Natural Science Forum / Physics / Relativity / July 2009

Lrt us consider wave omni-directaional emitting wave source moving in a
medium configured as a Michelson setup.

               o___________________________________|  
          Source<-------------  d  ---------------->Reflector

In any mundane medium, the speed of the wave [c] is fixed by it's properties
(modulus and density), and is independent of any motion of a source.  
Therefore, in the case of a round trip, traveled from any moving emitting
source from a co-moving reflector separated by a 'fixed' distance [d]
<see figure above>, the distances traveled can be described as follows.  For
the 'path' taken in the direction of motion the time required to complete
the trip is:

                 t =  d/(c - v) + d/(c + v)                       ... {1}

This is strictly due to the so-called Galilean transformations.  We note that
when v = c the time it takes for the wave to get to the reflector is infinite,
that is, it WILL never catch the reflector which is moving away just as fast
as the wave can propagate.  Now doing some simple algebra the round trip, we
get:

                       d(c + v) + d(c - v)
                  t = --------------------                        ... {2}
                         (c - v)(c + v)

which resolves to:

                              2d
                     t = --------------                           ... {3}
                         c[1 - (v/c)^2]

Thus the actual distance [L], or path of the wave for the round trip becomes:

                              2d
                     L = -------------                            ... {4}
                         [1 - (v/c)^2]

Now if we evaluate the orientation of the assembly at 90 degrees relative to
the motion we get the time as:

                             2d
                 t = ------------------                           ... {5}
                     Sqrt[1 - (v/c)^2]c

This comes about from the fact that the actual path taken by the wave is not
d, but the hypotenuse of a triangle illustrated as follows:
                             ___
                             / \
                            / | \
                           /  |  \
                          c   |   c
                         /    d    \
                        /     |     \  
                       o___v__|___v__o

And the actual distance as:

                            2d          
                 L = -----------------                            ... {6}
                     Sqrt[1 - (v/c)^2]

Which is different from the parallel direction by a factor of exactly:

                       Sqrt(1 - (v/c)^2) ['gamma']                ... {7}

This was the expected difference which would have resulted in the fringes
predicted to be seen in the MMX.

The lack of which is why, in 1889 Fitzgearld and in 1895 Lorentz suggested that
matter contracted by this factor along the axis of motion when moving through
the medium, since this results in both round trip paths becoming of equal length.

Now with the Lorentz/Fitzgerald contraction we find that for the local (moving),
system

                              dx    L
                          c = -- = --                            ... {8}
                              dt   dt

and the defintions of dt as,

                                2d
                        ------------------
                        Sqrt[1 - (v/c)^2]c

Then,

                            2d           Sqrt[1 - (v/c)^2]c  
                 c = ----------------- x ------------------      ... {9}
                     Sqrt[1 - (v/c)^2]           2d

or simply,

                              c = c

Now it should be clear to anyone that gamma disappears and for any given
local value of d and the definitions,

                              c = dx/dt

and

                              dx = 2d

and

                               dt = 2d/c  

the wave speed c will be measured as the same value regardless of v, as long
as v < c...

If however the actual round trip paths are L (Eq. 6) and not 2d (the local path)
then the actual travel times are expressed by Eq. 5, and not simply local time
2d/c and all paths MUST travel per Eq. 6, one can set up local coordinates
based upon measured speed c and a local time t' BUT the relationship of different
lcoal times at different speeds v leads direct back to Lorentz's 1904 paper known
as LET.

From the above one can get out the posulates that for the functional basis
of Special Relativity.  All physical processes would be constrained to behave
in a manner consistent with the tenets of SR.  However, the very existence of
observable time distortions (time dilation) between moving inertial frames can
be considered strong evidence that light's speed is universal invariance and an
underlying medium is what regulates this behavior and give rise to its very
existence.