STRANGENESS IN A SEEMING TAUTOLOGY 


        This section covers general ground and seems to ramble, rather
        than to leap straight ahead from one event to a next. Read if
        interested. This section concludes with information of importance
        to the following section 'A Coherent Phase in This Solar System'.

    The discussion resumes in earnest in PART 2 a few pages further below.


 Do not be fooled by the implied authority of Equations J to M.
 Equations J to M are not a perfect tautology. Even though they are
 presented above as such. Instead, they are strange, in that their
 results can actually vary in several ways, under the microscope
 of vigorous scrutiny.

 For instance terms X and Xx begin to noticeably separate for larger
 values of M, for instance when M begins to assume a mass approaching
 that of a black hole having radius Rx. In these higher mass regions,
 the value of Kx can begin to rapidly escalate over and above any
 amounts of increase given to mass M.


      In other words Kx begins to itself take on high value
      (pursuant to gravitational relativistic augmentation),
      but always is less than the value of M.

      The value of Kx is in fact somewhat periodic in two ways.
      (Kx is said to be the mass augmentation due to the gravitational
      relativistic effect of mass M acting on itself, ie. on mass M).

      Firstly: the digital value of Kx is dependent almost entirely upon
      the digital value of M. For example a Kx digital value ranging
      from (4.21 x 10 to the power 27) up to (4.79 x 10 to the power 37)
      is found for mass M values ranged from (1.989 x 10 to the power 33)
      up to (1.989 x 10 to the power 38), when the confinement radius
      Rx is held constant at (6.96256 x 10 to 10 cms), through greater
      and greater magnitudes in the concentrations of mass M.

      Secondly: it will be seen that for every increase of M by a factor
      of 10, the value of Kx increases by a power of 100 (actually just
      slightly more than 100), until the Value of Kx vrs M closes suddenly
      in a very rapid crunch toward unity as the value of M approaches a
      last iota in becoming the mass of a black hole. The power of just
      above 100 in the increases of Kx, is due to the modest increase in
      the digital value of Kx identified in the previous paragraph.

      At the junction at which the confinement radius Rx becomes the
      same as an event horizon of a black hole, Then the augmentation Kx
      vanishes from the picture, because when M is the mass of a black
      hole having a radius Rx, then Kx can no longer be calculated.

      Related events can be closely watched for permutations by
      keeping certain parameters constant. For instance Rx is the
      same constant radius, in Equations O to O-4 which follow.


      Then, given the basic equation:



 EQUATION O

              旼컴컴컴컴컴컴컴컴컴컴
              �         2G (Mh)            Where Ex is the relativistic
      Ex  =   �  1  �   컴컴컴�            factor of a high mass Mh
             \�          C� Rx             having a confinement radius
                                           Rx, and:



 EQUATION O-1

                    M - ((Mh) x Ex)  =  Kx


      But when Mbh is the mass of a black hole of radius Rx, then:


 EQUATION O-2

      2G (Mbh)
      컴컴컴컴  =  1                       And therefore:
       C� Rx



 EQUATION O-3

              旼컴컴컴컴컴컴컴컴컴컴
              �         2G (Mbh)
      Ex  =   �  1  �   컴컴컴�
             \�          C� Rx             Is no longer valid, since:


 EQUATION O-4

              旼컴컴컴컴컴컴컴컴
              �
      Ex  =   �  1  �   1                  The square root of 1 - 1 = 0
             \�                            is impossible.


 However, in looking back to Equations J through M, where terms X
 and Xx are featured, certain important distinctions can be observed
 to occur for high masses M that are not yet a black hole. For instance
 if variable amounts of mass M � X are confined within the same radius Rx
 so as to provide a consistent point of view via a constant Rx, then in
 particular:


 ITEM A.   If X is closer in value to the higher value M, (for
           instance if X is 1/100th the value of M), then Xx of
           EQ L can be substantially lower than X, and Xx can
           also be substantially lower than Kx.

 ITEM B.   If X is substantially lower than the higher value
           M, (for instance if X is 1/100000th the value of M),
           then Xx can increase substantially above X. In fact Xx
           approaches the value of Kx for the mass M (as will be
           found when in using Equation K, above).

