A TEST CASE:                             
     GOLDEN HARMONIC RATIO IN THE TWO MODES OF RELATIVITY     


       Let's look at the critical limit situation in more detail.

       An apparent mass aggregate Mk contains an original mass, plus
       an augmentation in mass due to gravitational relativity. And
       so let the originating mass be Mo, the augmenting mass be Ko,
       and the resulting mass be Mk. And therefore:


 For Gravity relativity 


 EQUATION Z-2

              ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
              ³         2G (Mo)               Mo is an original mass
      Eg  =   ³  1  Ä   ÄÄÄÄÄÄÄ               before augmentation
             \³          Cý R


 
 EQUATION Z-3

      (Mo x 1/Eg) - Mo = Ko          Ko is the mass augmentation
                                     on Mo, due to effect 1/Eg

 EQUATION Z-4

      Mo + Ko = Mk                   Mk is the measured (apparent)
                                     mass, consisting of original
                                     plus augmentive masses


 EQUATION Z-5

      When Mo = Mc = Mk/GH then:         Where Mc is a critical mass
                                         value for original mass Mo
            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³          2G Mk
    Eg  =   ³  1  Ä    ÄÄÄÄÄÄ            Mk is black hole mass with
            ³            GH              horizon radius Rbh, and GH is
            ³       ÄÄÄÄÄÄÄÄÄÄÄÄ         the Golden Harmonic Ratio equal
           \³          Cý Rbh            to the number 1.61803398875


 EQUATION Z-5-1

            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ     Mass Mbh is the same as mass
            ³          2G Mbh            aggregate Mk.
    Eg  =   ³  1  Ä    ÄÄÄÄÄÄ
            ³            Ng              Ng is ratio Nx when the value
            ³       ÄÄÄÄÄÄÄÄÄÄÄÄ         of Nx is GH, which is the
           \³          Cý Rbh            Golden Harmonic Ratio




 EQUATION Z-6

      With digits substituted for GH, then:

                           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                           ³          2G Mbh
   Eg  = .61803398875  =   ³  1  Ä    ÄÄÄÄÄÄ              =      1
                           ³       1.61803398875            ÄÄÄÄÄÄÄÄÄÄÄÄÄ
                           ³       ÄÄÄÄÄÄÄÄÄÄÄÄÄ            1.61803398875
                          \³          Cý Rbh


 EQUATION Z-7


    because:

               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ             When and only when Nx = GH.
       1       ³         1                   The Golden Ratio contains
      ÄÄÄ  =   ³  1  Ä  ÄÄÄ                  this self appreciating
       Nx     \³         Nx                  mathematical property




    and so:

               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
       1       ³          1                  GH is the Golden Ratio
      ÄÄÄ  =   ³  1  Ä   ÄÄÄ                 1.61803398875
       GH     \³          GH




ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ For Special relativity ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ



 EQUATION Z-8


            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³        ÚÄ        Ä¿ý                ³          (Vc)ý
    Es  =   ³        ³    C     ³            =    ³  1  Ä   ÄÄÄÄÄÄ
            ³  1  Ä  ³ ÄÄÄÄÄÄÄÄ ³                \³           cý
            ³        ³   ÚÄÄÄÄ  ³
            ³        ³  \³ Nx   ³
            ³        ÀÄ        ÄÙ
            ³       ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
           \³             Cý







 EQUATION Z-9

            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ           ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³        ÚÄ        Ä¿ý                ³          (Vc)ý
    Es  =   ³        ³    C     ³            =    ³  1  Ä   ÄÄÄÄÄÄ
            ³  1  Ä  ³ ÄÄÄÄÄÄÄÄ ³                \³           cý
            ³        ³   ÚÄÄÄÄ  ³
            ³        ³  \³ GH   ³
            ³        ÀÄ        ÄÙ
            ³       ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
           \³             Cý




 EQUATION Z-9-A     And so:


       (Mc x 1/Es)  =  (Mc x GH)  =  Mbh,   because  (Es = 1/GH)

       when 1/Es is the special relativitistic effect on
       mass Mc which is moving at velocity Vc of EQ Z-9


 EQUATION Z-10      As in:


                    ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                    ³        ÚÄ                  Ä¿ý
   .61803398875 =   ³        ³          C         ³
                    ³  1  Ä  ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³
                    ³        ³   ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ  ³
                    ³        ³  \³ 1.61803398875  ³
                    ³        ÀÄ                  ÄÙ
                    ³       ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                   \³                   Cý

  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´     FOR SPECIAL RELATIVITY EFFECT ON BOTH MASS AND RADIUS    ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

    There is yet another factor to consider. In special relativity
    the radius of a mass contracts in reciprocal proportion to the
    enhancement of mass. In this regard, when the radius is contracted,
    less mass will be required to form a black hole in the relativist-
    ically reduced radius.


        How does this effect the status of the critical limit Mc,
        where the original mass Mo is the black hole mass divided
        by the Golden Ratio?

        Specifically, what mass will now form the black hole,
        when the original mass's radius is concomitantly reduced
        by special relativity's effect?


    The new mass is easy to find.


    EQ Z-9 is abruptly rewritten to accommodate both a reduction in
    radius, and expansion in mass, upon original (critical) mass Mc.
    The correct velocity for mass Mc can be labelled as (Vbh), as in
    'Velocity for black hole', and is easy to find. It turns out to be:

                  Vbh = (C / GH)           Given as:






 EQUATION Z-11

            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ        ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³        ÚÄ     Ä¿ý                ³          (Vbh)ý
    Es  =   ³        ³   C   ³            =    ³  1  Ä   ÄÄÄÄÄÄ
            ³  1  Ä  ³  ÄÄÄÄ ³                \³           cý
            ³        ³   GH  ³
            ³        ÀÄ     ÄÙ
            ³       ÄÄÄÄÄÄÄÄÄÄÄ
           \³            Cý


    Es turns out to be the reciprocal of the square root
    of the Golden Harmonic. That is;  Es = (1/ûGH).

