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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 ?

Qmechanic
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Frobenius
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1 Answers1

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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.

MaxSchambach
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  • 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? – MaxSchambach May 08 '16 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. – MaxSchambach May 09 '16 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: – MaxSchambach May 09 '16 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? – MaxSchambach May 09 '16 at 14:42