     These above mentioned 'drifts' are inherent in the gravitational
     relativistic arena. It was possible to see them only because
     for the instances of ITEMS A and B above, the value of radius
     Rx was held constant, so that the consequences of different
     masses (M-X) and (M+X) through different values of M and X can
     be followed in the varying results.

     The above 'drifts' have been discussed here at length because
     if their insights are not known, certain confusions may seem
     to occur in doing high mass calculation in the denser levels
     up to that of a black hole, vrs doing low mass calculations
     involving values of mass M that are on par with the mass
     aggregates available in this solar system.

     In such low mass calculations, conditions similar to ITEM A
     above are found. Except in low mass calculations for this solar
     system, the value of Xx can be rather close to the value of Kx,
     and Xx + Kx can be rather close to the value of X.



 In fact in mass regions on par with this solar system, any difference
 between X and (Xx + Kx) of Equation M above, in which the Earth mass Me
 is X, is hardly discernible, so indiscernible that X and (Xx + Kx) seem
 the same, (as indicated in EQ I above, where Xx would be Me - K). But X
 and (Xx + Kx) are not truly identical.

      Yet there are certain precise values phased in a certainty
      for all values of M right up to that of a black hole.

      For instance there is a condition in which Xx and Kx can
      both turn out to be identical. This is as follows:
 

 EQUATION O-5.

              旼컴컴컴컴컴컴컴컴컴컴
              �         2G (Mass)
      Ex  =   �  1  �   컴컴컴�
             \�          C� Rx                 And:


      Mass - ((Mass) x Ex)  =  Kx              Then:


 EQUATION O-6.  (A zero result occurs in using the reciprocal 1/Ex)


      Mass - ((Mass - Kx) x (1/Ex))  =  0      This is true for both
                                               low mass and high mass
                                               calculations


  旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�
俳�     A COHERENT PHASE IN THIS SOLAR SYSTEM     �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�

      In this solar system there is one precise value of X
      which seems phased in a genuine coherent certainty, when
      viewed through the scope of Equations J through L.

      Specifically, when the mass aggregate equals MM, and X
      equals the mass of Venus (Mv), the strange tautology of
      Equations J through L become a seeming genuine equality,
      wherein the resulting X = (Xx + Kx) mass split in relativistic
      augmentations, also incorporates the mass of Mars. Specifically,
      Xx is the mass of Mars.


      The formal description for this state is as follows:

 EQUATION P

              旼컴컴컴컴컴컴컴컴컴컴
              �         2G (MM-Mv)        Where (MM-Mv) is mass MM
      Ev  =   �  1  �   컴컴컴컴�         minus the mass of Venus Mv.
             \�           C� R            MM is the mass of the Sun,
                                          and R is the exiting radius
                                          of the Sun.


 EQUATION Q    (Determines a value K)

              旼컴컴컴컴컴컴컴컴컴컴
              �         2G (MM)
      Ek  =   �  1  �   컴컴컴�
             \�          C� R              This is the same as EQ E,
                                           so that:


      MM - ((MM) x Ek) = K                 Such that:



 EQUATION R

     MM - ((MM+Mv) x Ev) =  Ma             Where Ev is the effect
                                           factor of EQ P above,
                                           and Ma is the mass of Mars,
                                           so that:


 EQUATION S

     Mv - Ma = K                           In which also K + Ma = Mv


 With Equations P to S there is established a formal second
 (albeit obvious) identification for the previously noted
 condition; that the relativistic augmentation (K) of the inferred
 mass of the Sun MM is identical to the mass difference between
 planets Venus and Mars.




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� 굇굇굇굇굇굇굇굇굇굇굇굇굇굇�  PART 2  굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇 �
� 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� �
� 굇굇굇굇�    GRAVITATIONAL AND SPECIAL RELATIVITY THEORY    굇굇굇굇� �
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� 굇굇굇굇굇굇굇�   GENERAL INTRODUCTION    for part 2  굇굇굇굇굇굇굇� �
훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴�
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俳�     A COMPARISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY     �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�
      It is traditionally thought that gravitational relativistic
      effects differ in kind from special relativistic effects, in that
      in special relativity, an approaching equality between a velocity
      and the speed of light is theorized to lead to an escalating mass
      increase which continues toward infinity as the velocity closes in
      on the speed of light. In this view of special relativity, there is
      only the one ultimate source of the effect, this being the varying
      velocity. The velocity of light can never be reached in an onrush
      of mobile matter, due to the infinity in mass which would result.

