Deriving Lorentz Transforms: Unnecessary Step?

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Discussion Overview

The discussion revolves around the derivation of the Lorentz transformations as presented in Ray D'Iverno's "Introducing Einstein's Relativity." Participants are examining a specific step in the derivation process, questioning its necessity and the underlying mathematical rigor involved.

Discussion Character

  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant expresses confusion about a step in the derivation where it is concluded that I=I' based on the condition that I=0 and I'=0, questioning the necessity of the preceding reasoning involving the transformation factor n.
  • Another participant argues that knowing I=0 and I'=0 does not imply that I=I' for all values, suggesting that additional reasoning is required to establish the relationship when I and I' are not zero.
  • A follow-up response reiterates the concern that without further justification, one could assume a relationship such as I=-3I', which would contradict the conclusion that I=I'.
  • Participants engage in a back-and-forth, with one acknowledging the need for the additional mathematical rigor after considering the implications of the earlier statements.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the necessity of the step in question. There is an acknowledgment of differing views on the implications of the conditions set for I and I' and the need for further justification in the derivation process.

Contextual Notes

The discussion highlights the potential limitations in the reasoning presented, particularly regarding the assumptions made about the relationship between I and I' beyond the zero condition.

kmm
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I'm going through Ray D'Iverno's "Introducing Einstein's Relativity", and there is a step he makes in deriving the Lorentz transformations that doesn't seem necessary to me. So I'm not sure what I'm missing. He derives them from Einsteins postulates of relativity. From the postulate that the speed of light is the same in all reference frames, we take a rest frame S and a frame moving with respect to S, S'. When their origins meet, a pulse of light moves out from their origins as a sphere. The events constituting this sphere satisfy the equations I=x2+y2+z2-c2t2=0 for the rest frame and I'=x'2+y'2+z'2-c2t'2=0 for the moving frame. He sets these equations equal and derives the Lorentz transformations, which is pretty straight forward. The step that I'm not sure about is before setting I=I', he says, "..it follows that under a transformation connecting S and S', I=0 ⇔ I'=0, and since the transformation is linear.." (by the first postulate) "we may conclude I=nI'." He then goes on to show how we can reverse the roles of S and S' giving I'=nI and combining the equations gives, n2=1 ⇒ n=±1, and in the limit as v→0 the two frames coincide so I' → I so we must take n=1. It's at this point that he sets I=I'. That's fine with me, I just don't understand why all of that was necessary in showing I=I'. Why couldn't we have just concluded that I=I' when we said earlier that I=0 and I'=0? I'm suspicious that there is some necessary mathematical rigor that I'm missing, in this step.
 
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kmm said:
Why couldn't we have just concluded that I=I' when we said earlier that I=0 and I'=0?

Because just knowing that l=l' when both are zero does not guarantee that l=l' when both are not zero.
 
kmm said:
Why couldn't we have just concluded that I=I' when we said earlier that I=0 and I'=0? I'm suspicious that there is some necessary mathematical rigor that I'm missing, in this step.
What other reason would you have had for concluding that I'=I? Given that I=0 iff I'=0, you could have, for example, I=-3I'.
 
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PeterDonis said:
Because just knowing that l=l' when both are zero does not guarantee that l=l' when both are not zero.

bcrowell said:
What other reason would you have had for concluding that I'=I? Given that I=0 iff I'=0, you could have, for example, I=-3I'.

Ah of course, thanks.
 

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