    It means that when a mass Mc is physically moving at velocity
    Vbh relative to a stationary observer, its radius Rbh contracts
    by (1/ûGH), as its rest mass Mc expands by (ûGH), with the result
    that a new black hole is formed, having a lesser mass equal to
    (Mc x ûGH), and a lesser radius equal to (Rbh x 1/ûGH).

        As already said, this occurs when velocity Vbh is equal
        to the speed of light divided by the Golden Harmonic Ratio.

    The new mass can be labelled as Mbh-, which is less than the
    gravitational black hole mass Mbh, by a factor of ûGH. As already
    indicated, Mbh/Mc = GH, but the special relativistic mass result
    Mbh- is not the same as Mbh. There is a series:


 EQUATION Z-12

        Mc  x  ûGH  =  Mbh-  x  ûGH  =  Mbh

    It means that a visible mass cannot expand to infinity,
    because velocities can approach but can never reach the speed
    of light, due to built in limiting factors. This statement
    is true specifically for visible masses.

    For instance, the maximum velocity possible for mass Mc is Vbh
    which is C/GH, but this is only when the original mass Mo is at
    the critical mass limit Mc which is a black hole mass Mbh divided
    by GH. Whereupon the mass becomes a new black hole of mass Mbh-
    and disappears from view, relative to a stationary observer.

         The ratio C/GH is (C / 1.61803398875)


    (The preceding does not take into account any effect that
    gravity might have to relativistically reduce the radius of the
    mass causing the gravity's relativistic effect. It is realized
    that if a reduction in gravitational radius is also needed as a
    key term, than the parameters of the critical mass limit Mc regards
    the black hole final limit Mbh, will adjust accordingly, as will
    the exact factors related to the Golden Harmonic Ratio).

    (The question of such possible adjusting is not addressed in
    this disclosure, whose prime intention is to simply show that
    certain critical limits and equalities do synonymously exist
    in the domains of gravitational and special relativity. And
    that the Golden Harmonic Ratio is a fundamental primary term).

        ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
        ³     A REMARK     ³
        ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

        The Golden Ratio was not a term pulled with a sleazy wink from
        a magician's hat to fit an idea. The Golden Ratio turned out
        to be a resulting term that provided a theory; whose gist is
        as follows:


        How can a limiting velocity (thus a universal barrier to infinite
        expansion of visible mass relative to a stationary observer), be
        determined for any visible mass, in special relativity?

        The answer to this is straight forward and demonstrates that
        a visible mass can never expand to infinity. A discussion
        regards this answer begins further below under:

           'Special Relativistic Effects on any Mass and Radius'.


  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´     SUPPLEMENTAL REMARKS     ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

    The following remarks are included to complete the discussion
    regards relativity theories and the Golden Harmonic Ratio. These
    supplemental remarks cover the subject of how the Golden Ratio
    was found to be a constant in critical limit situations.

    The remarks discuss the issue from firstly; effects on the critical
    mass only; and secondly for effects on the critical mass and radius.

  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³ Golden Harmonic Relativistic Effects on Mass Only ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

     How was the Golden Harmonic found to be the critical
     ratio factor Ng for Nx in Equations Z-5 and Z-5-1 ?

          A value of (square root of 2) was first tried for Nx, yielding
          a mass augmentation result (1/Eg x Mo), which was greater
          than mass Mbh, when root 2 for Nx was ratio (Mbh/Mo = Nx).

          In intuitional trial and error, an Nx value arbitrarily
          selected as 1.8 was next tried. It yielded an (1/Eg x Mo)
          value which was slightly less than mass Mbh.

          So the two Nx values were averaged as in 1/2(û2 + 1.8)
          to yield a value of 1.608. Since this number was close to a
          known number (1.61803398875), this known number was tried to
          see how close the Es result (1/Es x Mo) came to Mbh, using
          this familiar number as Nx for a point of reference.

          It turned out that 1.61803398875 happened to be the very
          term wanted, because the result was perfect. This fast
          found number was given the label GH.

          When GH was Nx, then (1/Es x Mo) = Mbh.

              And so this particular Nx was
              labelled Ng (for Golden Ratio).

              And Mo was understood to be
              the same value as mass Mc.

          Equations Z-6 and Z-7 show why Ng is a constant. The
          set of Equations Z to Z-10 followed as a consequence
          of knowing this.

  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³ Golden Harmonic Relativistic Effects on Mass and Radius ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

          But Equations Z to Z-10 consider only the special relativistic
          effect on mass, and left unanswered another question which was:

              'What modifications would occur in the parameters of
               mass when the radius of the mass is also conjointly
               changed by special relativity effects'.

          The answer to this was also quickly forthcoming, but
          in hindsight seems to reflect a very fortuitous guess.

          Trial and error was started again. A velocity was needed,
          to determine at what rate mass Mc would be travelling to
          relativistically increase to mass Mbh-, when radius Rbh
          of mass Mc was conjointly contracted to radius Rbh-.
          In this thought balloon, Mbh- and Rbh- would be the
          parameters forming a new black hole when mass Mo was
          travelling at sufficient high velocity.

          At this point the rate of joint contraction on mass Mbh
          and radius Rbh was not known. And neither was the velocity.

              The intention was to find what term Nx is
              divided into C to yield the significant velocity.

              In a remarkably lucky guess, the first Nx
              term tried was GH itself, (in EQ Z-11).

