Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
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From: Sue... on 6 Sep 2005 03:36 Thomas Smid wrote: > 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? Let's > for example have a look at Einsteins own derivation (from his book > 'Relativity: The Special and General Theory') given at > http://www.bartleby.com/173/a1.html which seems to be a very elegant > way of deriving the Lorentz transformation. > > It is only necessary here to examine the initial equations for this, > which describe the 'equations of motion of a light signal' in the > unprimed and primed reference frames, i.e. > > (1) x-ct=0 > (2) x'-ct'=0 > where c is the speed of light (which obviously has to be a constant >0) > > In the same way, the propagation of a signal in the opposite direction > yields > (3) x+ct=0 > (4) x'+ct'=0 > (note that these equations are not written explicitly in Einstein's > derivation). > > >From equations (1)-(4), the Lorentz transformation is then derived by > some algebraic manipulations. > > But are the above equations mathematically consistent at all? Let's > subtract equation (1) from (3), which yields > (5) 2ct=0 > which means that for any time t>0 > (6) c=0, > in contradiction to the requirement that c>0. > > This shows that the equations used to derive the Lorentz transformation > are mathematically inconsistent. The fact that the Lorentz > transformation itself seems to be mathematically consistent only > demonstrates that the 'length contractions' and 'time dilations' > involved in the completion of the derivation are not ony physically > unacceptable (as argued on my page > http://www.physicsmyths.org.uk/lightspeed.htm ) but also mathematically > inconsistent as they contradict the initial definitions. > > Thomas The equations are constistant to the degree they extend the spatial relations of Pythagoras rules. They are inconsistant in the assumption about time and velocity where they ignore the contribution of matter to the permeability and permittivity of the so called *vacuum*. http://physics.nist.gov/cuu/Images/alphaeq.gif http://physics.nist.gov/cuu/Constants/alpha.html Sue...
From: Thomas Smid on 6 Sep 2005 04:53 Todd wrote: > "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message > news:1125952408.242594.268300(a)g43g2000cwa.googlegroups.com... > > > Now consider both wavefronts at the same time in each reference frame > > i.e. for t(e2)=t(e1) and t'(e2)=t'(e1). > > If e1 and e2 are simultaneous in the unprimed frame, then you can't assume > that they will also be simultaneous in the primed frame. Sure I can assume it. Consider a spherical wavefront emitted from the origin of both coordinate systems at the moment these coincide. The invariance of c requires that observers at positions x and -x in the unprimed frame observe the signal at the same time t and observers at positions x' and -x' in the primed frame observe the signal at the same time t' (x(e2)=-x(e1) requires t(e2)=t(e1) and x'(e2)=-x'(e1) requires t'(e2)=t'(e1)). Thomas
From: Thomas Smid on 6 Sep 2005 05:15 Daryl McCullough wrote: > Thomas Smid says... > > >OK, with A=E and B=D we have then > > Wait a minute. Are you now in agreement that you > were wrong about your *original* reason for saying > that Einstein's derivation is inconsistent? You > seem to be moving onto a completely different > argument. > > We've discovered algebraic mistakes that you've > made, and algebraic mistakes that I've made, but > we have not yet found an algebraic mistake that > Einstein made. I have never claimed that Einstein made an algebraic mistake but that his equations are mathematically inconsistent as he worked from the equations x-ct=0 x+ct=0 which is inconsistent unless everything is identically zero. If I had to review a paper containing these equations, I would reject it out of hand (whether it was written by Einstein or anybody else). Your approach is in fact the reverse of Einstein's ( http://www.bartleby.com/173/a1.html ): you start off with the transformation involving A and B and derive the equations involving lambda and mu and you avoid Einstein's inconsistency by introducing different variables for the two light signals, but you still haven't derived values for tau and mu (or A and B) yet, whereas I showed above that B=0 and A=1 and thus x'=x and t'=t (which in fact satisfies the invariance of c). Thomas
From: Thomas Smid on 6 Sep 2005 05:24 Daryl McCullough wrote: > Thomas Smid says... > > >OK, with A=E and B=D we have then > > Wait a minute. Are you now in agreement that you > were wrong about your *original* reason for saying > that Einstein's derivation is inconsistent? You > seem to be moving onto a completely different > argument. > > We've discovered algebraic mistakes that you've > made, and algebraic mistakes that I've made, but > we have not yet found an algebraic mistake that > Einstein made. I have never claimed that Einstein made an algebraic mistake but that his equations are mathematically inconsistent as he worked from the equations x-ct=0 x+ct=0 which is inconsistent unless everything is identically zero. If I had to review a paper containing these equations, I would reject it out of hand (whether it was written by Einstein or anybody else). Your approach is in fact the reverse of Einstein's ( http://www.bartleby.com/173/a1.html ): you start off with the transformation involving A and B and derive the equations involving lambda and mu and you avoid Einstein's inconsistency by introducing different variables for the two light signals, but you still haven't derived values for lambda and mu (or A and B) yet, whereas I showed above that B=0 and A=1 and thus x'=x and t'=t (which in fact satisfies the invariance of c). Thomas
From: Daryl McCullough on 6 Sep 2005 07:26
Thomas Smid says... >I have never claimed that Einstein made an algebraic mistake but that >his equations are mathematically inconsistent as he worked from the >equations >x-ct=0 >x+ct=0 >which is inconsistent I've explained to you that you are wrong about that. Einstein did not assume that x-ct = 0. As I explained, those two equations refer to *different* events: x(e1) - ct(e1) = 0 x(e2) + ct(e2) = 0 >unless everything is identically zero. If I had >to review a paper containing these equations, I would >reject it out of hand (whether it was written by Einstein >or anybody else). But you haven't pointed out any mistakes in Einstein's derivation. You've only pointed out mistakes in your *own* understanding. >Your approach is in fact the reverse of Einstein's ( >http://www.bartleby.com/173/a1.html ): you start off with the >transformation involving A and B and derive the equations involving >lambda and mu and you avoid Einstein's inconsistency by introducing >different variables for the two light signals. No, Einstein assumed (wrongly in your case) that the reader would understand that x=ct and x=-ct referred to *different* circumstances: the first circumstance was a light signal travelling in the +x direction, the second was a light signal travelling in the -x direction. Obviously, it's not the same x, and not the same t. >but you still haven't derived values for lambda and mu >(or A and B) yet, whereas I showed above that B=0 and A=1 No, you didn't! You showed that *if* (condition of absolute simultaneity) t(e1) = t(e2) and t'(e1) = t'(e2) then B=0. That is, if events that are simultaneous in one frame are simultaneous in another frame, then B must equal 0. But there is no reason for assuming absolute simultaneity. Once again, you've pointed out errors of yours, not Einstein's. -- Daryl McCullough Ithaca, NY |