Einstein's sigh: hidden symmetry in Einstein's derivation of the Lorentz transformation

• Published in: European journal of physics 2017-03, Vol.38 (2), p.25601
• Main Author: Chao, Sheng D
• Format: Article
• Language: English
• Subjects: Lorentz transformation; special relativity; symmetry principle
• ISSN: 0143-0807; 1361-6404
• DOI: 10.1088/1361-6404/aa5392

'Das hätte ich einfacher sagen können (I could have said that more simply).' was Einstein's sigh when he had a chance to remark on his own derivation of the Lorentz transformation (LT) in the 1905 seminal paper. In fact, in a popular science exposition of the theory of relativity Einstein did provide such a simple derivation of the LT. It is a curious historical fact that the latter derivation was presented in 1916, while Einstein's remark was made in 1943. Was the 1916 derivation simple enough to relieve his sigh? Had he expected an even simpler derivation beyond his thoughts? In this paper, Einstein's simple derivation of the LT is revisited and analysed. We show that the LT can be obtained from a symmetry principle hidden in Einstein's logical reasoning. First, the relativity principle can be restated as a mirror principle based on the space-time exchange-inversion operation. Second, the assumed constancy of the speed of light (Einstein's second postulate) can be derived by using the velocity reciprocity property, which is a deductive result of the space-time homogeneity and the space isotropy. Therefore, Einstein could have presented his derivation of the LT more simply, thus turning Einstein's sigh of regret into a sigh of relief.