From: abian@iastate.edu (Alexander Abian)
Subject: TIME HAS INERTIA. RELATIVISTIC ADDITION OF VELOCITIES part 4
Organization: Iowa State University, Ames IA
Date: Mon, 21 Dec 1992 03:00:46 GMT

               
   
CONSTRUCTION OF NEW ALGEBRAS FROM THE OLD ALGEBRAS  (12-20-92- part 4)


   I have promised to derive the formula for Relativistic addition of
velocities based solely on the elementary algebra and the results of 
Michelson-Morley experiment (to which I do not necessarily subscribe).

   I have given 3 preliminary postings just to motivate the reader for
developing an abstract approach in connection with algebraic binary
operations defined on a set.  
   
   The essential point is that a binary operation  *(x,y), - customarily  
written as  x*y - is very conveniently defined when its table is 
explicitly given.  
   
   Of course the difficulties arise when  the set on which  *  is defined  
has a large finite number or an infinite number of elements and also
when several (usually two) binary operations are defined on the same set.
The usual examples of which are the algebras of real numbers, of complex
numbers, of quaternions, of matrices, etc., etc.
  
   None of the three previous postings are needed for my derivation of 
the formula for Relativistic addition of velocities.  However, since
Relativistic addition (of velocities) is  a new type addition, i.e.,
is a new algebra (where 3 + 4 is not equal to 7), I thought that it
would be helpful to introduce the notion and examples of abstract 
unintuitive algebras.

   In part 2, I have given via (6) and (7)  an example of a very abstract
   algebra where addition and multiplication are not commutative
   are not associative, etc., In contrast in part 2, via (14) and (15) 
   I have given an example of a rather decent algebra.

 Let me give another example of a decent algebra. In fact an example of
a FIELD where addition forms a commutative group and where multipli-
cation of nonzero elements also forms a commutative group and multi-
plication is distributive  w.r.t. addition.  Thus this example
satisfies all the desirable properties of the usual arithmetic of the
real numbers.

       + |  0   1   2   3   4            * |  0   1   2   3   4                
     ----|---------------------         ---|--------------------- 
       0 |  0   1   2   3   4            0 |  0   0   0   0   0   
       1 |  1   2   3   4   0            1 |  0   1   2   3   4
 (23)  2 |  2   3   4   0   1    (24)    2 |  0   2   4   1   3
       3 |  3   4   0   1   2            3 |  0   3   1   4   2
       4 |  4   0   1   2   3            4 |  0   4   3   2   1
         
The above example is the familiar FIELD of Integers mod. 5
The example given by (23), (24)  very much resembles the algebra of
the real numbers (extensively used in Calculus, Physics, etc)    

Every elements  x  has a  additive inverse  -x ,  where  x+(-x)=0

For instance,

(25)    -0 = 0,    -1 = 4,    -2 = 3,    -3 = 2,   -4 = 1
    
Every nonzero element x has a multiplicative inverse  1/x  or  x^(-1)

where  (1/x)*x = 1 = (x^(-1))*x 

For instance

(26)    1/1 = 1,    1/2 = 3,   1/3 = 2,    1/4 = 4

  Now, let us call  (23) and (24) an OLD algebra. New algebras most
conveniently can be created based on OLD algebras in variety of ways.
This is a very common practice is Mathematics and has the advantage 
of preserving some of the (Time Tested) valuable  properties of  OLD 
algebras.  
  For instance, we may create a NEW algebra from an OLD algebra by
preserving the OLD multiplicative table and by defining a NEW addition
by a suitable formula.
  
   Example 1  of a NEW algebra  using the OLD algebra (23),(24)

  Let us create a New algebra on the set {0,1,2,3,4} based on the Old algebra
(23), (24)  by preserving the OLD multiplication and by defining NEW addi-
tion  (+)  via the formula

(27)    x (+) y  =   3*x - 2*y      
  
  Accordingly in our NEW  algebra  we will have

        4 (+) 2  =  3*4 -2*2  which by (23) and (24)  yields  
                 =   2 + 1 = 3
  So in our NEW algebra
        
        4 (+) 2 = 3     whereas in the OLD algebra    4 + 2 = 1
 
 Again, in our NEW algebra

       (3(+)1) * ( 2(+)2 ) = (3*3-2*1) * (3*2-2*2) 
                           =   2 * 2 = 4

  Example 2.  of a NEW algebra  based on the OLD algebra (23),(24)
              where we impose on  NEW addition (+)  a certain condition

     I want to create a NEW algebra on  {0,1,2,3,4} while preserving the 
OLD multiplication * (24)  but I want a recipe for NEW addition (+)
in such a way that  3  plays almost the role oft he  speed of LIGHT, i.e.,
where I impose on  3 the property:

(28)   3 (+) y  =  3   and  x (+) 3 = 3  for every  x,y = 0,1,2,3,4      
   
   Now, how to devise a recipe for   x (+) y  ?
  
   Well, I say, let me try  the following linear combination for (+)
       
(29)   x (+) y  =  a*x + y         where  * and + are OLD addition and 
                                   multiplication given by (23) and (24)

  and let us determine  a  in (29) so that (28) is satisfied, i.e.

      3 (+) y  =  a*3+y = 3    Thus,   a = (3 -y)/3 = 2*(3 -y)

replacing this  a  in (29), we have

(30)    x (+) y  =   2*(3-y)*x + y

  SO, OUR NEW ALGEBRA with  (+)  and  *    is defined by

 (31)   x (+) y  =  2*(3-y)*x +y    and    x*y = x*y   
                    where  +  and  *  refers to OLD algebra  (23), (24)

   But now, in our NEW algebra   3  acts almost like the speed of light
   Indeed,
       3 (+)4 = 2*(3-4)*3 +4 = 3
       3 (+)3 = 2*(3-3)*3 +3 = 3
       3 (+)2 = 2*(3-2)*3 +2 = 3     because   3(+)y = 3 - y + y = 3 
       3 (+)1 = 2*(3-1)*3 +1 = 3
       3 (+)0 = 2*(3-0)*3 +0 = 3
                            and also  because   x(+)3 = 2*(3-3)*x + 3 = 3

     NEXT TIME I WILL DERIVE  THE FORMULA FOR THE RELATIVISTIC
     ADDITION OF LIGHT AND FINISH  THE WHOLE THING.

P.S   (A2)   i.e.,   TIME HAS INERTIA  is the indispensable principle for
                     Physics.  NO avoiding it . Its acceptance is 
                     INEVITABLE.
                                            Alexander Abian
                                                   






-- 
   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)