Lorentz-Conformal Transformations in the Plane

TL;DR

This paper explores Lorentz-conformal transformations in the plane, revealing how they map geometric objects like quadrilaterals and curves, with a focus on their properties, symmetries, and explicit functional descriptions.

Contribution

It characterizes Lorentz-conformal maps using symmetries, describes how simple shapes transform, and introduces a geometric 'rectangle rule' for quadrilaterals.

Findings

Squares transform into curvilinear quadrilaterals with a geometric rule.

Explicit functional degrees of freedom exist for mappings.

Symmetry groups classify Lorentz-conformal maps.

Abstract

While conformal transformations of the plane preserve Laplace's equation, Lorentz-conformal mappings preserve the wave equation. We discover how simple geometric objects, such as quadrilaterals and pairs of crossing curves, are transformed under nonlinear Lorentz-conformal mappings. Squares are transformed into curvilinear quadrilaterals where three sides determine the fourth by a geometric "rectangle rule," which can be expressed also by functional formulas. There is an explicit functional degree of freedom in choosing the mapping taking the square to a given quadrilateral. We characterize classes of Lorentz-conformal maps by their symmetries under subgroups of the dihedral group of order eight. Unfoldings of non-invertible mappings into invertible ones are reflected in a change of the symmetry group. The questions are simple; but the answers are not obvious, yet have beautiful geometric, algebraic, and functional descriptions and proofs. This is due to the very simple form of nonlinear Lorentz-conformal transformations in dimension 1+1, provided by characteristic coordinates.