Monday, July 27, 2009

Lorentz Transform and Related Concepts

Review: Lorentz Transform and Related Concepts

Lorentz Transform (LT) measures relativistic growth of particles due to very high speeds, significant portion of c, light speed.

mr =mo
√(1 - v2/c2)

where relativistic mass, mr, is traveling at high speed, v.

mr growth is predicted by LT, where is original mass size is mo.

c, light speed in vacuum, has been shown by Einstein to be constant.

Einstein also used "thought experiments" to demonstrate many of his concepts. In a small way, we'll emulate this method and construct a spacship which uses a particle accelerator for propulsion. Our experiment will use LT to describe particle growth due to high velocity.

ffExh =ffSec
√(1 - v2Exh/c2)
ffsec, orginal fuel particles at relative rest with spaceship. The subtag "sec" is chosen because this quantity represents the amount of particles needed to sustain fuel flow for one second.

ffExh, same fuel particles as for ffsec, except they have accelerated to near light speed prior to expulsion from spaceship where their momentum propels spaceship in opposite direction. Spaceship momentum gains from exhaust particle's extreme speed as well as relativistic mass growth.

vExh, exhaust particle's speed.

c, light speed through a vacuum, is a constant 299,792,458 metres per second which is often rounded to 300 million m/s.

Concept of n, multiple of ffsec

Let n be a multiple of ffsec such that
ffExh = n * ffsec
In that case,
n = 1/√(1 - v2Exh/c2)
Tables simplified this concept by restricting discussion to a small subset of n, all integers such n > 1.

Concept of d, decimal c.

Let d be a coefficient of c, such that
vExh = d * c
Further, let d be a decimal number between 0 and 1 .
Then, v2Exh/c2 = d2 c2 / c2
Which reduces to simply: d2

Relationship of n to d
n = 1/√(1 - v2Exh/c2)
changes to
n = 1/√(1 - d2)
n2 = 1/(1 - d2)
1 - d2 = 1 / n2
d2 = 1 -( 1 / n2)
d2 = (n2 - 1) / n2
d = √(n2 - 1)/n

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