From: abian@iastate.edu (Alexander Abian)
Subject: TIME HAS INERTIA. RELATIVISTIC ADDITION OF VELOCITIES final part 5
Summary: Relativistic addition derived solely based on M.M. experiment
Keywords: Michelson-Morley
Organization: Iowa State University, Ames IA
Date: Sat, 26 Dec 1992 17:42:29 GMT


DERIVATION OF RELATIVISTIC ADDITION OF VELOCITIES BASED SOLELY ON
  MICHELSON-MORLEY EXPERIMENTAL RESULTS     (12-26-92 part 5)

  As mentioned earlier, none of the 4 preceding parts are needed for 
this final part 5.  This final part 5  is self-contained.

  The previous parts were given to show examples of forming new arith-
metics (or algebras) from old arithmetics (or algebras).  The advantage
of doing this is that many of the desirable and time-tested properties 
of old arithmetics are preserved and slight modifications are introduced
in forming the new arithmetics to serve our purposes.

  Let  (R,+,.)  be the usual arithmetic of the real numbers.  As shown
the usual addition is denoted by +  and the usual multiplication is de-
noted by  .  that, as usual we shall omit.

  We may create a new arithmetic  (R,(+),.)  preserving the old multi-
plication but defining the new addition (+) in variety of ways.

For example, we may define the new addition (+)  by:

   x (+) y = y cos x + xy sin(x+3y)      or     x (+) y  = x^y - y^x

and preserve the old multiplication and thus obtain two new arithmetics.

  A simple example would be to define  x (+) y  as a linear combination:

(32)   x (+) y = x3 + y6      with     multiplication as usual 

Then in the Arithmetic (32), we will have the following:

   5 (+) 7  =  57,     1 (+) 1 = 9,      3(6 (+) 2) / (1 (+) 1) = 10

Now, to derive the Relativistic addition of Velocities, solely based
on the Michelson-Morley experimental results, we choose for our new
addition  (+)  of velocities  v  and  w, the following form:

(33)       v (+) w  =  vx + wy 

and we determine  x  and  y  in (33)  in such a way as to satisfy the
Michelson-Morley experimental results, i.e.,
 
(34)     c (+) w = c        and       -v (+) c = c

where  c  is the velocity of Light and  " -"  is the old minus sign.

  Our aim is to determine  x  and  y  in (33) so that (34) is satisfied.
  
  To this end, we replace (34) in (33) and obtain

(35)   cx + wy  =  c          and       -vx + cy  =  c

Solving the two equations given in (35)  for  x  and  y, we have:

(36)     x =  (c^2-cw)/(c^2+vw)   and   y = (c^2+cv)/(c^2+vw)

substituting  x  and  y  given in (36) into (33), we derive

(37)     v (+) w  =  (v + w) / (1 + vw/c^2)

which is the formula for the Relativistic addition of velocities.

  I know the question !!  Why did I choose the form (33) for the
new addition ?  My answer is that a "COMBINATION" of the form (33) is
SIMPLE and the nature loves INGENIOUS SIMPLICITY.  So, I think my choice 
of the form (33) is quite reasonable - just as Einstein's choice of his
assumptions is accepted by many  (not by all) as reasonable.

  I do not prescribe to (37) or to SR (Special Relativity).  I simply 
derived (37) assuming the Michelson-Morley experimental results are valid 
and that my choice of (33) is a reasonable one.

In my derivation of (37), I did not use the usual Lorentz Transfor-
mations or the assumptions of the Einstein's SR  theory which imply
Lorentz Transformations.  Thus, it is conceivable to have models
where velocities are  "added"  according to (37)  whereas neither the
contraction of lengths nor the dilatation of Time  occurs, as suggested
by Lorentz Transformations.  This circumstance fortifies my:


(A3)     TIME HAS INERTIA  and some energy is irretrievably spent 
         for moving TIME forward, according to:

                            dE = -A E(t) dt

I hope that somebody has benefited from my 5 part exposition (any comments?)
Perhaps my derivation is so old that it was known to Adam and Eve or 
to Einstein  who however reportedly denied knowing the Michelson-Morley's 
result (true or false).

   With best wishes and regards,
                                            Alexander Abian

I post and read almost exclusively in  TIME HAS INERTIA  subject.

-- 
   The tendency of maintaining the status-quo, Reaction to provocation and
                The tendency of maintaining again a status-quo.  
    TIME HAS INERTIA  and some energy is lost to move Time forward  
  E = mcc  (Einstein)    must be replaced by    E = m(0) exp(-At) (Abian)
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