          To begin, radius Rbh was modified by (Es x Rbh) as gained
          from (EQ Z-11) with Nx equal to GH in the ratio C/GH, to give
          contracted radius Rbh-. Then, using EQ 5 of APPENDIX B below
          to find the mass of a black hole formed in radius (Es x Rbh-),
          a new mass Mbh- was the result. It turned out that the ratios
          of masses (Mbh/Mbh-) and (Mbh-/Mc) both equaled the square
          root of ratio GH.

              It had thus been found that when (C/GH = Vbh), then
              EQ Z-11 yielded the square root of GH as the Es value.

          The result is that with Es equaling the reciprocal of the
          square root of the Golden Ratio, when Rbh is multiplied by
          Es to yield radius Rbh-, and mass Mc is multiplied by the
          reciprocal of Es to yield mass Mbh-, then radius Rbh- and
          mass Mbh- are the correct parameters to form a new black
          hole from the special relativity effects on both mass Mc
          and radius Rbh, when Mc is travelling at a (C/GH) velocity.


         ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
         ³ How was this verified ? ³
         ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

          The 'dual effect' event was easily
          verified by the following:



          A.   Radius Rbh- was found from radius Rbh,
               by using the Es effect of EQ Z-11 in:

                   Rbh x Es = Rbh-

          B.   Using radius Rbh- to find mass Mbh- in:

                   Cý Rbh-               Finding mass Mbh- needed for a
     Mbh- =   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ            black hole whose Schwarzschild
                     2G                  radius is given as Rbh-


          C.   Mbh- turned out to be mass Mbh / (1/ûGH)
               when effect Es (of EQ Z-11) was 1/GH.

          D.   It meant mass Mbh- and radius Rbh- form a new black hole,
               which is less than a black hole of mass Mbh and radius Rbh,
               by a factor of the square root of the Golden Ratio for
               both Mbh- and Rbh-.

          E.   This is true when mass Mc is travelling in special
               relativity, at a reduced velocity Vbh, as gained
               from EQ Z-11.

          F.   The synonymous special relativistic 'dual effect' event
               for a gravitational relativistic event at the critical
               mass limit Mc, is gained by using term Nb = GH (as used
               in EQ Z-5-1), to find velocity Vbh in EQ Z-11.


ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ
º SPECIAL RELATIVISTIC EFFECTS ON ANY MASS AND RADIUS º
ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ

           Only certain critical limit cases
           (for masses Mo and Mc = black hole mass Mbh/GH)
           have so far been considered.

   ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
   ³     QUESTIONS     ³
   ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

    What if instead of Mc there is given any general mass Mo,
    having a radius said to be Ro. Would there still be critical
    limits involving Golden Harmonic factors that would limit a
    general test case to a state that is less than infinite mass,
    at a velocity which can never tightly approach the speed of light?

          For that matter are other, more general, limits possible,
          besides those already shown to be related to the Golden Ratio?

          And if general limits are in the fabrics of physics, how to
          determine them, given a general mass quantity that to begin
          with is not known to be related to anything else, especially
          when it is NOT RELATED to the Golden Ratio ?

   ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
   ³     ANSWER     ³
   ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

    This questioning also came to a quick answer, although
    the finding of the answer was not all that straightforward.

    The answer demonstrates that any visible mass travelling at a
    relativistic velocity in special relativity, reaches a limiting
    barrier, beyond which the mass does not visibly increase any further
    toward infinity, and its velocity closes no further toward equaling
    the speed of light.


          The first insight is that any entity (in its most general
          sense) comprises a mass and a radius. With mass is some
          gravity. For instance a typical Sun sized star is an
          ideal test case entity.

          For example, the ratio of the Sun's existing mass M over
          the Sun's existing radius R is its (mass/radius) ratio,
          ie., M/R

               (Note that Mo would be the Sun's original mass before any
               mass augmentation effect due to gravitational relativity.
               The Sun's original mass Mo is less than its existing
               mass M, since the existing mass as physically measured
               is assumed to include a mass augmentation upon mass Mo).



          The Sun's black hole Mbh mass  (silent partner mass)   
          is easily found by:





 EQUATION Z-13



                       Cý R              Finding mass Mbh needed for a
          Mbh =   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ        black hole whose Schwarzschild
                        2G               radius is given as R when
                                         R is the radius of the Sun

                 so that another ratio is found, this being (Mbh/R)
                 which is the Sun's (black hole mass/radius) ratio.

          But actually, term Mbh of EQ Z-13 is worthless. What
          we really want to find is what (Mbh-/R-) ratio forms a
          black hole out of the original Mo/R parameters, when Mo is
          travelling at increasingly faster velocities approaching the
          speed of light.

          We need a comparative term, to study any differences between
          the Sun when standing still, and when moving at a relativistic
          velocity. The comparative term we want to know is found as:



 EQUATION Z-14


          Mbh      Cý                  Where ratio Cý/2G is a constant,
          ÄÄÄ  =  ÄÄÄÄ                 when C is the speed of light, and
           R       2G                  G is the universal gravitational
                                       constant.

               R is the original radius of original mass Mo

               Mass Mbh is instantly found from EQ Z-13.

               The logical argument formed in advance, was that
               any mass result M+, and radius result R-, ensuing
               from special relativistic effects on original states
               Mo and Ro, should also equal the black hole constant
               ratio Cý/2G, if mass M+ and R- were relativistically
               altered sufficiently to form a new black hole.

          Ratio Cý/2G can be labeled ratio CR (for 'constant ratio') and
          has the value of (6.735275620 x 10 to 27 grs/cm), given a speed
          of light whose digital value is 2.99792458, and a gravitational
          constant whose digital value is 6.6720 x 10 to -8.


               Ratio Cý/2G is known as a constant
               for the given values of C and G.


