Lorentz transformations in 1+1 dimensional spacetime: mainly the superluminal case

TL;DR

The paper presents a $1+1$-dimensional Lorentz transformation framework tailored to superluminal reference frames by introducing a relativistic $2$-velocity $U$ and a clockwork postulate to assign consistent temporal orientation to primed axes. It derives the $2$-velocity addition law, recovers the standard velocity addition in the subluminal limit, and analyzes length contraction and time dilation under this scheme, revealing nonstandard behavior for large $|u|$. It further discusses restricted transformations, the group property, and the reinterpretation principle, contrasting its approach with Parker–Dragan formulations and highlighting counterintuitive features. The work lays groundwork for extending the formalism to higher dimensions and for deeper discussions of causality in tachyonic kinematics.

Abstract

We discuss the most general form of the Lorentz transformation in 1+1 dimensional spacetime, focusing mainly on its superluminal branch. For this purpose, we introduce the 2-velocity of a reference frame and the clockwork postulate. Basic special relativity effects are discussed in the proposed framework. Different forms of the superluminal Lorentz transformation, which were studied in the literature, are critically examined from the perspective of our formalism. Counterintuitive features of the superluminal Lorentz transformation are identified both in our approach and in earlier studies.

Figures (2)

• Figure 1: The schematic illustration of four possibilities for orientation of the axes of the superluminal reference frames. The red arrows display $2$-velocities $U_{\alpha\beta}=(\alpha\gamma(u),\beta\gamma(u)|u|)$, where $\alpha,\beta=\pm$ and $|u|>1$. The light cones are plotted with blue dashed lines.
• Figure 2: The schematic plot illustrating the discussion of length "contraction" (see Sec. \ref{['length_sec']} for the definition of the events $A$, $B$, $C$, and $D$). The rod is depicted via the thick green line. Its endpoints are marked by blue and red dots (worldlines of the rod's ends are shown in the same colors). While both panels are prepared for the same $U^1/U^0=u>1$, $U^0$ is larger (smaller) than zero in the left (right) panel. The rod is simultaneously at rest in both primed reference frames. Its appearance in the unprimed reference frame does not depend on whether the primed reference frame moves forwards or backwards in time.