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PLAN OF DB SPECIAL RELATIVITY
 dba axioms of special relativity 
 dbb lorentz transformations 
 dbc length contraction and time dilation 
 dbd speed cumulation 
 dbe E=MC^2 
 dbf paradox of langevins traveller 

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DBC.LENGTH CONTRACTION AND TIME DILATION

CONTEXT


NOTE: neologies and ambiguous terms clarified in GLOSSARY
are marked "[G]".

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[G] 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 "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]


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".
(Einstein's Covering Principle [G])
dh1=gamma*dn1                [3]
and
dh1 > dn1

Lighttime distance corresponding to the elementary period 
between two clock ticks, is measured. as any distance, with 
those Rods, which contract in the N (non-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.