Two-dimensional Minkowski causal automorphisms and conformal maps

TL;DR

This paper characterizes causal automorphisms in two-dimensional Minkowski space using conformal maps and observer classes, providing insights into their structure, wave equation characterization, and applications like the twin paradox.

Contribution

It offers a novel characterization of 2D Minkowski causal automorphisms via conformal maps and observer classes, and addresses wave equation characterization questions.

Findings

Causal automorphisms are M"{a}rzke-Wheeler maps of certain observers.

Differentiable causal automorphisms are Minkowski conformal maps with specific restrictions.

A proper time formula for accelerated observers is derived, resolving the twin paradox.

Abstract

Treating the two-dimensional Minkowski space as a Wick rotated version of the complex plane, we characterize the causal automorphisms in two-dimensional Minkowski space as the M\"{a}rzke-Wheeler maps of a certain class of observers. We also characterize the differentiable causal automorphisms of this space as the Minkowski conformal maps whose restriction to the time axis belongs to the class of observers mentioned above. We answer a recently raised question about whether causal automorphisms are characterized by their wave equation. As another application of the theory, we give a proper time formula for accelerated observers which solves the twin paradox in two-dimensional Minkowski spacetime.