# Parallel axes between inertial frames in Special Relativity

In "CLASSICAL ELECTRODYNAMICS" by J.D.Jackson, 3rd Edition , $\S$ 11.3, the author gives in equation (11.19) a generalization of Lorentz transformation as follows :

If the axes in K and K' remain parallel, but the velocity $\:\mathbf{v}\:$ of the frame K' in frame K is in an arbitrary direction, the generalization of (11.16) is

\begin{align} x'_{0} & =\gamma\left(x_{0}-\boldsymbol{\beta}\boldsymbol{\cdot}\mathbf{x}\right)\\ \mathbf{x}^{\prime} & = \mathbf{x} +\dfrac{\left(\gamma-1\right)}{\beta^{2}}\left(\boldsymbol{\beta}\boldsymbol{\cdot}\mathbf{x} \right)\boldsymbol{\beta}-\gamma\boldsymbol{\beta}x_{0} \end{align} \Biggr\} \tag{11.19} where \begin{align} \boldsymbol{\beta} & = \dfrac{\mathbf{v}}{c}\; \qquad \beta=|\boldsymbol{\beta}| \\ \gamma &=\left(1-\beta^2 \right)^{-1/2} \end{align} \tag{11.17}

and \begin{align} x'_{0} & =\gamma\left(x_{0}-\beta x_{1}\right)\\ x'_{1} & =\gamma\left(x_{1}-\beta x_{0}\right)\\ x'_{2} & =x_{2}\\ x'_{3} & =x_{3} \end{align} \Biggr\} \tag{11.16} the Lorentz Transformation with the velocity $\:\mathbf{v}\:$ parallel to the common $\:x-x'\:$ axis.

In case (11-16) it's permissible to talk about parallel axes. But in the generalized case (11-19) is it permissible to talk about parallel axes ? What is the meaning of parallelism in this later case ?

• Relevant paper: aapt.scitation.org/doi/abs/10.1119/1.11632?journalCode=ajp Oct 26, 2021 at 17:45
• I once said that Lorentz boosts are interpreted as the coordinate transformations between observers with the same axes and you corrected me and said that there is no sense in "...between observers with the same axes..." generally. Would you be willing to elaborate if I post a question regarding this? Aug 19, 2022 at 19:15
• @Filippo : Take a look in my answer here How to represent a pair of inertial frames in relativity?. Aug 19, 2022 at 19:25
• @Frobenius Thank you! Aug 19, 2022 at 19:38
• @Filippo : Welcome !!! Aug 19, 2022 at 19:50

In general, Lorentz (or rather Poincare) transformation are those transformations that keep the speed of light in any reference frame the same. They can be decomposed into the following:

1.) Translations
2.) Rotations and
3.) Boosts

Translations and Rotations are defined just like in the Galileo case and don't make up the "interesting" physics of special relativity. Therefore, one often only really talks about the boosts when talking about Lorentz transformations. These boosts can be done in some coordinate direction, most easily in the direction of an coordinate axis, for example the $x$-axes. The corresponding transformation is written in your equation (11.16). Now Jackson talks about a generalization, by that he means here to write down the transformation for a boost in $\overrightarrow{v}$-direction which he does in (11.19).
In principle it's the same kind of transformation. So in the same fashion as for the boost in $x_1$ direction, the spatial coordinate axes are parallel, i.e. the transformation does not include any rotation. If you were to shoot an arrow in $\overrightarrow{x}_i$-direction, $i=1,2,3$, the observer in the reference frame with primed coordinates would also observe the arrow going in $\overrightarrow{x}'_i$-direction.
The parallelity refers to the spatial coordinates only.

• well, try to solve it yourself: suppose you have a set of events given in coordinates of $K$, maybe as a finite number of lights, say ten, blinking at time $t=0$ placed along the $x_0$ axis at a distance of 1, $\{(0,n,0,0) : n=1,2,...,10\}$. What do the events look like in reference frame $K'$? Will the primed observer see the lights flashing at the same time? May 8, 2016 at 20:56
• For parallelity, recall the example of the arrow I have given. I think your problem lies with the idea of "simultanous". Given the set I specified before, the events representing a array of lights flashing at $t=0$. As you said, they will not flash simultanously in the primed reference frame, they will however still be aligned along the $x'_1$-axis. Even though the do not blink simultanously in the primed reference system, you can think of a scenario where they turn on at the specified events. So they will appear, one after one another, and stay on, spatially parallel to the coordinate axis. May 9, 2016 at 12:20
• If you want to further discuss this, I'd prefer a more respectful writing. Being downvoted and writtenly yelled at, I feel more ad more discourage to spend time thinking about this with you and trying to formulate my thoughts. As what my example concerns, yes you are right, I only thought about boosts in $x$ direction. As what the parallelity regards, I think what is ment is: May 9, 2016 at 14:42
• Both (spatial) reference frames are embedded in $\mathbb{R}^3$ with unit vectors given by $\overrightarrow{e}_i$, e.g. $\overrightarrow{e}_1=(1,0,0)$. The same holds vor the primed $\overrightarrow{e}'_i$. In that sense, the axes are parallel, even though from the point of the observer, the other reference frame's axes might not seem parallel to the own ones. Might that be it? May 9, 2016 at 14:42