          What we can do is follow special relativistic changes upon
          both Mo and Ro through successively greater velocities, until
          the combined ratios (1/Es x Mo) / (Es x Ro)  equals the ratio
          Cý/2G, as in:



 EQUATION Z-14A



          ((1/Es x Mo) / (Es x Ro))  =  (M+/R-)  =  (Cý/2G)

          where Es is the special relativistic effect.





  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³ Finding a significant Velocity value, which results in ratio CR ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

    It was useful that a good test model was available in the
    solar system's Sun, where given the Sun's existing mass as M,
    and existing radius as R. The Sun has to be accelerated to such
    an extent that through the parameters of special relativity, the
    Sun's modified mass M+ and radius R- reach a point where they
    transfigure into conditions which form a new black hole.

        It was assumed that such a transfiguration should
        occur, and that the transfigurating velocity in
        special relativity could be inferred.

    How could the velocity needed for the transfiguration, be
    determined for an arbitrary general case such as the Sun ?

    At this point, some intuitively lucky guesswork again prevailed;
    a 'seeing around corners' so to speak. To make a long story short,
    it is easy to predetermine the prerequisite velocity.  How, is
    outlined as follows:


      1.   Given an existing Sun mass M of 1.99099305 x 10 to 33 gms
           (mass MM from Part 1 above)

      1A.  Given a Sun radius R of 6.96265 x 10 to 10 cms

      1B.  Given constant ratio CR = Cý/2G
                                   = 6.735275620 x 10 to 27 grms/cms

      2.   Given the black hole radius parameter
           of EQ 4 of APPENDIX B, as:



 EQUATION Z-14-1

                      2G M                 Finding the Schwarzschild
          R' =   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ           radius R' of a black hole's
                       Cý                  event horizon, when given
                                           mass M


      3.   And given Equation 5 of APPENDIX B, rewritten as:


 EQUATION Z-14-2



                       Cý R               Finding mass Mbh needed for a
          Mbh  =   ÄÄÄÄÄÄÄÄÄÄÄÄÄ          black hole whose Schwarzschild
                        2G                radius is given as R

                                       Mass Mbh is the black hole silent
                                       partner mass for any given mass M.



      4.  Given Equation Z-8 above for special relativistic effect
          on both an original rest mass and its original radius, based
          on a term Nx to determine a velocity, so that:








 EQUATION Z-15


            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ        ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³        ÚÄ     Ä¿ý                ³          (Vx)ý
    Es  =   ³        ³   C   ³            =    ³  1  Ä   ÄÄÄÄÄÄ
            ³  1  Ä  ³  ÄÄÄÄ ³                \³           Cý
            ³        ³   Nx  ³
            ³        ÀÄ     ÄÙ
            ³       ÄÄÄÄÄÄÄÄÄÄÄ
           \³            Cý



      5.  Given that (1/Es x M) = M+

      6.  Given that (Es x R)   = R-

      7.  Given that (1/Es x M+) / (Es x R-) = Cý/2G  =  M+/R-

      8.  Then it should be possible to find a velocity for EQ Z-15-1
          below such that the resulting (M+/R-) ratio = Cý/2G


      9.  A first arbitrary value for Nx was tried, being 1.0001, which
          produced results that were too low for the above Item 7 to be
          correct.

     10.  A second arbitrary value for Nx was tried in EQ Z-15, being
          1.00001, which was of the right magnitude for a mass M+, but
          Item 7 was still not correct.

     11.  However, it was noticed that 1/1.00001 by itself was in the
          magnitude range of gravitational relativistic effect Eg from
          the Sun's mass, as determined in EQ C of Part 1 further above.
          (MM in EQ C is the same value as Sun mass Mo given in EQ Z-2,
          and immediately above in Item 1. And Eg of EQ Z-2 is the same
          as Eg used immediately below in Item 12).

     12.  And so Eg was determined for the Sun's mass M = MM = Mo in
          EQ Z-2, and conveniently labelled Egs (for 'effect gravity Sun
          mass'), and was substituted as term 1/Nx in EQ Z-15 immediately
          above, to give:





 EQUATION Z-15-1

            ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³         ÚÄ         Ä¿ý              ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
            ³         ³  C x Egs  ³               ³          (Vx)ý
    Ess =   ³         À           Ù          =    ³  1  Ä   ÄÄÄÄÄÄ
            ³  1  Ä  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ             \³           Cý
           \³             Cý

                                    where velocity Vx is (C x Egs),
                                    and special effect Ess conveniently
                                    means an Es effect related to the
                                    gravitational mass via term Egs.


     13.  Then;  Sun mass M in (M x 1/Ess) = M+

     14.  And;   Sun radius R in (R x Ess)  = R-

     15.  And;   ratio  (M+/R-) =  6.73527458 x 10 to 27 grms/cms

                 As found in:


 EQUATION Z-15-2

         (M x 1/Ess) / (R x Ess)  =  CR  =  (M+/R-)


     16.  Which turned out to be an excellent approximation of ratio
          CR (being Cý/2G as created in Item 1B immediately above)

              Well, this was very good for a first found attempt. How
              about for other masses, and how did the ratio result of
              Item 15 favorably equate in truth to Item 1B above, in
              that the CR result in Item 15 is marginally below the
              CR constant in Item 1B ?

     17.  The mass of the Sun was arbitrarily raised by a factor
          of 1000, so that now M = 1.99099305 x 10 to 36 grms

     18.  A new Egs effect factor was determined using the
          larger mass of Item 17, in EQ Z-2 above

     19.  The new Egs factor was substituted in EQ Z-15-1
          to give a new Ess factor


     20.  The new Ess factor was substituted in the
          terms of Items 13, 14, and 15

     21.  The result M+/R- = 6.735275620 x 10 to 27 gms/cms  =  CR,
          which is exactly the constant of Item 1B

              Two things were instantly made clear.