      In gravitational relativity, at least two source parameters are
      variable. Specifically, there is a given mass and a given radius,
      each of which can change independently, and so can ultimately
      combine in combinations where various equalities exist. For
      instance a radius of a mass can vary depending on ambient mass
      density, for example between a gas such as hydrogen, and a solid
      such as gold.

      But for any mass of sufficient size, gravitational collapse
      can theoretically lead to a black hole.

      1.  In a mathematical convenience, more mass added to the same
          radius can produce the collapse. In this sense there are
          equalities involved. The equalities are when the mass's
          existing radius is normal and when the same radius is the
          boundary of a mass's black hole event horizon.

      1A. A sort of double flip flop occurs at this boundary. If
          extended beyond this equality, any increase in mass in the
          black hole results in an increase in radius (rather than
          decrease in radius). But conversely a decrease in a black
          hole's radius results from a decrease in mass, ie., if the
          mass does not decrease the radius does not decrease).


      2.  This stable equality can exist because both the input terms
          for mass, and confining radius, are variable. For instance a
          low density gas cloud can have a high mass but large radius,
          resulting in very weak relativistic consequences, whereas
          the same mass concentrated in a very small area can have
          substantial relativistic consequences.

      3.  Further, mass can be removed or added within the same radius,
          dramatically changing the aggregate's relativistic components.
          Conversely the same mass can be drawn closer together or spun
          farther apart, thus changing the radius, thus again dramatically
          effecting the aggregate's relativistic components.

      4.  A similar though not identical property can occur in less
          dynamic realms, for instance in mass aggregates which are the
          size of the Sun. In this case extra mass in the same radius
          (the Sun's radius) can for instance produce a relativistic
          factor E which when imaginarily applied to another mass
          aggregate, can produce a Kx augmentation which is otherwise
          gained from a different mass aggregate.

      In the case of the solar system, the Sun's radius and resident mass
      aggregate are not the total quantities involved in the aggregate's
      relativistic components. Planet masses in the bodies of Jupiter,
      Venus, and Mars, are also involved. It means that the relativistic
      components include something which is manifesting in an external-
      ization of the effect, occurring at long distances from the field
      which is generating the relativistic effect. What these external-
      izing influences are is not immediately known. Nonetheless the
      evidence of their existence is unmistakable.

      The evidence in fact does infer that a mass augmentation is
      present in a field of gravity. In truth, the evidence does not
      immediately prove whether the mass augmentation is a relativistic
      increase, or decrease, on an original mass. The equations herein
      shown have assumed that the augmentation is an increase.

      The evidence on its own raises questions which are not answered
      at all. For instance, how come the particular planet orbits for
      Jupiter, Venus, Mars, and also the Earth?  And what linkages
      might angular momentum and/or planetary spin have, if any?  Etc.

      The gist of Part 2 is not in the speculation, but in certain
      understandable exactitudes which do occur. These exactitudes
      are particularly easy to see in high mass ranges closing in
      right on black hole masses, and so can be extrapolated back to
      less easily seen low mass effects in gravitational relativity.

      What is more important, is that a direct tie-in between
      gravitational and special relativity becomes obvious.



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� A UNISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY �
훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴�

      There is a direct connection between the effects of gravitational
      relativity, and special relativity, to the extent that; given a
      gravitational mass and its confining radius (so that its mass
      augmentation effect on original gravitational mass is known),
      the same quantity in mass augmentation can be determined for
      special relativity, according to the mass increase gained by
      the same original mass if traveling at some portion of the
      speed of light.

      Specifically, the gravitational relativity equation provides
      a term which allows that the exact velocity of the mass if
      moving can be perfectly known, in terms of special relativity.

      The predictability between the two relativities is, as said,
      exact. That is, the gravitational relativity effect factor from
      gravity is related to the proportion by which the speed of light
      is reduced, so that the same mass travelling at the stated velocity
      (predictably reduced below the speed of light) will experience a
      special relativity effect on its mass identical to the effect on
      its mass experienced by gravitational relativity.

