Timelike curves: homotopies and domain of determinacy

TL;DR

This paper investigates the domains of determination for solutions to hyperbolic PDEs, establishing connections between deformations of hypersurfaces, homotopies of timelike arcs, and influence curves, with applications to D'Alembert's equation.

Contribution

It introduces a novel timelike homotopy criterion linking hypersurface deformations and homotopies of timelike arcs for determining solution domains.

Findings

Points reachable by noncharacteristic hypersurface deformations match those by timelike arc homotopies.

The future-intersect-past set is an exact domain of determination for D'Alembert's equation.

Examples show the domain of determination can be larger or smaller than the future-intersect-past candidate.

Abstract

This paper studies domains of determination of linear strictly hyperbolic second order operators $P$. For an open set $\mathcal O$, a set $Z$ is a domain of determination when the values of solutions of the differential equation $Pu=0$ are determined on $Z$ by their values in $\mathcal O$. Fritz John's global H\"olmgren theorem implies that points that can be reached by deformations of noncharacteristic hypersufaces with initial surface and boundaries in $\mathcal O$ belong to a domain of determination provided that local uniqueness holds at noncharacteristic surfaces. Using spacelike hypersurfaces yields sharp finite speed results whose domains of determination are described in terms of influence curves that never exceed the local speed of propagation. This paper studies deformations of noncharacteristic nonspacelike hypersurfaces. We prove that points reachable by (repeated) deformations by noncharacteristic nonspacelike hypersurfaces coincide exactly with the set of points reachable by (repeated) homotopies of timelike arcs whose initial curves and endpoints belong to $\mathcal O$. When the set $\mathcal O$ is a small neighborhood of a forward timelike arc connecting $a$ to $b$, a natural candidate for $Z$ is the intesection of the future of $a$ with the past of $b$. This candidate is exact for D'Alembert's equation. We prove that it is also exact when $a,b$ are points close together on a fixed timelike arc. The timelike homotopy criterion fuels the construction of surprising examples for which the domain of determination is strictly larger (resp. strictly smaller) than the future-intersect-past candidate.