              It is clearly evident that Equations Z-15,  Z-15-1,
              and Z-15-2, are correct for any mass, to yield (M+/R-)
              ratios equal to Cý/2G.

              It is clearly evident that ratio (M+/R-) closes
              in on ratio Cý/2G, the closer that given original
              mass M is to the black hole silent partner mass Mbh
              as determined in EQ Z-14-2

             (It is also clear from preceding explorations, that
             when relativistic effects are to act upon an original
             mass, the original mass M can never approach its black
             hole silent partner equivalent Mbh any closer than by
             Mbh divided by factors of the Golden Ratio).

  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³ Finding that terms M+ and R- are properties of a black hole ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

     At this point we are still not finished. The final question is;
     are terms M+ and R- (as determined by Equations Z-15-1 and Z-15-2),
     in fact the terms of a new black hole whose mass is M+ and whose
     radius is R- ?

     This final question was very easy to test by a double check:

     22.  The value of M+ from Equation Z-15-1 and Item 13 for the
          Sun mass arbitrarily increased by a factor of 1000, as in
          Item 17, yielded an Ess value in Item 19, which as applied
          to Item 13, was:
                            3.055623494 x 10 to 27 grms

     23.  The value of R- from the same Ess in Item 19, applied
          to Item 14, was:
                            4.536746031 x 10 to 9 cms

     24.  Looking to Equations Z-14-1 and Z-14-2, it was found in
          EQ Z-14-2  (given mass M+ of Item 22), and found in EQ Z-14-1
          (given radius R- of Item 23), that (M+/R-) = CR. This is shown
          in the following three equations:


 EQUATION Z-15-3

                      2G M+                Finding the Schwarzschild
          R' =   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ            radius R' of a black hole's
                       Cý                  event horizon, when given
                                           mass M+

          R'  was  4.536746031 x 10 to 9 cms,
              exactly the same as R- in Item 23



 EQUATION Z-15-4

                      Cý R-                Finding mass M' needed for a
          M' =   ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ           black hole whose Schwarzschild
                       2G                  radius is given as R-

          M'  was 3.055623493 x 10 to 27 grms,
              exactly the same as M+ in Item 22

 
 EQUATION Z-15-5


          And so:   M' of EQ Z-15-4, divided by R' of EQ Z-15-3, = CR
          as in:    (M'/R') = CR
          where:    CR is the constant of Item 1B
          proving:  that M+ of Item 22 and R- of Item 23 are the
                    correct parameters of a new black hole created
                    by relativistic effect Ess of Item 19, on higher
                    mass M of Item 17, using EQ Z-15-1 to determine
                    Ess, after using EQ Z-16 to determine Egs.



  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´     SUMMARY EQUATIONS     ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

      The delineations of Items 1 to 23, and Equations Z-14 to Z-15-5,
      once understood, resolve into a quick series of steps, used to
      determine a relativistic barrier for any given mass M and its
      radius R, as in:


 EQUATION Z-16

             ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
             ³         2G M                M is any mass, R is its
    Egs  =   ³  1  Ä  ÄÄÄÄÄÄ               radius, and Egs is the
            \³         Cý R                gravitational relativistic
                                           effect of mass M


 EQUATION Z-16-1

             ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
             ³         ÚÄ         Ä¿ý              ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
             ³         ³  C x Egs  ³               ³          (Vx)ý
    Ess  =   ³         ÀÄ         ÄÙ          =    ³  1  Ä   ÄÄÄÄÄÄ
             ³  1  Ä  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ             \³           Cý
            \³              Cý

             Ess is the special relativistic effect ensuing from
             velocity Vx, determined as the direct consequence of
             the speed of light reduced by the mass's gravitational
             relativistic effect Egs.


 EQUATION Z-16-2

             (M x 1/Ess) = M+


 EQUATION Z-16-3

              (R x Ess)  = R-


 EQUATION Z-16-4

                (M+/R-)  =   Cý    =  CR
                            ÄÄÄÄ
                             2G

       and mass M+ and radius R- are a relativistic transfiguration of
       M and R into the parameters of a black hole, when ratio (M+/R-) = CR.

                  CR is a physical constant in black holes,
                  whose value is given as the speed of light squared
                  divided by twice the gravitational constant, and
                  whose value is 6.735275620 x 10 to 27 gms/cms.

 EQUATION Z-16-5

             And ultimately, Ess can be determined directly
             from Egs, by:

                  Essý  =  1 - (Egs)ý


             Ess is not the same value as Egs. Ess can be higher
             or lower than Egs. The exact relationship between the
             value of Egs and Ess is known by:



 EQUATION Z-16-6
                          ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                  Ess  = \³  1 - (Egs)ý

                          ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
                  Egs  = \³  1 - (Ess)ý




             Why this relationship occurs is explained
             further below, beginning with EQ Z-17),
             and explicitly in EQ Z-19.


 In a nutshell, Equations Z-16 to Z-16-6 fully show that
 fundamental terms in both gravitational (stationary) and
 special (moving) modes of relativity are synonymous.
 


      ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
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º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
º                   UNIFIED EFFECTS IN FIELD BEHAVIOR                   º
º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
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      ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ

ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
º ±±±±±±±  GENERAL INTRODUCTION   for part 4   Unified Fields   ±±±±±±± º
ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ


     'The best information seems to come after you think you
      have it wrapped up and have stopped thinking about it'.

         'For example, the following floated into
          consciousness as an afterthought'.

 In a broad sense, relativity synonymy evokes innuendoes
 of unified behavior between the fields of gravity and
 electromagnetism (a unified field theory).

