Dynamics in Non-Globally-Hyperbolic Static Spacetimes II: General Analysis of Prescriptions for Dynamics

TL;DR

This paper proves that in static, non-globally-hyperbolic spacetimes, the only acceptable way to define scalar field dynamics is via positive self-adjoint extensions of the spatial operator, ensuring uniqueness under certain physical conditions.

Contribution

It establishes the uniqueness of the prescription for defining scalar field dynamics in such spacetimes, ruling out alternative methods under specified criteria.

Findings

The dynamics must correspond to a positive self-adjoint extension of the spatial operator.

Other prescriptions do not satisfy the local, energy, and additional conditions.

The result constrains boundary conditions in anti-de Sitter spacetime for well-defined dynamics.

Abstract

It was previously shown by one of us that in any static, non-globally-hyperbolic, spacetime it is always possible to define a sensible dynamics for a Klein-Gordon scalar field. The prescription proposed for doing so involved viewing the spatial derivative part, $A$, of the wave operator as an operator on a certain $L^2$ Hilbert space $\mathcal H$ and then defining a positive, self-adjoint operator on $\mathcal H$ by taking the Friedrichs extension (or other positive extension) of $A$. However, this analysis left open the possibility that there could be other inequivalent prescriptions of a completely different nature that might also yield satisfactory definitions of the dynamics of a scalar field. We show here that this is not the case. Specifically, we show that if the dynamics agrees locally with the dynamics defined by the wave equation, if it admits a suitable conserved energy, and if it satisfies certain other specified conditions, then it must correspond to the dynamics defined by choosing some positive, self-adjoint extension of $A$ on $\mathcal H$. Thus, subject to our requirements, the previously given prescription is the only possible way of defining the dynamics of a scalar field in a static, non-globally-hyperbolic, spacetime. In a subsequent paper, this result will be applied to the analysis of scalar, electromagnetic, and gravitational perturbations of anti-de Sitter spacetime. By doing so, we will determine all possible choices of boundary conditions at infinity in anti-de Sitter spacetime that give rise to sensible dynamics.