PARENTS:
 LORENTZ TRANSFORMATIONS 
 RELATIVITY WORKSHOP 
 WORKSHOPS 
 ARCHIVE   
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Derivation of Lorentz Transformation (Georges)
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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
would not be 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
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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PARENTS:
  LORENTZ TRANSFORMATIONS 
 RELATIVITY WORKSHOP 
 WORKSHOPS 
 ARCHIVE