Mathematical Inconsistencies in Einstein's Derivation of the Lorentz Transformation
in [Relativity]
|
From: Daryl McCullough on 3 Sep 2005 13:50 Thomas Smid says... > >Daryl McCullough wrote: >> 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? The principle of relativity. Consider an object moving in the absence of any forces. Then the object must travel at a constant velocity. So the relationship between the two coordinate systems must be such that constant-velocity motion transforms to constant-velocity motion. That implies that the transformation is linear. >>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) [stuff deleted] >> 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 No, it's not for all times! It's for one specific time, which is the time of event e1. You are confusing constants with variables. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 3 Sep 2005 13:58 Thomas Smid says... >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 No. Since x(e1) = ct(e1), it follows that A ct(e1) + B ct(e1) = D ct(e1) + E ct(e1) which implies that A + B = D + E -- Daryl McCullough Ithaca, NY
From: Thomas Smid on 3 Sep 2005 14:16 Todd wrote: > "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. Yes, I noticed this already and I had already deleted my post before you posted it. I shall be posting a revised version of this. Thomas
From: Ken S. Tucker on 3 Sep 2005 16:09 Daryl McCullough wrote: > Ken S. Tucker says... > > >Bilge (potato head) wrote: > >> Do you believe the coordinate transformation, > >> > >> x' = x cos(A) - y sin(A) > >> y' = y cos(A) + x sin(A) > >> > >> is mathematically inconsistent? If not, then your argument against > >> the transformation, > >> > >> t' = t cosh(A) - x sinh(A) > >> x' = x cosh(A) - t sinh(A) > >> > >> is inconsistent. > > > >Ding-bat, x and x' are defined parallel, no > >relative rotation occurs. > > Bilge is talking about a generalized spacetime rotation, > using hyperbolic trigonometric functions instead of ordinary > trigonometric functions. His equations explain the analogy > very well. The parameter A is defined by tanh(A) = v/c. Then > cosh(A) = square-root(1/(1-tanh^2(A))) = gamma. > sinh(A) = tanh(A) cosh(A) = gamma v/c. So his > "rotation" equations are equivalent to the usual > Lorentz transformations. > Daryl McCullough >From Androcles post...quote AE, "let a spherical wave be emitted therefrom, and be propagated with the velocity c in system K. If (x, y, z) be a point just attained by this wave, then x2+y2+z2=c2t2. Transforming this equation with the aid of our equations of transformation we obtain after a simple calculation"... AE Why use an incorrect complex analogy if the simple physics is sufficient. Setting x and x' in differing directions has bugged me since HS, because it's WRONG, it's a crumby theoretical inventions that fucks reality. AE's spherical wave gives, 0 = ds^2 = (cdt)^2 - dr^2 and then generalized to, 0 = g_uv dx^u dx^v . See that, evolved from SR threw Minkowski SpaceTime to GR, - no silly x vs x' rotations analogy's - it's the Lorentz Transform in a nutshell. Regards Ken S. Tucker PS: Androcles, thanks for the link.
From: Dirk Van de moortel on 3 Sep 2005 16:53
"Daryl McCullough" <stevendaryl3016(a)yahoo.com> wrote in message news:dfcnps02504(a)drn.newsguy.com... > Thomas Smid says... > > > >Daryl McCullough wrote: > > >> 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? > > The principle of relativity. Consider an object moving > in the absence of any forces. Then the object must travel > at a constant velocity. So the relationship between the > two coordinate systems must be such that constant-velocity > motion transforms to constant-velocity motion. That implies > that the transformation is linear. > > >>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) > > [stuff deleted] > > >> 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 > > No, it's not for all times! It's for one specific > time, which is the time of event e1. > > You are confusing constants with variables. Actually, I think he is confusing the contents of his skull with brains :-) Dirk Vdm |