Wave-Type PDE

The split-complex Cauchy-Riemann equations yield the hyperbolic wave equation. The PDE is hyperbolic (not elliptic), with two families of real characteristics.

Wave-Type PDE

Summary

The split-complex Cauchy-Riemann equations yield the hyperbolic wave equation. The PDE is hyperbolic (not elliptic), with two families of real characteristics.

Statement

%
\label{def:wave-type-pde}
The split-complex Cauchy--Riemann equations
on~$H_\tau$ yield the \textbf{wave equation}:
\[
    \boxed{%
    \frac{\partial^2 u}{\partial x^2}
    \;-\;
    \frac{\partial^2 u}{\partial y^2}
    \;=\; 0.}
\]
The sign between the second-order terms is \textbf{negative}.
This PDE is \textbf{hyperbolic}, not elliptic.
The split-complex holomorphic condition
$\partial f / \partial x + \jj \cdot \partial f / \partial y = 0$
is equivalent to the wave equation for each component.

Proof / Justification

This item is definitional. No manuscript proof is required.

Source Context

• Registry source: book-02.jsonl line 55
• Manuscript source: 2nd-edition/book-ii-categorical-holomorphy/02_mainmatter/part04/ch21-wave-causal.tex lines 141-158

Lean / Formalization Notes

• Formalization: formalized
• Module: TauLib.BookII.Geometry.CausalStructure
• Name: CausalClass

Dependencies

• Canonical: I.D20, I.T10

Related Results

Generated by later projection phases.

Related Publications

Generated by later projection phases.

Revision Notes

• 2026-04-24: Initial pilot migration.