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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 ?

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 ?

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 the later case ?

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 ?