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Is math valid in step 5 of Simple Derivation of the Lorentz Transformation

  1. Mar 9, 2012 #1
    I started to study Simple Derivation of the Lorentz Transformation

    In this derivation, I gather that there are two rays of light being tracked--one going to the right of the origin and one going to the left of the origin.

    The right-going light is described by x=ct and x'=ct'.
    The left-going light is inferred to be described by x=-ct and x'=-ct'.

    This is bothersome that both rays are described by the same symbol x. Perhaps that could be overlooked as long as the statements are kept in context of right-going light and left-going light, but in step 5 the left and right x's are mixed together as the same. I think this means that from step 5 forward, the symbol x represents an intersection of the rays at time t. I think that the initial constraints on x,t,x',t' are required to continue to hold. In particular, I think that from step 5 forward, the intersection x is such that both initial constraints x=ct and x=-ct must be satisfied, which in turn implies that both x and t can only be zero when c is non-zero. Likewise x' and t' must equal zero to satisfy the initial conditions.

    Wouldn't justification for step 5 also require that it be accompanied by the initial constraints which would collectively imply that x=0, t=0, x'=0, t'=0? I think this may have been just an everyday error, but maybe there is something silly or deep that I'm missing.
    Last edited: Mar 9, 2012
  2. jcsd
  3. Mar 12, 2012 #2
    I like the way you think Kwestion :approve:

    The symbols make me wonder if the presentation was made by the author first working the problem backwards on scratch paper, then reversing the thought process for the published presentation.

    Maybe behind the scenes, the math was performed in this scenario: "If I want this outcome, what must the initial conditions be?" The answer to that question may have been x=+/-ct. Reversing the process to make a presentation in the style of "If I have these initial conditions, what must the outcome be?" could have difficulties. From the intended x=+/-ct, it might be natural to attempt the reverse question by posing this: "If I have these initial conditions: x=ct and x=-ct, what must the outcome be?", but that doesn't always work.

    Here's an example of how a reversal of steps might fail particularywhen +/- is involved. I might have in mind that 25=5x5 and ask if "If NxN=M, what is N?" I might get the answer "N=+/-√M"
    I might then, naively try to reverse the process and ask
    "If N=√M and N=-√M then what is M?"
    I get the response from my readers that M=0.
    My bad. I should not have treated "N=+/-√M" as coexisting conditions. In this case I should have said "If N=√M and N=-√M then what is M?"
    On the other hand if I had in mind -25=5x(-5), and I really did want the reader to consider two different co-existing numbers, then I should not use the same label N for both the 5 and the -5 in the same context.

    Anyhow, it's just an idea that I might look at closer, or maybe it'd be better to refer to a different author.
    Last edited: Mar 12, 2012
  4. Mar 14, 2012 #3
    Okay, that makes a lot of sense. I can see that a person might interpret an answer of x=+/-ct as two conditions--for the light on the positive axis x=ct; for light on the negative axis x=-ct, as pointed out by the author. This suggests to the reader that we are talking about light that proceeds away from the origin as time increases, never crossing the origin. However, more broadly, x=(+/-)ct could be interpreted as x=ct for right-going light and x=-ct for left-going light which does not restrict light from going from one axis to the other. It's possible that this more broad case was intended, but not clearly stated.

    Additionally, even if the conditions of t>0 was intended
    x=ct, x'=ct' for t,t'>=0 x,x'>=0
    we can rigorously derive:
    (x-ct)=λ(x'-ct') for x,t>=0; x',t'>=0
    Now although this seems to be derived with non-negative x,t, x',t' in mind, it might also be true for negative x, x' where negative x is in the form -ct and -ct'.
    Let me rephrase that more generally
    Now although this was derived with right-going light' in mind, it might also be true for left-going light where left-going x, x' is in the form -ct and -ct'.
    In that case the above states that -2ct=λ(-2ct'), so yes, its plausible. We haven't intentionally derived this portion from a proof, but it's there as a plausible truth or plausible theory that can be explored.
    So, if we can convert the fact that:
    (x-ct)=λ(x'-ct') for x,t>=0; x',t'>=0
    to consider a theory that:
    (x-ct)=λ(x'-ct') for x<0,t>=0; x'<0,t'>=0
    Therefore together, mixing fact and plausible truth together we have the broad statement:
    (x-ct)=λ(x'-ct') (no restrictions on x, x')
    which I think represents the author's equation (3)

    A similar discussion could be had for the author's equation (4) which works off of a truth about x=-ct, but supplies a plausible truth for x=-ct.

    Okay, so 3 and 4, in the land of plausible truth, have a different domain than the domain of the proven fact. In consideration that this is an exploration of possible truth, or theory, and even a back-calculation from something that is "known" to be true, this seems okay.

    Now that 3 and 4 have overlapping domains of plausible truth, this opens up their combination in author's (5) to also have a large domain of plausible truth. If not for (3) and (4) being open to explore new ground, (5) would be limited to be exploring the intersection of (x>=0) and (x<=0), or just zero.

    This sounds reasonable. If this is the situation, I wish the author would have a) started without the misleading statements about x,c,t, positive, negative axis that provide imagery of light moving away from the origin, and b) drawn out that the paper was exploring plausible truth in (3) and (4) instead of just saying that we have them and be silent as to the big deal about expanding the domain of exploration in those equations and how it is justifiable.
    Last edited: Mar 14, 2012
  5. Mar 14, 2012 #4
    Are you talking to yourself on an online forum?
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