Simultaneity and the Constancy of the Speed of Light: Normalization of Space-time Vectors in the Lorentz Transformation

One of the main consequences of Einstein’s derivation of the Lorentz transformation (LT) was the conclusion that a given event may not occur simultaneously for two observers in relative motion. It is pointed out that the constancy of the speed of light in free space is not sufficient in itself to completely specify the relationship between space-time vectors in different rest frames, as first noted by Lorentz in 1899. The LT results by assuming that distances measured perpendicular to the line of two observers must be equal ( d d ' y y = and d d ' z z = ), and it is this assumption that produces the non-simultaneity characteristic of the resulting equations. If one forgoes the latter assumption, it is possible to impose simultaneity ( d d ' t t = ) as an alternative normalization condition for the space-time vectors. The