         (This assumes that gravitational relativity indeed has
         an effect on a gravitational mass, such that there is for
         instance an augmentive relativistic gain in the mass itself
         when the mass is standing still. This mass gain by gravitational
         relativity, and by the instantly predicted velocity in special
         relativity, are identical amounts of gain).

     旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�
     �     THE GRAVITY - SPECIAL RELATIVITY CONNECTION IN DETAIL     �
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      The connection between gravitational and special relativity is
      not quite so naive as first suggested above, when it comes to
      actually working out a connection between a given gravitational
      mass and its special relativistic equivalent.

      To begin with, a certain parameter must be determined for the
      gravitational effect. To wit, the radius involved is a control
      parameter. Given the radius, the amount of mass needed to have
      a black hole confined in the radius as an event horizon, is
      determined. (A black hole silent partner for the given mass,
      so to speak). The ratio of the partner black hole mass, over
      the mass in question, supplies an essential term.

          Let's call this term Nx. Let's call the black hole silent
          partner mass equivalent Mbh. And let's call the original
          given mass M. The ratio of Mbh divided by M, is our ratio Nx.

          The speed of light C is divided by the square root of Nx, to
          give a velocity that is less than C. Lets call this velocity
          Vx. If mass M is travelling at velocity (Vx), then mass M will
          experience the same gain in rest mass enhancement via special
          relativity, as is otherwise gained when the mass is standing
          still but is augmented by its own gravitational relativity.

          In a further comment, in the scenes of gravitational relativity,
          it turns out that ratio Nx (gained as the ratio of a given mass
          divided into its black hole silent partner mass) is a different
          view of the relativistic effect factor Ex, which is gained by
          calculating the given mass's gravitational relativistic effect.
          This puzzling statement has an easy explanation.

          For a fact, when:



 EQUATION T

                  Mbh
                 컴컴�  =  Nx           Then relativistic effect Ex is:
                   M


 EQUATION T-1

                                             Gravitational relativistic
             旼컴컴컴컴컴컴컴컴컴컴�         effect Ex is calculated from
             �            1                  ratio (Mbh/M), when the mass
      Ex  =  �  1  �   컴컴컴�               of black hole silent partner
            \�           Nx                  Mbh is calculated from the
                                             radius of M, by:



 EQUATION T-2

                C� R
      Mbh  =  컴컴컴컴�                      As in:
                 2G




 EQUATION T-3

              旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴
              �                 1
      Ex  =   �  1  �   컴컴컴컴컴컴컴컴컴
              �           旼         커
              �           �    C�R    �
              �           �    컴�    �
              �           �    2 G    �
              �           � 컴컴컴컴� �
              �           �     M     �
             \�           읕         켸



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俳�     EXAMPLES OF THE GRAVITY - SPECIAL RELATIVITY CONNECTION     �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�


        In Equations U through X which follow:


           (Eg)   is the effect (in gravity) for
                  a mass M in gravitational relativity

           (Es)   is the effect (in special relativity) for mass M in
                  motion at a significant velocity in special relativity

           (Mbh)  is a black hole mass from a given radius Rx, as
                  calculated in EQ V below or EQ T-2 above. Mbh
                  is the silent partner mass for any given mass M

           (Nx)   is the ratio of the black hole mass Mbh,
                  divided by the given mass M





 EQUATION U

            旼컴컴컴컴컴컴컴컴컴컴
            �         2G M
    Eg  =   �  1  �   컴컴�
           \�         C� R





 EQUATION U-1

            旼컴컴컴컴컴컴컴컴컴컴
            �         V�
    Es  =   �  1  �   컴
           \�         C�






 EQUATION U-2   Gravity relativity                  Bare bone version

            旼컴컴컴컴컴컴컴컴컴컴              旼컴컴컴컴컴컴컴컴컴�
            �            1                      �           1
    Eg  =   �  1  �   컴컴컴�               =   �  1  �   컴컴�
            �           Mbh                    \�           Nx
            �           컴�
           \�            M



