contribution

It introduces a first order operator formalism for stationary spacetimes, generalizing previous static spacetime results, and maintains key properties like causality and smoothness.

Findings

Formalism preserves Cauchy evolution within the domain of dependence.

Smooth, compactly supported data lead to smooth solutions.

Application to field quantization in non-globally hyperbolic spacetimes.

Abstract

Classically, the dynamics in a non-globally hyperbolic spacetime is ill posed. Previously, a prescription was given for defining dynamics in static spacetimes in terms of a second order operator acting on a Hilbert space defined on static slices. The present work extends this result by giving a similar prescription for defining dynamics in stationary spacetimes obeying certain mild assumptions. The prescription is defined in terms of a first order operator acting on a different Hilbert space from the one used in the static prescription. It preserves the important properties of the earlier one: the formal solution agrees with the Cauchy evolution within the domain of dependence, and smooth data of compact support always give rise to smooth solutions. In the static case, the first order formalism agrees with second order formalism (using specifically the Friedrichs extension). Applications to field quantization are also discussed.