I hope I have finally understood the one page article "World's Fastest
Derivation of the Lorentz Transformation" found at
http://faculty.luther.edu/~macdonal/LorentzT.pdf

I am posting a rewrite in, I think, more intelligible language. If it
gets a favorable reception here, I might have the nerve to send it to
Prof. Macdonald. It is interesting in that the linearity of the
transformation and the fact that when considering the inverse, X'=0
gets transformed into X=-vT comes from the proof and is not an
assumption. However, he assumes that time dilation has been
established on the wordline for the origin of the moving frame. I
will try to use his notation as far as possible.

Only the 2D case is discussed in the paper. I, I' are the rest and
moving frames. The coordinates of an event E in I are (X(E), T(E)); in
I' (X'(E), T'(E)). We are seeking a transformation that maps
(X(E),T(E)) into (X'(E),T'(E)) for all events E. The worldline of the
moving origin of I' is referred to as X=vT, O, or X'=0. The two
branches in I of the lightcone at E are I, L-: T+X=T(E)+X(E), L+:
T-X=T(E)-X(E), so c=1.

Assumptions:
(A) The equations for the lightcones at E in I' are L'-:
T'+X'=T'(E)+X'(E), L'+: T'-X'=T'(E)-X'(E) and L-, L+ get mapped into
L'-, L'+ respectively by the sought for transformation.
(B) On O, T=Y*T', where I use Y for gamma. It is assumed that gamma is
a function only of |v|, v<1, and gamma is positive. The last is not
stated by Macdonald but he needs it when he takes the positive square
root to find gamma.

On O,
T+X=Y(1+v)(T'+X'), (1)
T-X=Y(1-v)(T'-X'). (2)
The intersections of O with the lightcones for any event E satisfy:
T(E)+X(E)=Y(1+v)(T'(E)+X'(E)),
T(E)-X(E)=Y(1-v)(T'(E)-X'(E)) yielding by addition and subtraction:
T(E)=Y(T'(E)+vX'(E))) (3)
X(E)=Y(vT'(E)+X('E)). (4).

For X(E)=0, the last equation gives X'(E)=-vT'(E). Thus we can use
the previous reasoning to get (1) again with primes reversed, -v
instead of v and the explicit reference to E dropped:
T'+X'=Y(1-v)(T+X). Then plugging this into (1), we get
Y²(1-v²)=1 which completes the proof.

Comments?

On Saturday, 25 June 2022 at 22:19:16 UTC+2, Ricardo Jimenez wrote:
> I hope I have finally understood the one page article "World's Fastest
> Derivation of the Lorentz Transformation" found at
> http://faculty.luther.edu/~macdonal/LorentzT.pdf
>
> I am posting a rewrite in, I think, more intelligible language. If it
> gets a favorable reception here, I might have the nerve to send it to
> Prof. Macdonald. It is interesting in that the linearity of the
> transformation and the fact that when considering the inverse, X'=0
> gets transformed into X=-vT comes from the proof and is not an
> assumption. However, he assumes that time dilation has been
> established on the wordline for the origin of the moving frame. I
> will try to use his notation as far as possible.

In the meantime in the real world, "improper" clocks of
GPS and TAI keep measuring t'=t, just like all serious clocks
always did.