Abstract

In this paper, we define a gyrogeometric mean on the Einstein gyrovector space. It satisfies several properties one would expect for means. For example, it is permutation-invariant and left-translation invariant. It is already known that the Einstein gyrogroup is a gyrocommutative gyrogroup. We give an alternative proof which depends only on an elementary calculation.

Highlights

  • Einstein addition is a binary operation that stems from his velocity composition law of relativistic admissible velocities

  • Ungar initiated the study of gyrogroups and gyrovector spaces [1] associated with the Einstein addition in the theory of special relativity

  • We propose an alternative definition of the mean of three or more elements, the gyrogeometric mean, and show that it has several properties one would expect for means

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Summary

Introduction

Einstein addition is a binary operation that stems from his velocity composition law of relativistic admissible velocities. Ungar initiated the study of gyrogroups and gyrovector spaces [1] associated with the Einstein addition in the theory of special relativity. The Einstein addition ⊕ E on Vs is a binary operation on Vs given by the equation u ⊕E v =. By the definition of ⊕ E , u ⊕ E v ∈ Vs for every pair u, v ∈ Vs by Theorem 3.46 and the identity (3.189) in [1]. It is symmetric in the sense that permutation-invariant by the definition of the gyrogeometric mean The gyrometric needs not be a metric To begin with the proof of the inequality (2), we show an equation related to the gyrometric and gamma factor.

The Gyroconvex Set and the Gyroconvex Hull in a Gyrovector Space
The Gyrogeometric Mean
Properties of the Gyrogeometric Mean
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