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LENGTH CONTRACTION AND TIME DILATION


NOTE: The upper case "SPACE" denotes the abstract mathematical 
construct, in order to distinguish it from the "space" of 
perception. 


CONTEXT


Let X,Y lighttimeSpace referentials of coordinates x(i),y(i), 
moving relativly at the speed V along x(2),y(2), chosen as 
parallel. 
For simplicity's sake we shall disregard the trivial space
dimensions x3,y3,x4,y4 and consider X,Y in 2D Minkowski 
SPACE with Lighttime coordinates x1=ctx, y1=cty, space 
coordinates x2,y2 and signature: [-1 0,0 1].
 
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 "not-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]


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]
which implies:
dn2=dh2/gamma              [1c]

We see that dn2 < dh2    
 
[1c] is Lorentz Space Contraction.


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]  
implying differentials or "elementary covering Rods".
(see Einstein's Covering Principle in "NATURAL MODEL")
dh1=gamma*dn1                [3]
and
dh1 > dn1

Lighttime distance corresponding to the elementary period 
between two clock ticks, is measured. as any distance, with 
Rods, which contract in the N (not-home, relatively moving) 
Referential. Thus, the elementary period of N takes 
more Rods, or is longer than that of H (home Referential).
The N-clock ticks slower than H-clock. Calling elementary
periods of H and N respectively dth and dtn, we get:

dtn > dth               [4]   

Whatever may be the metaphysical vision of time, we consider 
in Physics time as that what is measured by intervals between 
clock ticks, dismissing hypothesis of its nature. We say 
that time "dilates" when the clock ticks slower and that 
time dilates in the N (relatively moving) Referential with 
respect to the relatively stationary H.

This effect of clocks relatively slowing down in relatively
moving Referentials is known as Lorentz Time Dilation.