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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

Qmechanic
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Shashaank
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    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. – WillO May 10 '16 at 22:03
  • 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 – Shashaank May 10 '16 at 22:06
  • You can't derive the Lorentz transformation from anything. They are a simple fit to experiments, just like the rest of physics. – CuriousOne May 10 '16 at 23:41
  • @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? – WillO May 11 '16 at 00:11
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    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. – WillO May 11 '16 at 00:15
  • @WillO: Historically the Lorentz transformations were derived from Maxwell's equations, which were experimentally tested ad nauseam at the time. Einstein's beautifully simple synopsis doesn't actually derive them. He knew what nature demanded, the only question left was how classical mechanics fit into the experimentally verified facts of electrodynamics. Be careful with dismissing people. – CuriousOne May 11 '16 at 00:22
  • @CuriousOne: Is it your position that A) the derivation of momentum conservation from translational symmetry via Noether's theorem is a simple fit to experiments, or B) that the the derivation of momentum conservation from translation symmetry via Noether's theorem is not part of "the rest of physics", or C) that somehow A) and B) can both be false, yet it is still true that all of "the rest of physics" consists of simple fits to experiments? – WillO May 11 '16 at 00:27
  • @WillO: Everything in physics is made to fit experiments. That's what natural sciences are: empirical research. If you want to see what theorists without observational data look like, look not further than string theory. Close to half a century of mathematical guesswork without a single result worthy to be called science. – CuriousOne May 11 '16 at 00:30
  • @CuriousOne: I still can't figure out whether your position is A), B), or C), and I also can't figure out why you don't want to tell me. But in response to the automated request to avoid extended discussions in comments, I'll let this drop here. – WillO May 11 '16 at 00:31
  • Let us continue this discussion in chat. – CuriousOne May 11 '16 at 00:32

2 Answers2

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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.

Mark H
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The fastest way to derive the Lorentz factor that I can think of is to say: "We want to construct a theory that generates the same results regardless of whether lightspeed is fixed with respect to the observer's or the emitter's frame."

  • A – If it's fixed in the observer's frame, the Doppler and visible length-change effects are E'/E = Len'/Len = c/(c+v) for recession and E'/E = 1 for transverse motion.

  • B – If it's fixed in the emitter's frame, the Doppler and visible length-change effects are E'/E = Len'/Len = (c-v)/c for recession and E'/E = 1 - v2/c2 for transverse motion (Lodge, 1909).

So to create a hybrid that gives the same answer in both cases, we can multiply both conflicting predictions together and square root to get their geometric mean. That gives us the predictions of special relativity: $SR = \sqrt{A×B}$

Alternatively, since the new intermediate prediction must always differ from both the previous predictions by exactly the same proportion, we can derive this proportion by dividing B by A (to find the original discrepancy that we wanted to get rid of), and then square rooting (to find the difference between the new prediction and both the old predictions). This can then be applied as a "correction factor" to either of the original predictions to get the new result.

This necessary correction factor, to be multiplied against A or divided out of B, comes out as $\sqrt{B/A} = \sqrt{1-v^2/c^2}$ In other words, the Lorentz factor.

The derivation is probably nonstandard, but it's a heck of a lot shorter than Einstein's, and it doesn't use any of the mathematical overhead that people often try to attach to SR. Hyperbolic functions?!? Pfft.

Eric Baird
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