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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 BACK TO SITE PLAN Site Plan :
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.