Timelike curves: homotopies and domain of determinacy

TL;DR

This work analyzes domains of determinacy for linear strictly hyperbolic operators $P$ on Lorentzian space-times, introducing $Z_{\mathcal{O}}$ and its endpoint-extended form $Z_A$ to characterize how solution data on $\mathcal{O}$ determines $Pu=0$-solutions on larger sets. A central contribution is the Timelike Homotopy Theorem, which shows that points reachable by iterated noncharacteristic nonspacelike deformations coincide with those reachable by iterated timelike homotopies, and, under unique continuation for noncharacteristic nonspacelike hypersurfaces, these sets sit inside the domain of determinacy $Z_{\mathcal{O}}$, providing a robust reachability criterion. The paper then leverages pseudo-Riemannian geometry, exponential maps, and arrival-time analysis to relate reachability to double-cone structures; for small double-cones the domain of determinacy matches the double-cone, and exact results are obtained for the D'Alembert operator. Through carefully constructed doldrums examples, the authors demonstrate that $Z_{\gamma(]a,b[)}$ can be strictly larger or smaller than the natural double-cone candidate, highlighting subtle propagation phenomena and observability implications in hyperbolic PDEs.

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 Hölmgren 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.

Figures (24)

• Figure 1: Deformation of a tube on the left and disk on the right. ${\mathcal{O}}$ is blue.
• Figure 2: A homotopy of timelike curves as in Definition \ref{['def: reachable point -\n homotopy']}. ${\mathcal{O}}$ is blue.
• Figure 3: Three double-cones in the Minkowski space.
• Figure 4: Sketch of a plane wave moving upward and interacting with a low speed region. The bluer the region, the smaller the wave speed.
• Figure 5: Example where $\mathcal{h}^1 \mathcal{O} \subsetneq \mathcal{h}^2 \mathcal{O}$. $\mathcal{O}$ is blue.

Theorems & Definitions (104)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Definition 1.5
• Theorem 1.6: John's Global Hölmgren
• Definition 1.7
• Remark 1.8
• Definition 1.9
• Definition 1.10