Derive Special Relativity: Alternative Paths

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Discussion Overview

The discussion revolves around alternative methods to derive special relativity beyond the traditional approach of starting with Einstein's two postulates. Participants explore various derivations, principles, and implications related to the theory, touching on concepts of spacetime, wave equations, and transformations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that the standard derivation of special relativity begins with the two postulates and the Lorentz transformations, questioning if alternative methods exist.
  • Another participant references a previous discussion that may provide insights into the independence of Einstein's axioms.
  • A different contribution presents a derivation based solely on the homogeneity and isotropy of spacetime, linking to an external resource.
  • One participant proposes starting from the principle of relativity and asserting the invariance of the interval to derive the transformations, while also noting the inadequacy of Galilean relativity.
  • Another participant shares a resource that discusses transformation laws in isotropic and homogeneous spacetime, indicating that one can derive either Galilean or special relativity based on specific conditions.
  • A participant explains the necessity of special relativity by discussing the invariance of wave equations under Galilean transformations, emphasizing that light does not require a medium and thus presents unique challenges to classical physics.
  • There is a suggestion for a quick treatment of relativistic waves in a medium, linking to an external document.
  • One participant offers a written explanation of their derivation, expressing hope that it is free of mistakes.

Areas of Agreement / Disagreement

Participants present multiple competing views and derivations regarding the foundations and implications of special relativity, indicating that the discussion remains unresolved with no consensus on a singular alternative derivation method.

Contextual Notes

Some claims depend on specific assumptions about spacetime and the nature of wave equations, and there are unresolved mathematical steps in the proposed derivations.

user_12345
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The only way I know of to derive special relativity is to start with the two postulates, derive the Lorentz transformations, and rewrite the laws of physics consistent with those transformations.
Are there alternative ways to derive special relativity?
Thank you.
 
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State the interval and assert it is invariant; deduce the transforms that make it so.

Start from the principle of relativity and derive Galilean or Einsteinian relativity. Assert that we don't live in a Galilean universe.
 
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I found this colelction here on physicsforums.com from a suggestion from one of the respected members (can't remember who for credit, sadly), and I love this particular derivation because it derives a general case for transformation laws in an isotropic and homogeneous spacetime, and then you can select Galilean relativity or special relativity based upon whether you set a particular constant equal to 0 or 1.

https://www.mathpages.com/rr/s1-07/1-07.htmEDIT - regarding the source, all I can say is it's recommended by Fields Medal winner Timothy Gowers if that carries any weight, and as I said, someone here with one of those icons next to their name showed me this.
 
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The reason why we need special relativity is, that wave equations are not invariant under galilean transformations. The reason for this is, that the veolcity of a wave is determined by the medium. That means that an observer with a relative speed to the medium v>0, simply observes the wave going at the wrong speed. So he cannot describe the wave correctly. Now, with a medium, one can always get away with saying that the medium is the relevant reference frame and that therefore we don't have to worry about galilean transformations.
With light, things are different, since light doesn't require a medium. Hence, either the wave equation of light is wrong or the galilean transformation.
c0202839-800px-wm.jpg

Since the partial derivatives are coefficients of a dual vector, the transformation that will leave this equation invariant, will be orthogonal with the metric of special relativity (due to the relative minus sign between the time and space components).
One can now simply generalize the galilean transformation, by putting variables in front of each term, and then use the chain rule of the partial derivatives to work out the Lorentz transformation.

Only the wave equation with velocity c is then invariant under the so found Lorentz transformation. All wave equation with speeds different to c are still not invariant. This means that the speed of light has to be the same for all observers, or otherwise we still wouldn't have an invariant description of light.

I hope this helps.
 
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Or in written form, hoping that I didn't make mistakes...
 

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