Added the condition for dk2/du'=0 when m is a function of u' in the preprint.
Although traditional derivations of the Lorentz transformation rely on both postulates of relativity and constancy of the speed of light, some researchers have argued that linear transformations per se would naturally lead to a Lorentz type transformation with an unspecified velocity limit. The present study analyzes such a published derivation and shows that linear transformations do not naturally lead to invariant velocities. In the derivation analyzed by this study, in addition to its mathematic mistakes, its author’s interpretation of the derived results is also logically invalid. Its author and some researchers overlooked other consequences than the ones that appear to imply an invariant velocity or a velocity limit, when they were deriving parameters for linear transformations. Therefore, imposing at least two postulates is necessary for deriving the Lorentz transformation or its family of linear transformations. Linear transformations conforming to the principle of relativity alone are not sufficient to ensure a velocity limit or an invariant velocity.
EINSTEIN A.. On the Electrodynamics of Moving BodiesSpecial Theory of Relativity. p. 187–218. 1970. Elsevier. [Cross Ref]
Rindler Wolfgang. Essential Relativity. 1977. Springer Berlin Heidelberg. [Cross Ref]
Berzi Vittorio, Gorini Vittorio. Reciprocity Principle and the Lorentz Transformations. Journal of Mathematical Physics. Vol. 10(8):1518–1524. 1969. AIP Publishing. [Cross Ref]
Coleman B. A dual first-postulate basis for special relativity. European Journal of Physics. Vol. 24(4)2003. IOP Publishing. [Cross Ref]
Shen Jian Qi. Generalized Edwards Transformation and Principle of Permutation Invariance. International Journal of Theoretical Physics. Vol. 47(3):751–764. 2008. Springer Science and Business Media LLC. [Cross Ref]
Sela Or, Tamir Boaz, Dolev Shahar, Elitzur Avshalom C.. Can Special Relativity Be Derived from Galilean Mechanics Alone? Foundations of Physics. Vol. 39(5):499–509. 2009. Springer Science and Business Media LLC. [Cross Ref]
Drory Alon. Special Relativity Cannot Be Derived from Galilean Mechanics Alone. Foundations of Physics. Vol. 43(5):665–684. 2013. Springer Science and Business Media LLC. [Cross Ref]