Q : What happens in mismatched reference frame with the 
    necessary condition of simultaneity in the 
    interpretation of relativistic length measurement ?

A : To answer this question requires a historic debugging:

          I. Debugging a Historic Bug

     Historically the FitzGerald-Lorentz contraction has its
     origins in the assumed "aether" in which a moving stick
     suppopsedly contracted in length to keep the speed of
     light invariant. Therefore, the reference frame for this
     assumed contraction is stationary with the assumed "aether"
     and with the observer.

     Lorentz published his transformation in 1904 [1] in
     order to keep Maxwell's equations valid in all force-
     free inertial reference frames which were assumed to form
     the "aether" frame. Within a year, Einstein found the same
     transformation [2] independently and without any reference
     to the "aether", i.e., it does not matter whether "aether"
     exists or not.

     The length measurement is usually interpretated in his
     own words by Einstein [2]:
     "...the x-dimension appears shortened in the ratio of
     1 : 1 / Sqrt(1 - v ^2/ c^2).... viewed from the 'stationary'
     system........". Thus the relativistic length measurement is
     usually expressed as:

          Delta(X') = Delta(X) / r ............ (1)

     The relativistic time interval measurement is expressed as:

          Delta(T') / r = Delta(T) ............ (2)
     Where r = ( 1 - v^2 / c^2 ) ^(-1/2), the Lorentz factor.

     Seeing this pair of equations (1)&(2), one is likely to ask
     a seemingly silly question: Why from the same reference frame,
     the Lorentz factor acts differently for these two equations,
     if the dimensional ratio of speed, i.e., the ratio of length
     to time interval to remain constant before and after the
     transformation?

     In 1908, Abraham asked exactly the similar question and made
     an analysis on the relativistic measurements. He warned that
     the speed of light might appear to be covarying with relative
     speed [3]. This was serious enough and caused Pauli to ask
     Einstein himself [4], however, the answer seemed unclear.

     In 1959, Terrell interpreted for the invisibility of
     Lorentz contraction as rotation [5]. Penrose had the similar
     idea [6]. The conundrum of interpretation in the relativistic
     measurement has been lingering [7] even in 1996.

     In 1997, some new findings show reciprocity between
     Lorentz transformation and its Inverse transformation
     to bridge relative reference frames. To preserve the space-
     time invariance, an automatic covarying scale conversion
     was found necessary with Lorentz factor as the scale converter
     between two relative frames. This newly found inseparability
     between relativistic interval measurement and covarying
     scale [8] sheds new light onto the old conundrums.

     Historically, for the Special Theory of Relativity, the
     ellegant algebra has been used. Somehow, the necessary
     conditions were not clearly checked. With this understanding,
     we will show why the relativistic length measurement was so
     mysterious, because the reference frame and the necessary
     condition for measurement were mismatched.

     Of course this over-looked historical paradigm of interpretation
     for the relativistic length measurement will not harm the Special
     Theory of Relativity itself. The theory is safe and sound. We are
     glad to show that a bug has now been detected.


              II. The Mismatch

     The Lorentz Transformation pair between the frames F(X,T)
     and F'(X',T'), with the speed of light c set to unity, can
     be written as:

        dX' = r (dX - VdT) .....................(3)  
        dT' = r (dT - VdX) .....................(4)

     The inverse transformation, similarly, can be found as:

        dX = r (dX' + VdT') ....................(5)
        dT = r (dT' + VdX') ....................(6)

     Where r is the Lorentz Factor, and V is the relative speed in
     fractions of light speed which is set to unity. The reference
     frame F(X,T) is chosen at rest w.r.t.the observer as shown in
     Figure 1 in the paper.

     From the above equations, there are two possible cases of
     relativistic length measurements:

     (a) Simultaneity in the frame F(X,T) at rest w.r.t. to the
         observer is the necessary condition, i.e., dT=0 in the
         equation (3), then

            dX' = rdX ...........................(7)

     (b) Simultaneity in the other relative frame F'(X',T'),
         i.e., dT'=0 in the equation (5), then

             dX = rdX'............................(8)

     We see the reciprocity in between equations (7) and (8).
     Keep in mind that the Lorentz factor r is always greater
     than unity, i.e.,

          r = ( 1 - v^2 / c^2 ) ^(-1/2) > 1 .......(9)

     where v is the relative speed, and c is the speed of light,
     i.e., V=v/c.

     Eq(7) shows the relativistic length measurement from the
     point of view in the reference frame F(X,T) at rest with
     respect to the observer with the necessary condition of
     simultaneity.

     Comparing eqs(7)&(8) with eq(1), we see the historical
     paradigm shown in eq(1) has mismatched reference frame
     with the necessary condition of simultaneity viewed
     from the observer. No wonder that in 1908 Minkowski
     talked about the length contraction : "........This 
     hypothesis sounds extremely fantastical,........but 
     simply as a gift from above,......." [9]. We now can
     appreciate his helpless humor, because Minkowski knew
     then something illogical, and yet he could not find it.
     People tend to find difficult things, but the reality
     can be very simple, such as a pure simple mismatching
     of the necessary condition with the refernece frame as 
     we find in this case.  
		

     References
 
      [1] H.A. Lorentz, Proc. Ak. Amsterdam, 6, 809, 1904
          (Reprint in English translation, Dover Books, N.Y.)
      [2] A. Einstein, Ann. der Phys., 17, 891, 1905
          (Reprint in English translation, Dover Books, N.Y.)
      [3] M.Abraham,Theorie der Elektrizitaet,2,367,Leipzig,1908
      [4] W.Pauli, Theory of Relativity, p.14, Pergamon, 1958
      [5] J.Terrell, Phys.Rev., Vol.116, No.4, 1041-1045, 1959
      [6] R.Penrose, Proc. Camb. Phil. Soc., Vol.55, Jul 1958
      [7] N.D.Mermin, Physics Today, Vol.49,No 4,P.11, Apr 1996
      [8] C.Y.Yang, Causality and Locality in Modern Physics, 253,
          Kluwer Acad. Pub., the Netherlands, 1998
      [9] H. Minkowski, Proc. Ass. Ger. Nat.Sci.& Phys.,1908
          (Reprint in English translation, Dover Books, N.Y.)
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           *G.R.A.A.= Goddard Retirees Alumni Association
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