Initial Value Problems and Signature Change

TL;DR

This paper rigorously analyzes the initial value problem for the Klein-Gordon equation on a 2D signature-changing spacetime, revealing instability and divergence issues through operator theory techniques.

Contribution

It introduces a method to determine boundary conditions at signature change surfaces via self-adjoint extensions of the Hamiltonian, highlighting the ill-posedness of the problem.

Findings

Solutions are unstable and diverge in norm after finite time.

Boundary conditions at signature change are characterized by self-adjoint extensions.

The initial value problem is ill-posed in the studied setting.

Abstract

We make a rigorous study of classical field equations on a 2-dimensional signature changing spacetime using the techniques of operator theory. Boundary conditions at the surface of signature change are determined by forming self-adjoint extensions of the Schr\"odinger Hamiltonian. We show that the initial value problem for the Klein--Gordon equation on this spacetime is ill-posed in the sense that its solutions are unstable. Furthermore, if the initial data is smooth and compactly supported away from the surface of signature change, the solution has divergent $L^2$-norm after finite time.