Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
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From: Daryl McCullough on 6 Sep 2005 07:43 Thomas Smid says... > >Todd wrote: >> 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)). You cannot assume that x'(e2)=-x'(e1)! Can you please acknowledge that this is a *new* objection to Einstein's derivation? It has nothing to do with your *original* reason for saying that Einstein's derivation was inconsistent. More generally, let a spherical wavefront spread out from the point x=0 at time t=0. Let e1, e2 and e3 be such that 1. x(e1) = c t(e1) 2. x(e2) = - c t(e2) 3. t(e2) = t(e1) 4. x'(e1) = c t'(e1) 5. x'(e3) = - c t'(e3) 6. t'(e3) = t'(e1) 7. x(e3) = - c t(e3) 8. x'(e2) = - c t(e2) You are assuming that e2 and e3 are the same event. But there is no reason for assuming that events that are simultaneous in one frame are simultaneous in another frame. e1 and e2 are simultaneous in the (x,t) coordinate system, but not in (x',t') coordinate system, and e1 and e3 are simultaneous in the (x',t') coordinate system, but not in the (x,t) coordinate system. -- Daryl McCullough Ithaca, NY
From: Thomas Smid on 6 Sep 2005 10:52 Daryl McCullough wrote: > Thomas Smid says... > > > >Todd wrote: > > >> 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)). > > You cannot assume that x'(e2)=-x'(e1)! > > Can you please acknowledge that this is a *new* > objection to Einstein's derivation? It has nothing to do with > your *original* reason for saying that Einstein's derivation > was inconsistent. > > More generally, let a spherical wavefront spread out from the point > x=0 at time t=0. Let e1, e2 and e3 be such that > > 1. x(e1) = c t(e1) > 2. x(e2) = - c t(e2) > 3. t(e2) = t(e1) > > 4. x'(e1) = c t'(e1) > 5. x'(e3) = - c t'(e3) > 6. t'(e3) = t'(e1) > > 7. x(e3) = - c t(e3) > 8. x'(e2) = - c t(e2) > > You are assuming that e2 and e3 are the same event. But > there is no reason for assuming that events that are > simultaneous in one frame are simultaneous in another > frame. e1 and e2 are simultaneous in the (x,t) coordinate > system, but not in (x',t') coordinate system, and e1 and e3 > are simultaneous in the (x',t') coordinate system, but not > in the (x,t) coordinate system. The idea that the equations x=ct and x'=ct' describe events at a certain point in time and space is misleading here. These equations do in fact represent the location of the wavefront in both reference frames (x,x') as a function of time (t,t') and the equations must hold true for all values of the independent variables (t,t'). In order to emphasize the fact that we are dealing with continuous variables here, it is better if we drop the arguments (e1) and (e2) that you introduced earlier and merely use x1,x2 and x1',x2' and t and t' (obviously t and t' are scalars and we don't need to distinguish regards direction here) My sets of equations (3) and (4) in my post 74 (by date) then read (3a) x1=ct (3b) x1'-c t' = (A-B)(x1-ct) (= 0) (3c) x1'+c t' = (A+B)(x1+ct) and (4a) x2=-ct (4b) x2'-c t' = (A-B)(x2-ct) (4c) x2'+c t' = (A+B)(x2+ct) (= 0) Now from Eqs.(3c) and (4b) (and taking the other equations into account) we have thus immediately without making any further assumptions (5) 2ct'=(A+B)2ct (6) 2ct'=(A-B)2ct i.e. (7) B=0 Thomas
From: Thomas Smid on 6 Sep 2005 10:56 Daryl McCullough wrote: > Thomas Smid says... > > > >Todd wrote: > > >> 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)). > > You cannot assume that x'(e2)=-x'(e1)! > > Can you please acknowledge that this is a *new* > objection to Einstein's derivation? It has nothing to do with > your *original* reason for saying that Einstein's derivation > was inconsistent. > > More generally, let a spherical wavefront spread out from the point > x=0 at time t=0. Let e1, e2 and e3 be such that > > 1. x(e1) = c t(e1) > 2. x(e2) = - c t(e2) > 3. t(e2) = t(e1) > > 4. x'(e1) = c t'(e1) > 5. x'(e3) = - c t'(e3) > 6. t'(e3) = t'(e1) > > 7. x(e3) = - c t(e3) > 8. x'(e2) = - c t(e2) > > You are assuming that e2 and e3 are the same event. But > there is no reason for assuming that events that are > simultaneous in one frame are simultaneous in another > frame. e1 and e2 are simultaneous in the (x,t) coordinate > system, but not in (x',t') coordinate system, and e1 and e3 > are simultaneous in the (x',t') coordinate system, but not > in the (x,t) coordinate system. The idea that the equations x=ct and x'=ct' describe events at a certain point in time and space is misleading here. These equations do in fact represent the location of the wavefront in both reference frames (x,x') as a function of time (t,t') and the equations must hold true for all values of the independent variables (t,t'). In order to emphasize the fact that we are dealing with continuous variables here, it is better if we drop the arguments (e1) and (e2) that you introduced earlier and merely use x1,x2 and x1',x2' and t and t' (obviously t and t' are scalars and we don't need to distinguish regards direction here) My sets of equations (3) and (4) in my post 74 (by date) then read (3a) x1=ct (3b) x1'-c t' = (A-B)(x1-ct) (= 0) (3c) x1'+c t' = (A+B)(x1+ct) and (4a) x2=-ct (4b) x2'-c t' = (A-B)(x2-ct) (4c) x2'+c t' = (A+B)(x2+ct) (= 0) Now from Eqs.(3c) and (4b) (and taking the other equations into account) we have thus immediately without making any further assumptions (5) 2ct'=(A+B)2ct (6) 2ct'=(A-B)2ct i.e. (7) B=0 Thomas P.S.: This is not a new objection to Einstein's derivation (not a direct one anyway) but merely an attempt to continue your derivation where you left it off.
From: Thomas Smid on 6 Sep 2005 11:02 Daryl McCullough wrote: > 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. What he assumed or not is irrelevant for the mathematical consistency of the equations. I can only judge the derivation from the mathematical symbols he uses and when he uses the equations x-ct=0 and x+ct=0 in the same context, they *are* mathematically inconsistent. Thomas
From: Dirk Van de moortel on 6 Sep 2005 12:05
"Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125998692.023918.86680(a)g14g2000cwa.googlegroups.com... > > 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). Nice. "If I had to review a paper": http://users.pandora.be/vdmoortel/dirk/Physics/Fumbles/ReviewPaper.html Don't stop, Daryl ;-) Dirk Vdm |