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DERIVATION OF LORENTZ TRANSFORMATIONS
Some notational conventions
Lower and upper indexes indicate co- and contravariant
components of vectors or tensors.
j
Thus A and B denote the i-th covariant component of A
i
and the j-th contravariant component of B.
In cases when the variance is not yet defined we shall
write C(k) as the k-th component of C of unknown variance.
We shall write derivative of y with respect to x as: d(y)/d(x)
and partial derivative of u with respect to v: &part(u)/&part(v)
Einstein's indexing notation implies summation over each index
repeated within a monome as upper and lower one.
Thus, for 3d:
i 1 2 3
A B = A B + A B + A B
i 1 2 3
This convention applies also to partial derivatives as follows:
i j j
(&part(x )/&part(y ))dy =
i 1 1
(&part(x )/&part(y ))dy +
i 2 2
(&part(x )/&part(y ))dy +
i 3 3
(&part(x )/&part(y ))dy
Derivation
Let SpaceTime be a 4D SPACE of space coordinates
1 2 3
u ,u ,u and time coordinate t, affine between
space and time, which don't have a common measure.
i
Spherical light wave emitted origin of coordinates u ,t
i
reaches after time dt points du =Cdt where C= speed of
light.
Let's introduce the Fundamental Euclidean Tensor δ,
known as Kronecker Symbol:
δ =1 for i=j
ij
δ =0 for i!=j
ij
or in matrix form:
δ =[100,010,001]
ij
In Einstein's indexing notation a radius dr of the sphere
is given by:
i j
dr^2=δ du*du
ij
where dr=Cdt, C being the speed of light.
Thus:
i j
(Cdt)^2 = δ du*du, or
ij
i j
(Cdt)^2 - δ du*du = 0 [1]
ij
We find ourselves here at cross-roads.
A.We may continue to consider two distinct affine subspaces:
1.time (dt),
i j
2.space δ du*du
ij
i j
B.We may take advantage of Cdt and δ du*du
ij
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 the SPACE of SR and we shall
continue the LT derivation within MinSp.
Let's recall some basic concepts of MinSp:
Fundamental Tensor μ:
ij
μ = -1 for i=j=1
ij
μ = 1 for i=j=2,3,4
ij
μ = 0 for i!=j
ij
or in matrix form:
μ=[-1000 0100 0010 0001]
ij
Base vectors: e1=i em(m=2,3,4)=1 [i=sqrt(-1)]
1 m
and ek=0 for l!=k.
l
In SR instance of Minkowski SPACE:
1
x=Ct (LightTime)
m
x(m=2,3,4) space dimensions.
NOTE: the fundamental difference between the Pre-SR "(t,x)"
m
4D SPACE (t,x(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 the LightTime Ct).
Consequently, SR SPACE admits metric as described above and
rotation-type transformation, namely pseudo-rotation in the
m
pseudo-orthogonal complex plane Ct / x. This
pseudo-rotation is equivalent with Lorentz Transformation,
as will be shown below.
The invariant form ds^2 in SR SPACE:
1 m
ds^2=(dx)^2-Σdx^2(m=2,3,4)
or
m
ds^2=(Ct)^2-Σdx^2(m=2,3,4)
2
Pseudo-rotation transforming x,t to X,T moving along x,
keeping invariant ds^2:
1
Ct = Xsh(θ) * CT ch(θ)
2 2
x = X ch(θ) + CT sh(θ)
3 3 4 4
x = X x = X
where sh, ch are hyperbolic functions.
Putting th(θ) = v/c:
2
t = (T + (v/c^2)X) / sqrt(1 - v^2/c^2)
2 2
x = (X + vT) / sqrt(1 - v^2/c^2)
3 3 4 4
x = X x = X
Which are the Lorentz Transformations.
