strangerep
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Then take a few steps back, a deep breath, and listen carefully...Saw said:I concede that the title of the thread may be misleading, because I mentioned LTs, but I think that the OP is clear in that my concern is only about the way to derive the *ST interval*.
This is not the usual, most physically fundamental, meaning of the Relativity Postulate.Saw said:
From Rindler** :
Principle of Relativity:
The laws of physics are identical in all inertial frames,
or, equivalently,
the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame.
One then searches for the maximal group of coordinate transformations ##(t,x^i) \to (t',x'^i)## which ensures that every inertial (i.e., non-accelerating) frame is transformed into another inertial frame, i.e., $$ \frac{d^2 x^i}{dt^2} ~=~ 0 ~~~~ \Leftrightarrow ~~~~ \frac{d^2 x'^i}{dt'^2} ~=~ 0 ~. $$ This determining equation already has spatial isotropy inherent within it, therefore we may use this (i.e., without an extra separate assumption) when searching for the coordinate transformations. Similarly for spatiotemporal translation invariance.
From this (long, tedious) analysis, looking especially for a subgroup corresponding to velocity boosts such that velocity boosts in along any given spatial direction form a 1-parameter Lie group, one finds the Lorentz group, together with a universal constant with dimensions of inverse velocity squared that emerges unbidden from the analysis. By comparison with experiment, one realizes that this constant corresponds to ##c^{-2}##. But one does not mention anything about light at the beginning.
Having the Lorentz transformations, one then notices that it preserves a particular quadratic form in spacetime. I.e., by this route, the invariant spacetime interval is derived from the Relativity Principle.
Importantly, the usual Light Postulate (i.e., assume invariance of the speed of light) is not actually necessary. Similarly, anything to with spherical wavefronts of light is not needed up front. All such things are simply shortcuts to make the LT derivation quicker, hence more palatable to the average student. But if you seek the deepest foundation that we know of, then study the so-called "1-postulate" derivation sketched above.
Here are some older PF threads where I talked about this stuff:
https://www.physicsforums.com/threa...-1-spacetime-only.1000831/page-2#post-6468652
Post #44.
https://www.physicsforums.com/threads/linearity-of-the-lorentz-transformations.975920/#post-6219004
Post #6.
https://www.physicsforums.com/threads/derivation-of-the-lorentz-transformations.974098/
Post #26.
Other PF members have also talked about it. Try searching for "1-postulate" and/or for ways of deriving the LTs.
Textbook Reference:
**) Rindler, Introduction to Special Relativity,
Oxford University Press, 1991 (2nd Ed.), ISBN 0-19-85395-2-5.