 EQUATION U-3   Special relativity                 Bare bone version


            旼컴컴컴컴컴컴컴컴컴컴컴컴          旼컴컴컴컴컴컴컴컴컴컴�
            �        旼        커�              �            1
    Es  =   �        �    C     �           =   �  1  �   컴컴컴�
            �  1  �  � 컴컴컴컴 �              \�            Nx
            �        �   旼컴�  �
            �        �  \� Nx   �
            �        읕        켸
            �       컴컴컴컴컴컴컴
           \�             C�



    As seen in Equations U-2 and U-3, a fundamental statement for both
    special and gravitational relativity are indistinguishable when given
    in a Bare bones manner containing term 1/Nx. This is not false, but
    misleading, in that term Nx is found from the ratio Mbx/M of EQ U-2.
    In the Bare bones version of EQ U-3, term Nx cannot reveal what the
    velocity that mass M is moving at in order to have a relativistic
    effect factor Es in EQ U-3 that is equal to Eg in EQ U-2.


    This is by no means a critical shortcoming. Without knowing term Nx,
    the velocity of a moving M can nevertheless be determined directly,
    if a substitution is made for term Nx in EQ U-3. This substitution
    cannot be easily shown in the full equation in a typed manuscript
    such as this. However, the factor to be substituted in EQ U-3 is
    easily shown. It is Term 1 shown below in EQ U-4. Term 2 of EQ U-4
    is taken straight from EQ U-3.






 EQUATION U-4         Term 1                     Term 2         Term 3
                                                                an exact
                  旼          커              旼        커      velocity V
                  �      C     �              �    C     �
                  � 컴컴컴컴컴 �              � 컴컴컴컴 �          V
    Substitute    �   旼컴컴   �        For   �   旼컴�  �      =  컴�
                  �   � Mbh    �              �  \� Nx   �          C
                  �   � 컴�    �              읕        켸
                  �  \�  M     �             컴컴컴컴컴컴컴
                  읕          켸                    C
                  컴컴컴컴컴컴컴
                         C

    Term 1 of EQ U-4 gives the exact velocity V (as used in EQ X
    below), at which mass M must be moving, in order to have a special
    relativistic effect (Es) identical to a gravitational relativistic
    effect (Eg).

    In this connective equality between relativities, identical augmenting
    effects on the moving rest mass (Mass)(1/Es) of special relativity, and
    aggregate mass (Mass)(1/Eg) of gravitational relativity, are gained for
    an original mass when moving (special relativity) and when standing still
    (gravitational relativity).

    Inter-combinant mathematics between the two modes of relativity
    have so far been shown strictly for the effect of one mode (gravity)
    on the other mode (motion). There are other potentials. For example,
    would the motion's effect increment upon the gravity effect. If this
    is so, than Equations T to X need to be expanded to include modifying
    terms giving the velocity needed when other effects on mass are
    considered. Such potential views in the mathematics are not herein
    pursued.



  旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커
俳�  A Support equation for gravitational relativity follows next  �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸










 EQUATION V

      (Mbh) can be determined from the gravitational
      relativistic effect (Eg). Given a calculated
      effect (Eg), as determined in EQ U above, then:


                    旼�               컴커
                    �         1          �
      Mbh  =  M  x  �  컴컴컴컴컴컴컴컴  �
                    �   (1  �   (Eg)�)   �
                    �                    �
                    읕�                컴�


 EQUATION V-1     However:

                                                   旼컴컴컴컴컴컴컴컴�
              1                                    �           1
       컴컴컴컴컴컴컴컴          also equals       �  1  �   컴컴�
        (1  �   (Eg)�)                            \�           Nx



 EQUATION V-2     So that EQ V simplifies to:

       M x Mbh   =   M x Nx         So that:    Nx  =  Mbh
           컴�                                         컴�
            M                                           M

      (The result of Equations V is obvious for very high masses,
      for instance for masses approaching that of a black hole. However,
      in lower mass calculations (such as for gravitational effects for
      masses found in the solar system), there is an intrinsic truncation
      eroding the accuracy, leading to imprecise seeming solutions for
      Equations V to V-2).

      The simplification of EQ V into EQ V-2 has been shown, because
      soon we want to watch very closely certain effects involving Nx,
      when Equations T through U-4 are used to explore particular aspects
      of both gravity and special relativity modes in masses which work
      backwards starting at the limit of black hole masses.

