Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
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From: Daryl McCullough on 6 Sep 2005 11:50 Thomas Smid says... > >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. What Einstein clearly meant was that forall x, forall t, if x=ct, then x'=ct' forall x, forall t, if x=-ct, then x'=-ct' As is usually, the case, he left off the explicit quantifiers, assuming that the reader was competent to understand that they were implied. -- Daryl McCullough Ithaca, NY
From: Dirk Van de moortel on 6 Sep 2005 12:11 "Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1126018963.246681.126050(a)f14g2000cwb.googlegroups.com... > 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. That is because you don't understand the meaning of these equations, nor that you have got a clue about the context in which they are used. You are so stupid and/or such a malicious dishonest troll, that I would almost hope that - just because of the entertainment you provide this way - no one will ever manage to explain it to you. You are so good! Dirk Vdm
From: Daryl McCullough on 6 Sep 2005 12:11 Thomas Smid says... >The idea that the equations x=ct and x'=ct' describe events at a >certain point in time and space is misleading here. On the contrary, not distinguishing events leads to confusion. >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) If you don't want to explicitly mention events, that's fine, but you certainly need to explicitly mention time. Let t1 be the time at which the light signal reaches x1, and let t2 be the time at which the light signal reaches x2. You want to arrange things so that t1 = t2. That's fine, but it *won't* necessarily be the case that t1' = t2'. >My sets of equations (3) and (4) in my post 74 (by date) then read > >(3a) x1=ct No, that should be x1 = c t1 You need to keep the subscripts straight. >(3b) x1'-c t' = (A-B)(x1-ct) (= 0) No, that should be x1' - c t1' = (A-B)(x1 - c t1) (= 0) You need to keep the subscripts straight. >(3c) x1'+c t' = (A+B)(x1+ct) That should be x1'+c t1' = (A+B)(x1+c t1) >and > >(4a) x2=-ct No, that should be x2 = -c t2 >(4b) x2'-c t' = (A-B)(x2-ct) No, that should be x2'-c t2' = (A-B)(x2-ct2) >(4c) x2'+c t' = (A+B)(x2+ct) (= 0) No, that should be x2'+c t2' = (A+B)(x2+ct2) (= 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 You've already made the critical assumption, which is that t2' = t1'. There is no reason to believe that, and as a matter of fact, that's provably *false*, given the other assumptions that you've already made. -- Daryl McCullough Ithaca, NY
From: Thomas Smid on 6 Sep 2005 13:12 Daryl McCullough wrote: > Thomas Smid says... > > >The idea that the equations x=ct and x'=ct' describe events at a > >certain point in time and space is misleading here. > > On the contrary, not distinguishing events leads to confusion. > > >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) > > If you don't want to explicitly mention events, that's > fine, but you certainly need to explicitly mention time. > Let t1 be the time at which the light signal reaches x1, > and let t2 be the time at which the light signal reaches x2. I am sorry, but you still misinterprete the variables as specific coordinates. x1 and x2 are *not* coordinates but scalar variables for the two directions representing the vector variable x. Fully written it would be x1(t)=ct, x2(t)=-ct etc. I merely left out the argument (t) in my equations. > You've already made the critical assumption, which is that > t2' = t1'. There is no reason to believe that, and as a matter > of fact, that's provably *false*, given the other assumptions > that you've already made. What other assumptions? Before this point everything was identical with your own equations. I am just trying to push this a bit further now. Thomas
From: Daryl McCullough on 6 Sep 2005 16:39
Thomas Smid says... >I am sorry, but you still misinterprete the variables as specific >coordinates. x1 and x2 are *not* coordinates but scalar variables for >the two directions representing the vector variable x. Fully written it >would be x1(t)=ct, x2(t)=-ct etc. I merely left out the argument (t) in >my equations. That doesn't change my objections to your derivation. You wrote (3c) x1'+c t' = (A+B)(x1+ct) (4b) x2'-c t' = (A-B)(x2-ct) The point is that x' and t' are *functions* of x and t. Since x2(t) is unequal to x1(t), the corresponding value of t' in (4b) is *not* equal to the corresponding value of t' in (3c). Let's put in all the functional dependence explicitly. The transformation equations are these x'(x,t) + c t'(x,t) = (A+B) (x + ct) x'(x,t) - c t'(x,t) = (A-B) (x - ct) Now, you want to specialize to the case x1(t) = ct in the first equation, and specialize to the case x2(t) = -ct in the second equation. Fine. That gives us: x'(ct,t) + c t'(ct,t) = (A+B) (2ct) x'(-ct,t) - c t'(-ct,t) = (A-B) (-2ct) Now, we use the fact that x' = ct' in the top equation, and x' = -ct' in the bottome equation to get: 2 c t'(ct,t) = (A+B) (2ct) -2 c t'(-ct,t) = (A-B) (-2ct) Your mistake was assuming that t'(ct,t) = t'(-ct,t). -- Daryl McCullough Ithaca, NY |