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
Summary
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.
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