In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

  
      Derivation of Lorentz Transformation 

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Let u(/1),u(/2),u(/3) orthogonal coordinates of 3D Euclidian space and t, time.

A spherical wave emitted from origin of space coordinates u(/i),t reaches after 
time dt the points du(/i)=Cdt where C= speed of light.

Let's introduce del, the Kroeckner Symbol ot the Fundamental Tensor of the 
Euclidian #space:

del(ij/)=1 for i=j 
del(ij/)=0 for i!=j 
or in matrix form:
del(ij/)=[100,010,001].
   
Let's further introduce Einstein's indexing notation implying summation over 
each index repeated within a monome as upper and lower one.

Then a radius dr of the sphere is given by: dr^2=del(ij/)du(/i)du(/j) 
where dr=Cdt, C being the speed of light.
Thus:
(Cdt)^2 = del(ij/)du(/i)du(/j), or (Cdt)^2 - del(ij/)du(/i)du(/j) = 0  [1]

We find ourselves here at cross-roads.

A.We may continue to consider two distinct #spaces: 
1.time (dt), 
2.space (del(ij/)du(/i)du(/j))

B.We may take advantage of Cdt and del(ij/)du(/i)du(/j) having the same measure 
of distance, thus [1] implying a 4D metric Minkowski #space (MinSp).

Question arises: Could a theory supporting invariance of C be constructed upon 
the assumption A? Possibly, but such a theory has never been constructed and it
would not have been Einstein's SR, which is based upon B.

Consequently we shall consider as an additional axiom of SR the choice of MinSp
as #space of SR's MS. 

We shall consequently continue our LT derivation within MinSp.

Let's recall some basic concepts of MinSp:

Fundamental Tensor mu(ij/):
mu(ij/) = -1 for i=j=1
mu(ij/) =  1 for i=j=2,3,4
mu(ij/) =  0 for i!=j
or in matrix form: 
mu(ij/)=[-1000 0100 0010 0001]

Base vectors: e1(1/)=i em(m/)(m=2,3,4)=1  [i=sqrt(-1)]
and ek(l/)=0 for l!=k.

In SR instance of Minkowski #space:
x(/1)=Ct (light-time) 
x(/m)(m=2,3,4) space dimensions.

NOTE: the fundamental difference between the Pre-SR "(t,x)" 4D #space 
(t,x(/m)(m=2,3,4)) and SR's "(Ct,x)" 4D #space consists in the first 
being affine and the second - metric.
Indeed, there is no common measure between t and x in Pre-SR #space, while 
all coordinates of SR #space have the common measure of "distance" (including, 
of course, the light-time Ct).  Consequently, SR #space admits metric as
described above and rotation-type transformation, namely pseudo-rotation in 
the pseudo-orthogonal complex plane Ct / x(/m). This pseudo-rotation is 
equivalent with Lorentz Transformation as will be shown below.
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The invariant form ds^2 in SR #space:
ds^2=(dx(/1))^2-sigma((dx(/m))^2)(m=2,3,4)
or
ds^2=(Ct)^2-sigma((dx(/m))^2)(m=2,3,4)

Pseudo-rotation transforming x,t to X,T moving along x(/2), keeping invariant ds^2:

ct    = X(/1)sh(th) * cT ch(th) 
x(/2) = X(/2) ch(th) + cT sh(th)
x(/3) = X(/3)
x(/4) = X(/4)

where sh, ch are hyperbolic functions.

Putting th(th) = v/c:

t     = (T + (v/c^2)X(/2)) / sqrt(1 - v^2/c^2) 
x(/2) = (X(/2) + vT) / sqrt(1 - v^2/c^2) 
x(/3) = X(/3)
x(/4) = X(/4)

Which is the Lorentz Transformation.
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