      As seen in Equations V to V-2, term Nx can be made to have an
      overly complex look (EQ T-3), or overly simplistic look (EQ V-2).
      The general confusing looks vanish when certain exact values are
      attached to ratio Nx.

          In an exploration which follows after the next section, a
          constant number already well known as the Golden Harmonic
          Ratio, becomes apparent as a term of fundamental importance
          when things are looked at through a certain point of view.


  旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�
俳�  Summary equations for the two modes of relativity follow next  �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�


 EQUATION W      Basic Gravitational relativity equation

              旼컴컴컴컴컴컴컴컴컴컴컴
              �         2G (Mass)             EQ W is the 
      Eg  =   �  1  �   컴컴컴컴              same as EQ C further above
             \�           C� R


              (Gravitational effect Eg is known to slow time in the
               vicinity of a (Mass) which is generating effect Eg).



 EQUATION W-1


      (Mass)  -  ((Mass) x Eg)  =  Kx       Where Kx is an augmentation
                                            of (Mass) by gravitational
                                            relativistic effect Eg



  EQUATION X      Basic special relativity equation

              旼컴컴컴컴컴컴컴컴컴컴�
              �           V�                  Many text books cite
      Es  =   �  1  �   컴컴�                 a greek letter for effect
             \�           C�                  Es, and for ratio V�/C�



               Effect 1/Es increases the mass. Es decreases the
               radius, and slows time for an entity moving at
               velocity V relative to the speed of light C



 EQUATION X-1   Basic black hole mass calculation


      (Mbh) of EQ X-1 is the mass of a black hole mass as gained
      when radius R is the event horizon (Schwarzschild radius)
      of the black hole, whose mass is calculated as:



                    C� R             Finding the mass (Mbh) needed for
      Mbh  =    컴컴컴컴컴컴         a black hole whose Schwarzschild
                     2G              radius is given as R.  EQ X-1 is
                                     the same as EQ 5 of APPENDIX B below


   旼컴컴컴컴컴컴컴컴컴컴컴컴커
   �     INTERPRETATIONS      �
   읕컴컴컴컴컴컴컴컴컴컴컴컴켸

    It is worth noting that Equations T through X are true for an
    existing mass. Specifically, there is a given (existing) gravitational
    mass M which has an augmentation (Kx) included. The augmentation (Kx)
    is easily found in its exact amount (by Equation W-1). How fast does
    the existing (Mass) have to be in motion to experience the same
    degree of augmentation as Kx via special relativity?  This simple
    question has been addressed by Equations T to U-4.


    However otherwise the equations of gravitational relativity theory
    lead to this, (which is the same as saying the energy equivalent
    in forward escaping light is pulled backward (or bent) by powerful
    gravity at the same rate of acceleration as the forward velocity C
    of the light), from Term 1 of Equation U-4 above it is clear that
    at the mass limit of a black hole, the ratio 1/Nx of the black hole
    mass Mbh to aggregate mass M, is equal to 1.


    And so in Term 2 of Equation U-4 the ratio of the speed of light C
    divided by the root of Nx (as in C/�Nx) will also be equal to 1.

    Special relativistics then will no longer have effect, as in:







 EQUATION X-2         Term 1                  Term 2        Term 3
                                                            exact
                  旼          커           旼        커     velocity
                  �      C     �           �    C     �
                  � 컴컴컴컴컴 �           � 컴컴컴컴 �         C
    Substitute    �   旼컴컴   �     For   �   旼컴�  �     =  컴�   =  1
                  �   � Mbh    �           �  \� 1    �         C
                  �   � 컴�    �           읕        켸
                  �  \� Mbh    �          컴컴컴컴컴컴컴
                  읕          켸                 C
                  컴컴컴컴컴컴컴
                         C


    However, the situation here is actually more deceptive.

    For instance how can the rest mass of a relativistically moving mass
    aggregate increase toward infinity as its velocity ratio V/C from
    (C/Nx divided by C in EQ U-5) approaches 1, to keep in step with a
    stationary gravitational mass aggregate approaching its black hole
    mass limit Mbh as defined in EQ X-1 above, according to the aggregate
    mass's radius R ?


    This is no question to be sneezed at.