 But wait, this is not a fully fledged unified field theory. What
 is under review here are only parts of what appear to be a unified
 field theory environment. What is shown are exactitudes whereby
 gravitational effects of an assumed mass changing character on a
 body, result explicitly in equivalent special relativistic effects
 synonymous to the body moving at characteristic velocities.

 Certain rules of behavior define these two modes of relativity in
 their unified behavior. These rules are easy to understand, once
 clearly seen, but can be very confusing until their characteristics
 are shown in an obvious way. This next section (Part 4) explores
 the rules.


 To do the job, a particular environment is arbitrarily created. Exact
 test cases are followed to the nth degree. The created environment is
 in violation of certain conditions already outlined in Part 2 above;
 to wit:  that certain critical limits exist in the rate of mass
 expansion, where the maximum expansion oscillates between a black hole
 mass equivalent Mbh, and plateaus below this, articulated as functions
 of the Golden Harmonic Ratio 1.61803398875.

 For the test cases, it is desirable to see what happens
 mathematically for events which are right at the brink of
 a black hole mass, compared to masses well below the brink.
 The phenomenology is thus most easily watched in detail.

 For this, such masses are arbitrarily created, and assumed to exist
 in violation of the statements in Part 2 above (which delineate that
 a mass of black hole equivalent includes an original mass Mo, a mass
 augmentation unit Ko, and resultant mass aggregate which is that of a
 black hole or less. If the mass is that of a black hole, the original
 mass is at a critical mass limit Mc, and the ratio Mbh/Mc = Ng is a
 function of the Golden Ratio. For masses other than than Mc, ratio Ng
 is given the general label Nx).



 In the following, the cases for Mc and Ng parameters are ignored by
 conveniently looking the other way. In the test cases which follow,
 the existence of discrete portions denoted by terms such as Mo, Mc,
 and Ng, are expeditiously put aside, and a mass value is assumed which
 can be anything less than Mbh, even if less than Mbh by a few parts in
 a thousand. This is called a HIGH mass, for convenience.


ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
ÌÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ
º TEST CASE º
ÈÍÍÍÍÍÍÍÍÍÍͼ

     In a test case, a HIGH mass value is studied which hangs right
     below the mass of a black hole Mbh. This is in a deliberately
     selected HIGH mass range which as already said ignores properties
     such as a critical mass factor (Mc) outlined in Part 2 above.

     The intention this time is to follow test case examples in
     excruciating digital detail, so that the effects and their
     changes are unmistakable.


     The sole intention of the following, is to observe how certain
     properties are universally united in a general way through various
     transformations between gravity and electromagnetic field behaviors.

     And so a new study model is created, based on the arbitrary
     criteria that any job needed to do a certain job is good enough
     for the purpose intended.

     A HIGH mass gravitational event and a LOW mass event are thus
     arbitrarily created from the same Mbh term, which is the mass
     of a black hole confined in the Sun's radius. Mbh for the Sun's
     radius is (4.689536679 x 10 to 38 grms).

     The Sun's radius (6.96265 x 10 to 10 cms) has been chosen as an
     easily recognized radius for use as a constant to investigate
     the effects of different mass densities confined in a fixed
     (unchanged) area. Otherwise, the Sun's radius has no physical
     significance when tied to the following arbitrary mass aggregates.

     To supply the study, a small ratio Nx has been selected for a
     control in the study. Nx is meaningless other than its value
     is the charge to mass ratio of the hydrogen atom, ie.:


     ((Proton + electron) / electron)  =  1.000544617  = Nx.

             (The interpretation is that the negative electron charge
              of the lightweight electron influences the heavy proton
              by only 1.000544617 of the effect the proton has on the
              electron, since both particles have the same quantity of
              charge (opposite) despite widely divergent rest masses.
              This is mentioned only to satisfy curious minds. As said,
              the real value for the above ratio Nx has no intrinsic
              significance in the following).


 MASS1   In our study model, Mbh is arbitrarily reduced by the
         small ratio Nx to give a HIGH Mass1 term, which is very
         slightly below Mbh.

 MASS2   Mass1 is then arbitrarily reduced by a factor of 100,000 to
         give a LOW Mass2 term having the same digits but much lower
         magnitude then Mass1.

     The intention is to be able to follow certain relativistic field
     effects in detail by following the digital results of both the
     HIGH mass term (Mass1), and LOW mass term (Mass2), to more openly

     follow the unifying effects between the two fields (being gravity
     and electromagnetism).

     In the study model, as already said, the value of Nx has no
     significance except that it provides a convenient low value
     Nx ratio to arrive at a HIGH mass term for the study model.


 Nx      is given to 13 significant digits as gained from the
         ratio (P 938.2796 mev + E .5110034 mev) / (P 9382796 mev)
         =  1.000544617404













 TABLE 4-A
  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³   ARBITRARY STUDY MODEL DATA                             ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³                                                          ³
  ³      Nx   =  1.000544617404   =  (P + E) / E             ³
  ³      Mbh  =  4.689536679 x 10 to 38 grms                 ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³                                                          ³
  ³      HIGH  mass1   =  Mbh / Nx                           ³
  ³                    =  4.686984066 x 10 to 38 grms        ³
  ³               Nx   =  1.000544617404                     ³
  ³                                                          ³
  ³      LOW   mass2   =  Mass1 / 100,000                    ³
  ³                    =  4.686984066 x 10 to 33 grms        ³
  ³               Nx   =  100054.4617404                     ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³                                                          ³
  ³      In the following, Equations Z-17-1 and Z-17-3       ³
  ³      are the same as EQ Z-15-1 above, except, the real   ³
  ³      digit value of each Egs ratio is substituted for    ³
  ³      the algebraic term Egs.                             ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

 EQUATION Z-17      HIGH gravitational Mass1 results:



               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
               ³         2G  (4.686984066 x 10 to 38 grms)
      Egs  =   ³  1  Ä  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
              \³                   Cý R

                                                  Mass1 has been given in
                                                  terms of a real weight.

