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."

  
      Length Contraction and Time Dilation

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Length Contraction and Time Dilation.
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CONTEXT
We shall disregard for simplicity's sake the trivial dimensions x3, y3, x4, y4 
and consider a 2D Minkowski space with signature: [-1 0,0 1]
 
Let X,Y lighttime-space referentials with coordinates respectively x1,x2 and 
y1.y2 where x1=ctx, y1=cty and xn respectively parallel to yn.
 
Let them move with respect to one another at constant V along x2, y2.
 
Let beta=V/C, gamma=1/sqrt(1-beta^2)
 
Lorentz Transformations may be written:
 
y2=gamma*(x2-beta*x1)       [1]
y1=gamma*(x1-beta*x2)       [2]

Let's call "observed" and "observing" referentials respectively "home" (H) and 
"non-home" (N) and the coordinates of their systems "h" and "n".
 
We may rewrite [1],[2]:
 
h2=gamma*(n2-beta*n1)       [1a]
h1=gamma*(n1-beta*n2)       [2a]
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SPACE SUB-#SPACE
 
Let's consider space sub-#space and chose coordinates so that n1=0.
[1a] becomes:
h2=gamma*n2
or
n2=h2/gamma                [1b]
and
dn2=dh2/gamma              [1c]

We see that dn2 < dh2    
 
[1c] is Lorentz Space Contraction.
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LIGHTTIME SUB-#SPACE
 
Let's consider lighttime sub-#space and chose coordinates so that n2=0.

[2a] becomes:
h1=gamma*n1 or n1=h1/gamma              [2b]  

dn1=dh1/gamma  contraction of Ct rod

Let's take Ct components of distance in H, observed from H and N be SH and SN.

[2b] tells us that SN will need more contracted rods than SH. 
Thus: SN > SH     [3]
 
But SH = Ct(H)  and SN = Ct(N)

C being constant we get from [3]:

t(N) > t(H)         [4]

[4] expresses Lorentz Time Dilation.
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