Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
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From: Perspicacious on 3 Sep 2005 21:13 Igor wrote: > Congratulations! You've just discovered that the average speed of two > light rays moving in opposite directions vanishes. What this has to do > with inconsistencies in the Lorentz transformation, which you didn't > even get to, I have no idea. My first reaction was that this had to be > a big joke, since no one could be that stupid. But I could be wrong. The greatest riddle, with as yet an undiscovered rationale, is why so many kooks are drawn to special relativity. I believe the answer is that it special relativity is taught primarily by physicists who strive for sensationalism and not by mathematicians who prefer clarity and logical consistency. http://www.everythingimportant.org/relativity/special.pdf
From: Perspicacious on 3 Sep 2005 21:56 Perspicacious wrote: > Igor wrote: > > Congratulations! You've just discovered that the average speed of two > > light rays moving in opposite directions vanishes. What this has to do > > with inconsistencies in the Lorentz transformation, which you didn't > > even get to, I have no idea. My first reaction was that this had to be > > a big joke, since no one could be that stupid. But I could be wrong. > > The greatest riddle, with as yet an undiscovered rationale, is why so > many kooks are drawn to special relativity. I believe the answer is > that it special relativity is taught primarily by physicists who > strive for sensationalism and not by mathematicians who prefer > clarity and logical consistency. > > http://www.everythingimportant.org/relativity/special.pdf Consider this hypothesis: Physicists strive to teach special relativity in a way that suggests that modern physics is entirely irrational and seemingly contradictory. Once that sentiment is widely accepted by the predominantly unsophisticated mainstream and mathematically ignorant masses, it's then equally easy for conspiracy theorists to imagine otherwise and advocate the notion that they alone have common sense and that physicists and their unthinking followers are obviously mistaken and extremely deluded.
From: Bilge on 4 Sep 2005 01:28 Thomas Smid: >Bilge wrote: >> Thomas Smid: >> >Many people maintain that the Lorentz transformation is derived >> >mathematically consistently and that there is therefore no way to >> >challenge SR on internal consistency issues. Is this really so? >> >> Yes. It's really so. Lorentz boosts and spatial rotations >> are obtained in the same derivation. If the lorentz transforms >> are mathematically inconsistent, then so is euclidean geometry. >> I trust you haven't disproved the pythagorean theorem, since >> disproving the pythagoren theorem would be big news > >As far as I am aware the invariance of c was big news at the time, and Who cares? We aren't discussing the sociological effect on the victorian mindset. This is 2005, not 1905 and the topic is physics, not sociology. >it excatly amounts to the fact that the speed of light can not be >represented vectorially in the usual sense. It is a scalar that has a >fixed value in all reference frames. Everything else is just a futile >attempt to apply usual geometrical principles where they can not be >applied. You didn't answer my question. In fact, you snipped the question, so it would appear your entire line of argument has been conceived to prop up your own preconception of nature. I realize the question was straight forward and to the point, but I'd just as soon you not get side tracked. If you have no argument against the consistency of the coordinate transformations I wrote down, your objections are now null and void. If you do have a legitimate argument, I suggest that you skip usenet and concentrate on writing an article for a mathematics journalm since invalidating mathematics going back to euclid and pythagoreas will be a lot bigger news than you could have ever imagined relativity to be.
From: Bilge on 4 Sep 2005 01:42 Mike: >The problem with Al's derivation of LT is that he postulates a linear >transformation between the space time points in K and K': So, the same argument can be applied to rotations on a circle, so let's see how it applies: Translation: ``The problem with the derivation of spatial is that one postulates a linear transformation between the points in K and K.' >"A light-signal, which is proceeding along the positive axis of x, is >transmitted according to the equation x = ct >or > >x - ct = 0 (1) This statement is wrong, since all of the points on the x-axis (or any space axis) have a spacelike separation. The lightray can't propagate along the x-axis. It propagates along the light cone which is at 45 degrees to the x-axis in the x-t plane. Translation: ``A line lying along the x-axis is given by the equation x = by. or x - by = 0 Which is obviously silly, although that didn't stop you from saying the line x = ct is along the x-axis. > >Since the same light-signal has to be transmitted relative to k' with >the velocity c, the propagation relative to the system k' will be >represented by the analogous formula > >x' - ct' = 0 (2) > >Those space-time points (events) which satisfy (1) must also satisfy >(2). Obviously this will be the case when the relation > >(x'-ct') = lambda (x - ct) (3)" > >quoted from: http://www.bartleby.com/173/a1.html > >of course, no justification is provided for (3), except of the fact >that it fits the final objective of the derivation. > >But in general, lambda can be any function f of x, x', t, t', where f >not equal to infinity, and still (3) holds for all space time events >that satisfy (1) and (2). > >Now, this demonstrates that in order to derive the LT, arbitrary >assumption must be made about