Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
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From: Thomas Smid on 2 Sep 2005 05:06 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
From: Aage Andersen on 2 Sep 2005 05:26 "Thomas Smid" > which means that for any time t>0 > (6) c=0, > in contradiction to the requirement that c>0. If you let c be the speed of your car, you have thus shown that it never move. Aage
From: Curt on 2 Sep 2005 05:46 "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125652015.288928.309540(a)z14g2000cwz.googlegroups.com... ........ > 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 ........... > > 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. I am no expert in relativity, but I am aware that x, displacement, is a vector. In the following argument I take c>0, x>0. If I fire a photon in the positive x direction, from the origin of my frame of reference, then, after t, the photon has travelled x. Thus, x=ct, in accordance with (1).But how did you derive (3)? All you did was change the sign of c, neglecting to change the sign of x. If I fire a photon in the negative x direction, the photon, in my reference frame has -c velocity. After t seconds, the photon has has moved a distance x, but experienced a displacement of -x(displacement is a vector): -x = -ct Which IS seemingly consistent with (1): -1 * -x = -ct * -1 implies x=ct or x-ct=0. Is my logic flawed here? If it is please let me know so that I can remedy this fault ASAP. Curt
From: Bill Hobba on 2 Sep 2005 06:01 "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125652015.288928.309540(a)z14g2000cwz.googlegroups.com... > 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. Both equations can not be true at the same time because they consider two different situations - one is a light ray in the positive direction - the other is a light ray in the negative direction. Hence it is not valid to solver them simultaneously. Bill > > 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 >
From: "Androcles" <Androcles@ on 2 Sep 2005 06:21
"Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125652015.288928.309540(a)z14g2000cwz.googlegroups.com... | 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, they really do think they have mathematical consistency, the crazy nutters. 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) What's obvious about it? It's obvious time doesn't run backwards and obvious that you have a minus sign, therefore it is obvious that -c < 0. Androcles. | | 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 | |