*It is well known that the Lorentz transformations lead to Einstein’s velocity addition equation. Rybczyk however, does not consider a transformation, but does have an alternative velocity addition equation. In this post, we shall derive the transformations that are associated with the Rybczyk velocity addition equation.*

Consider two systems of reference, denoted as and , the velocity of with respect to is denoted as . A particular body is in motion and the velocity of body with respect to is denoted as and the velocity of body with respect to is denoted as . The Rybczyk addition equation can be written as

(1) (Rybczyk)

that can be found at (Millennium Relativity Velocity Composition, © 2002 Joseph A. Rybczyk). We ask the question, what is the co-ordinate transformation that is associated with the Rybczyk velocity addition? Such transformations I call Rybczyk transformations, named after Joseph A. Rybczyk, who proposed the Rybczyk velocity addition. The general linear co-ordinate transformation from to can be written as

(2)

where is defined for and is defined for . For the body we can write but also , thus

(3)

that is rewritten as

(4)

This has the form and this is true for any arbitrary value of (within a certain domain), thus , and . Therefore we obtain

(5)

The Rybczyk transformation can be written as

(6)

A quick verification gives

(7)

thus which is the Rybczyk addition equation. The Rybczyk transformation can also be written in the matrix-form as

(7)

It is clear that this has a different form then the Lorentz transformations.

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