[83] Here the reader may well question our right to argue as though velocities combined in the classical way, when the whole significance of the special relativity theory has been to deny the validity of the classical transformations. In point of fact the objection would be legitimate; and in all rigour the Einstein-Lorentz transformations should be applied for each successive instantaneous velocity of the enclosure. But it so happens that when the Einstein-Lorentz transformations are applied to a transverse beam of light, it is found that for low velocities of the enclosure the bending is practically the same as it would have turned out to be had we followed the classical rule of composition. In other words, had the motion of the enclosure been uniform, the transverse ray of light as measured in the enclosure would have been inclined much as in classical science. Indeed, were it not that the relativity transformations entailed a variation in the slant of a ray of light moving transversally, the theory would be incompatible with the well-known phenomenon of astronomical aberration.
[84] We are assuming that the field does not vary with time. When this is not the case, we must specify that our observations must also be conducted over a very short duration of time.
[85] Qualitatively at least. The precise quantitative justification will be furnished later.
[86] If the spatial mesh-system we are considering is one of straight lines, a Cartesian one, for example, the statement in the text is accurate, but if we consider the more general case of a curvilinear mesh-system, we must introduce certain restrictions. In this last instance, our rod must be of infinitesimal length, for were its length finite, its orientation as referred to the curvilinear mesh-system would vary from place to place. This would introduce complications. Hence it is preferable to restrict our attention to rods of infinitesimal length and consider orientation as defined at a point. With this restriction in force, there is nothing to change in the explanation given in the text.
[87] Thus, calling
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