Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
|
From: Thomas Smid on 3 Sep 2005 10:21 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 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. Thomas
From: Thomas Smid on 3 Sep 2005 10:40 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 and a > ticket to fame. As far as I am aware the experimental discovery of the invariance of c was big news at the time, and it exactly 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. Thomas
From: Daryl McCullough on 3 Sep 2005 11:15 Thomas Smid says... >As far as I am aware the experimental discovery of the invariance of c >was big news at the time, and it exactly 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. That's right. The geometry of spacetime is Minkowskian, not Euclidean. -- Daryl McCullough Ithaca, NY
From: Thomas Smid on 3 Sep 2005 11:44 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: What is the justification for such an assumption? >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) OK, with x'(e1) = ct'(e1) your original equations x'(e1) = A x(e1) + B ct(e1) ct'(e1) = D x(e1) + E ct(e1) result therefore in A x(e1) + B ct(e1) = D x(e1) + E ct(e1) and since this must for all times t D=A E=B which using your definitions above results in lambda=0 tau=0. Similarly for the ray travelling in the negative direction i.e. x'(e2) = -ct'(e2) we get from your original equations A x(e2) + B ct(e2) = -D x(e2) - E ct(e2) and thus D=-A E=-B and hence mu=0 sigma=0. So as with Einstein's approach, you again have the inconsistent equations x'-ct'=0 and x'+ct'=0. Thomas
From: Todd on 3 Sep 2005 13:29
"Thomas Smid" <thomas.smid(a)gmail.com> wrote in message news:1125762269.196256.173470(a)z14g2000cwz.googlegroups.com... > Daryl McCullough wrote: >>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) > OK, with x'(e1) = ct'(e1) your original equations > x'(e1) = A x(e1) + B ct(e1) > ct'(e1) = D x(e1) + E ct(e1) > result therefore in > A x(e1) + B ct(e1) = D x(e1) + E ct(e1) Yes, as long as you're considering events where x'(e1) = ct'(e1). > and since this must for all times t > D=A > E=B No, this is an incorrect conclusion. Think about it and see if you can see your mistake. Hint: If x'(e1) = ct'(e1), then how must x(e1) and ct(e1) be related? Todd |