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How can I derive the Lorentz transformations? I don't want to use hyperbolic functions and the fact that the light waves travel by forming spherical wavefronts. Is there a way to derive the Lorentz transformations applying the conditions I have mentioned. I was unable to understand the method given in Landau and lifshitz deeply. That's why I want a method other than the one using hyperbolic functions

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    $\begingroup$ From what do you wish to derive the Lorentz transformations? Given the invariance of the spacetime interval, the derivation is a simple calculation. Given some other starting point (e.g. the invariance of the speed of light?) there calculation will look different. $\endgroup$ – WillO May 10 '16 at 22:03
  • $\begingroup$ I want to derive the Lorentz transformations from any possible way but the hyperbolic functions...or ok can you please prove it from the invariance of speed of light $\endgroup$ – Shashaank May 10 '16 at 22:06
  • $\begingroup$ You can't derive the Lorentz transformation from anything. They are a simple fit to experiments, just like the rest of physics. $\endgroup$ – CuriousOne May 10 '16 at 23:41
  • $\begingroup$ @Shashaank : Hint: Start with the case of one space dimension. Setting $c=1$, a transformation in $SO(1,1)$ leaves the speed of light invariant if and only if it has $(1,1)^T$ as an eigenvector. Where can you go from here? $\endgroup$ – WillO May 11 '16 at 0:11
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    $\begingroup$ PS: I think you can safely dismiss @CuriousOne's stated position (that logical or mathematical derivations are never part of physics --- so that, for example, the derivation of momentum conservation from translational symmetry is not physics) as too idiosyncratic to take seriously. $\endgroup$ – WillO May 11 '16 at 0:15
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Here's a derivation that uses very basic properties of space and time (isotropy, homogeneity, the fact that two Lorentz boosts should compose into another valid Lorentz boost, etc.). The constant maximum speed through space (i.e., the speed of light) is a derived property, not an assumption.

One more derivation of the Lorentz transformation - Jean-Marc Levy-Leblond

Here's a similar one that uses linear algebra after deriving the fact that the transform is linear, with similar results.

Nothing but relativity - Palash B. Pal

These kinds of group-theory-based derivations go back to Vladimir Ignatowski in 1910.

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