AOH :: DOPPLER.TXT An attempt to derive the transvers Doppler formula from basic principles
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From: rose@garnet.acns.fsu.edu (Kermit Rose)
Subject: Transverse doppler effect.
Date: Thu Oct  7 15:06:21 1993

The following is an attempt to derive the transverse doppler formula from basic
principles, thus showing its validity.  The time and distance lorntz
transformations follow from the transverse doppler effect.

The basic principle is that we can not measure our velocity with respect to
space itself.  We can not grab a piece of space and say that it is moving or
is stationary with respect to ourselves.  Thus, if we are moving toward a
source of light, we can not say how much of the relative velocity is due to
our motion and how much of the relative velocity is due to the motion of the
source of light.  Any formula for a quantity depending on the velocity of
a source of light toward us will be in terms of the relative velocity only.
This is for the obvious reason that source velocity or observer velocity
cannot be measured separately.

Now imagine a source of light moving perpendicular to our line of sight.
Consider a wave train that is moving along our line of sight.
The transverse motion of the light source stretches out the wave train some.
This gives it a longer wavelength.

Draw a diagram.  Let the speed of the transverse  light source be V.

Then in time T, the light source would have moved distance VT.  The stretched
out light wave train would have traveled a diagonal distance cT.  The distance
measured in the reference frame of the light source of this same wave train
would be cT * sqrt(1 - V^2/c^2)

which we get by the pythagorean right triangle formula a^2 + b^2 = h^2.

Frequency * wave length is speed of light.
Let f' be observed frequence in our reference frame.  Let f be frequency in
reference frame of light source.

Wave length of train of light waves is (in our reference frame)
c/f' = ct  where t is the time for the light to travel distance of one
wave length.

c/f' = ct = [ct*sqrt(1-V^2/c^2)] / sqrt ( 1 - V^2/c^2)
= (c/f)/sqrt(1 - V^2/c^2)

c/f'   = c/( f*sqrt(1-v^2/c^2)

f' = f * sqrt(1 - V^2/c^2)

Thus the frequency observed is f*sqrt(1-V^2/c^2)  where f is the frequency of
the wave train in the reference frame of the light source.