# The isometry group of $n$-dimensional Einstein gyrogroup

## TL;DR

This paper characterizes the full group of distance-preserving transformations (isometries) of the space of relativistic velocities under Einstein addition, revealing its structure and composition law.

## Contribution

It provides a complete description of the isometry group of the Einstein gyrogroup equipped with the rapidity metric, a novel characterization in relativistic velocity space.

## Key findings

- The isometry group of the Einstein gyrogroup is fully described.
- The composition law of the isometry group is explicitly determined.
- The structure of the isometry group relates to the symmetries of relativistic velocity space.

## Abstract

The space of $n$-dimensional relativistic velocities normalized to $c = 1$, $$\mathbb{B} = \{\mathbf{v}\in\mathbb{R}^n\colon \|\mathbf{v}\| < 1\},$$ is naturally associated with Einstein velocity addition $\oplus_E$, which induces the rapidity metric $d_E$ on $\mathbb{B}$ given by $d_E(\mathbf{u}, \mathbf{v}) = \tanh^{-1}\|-\mathbf{u}\oplus_E\mathbf{v}\|$. This metric is also known as the Cayley-Klein metric. We give a complete description of the isometry group of $(\mathbb{B}, d_E)$, along with its composition law.