3. Dimensions, Clocks, and Rulers

The transform equations, Table 1, are dimensional equations. Therefore, the dimensions of the terms on both sides of the equations must match.
In the equation relating the space coordinates x and x’ the units of x and x’ must match; if x is measured in meters, then x’ must also be measured in the same meters. And the origin of the measurement must be the origin of the respective coordinate frame. If there are ‘measuring rods’ used to determine the values of x’, then those rods must have the same length as the rods used to measure x.
In the time equation, the units of t and t’ must match. If t is measured in seconds, then t’ must also be measured in the same seconds. And the origin of time must be the same so that when t = 0 also t’ = 0. And if there are clocks measuring t’ those clocks must count off time at the same rate as the clocks measuring t, and must be set so that t = 0 corresponds with t’ = 0. A clear description of this requirement is given in Ref. 5.
The light speed also has dimensional properties. . The requirement that light speed must be the same in both coordinate systems can be written mathematically as

light speed, c = x / t = x´ / t´.

This relation states that the numerical value of c and the units in which it is expressed must be the same for both systems. If c = x / t is 186,000 miles per second then x´ / t´ must give the same value. The relation can be
satisfied if x’ and t’ are larger or smaller than x and t, but the proportions must be the same and the units of measure, miles and seconds, must be the same for x’ and t’ as for x and t.
In the derivation of the transform equations, the length of the ‘ruler’ holding the mirror is taken to be the same by both observers, and in general the analysis of Fig 4 is done with the idea that space and time measures are the same for both coordinate frames.
This discussion of dimensions might be unnecessary or self-evident when using the Galilean transforms because the results can be supported by intuition and experience. But in the case of the Lorentz transforms, the results are new and strange so there is little experience and intuition is unreliable. Careful consideration is therefore required.