## Rybczyk transformations

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 $\mathcal{S}$ and $\mathcal{S}'$, the velocity of $\mathcal{S}'$ with respect to $\mathcal{S}$ is denoted as $v$. A particular body $\mathcal{B}$ is in motion and the velocity of body $\mathcal{B}$ with respect to $\mathcal{S}$ is denoted as $u_\mathcal{B}$ and the velocity of body $\mathcal{B}$ with respect to $\mathcal{S}'$ is denoted as $u'_\mathcal{B}$. The Rybczyk addition equation can be written as

(1) $u_\mathcal{B} = v + u'_\mathcal{B} \sqrt{1 - v^2/c^2}$ (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 $\mathcal{S}$ to $\mathcal{S}'$ can be written as

(2) $\left[\begin{array}{rcl}t' &=& k(v) t + l(v) x\\x' &=& p(v) t + q(v) x\end{array}\right.$

where $(t,x)$ is defined for $\mathcal{S}$ and $(t',x')$ is defined for $\mathcal{S}'$. For the body $\mathcal{B}$ we can write $x=u_\mathcal{B} t$ but also $x' = u'_\mathcal{B} t'$, thus

(3) $p(v) t + q(v) \left( v + u'_\mathcal{B} \sqrt{1 - v^2/c^2} \right) t = u'_\mathcal{B} \left( k(v) t + l(v) \left( v + u'_\mathcal{B} \sqrt{1 - v^2/c^2} \right) t \right)$

that is rewritten as

(4) $\left( l(v) \sqrt{1-v^2/c^2} \right) {u'_\mathcal{B}}^2 + \left( k(v) + l(v) - q(v) \sqrt{1-v^2/c^2} \right) u'_\mathcal{B} = \left( p(v) + v q(v) \right)$

This has the form $A(v) {u'_\mathcal{B}}^2 + B(v) u'_\mathcal{B} = C(v)$ and this is true for any arbitrary value of $u'_\mathcal{B}$ (within a certain domain), thus $A(v)=0$, $B(v)=0$ and $C(v)=0$. Therefore we obtain

(5) $\left[\begin{array}{rcl}l(v) &=& 0\\ k(v) &=& q(v) \sqrt{1-v^2/c^2}\\p(v) &=& - v(q)v\end{array}\right.$

The Rybczyk transformation can be written as

(6) $\left[\begin{array}{rcl}t' &=& q(v) \sqrt{1-v^2/c^2} t\\x' &=& q(v) (x-vt) \end{array}\right.$

A quick verification gives

(7) $\displaystyle u' = \frac{x'}{t'} = \frac{q(v) (x-vt)}{q(v) \sqrt{1-v^2/c^2} t} = \frac{u-v}{\sqrt{1-v^2/c^2}}$

thus $u = v + u' \sqrt{1-v^2/c^2}$ which is the Rybczyk addition equation. The Rybczyk transformation can also be written in the matrix-form as

(7) $\left(\begin{array}{c}t'\\x'\end{array}\right) = q(v) \left(\begin{array}{cc}\sqrt{1-v^2/c^2}&0\\-v&1\end{array}\right) \left(\begin{array}{c}t\\x\end{array}\right)$

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

Hope you like the post!