The Galileian transformation equations are:
x'=x-vt
y'=y
z'=z
t'=t
Two little equations that Einstein extracted from the Lorentz equations can be found on pages 33 and 34 of Relativity, the Special and General Theories, A. Einstein.
x=ct
x'=ct'
Einstein used these little equations to prove that the Lorentz equations satisfy the results of the Michelson-Morley experiment. What they mean is that light travels a distance of x in a time of t, and light travels a distance of x' in a time of t'.
However, comparing these two equations to the Galileian transformation equations shows that the second of these equations is saying something different than x' and t' in the Galileian transformation equations. Therefore, keeping x' as it exists in the Galileian transformation equations, we see that the time for light to go that distance is something other than t' because t'=t in the Galileian transformation equations.
We therefore use another term for this time.
x'=c(t2')
Now we solve for t2'.
x'=x-vt
c(t2')=ct-vt
(t2')=t(c-v)/c
Now we can derive the Lorentz equations from the Galileian transformation equations.
c=300,000 km/sec = x/t =x'/[t(c-v)/c]
=(x-vt)/[(ct-vt)/c] = (x-vt)/(t-vt/c)
= (x-vt)/(t-vx/c^2)
= (x-vt)gamma/(t-vx/c^2)gamma
= x'Lorentz/t'Lorentz
We see from this that x'Lorentz is a longer distance than x', and t'Lorentz is a longer time than t(c-v)/c. Consequently, if scientists substitute x'Lorentz directly in place of x' as they presently do, there has to be a "distance contraction" to compensate for the longer distance. Scientists change the distance x'Lorentz back to x' as it is in the Galilein transformation equations by multiplying by 1/gamma so that
(x-vt)gamma*1/gamma = x'
This operation is called by scientists the Lorentz-Fitzgerald contraction. I call it simple algebra. As it relates to the Galileian transformation equations, Einstein's second little equation should read
x'Lorentz=c(t'Lorentz)
instead of x'=ct'
Robert B. Winn |