Question on linearity of Lorentz transformations

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Discussion Overview

The discussion centers on the linearity of Lorentz transformations in the context of Special Relativity. Participants explore the implications of homogeneity and isotropy of space-time, the validity of traditional derivations, and the existence of rigorous demonstrations for linearity from first principles.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants argue that traditional derivations of Lorentz transformations rely on flawed assumptions about linearity and uniform motion.
  • Others propose that linearity is a consequence of homogeneity, suggesting that space and time intervals should not depend on a particular point.
  • A reference is made to a paper by Zeeman, which discusses causality and the Lorentz group.
  • Some participants cite Mariwalla's work, which claims that linearity can be established without appealing to homogeneity.
  • There is a contention regarding the applicability of linear Lorentz transformations to interacting particles, with references to the Currie-Jordan-Sudarshan theorem suggesting that these transformations may not hold in such cases.
  • One participant expresses uncertainty about the correctness of their earlier statements and seeks clarification on the arguments presented by others.
  • Another participant emphasizes that the absolute meaning in Special Relativity pertains to the invariant interval, rather than individual coordinate differentials.

Areas of Agreement / Disagreement

Participants express differing views on the validity of traditional derivations and the conditions under which linearity holds. There is no consensus on the existence of a rigorous demonstration for linearity from first principles, with some citing papers that address this while others remain skeptical.

Contextual Notes

The discussion highlights limitations in the assumptions made about linearity and the dependence on definitions of space-time intervals. The complexity of interactions among particles and their implications for Lorentz transformations are also noted.

  • #61
dx said:
2. Clock rates are uniform, i.e. intervals measured by clocks agree with the linear structure of V.

what is the meaning of time intervals agreeing with the linear structure of V.
 
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  • #62
facenian said:
It does when b_i=0 for i=0,1,2,3
You also have to set the c_i=0.
 
  • #63
Fredrik said:
You also have to set the c_i=0.

I don't see why. Setting b_i=0 for all i seems sufficient for T(0)=0
 
  • #64
facenian said:
I don't see why. Setting b_i=0 for all i seems sufficient for T(0)=0
Your function isn't smooth, or even defined for all x, because of the x in the denominator.
 
  • #65
I just started looking at the articles referenced on page 1. I haven't looked at George's reference yet (because it's not an online article), but all the others look interesting, especially the one atyy posted. It seems that the assumptions in my post #56 are much stronger than they need to be. I think I'm going to have to read that whole article soon.
 
  • #66
facenian said:
what is the meaning of time intervals agreeing with the linear structure of V?

If A - B = C - D, then a clock carried along AB will measure the same interval as it would if it were carried along CD. If A - B = λ(C - D), then a clock carried along AB will measure λ times the interval it would measure along CD.
 
  • #67
dx said:
If A - B = C - D, then a clock carried along AB will measure the same interval as it would if it were carried along CD. If A - B = λ(C - D), then a clock carried along AB will measure λ times the interval it would measure along CD.

Thank you dx. The fist condition seems to be consequence of homogeneity, and if A-B is understood only as distance could be consequence of isotropy.
 

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