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A problem with the "Voigt transformation"

Physics Asked by Mohammad Javanshiry on September 11, 2020

In the traditional Lorentz transformation when we derive unprimed variables in terms of the primed ones, and vice versa, we see that there is no change in the corresponding equations but the sign of velocity $v$. That is, the forms of the equations remain symmetric. However, in the Voigt transformation, I have an issue with the transverse directions, i.e., directions perpendicular to the velocity. In the recent transformation, we have:

$$y^prime=y/gamma space space and space space z^prime=z/gamma space .$$

However, as we attempt to derive unprimmed in terms of primed, we reach the following asymmetric equations:

$$y=y^prime gamma space space and space space z=z^primegamma space .$$

It seems that these equations mean that, say, the height of a moving object is measured increased by a factor $gamma$ from the viewpoint of the lab observer, and, from the viewpoint of the moving object, the height of the same object at rest in the lab frame of reference is measured decreased by a factor $1/gamma$. If this is the case, I just cannot follow why we have only two terms (primed and unprimed) for a transverse direction, whereas, to me, it would be more rational if we had at least three separate terms since the measurements are asymmetric. Let’s denote the height of the moving object $B$ measured by the lab observer $A$ as $y_{BA}$; the height measured by $B$ for the same object in the lab frame as $y_{AB}$, and the height measured in the rest frames of each $A$ & $B$ as $y_{AA}=y_{BB}$ (proper heights). Therefore, the Voigt transformation must be rewritten as follows:

$$y_{AB}=y_{BB}/gamma=y_{AA}/gamma space space and space space z_{AB}=z_{BB}/gamma=z_{AA}/gamma space ,$$

and, from the viewpoint of the lab observer, we have:

$$y_{BA}=y_{AA}gamma=y_{BB}gamma space space and space space z_{BA}=z_{AA}gamma=z_{BB}gamma space ,$$

However, in the Voigt transformation, it is not clear which of $y_{AB}$, $y_{BA}$ and $y_{AA}$ (or $y_{BB}$) relates to $y$ and $y^prime$. Can someone tell me where is the problem?

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