               Radius R is the radius of the Sun.
               Egs is the gravitational relativistic effect of Mass1


                                                       ÚÄÄÄÄÄÄÄÄÄÄÄÄ¿
    HIGH  gravity field effect                  Egs =  ³ .023330687 ³
    Egs   is closing toward 0                          ÀÄÄÄÄÄÄÄÄÄÄÄÄÙ





 EQUATION Z-17-1    Electromagnetic field effect results
                   (Ess is special relativistic effect)


               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
               ³         ÚÄ                Ä¿ý
               ³         ³  C x .023330687  ³                  Vý
      Ess  =   ³         À                  Ù              =  ÄÄÄÄ
               ³  1  Ä  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ                 Cý
              \³                  Cý

                                               .023330687 is effect Egs
                                                          of EQ Z-17

      Ess =     1 - (Egs)ý
      As in:    1 - (.023330687)ý = .999727802

                                                       ÚÄÄÄÄÄÄÄÄÄÄÄÄ¿
      LOW  special field effect                 Ess =  ³ .999727802 ³
      Ess  is closing toward 1                         ÀÄÄÄÄÄÄÄÄÄÄÄÄÙ
      V    velocity is starting to
           close toward 0


 EQUATION Z-17-2    LOW gravitational Mass2 results:


               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
               ³         2G  (4.686984066 x 10 to 33 grms)
      Egs  =   ³  1  Ä  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
              \³                    Cý R

                                                  Mass2 has been given in
                                                  terms of a real weight.


                                                       ÚÄÄÄÄÄÄÄÄÄÄÄÄ¿
      LOW  gravity field effect                 Egs =  ³ .999995002 ³
      Egs  is closing toward 1                         ÀÄÄÄÄÄÄÄÄÄÄÄÄÙ









 EQUATION Z-17-3    Electromagnetic field effect results
                   (Ess is special relativistic effect)


               ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
               ³         ÚÄ                Ä¿ý
               ³         ³  C x .999995002  ³                  Vý
      Ess  =   ³         À                  Ù              =  ÄÄÄÄ
               ³  1  Ä  ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ                 Cý
              \³                  Cý

                                               .999995002 is effect Egs
                                                          of EQ Z-17-2


      Ess =     1 - (Egs)ý
      As in:    1 - (.999995002)ý = .003161416

                                                       ÚÄÄÄÄÄÄÄÄÄÄÄÄ¿
      HIGH  special field effect                Ess =  ³ .003161416 ³
      Ess   is closing toward 0                        ÀÄÄÄÄÄÄÄÄÄÄÄÄÙ
      V     velocity is closing toward 1


  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´     COMPARING M+ AND R- RESULTS FOR HIGH AND LOW MASSES     ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

      As delineated in Items 22 to 24 above, and in Equations Z-15-3
      to Z-15-5 which immediately follow Items 22 to 24, two terms
      M+ and R- represent the enhanced mass and reduced radius on
      an object due to special relativistic results ensuing from the
      proper ratio of the speed of light divided by the proportionate
      relativistic effect of the object's gravity.

      And so the synonymity of related behaviors, (the resulting
      effects of Ess from Equations Z-17-1, and Z-17-3), when applied
      to the HIGH mass of EQ Z-17, and LOW mass of EQ Z-17-2, will yield
      appropriate M+ and R- terms for each of the masses. These are
      listed in the following:








 TABLE 5
  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³                                                          ³
  ³  HIGH MASS GRAVITY                                       ³
  ³                                                          ³
  ³            MASS1  =  (4.686984066 x 10 to 38 grms)       ³
  ³                                                          ³
  ³         RADIUS R  =   6.96265 x 10 to 10 cms             ³
  ³                                                          ³
  ³       Ess EFFECT  =   .999727802    ; from EQ Z-17-1     ³
  ³                                                          ³
  ³            M+  =  (Mass1 x 1/Ess)                        ³
  ³                =   4.688260199 x 10 to 38 grms           ³
  ³                                                          ³
  ³            R-  =  (radius R x Ess)                       ³
  ³                =   6.9607547839 x 10 to 10 cms           ³
  ³                                                          ³
  ³            CR  =  ratio (M+/R-)                          ³
  ³                =   6.735275620 x 10 to 27 grms/cm        ³
  ³                                                          ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ



 TABLE 6
  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³                                                          ³
  ³  LOW  MASS GRAVITY                                       ³
  ³                                                          ³
  ³            MASS2  =  (4.686984066 x 10 to 33 grms)       ³
  ³                                                          ³
  ³         RADIUS R  =   6.96265 x 10 to 10 cms             ³
  ³                                                          ³
  ³       Ess EFFECT  =   .003161416    ; from EQ Z-17-3     ³
  ³                                                          ³
  ³            M+  =  (Mass1 x 1/Ess)                        ³
  ³                =   1.482558107 x 10 to 36 grms           ³
  ³                                                          ³
  ³            R-  =  (radius R x Ess)                       ³
  ³                =   2.201183848 x 10 to 8 cms             ³
  ³                                                          ³
  ³            CR  =  ratio (M+/R-)                          ³
  ³                =   6.735276152 x 10 to 27 grms/cm        ³
  ³                                                          ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ



