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Physics
Special and General Relativity
Derivation of SR's time-dilatation in 1d?
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[QUOTE="Sagittarius A-Star, post: 6852366, member: 666305"] Einstein wrote in the[URL='https://en.wikisource.org/wiki/Relativity:_The_Special_and_General_Theory/Appendix#Appendix_I_-_Simple_Derivation_of_the_Lorentz_Transformation'] appendix of his popular book from 1916[/URL] transformation formulas, that satisfy the invariance of the speed of light signals, that move through the origin in (+x) direction and (-x) direction. The disappearance of the left sides involves the disappearance of the right sides: $$(x' - ct') = \lambda (x - ct)\ \ \ \ \ \ \ \ \ \ (3)$$$$(x' + ct') = \mu (x + ct)\ \ \ \ \ \ \ \ \ \ (4)$$From this stage, I continue differently than Einstein did. He derived the LT, I want to derive directly the invariance of the spacetime interval. I multiply equations (3) and (4) and get: $$(x'^2 - c^2t'^2) = \lambda\mu (x^2 - c^2t^2)$$From the 1st postulate (and assuming inertial frames are homogeneous and spatially isotropic), from the invariance of causality [I][edited] and when having a fixed unit of time[/I] follows reciprocity: ##\lambda\mu = +1##. Then using differentials: $$dx'^2 - c^2dt'^2 = dx^2 - c^2dt^2$$ [/QUOTE]
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Physics
Special and General Relativity
Derivation of SR's time-dilatation in 1d?
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