The necessity of the second postulate in special relativity

• Relativity without constancy of the speed of light yields generalized Lorentz transformations.
• These transformations contain an unknown parameter, k.
• Without postulating the value of k, the transformations do not represent any specific theory.
• In particular, the Newtonian case is a possibility, as the case k=0.
• An anisotropic version of special relativity exemplifies the problems with undefined constants.

Many authors noted that the principle of relativity together with space–time homogeneity and isotropy restricts the form of the coordinate transformations from one inertial frame to another to being Lorentz-like. The equations contain a free parameter, k (equal to c−2 in special relativity), whose value is claimed to be merely an empirical matter, so that special relativity does not need the postulate of constancy of the speed of light. I analyze this claim and argue that the distinction between the cases k  =0 and k≠0k≠0 is on the level of a postulate and that until we assume one or the other, we have an incomplete structure that leaves many fundamental questions undecided, including basic prerequisites of experimentation. I examine an analogous case in which isotropy is the postulate dropped and use it to illustrate the problem. Finally I analyze two attempts by Sfarti, and Behera and Mukhopadhyay to derive the constancy of the speed of light from the principle of relativity. I show that these attempts make hidden assumptions that are equivalent to the second postulate.