how the events are related in K and K'. >Or, in a better way, the events are forced to relate according to (3). > >The proper way is to show that (3) must hold uniquely and lambda is a >constant. Because by asserting (3), although the LT is derived, >restrictions are placed on the structure of spacetime. But this cannot >be done and it shows that the deduction is based on an explicit >hypothesis. > >But Al did not fully comprehend the laws of logic and mathematical >reasoning because he was a crank. In a mathematical derivation, any >assumptions made must be carried through and stated along with the >conclusion, to say the least. > >Of course, SR is corroborated by experiments but SR' is too. SR' is >any compensatory theory which involves some other function in place of >lambda in (3). the argument is of course, that SR is more elegant. I >know of some lunatics that dress very elegantly. > >Mike >
From: Thomas Smid on 4 Sep 2005 05:28
Daryl McCullough wrote: > Thomas Smid says... > > > >Daryl McCullough wrote: > > >> Einstein's equations were these > >> > >> (3) x' - ct' = lambda (x-ct) > >> (4) x' + ct' = mu (x+ct) > > > >So how did he get then to (4) in your opinion? (Hint: he got to (3) > >using his equations (1) and (2)) > > Yes, he used (1) and (2). Here's a more pains-taking explanation: > > For any event e, let x(e), t(e) be the location and time of e > in the first frame, and let x'(e) and t'(e) be the location > and time as measured in the other frame. We assume that these > coordinates are linearly related: There is some parameters > A,B,D,E that are functions of the relative velocity between > the two frames such that for all events e > > x'(e) = A x(e) + B ct(e) > ct'(e) = D x(e) + E ct(e) > > Now, these two equations can be rearranged into the equivalent > equations (I'm not going to write the dependence on e, to simplify > the appearance, but actually, x,t,x' and t' all depend on which > event e you are talking about) > > (0.1) x' - c t' = lambda (x - ct) + tau (x + ct) > (0.2) x' + c t' = mu (x + ct) + sigma (x - ct) > > where lambda, tau, mu, and sigma are linear combinations of > A, B, D, and E: > > lambda = 1/2 (A-D+B-E) > tau = 1/2 (A-D-B+E) > mu = 1/2 (A+D+B+E) > sigma = 1/2 (A+D-B-E) > > Okay, so what Einstein is arguing by considering light > signals is that tau = 0 and sigma = 0. Why does that > follow? Well, consider the following events: > > Let e0 be the event with coordinates x(e0) = 0, t(e0) = 0. > Let a light signal travelling in the +x direction be sent > from event e0 to some event e1. This event will have > x(e1) > 0, t(e1) > 0. > > Because light travels at speed c, we know, in the first frame: > > x(e1) = c * t(e1) > > or > > (1) x(e1) - c t(e1) = 0 > > > But light *also* travels at speed c in the second frame. So > we have: > > x'(e1) = c * t'(e1) > > or > (2) x'(e1) - c t(e1) = 0 > > By my equation (0.1) above, we know > > x'(e1) - c t'(e1) = lambda (x(e1) - c t(e1)) > + tau (x(e1) + c t(e1)) > > Using (1) and (2) to simplify this, we get: > > 0 = 0 + tau (x(e1) + c t(e1)) > > Since x(e1) and t(e1) are both positive, it follows that > > tau = 0 > > Putting this together with my equation (0.1) gives > > (3) x' - c t' = lambda (x - ct) > > Now, we go through the same sort of thing for a light > signal travelling in the -x direction: > > Let a light signal travelling in the -x direction be sent > from event e0 to some event e2. This event will have > x(e2) < 0, t(e2) > 0. Since light travels at speed c in > both frames, we have > > (1') x(e2) + ct(e2) = 0 > > and similarly > > (2') x'(e2) + ct'(e2) = 0 > > My equation 0.2 gives: > > x'(e2) + c t'(e2) = mu (x(e2) + ct(e2)) + sigma (x(e2) - ct(e2)) > > Using (1') and (2') to simplify equation (0.2) gives: > > 0 = 0 + sigma (x(e2) - ct(e2)) > > or > > sigma (x(e2) - ct(e2)) = 0 > > Since x(e2) is negative, and so is -ct(e2), it follows that > this is only possible if > > sigma = 0 > > Substituting this into my equation 0.2 gives > > (4) x' + c t' = mu (x + ct) > OK, but you should note that fully written your equations (3) and (4) read (1) x'(e1) - c t'(e1) = lambda (x(e1) - ct(e1)) (2) x'(e2) + c t'(e2) = mu (x(e2) + ct(e2)) so you can't use them to determine lambda and mu (and hence the Lorentz transformation) like Einstein did. But you can determine them in fact from your original equations (3) x'(e1) = A x(e1) + B ct(e1) ct'(e1) = D x(e1) + E ct(e1) (4) x'(e2) = A x(e2) + B ct(e2) ct'(e2) = D x(e2) + E ct(e2) which using your conditions (5) x(e1)=ct(e1); x'(e1)=ct'(e1) (6) x(e2)=-ct(e2); x'(e2)=-ct'(e2) yields (7) A+B=D+E (8) A-B=E-D. If you add and subtract (7) and (8), you get therefore (9) A=E; B=D. Now you have found previously that your constants tau and sigma are zero, so you from your defintions of tau and sigma: (10) A-D-B+E=0 (11) A+D-B-E=0 from which you get (12) A-B=D-E (13) A-B=E-D. Because the left hand sides of (12) and (13) are identical but the right hand side merely has the sign inverted, this means that (14) A=B; D=E. Now together with (9) you have therefore (15) A=B=D=E, which, going back to the definitions of your constants, tells you that not only tau and sigma but also lambda is zero. The only cnstant left to determine is mu , but from (2) and (6) we know that (16) 0=mu*0, so mu (and hence the numerical value of (15)) is arbitrary, which makes sense because the original equations (3) and (4) read now (17) x'(e1) = A x(e1) + A ct(e1) ct'(e1) = A x(e1) + A ct(e1) (18) x'(e2) = A x(e2) + A ct(e2) ct'(e2) = A x(e2) + A ct(e2) which using (5) and (6) yields (19) x'(e1)-ct'(e1)=0 (20) x'(e2)+ct'(e2)=0, which are just your separate equations for the two light signals all over again. This shows that a consistent set of equations (like yours) can not result in the Lorentz transformation. Thomas |