Lorentz-conformal transformations in the plane

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While conformal transformations of the plane preserve Laplace’s equation, Lorentz-conformal mappings preserve the wave equation. This property gives rise to a rich Lorentzian geometry in dimension 1+1. In elucidating the geometry of the Lorentzian plane, this work provides a window into the field of pseudo-Riemannian geometry, where the Lorentzian plane is of fundamental importance. In the Lorentzian plane, curvilinear quadrilaterals and pairs of crossing curves are transformed under nonlinear Lorentz-conformal mappings via geometric constructions that can be expressed also by functional formulas. Classes of Lorentz-conformal maps are characterized by their symmetries under subgroups of the dihedral group of order eight, and 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 simple form of nonlinear Lorentz-conformal transformations in dimension 1+1, provided by characteristic coordinates.