  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³   It is seen that results M+ , though higher than an     ³
  ³   originating mass, are lower than the ceiling mass Mbh  ³
  ³   in LOW mass results, and close in on ceiling mass Mbh  ³
  ³   in HIGH mass results. (Ceiling mass means a black      ³
  ³   hole mass equivalent Mbh formed in radius R.           ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³   In HIGH mass situations, M+ can look like the high     ³
  ³   mass itself, but in low mass situations, M+ is far     ³
  ³   removed from the low mass itself.                      ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³   Also, it is obvious that M+ of LOW mass results can    ³
  ³   gain substantially over the LOW mass itself, yet still ³
  ³   remain substantially below the final mass Mbh, whereas ³
  ³   M+ hardly gains over its originating HIGH mass, and    ³
  ³   can also look very much like final mass Mbh, when      ³
  ³   the HIGH mass itself looks closely like Mbh.           ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³   In real situations, the HIGH mass will be fixed at a   ³
  ³   maximum ceiling of critical limit Mc. In this current  ³
  ³   test case situation M+ looks neither like Mc, or Mbh.  ³
  ³   Yet M+ will be explicitly Mc x ûGH, and Mbh/ûGH, when  ³
  ³   GH the Golden Ratio 1.618034 is term Nx.               ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³      (Ratio CR in the LOW mass situation, is seen to be  ³
  ³      marginally more than CR = Cý/2G . This shift might  ³
  ³      be due to intrinsic truncations in the digital      ³
  ³      accuracy of the equations for lower mass densities. ³
  ³      It is hard to tell, in the scope of a digital       ³
  ³      accuracy limited to 13 significant figures).        ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ

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º FIRST INTERPRETATION º
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      Thus M+ can approach but never equal or exceed Mbh. As the Egs
      effect approaches 0 (greatest power in gravity field strength),
      the Ess effect approaches 1 (the least power, no effect), in
      velocity related relativistics.

      At the point where the gravity effect has its greatest value;
      at Egs = 0 ; the special relativistic effect ceases to exist
      (comes to a standstill), because there is no velocity, as when:


 EQUATION Z-17-4

      (C/0) / C  =  0/C  =  0 .


      This closes right in on a clear insight regards the question
      of how maximum potential relativistic gravity effect can
      contain light - effectively cancel the velocity of light.
      The velocity of light is not cancelled. The ability to have
      a velocity related to any special relativistic effect is
      cancelled. It appears this amounts to the same thing as a
      counteracting of the velocity of light.



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º DIRECT INTERPRETATION º
ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ




      A first interpretation of the consequences of Equations Z-17 to
      Z-17-3, is that a HIGH gravitational mass density results in a
      LOW special relativistic synonymity. And a LOW gravitational
      mass density results in a HIGH special relativistic synonymity.

      It has the immediate interpretation that things run faster in
      LOW gravitational events, and slower in HIGH gravitational events.

          It adds another picture to the experimentally
          confirmed property that proximity to gravity,
          relativistically causes time to slow.

      Intuitively, it answers a question as to how gravity at
      its highest can confine light. A see saw (or yin yang)
      characteristic in the works is summarized in the following:









 TABLE 7
  ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
  ³  HIGH mass gravity     Effect  Egs  approaches  1        ³
  ³                        Effect  Ess  approaches  0        ³
  ³                                                          ³
  ³  LOW  mass gravity     Effect  Egs  approaches  0        ³
  ³                        Effect  Ess  approaches  1        ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³       You can see at a glance how gravity can confine    ³
  ³       light. As gravity effect Egs closes in on 1,       ³
  ³       special effect Ess closes down toward 0 velocity.  ³
  ³       When Egs is right at 1, Ess is closed down right   ³
  ³       to 0 and the velocity of light C in a V/C ratio    ³
  ³       is vanished when 0/C = 0 .                         ³
  ³                                                          ³
  ³       Conversely, when Egs is low and closing down to 0, ³
  ³       effect Ess intensifies with a velocity approaching ³
  ³       1, which is equivalent to approaching the full     ³
  ³       speed of light.                                    ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³       In another sense, it is clearly seen that events   ³
  ³       are free to move more rapidly in activities of a   ³
  ³       HIGH velocity, in a LOW gravity field density.     ³
  ³                                                          ³
  ³       And in a HIGH gravity field density, events are    ³
  ³       constrained to low velocity activity approaching   ³
  ³       0 velocity, when the gravity field approaches the  ³
  ³       density of a black hole,  re: special relativity.  ³
  ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
  ³                                                          ³
  ³   Notes:                                                 ³
  ³                                                          ³
  ³       In real events, as summarized above in Part 2,     ³
  ³       if a mass augmentation is assumed for gravity      ³
  ³       effect Egs, then when a mass's density (without    ³
  ³       augmentation) reaches a critical mass factor Mc,   ³
  ³       the mass augmentation amount Ko is sufficient to   ³
  ³       jump the mass amalgamation in one whole bump to a  ³
  ³       black hole quantity Mbh, such that effect Egs = 1. ³
  ³       And thus effect Ess = 0; which is the equivalent   ³
  ³       of a 0 velocity for light.                         ³
  ³                                                          ³
  ³       The proportionate bump of mass Mc to Mbh is a      ³
  ³       function of the Golden Ratio 1.61803398875.        ³
  ³                                                          ³
  ³       It means there never is a situation where effects  ³
  ³       Egs and Ess slowly converge to 1 and 0, as is      ³
  ³       fictitiously indicated in Equations Z-17 and       ³
  ³       Z-17-1. As show in Part 2 further above, effects   ³
  ³       Egs and Ess will jump in a final leap to 1 and 0   ³
  ³       in a single bump via Golden Ratio functions, when  ³
  ³       the gravity mass density reaches Mc before         ³
  ³       reaching black hole mass Mbh.                      ³
  ³                                                          ³
  ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ






                       --  Continued in RELATIVE.4  --
                      
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