# Visualization of Thomas-Wigner rotations

## TL;DR

This paper visually illustrates how sequences of Lorentz boosts lead to Thomas-Wigner rotations, showing the relationship with simultaneity and event horizons, using a moving rigid object model.

## Contribution

It introduces a visual method to understand Thomas-Wigner rotations through a model of a rigid object undergoing multiple boosts, highlighting the role of acceleration and event horizons.

## Key findings

- At least five boosts are needed to return the object to its start position.
- The rotation angle depends solely on the maximum speed reached.
- Accelerated motion creates event horizons that bound the object.

## Abstract

It is well known that a sequence of two non-collinear pure Lorentz transformations (boosts) is not a boost again, but involves a spatial rotation, the Wigner or Thomas-Wigner rotation. The formation of this rotation is visually illustrated by moving a Born-rigid object on a closed trajectory in several sections. Within each section the boost's proper time duration is assumed to be the same and the object's centre accelerates uniformly. Born-rigidity implies that the stern of this object accelerates faster than its bow. It is shown that at least five boosts are required to return the object's centre to its start position. With these assumptions, the Thomas-Wigner rotation angle depends on a single parameter only, the maximum speed reached within each boost section. The visualization highlights the close relationship between the Thomas-Wigner rotation and the relativity of simultaneity. Furthermore, it is illustrated that accelerated motion implies the formation of an event horizon. The event horizons associated with the five boosts constitute a boundary to the rotated Born-rigid object and ensure its finite size.