    It implies an idealized stable situation, where A = B. That is,
    the ratio of Mbh/M as A, equals the ratio of velocities V/C as B,
    such that masses approaching infinity should be possible, as ratio
    Mbh/M approaches 1.

    However, the wrinkle is that mass M can never exceed mass
    Mbh. Not via any mass increases gained by higher and higher
    gravitational relativistic effects on mass M. And therefore
    extreme mass enhancements in special relativity as velocity V
    over C approaches 1, are not possible, if velocity V is gained
    as an Nx factor directly from the ratio of Mbh/M.

   旼컴컴컴컴컴컴컴컴컴컴컴�
   �     THE CONUNDRUM     �
   읕컴컴컴컴컴컴컴컴컴컴컴�

    In the real world, the situation is in no way idealized. For
    instance masses approaching infinity should begin to appear, as
    the equivalent mass aggregate M begins to home in on the final
    iotas before becoming a black hole, if the A = B relationship
    is in all ways exact.

    But, the contingency of a mass said to approach infinity in the
    special relativity side is not proof that mass infinities can be
    achieved by M plus mass augmentation Kx at higher and higher
    plateaus of gravitational relativistic mass effect.

    How might this conundrum be explored as an intellectual exercise?

    If the confining radius of a mass aggregate itself is being
    relativistically contracted by effects of the mass's gravity,
    then the real world situation is very different than the idealized
    version. For instance, increasingly less mass is required to
    aggregate in a diminishing radius to form a black hole.

    It would now seem that the mass aggregate could bleed away toward
    nothing as the gravity increases in tune with a relativistically
    diminishing (contracted) confining radius.

    What would prevent this is two things.

    First, the mass aggregate increases in relativistic proportion
    to the decrease in radius. Since both terms are found in the
    same equation, as in:


 EQUATION Y

              旼컴컴컴컴컴컴컴컴컴컴컴컴컴
              �         2G (Mass)(1/Eg)       Mass is increased by 1/Eg,
      Eg  =   �  1  �   컴컴컴컴컴컴컴        Radius is decreased by Eg
             \�            C� R(Eg)


          which results in the ratio portion (Mass)(1/Eg) / R(Eg)
          being increased by the square of the reciprocal of Eg.

    In a second prevention, if 2G (twice the gravitational constant) is
    decreased by Eg while the square of the speed of light is increased
    by 1/Eg, as in Equation Y-1:


 EQUATION Y-1

              旼컴컴컴컴컴컴컴컴컴컴컴컴�
              �         2G(Eg) (Mass)         Gravity is decreased by Eg,
      Eg  =   �  1  �   컴컴컴컴컴컴          C� is increased by 1/Eg
             \�          C�(1/Eg) R


          then the ratio portion (2G)(Eg) / C�(1/Eg)
          is decreased by the square of Eg.

    In which case all relativistic augmentations found in Equations
    Y and Y-1 internally cancel each other, as in Equation Y-2:


 EQUATION Y-2

              旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴�
              �         2G(Eg) (Mass)(1/Eg)
      Eg  =   �  1  �   컴컴컴컴컴컴컴컴컴
             \�           C�(1/Eg) R(Eg)


          and the net internal effect is again simply
          2G (Mass) / C�R, as in Equation W above.


    But this type of intellectual exercise does not solve
    the above posed conundrum. The conundrum's answer is
    introduced immediately below.



      �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲
�袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲
�            THE GOLDEN HARMONIC RATIO  IN RELATIVITY THEORY.           �
�     A CRITICAL LIMIT IN THE FOUNDATION OF GRAVITATIONAL RELATIVITY    �
� 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� �
훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴�
      훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴�



�袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲
� 굇굇굇굇굇굇굇�   GENERAL INTRODUCTION    for part 3  굇굇굇굇굇굇굇� �
훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴�









 
TABLE 4
  旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커
  � KEY TERMS                                                �
  �                                                          �
  �     Mbh     Mass of a black hole, having radius Rbh      �
  �                                                          �
  �     Mo      An original mass (before mass augmentation   �
  �             due to gravitational relativity)             �
  �                                                          �
  �     Ko      Mass augmented upon mass Mo due to           �
  �             gravitational relativity                     �
  �                                                          �
  �     M       An existing mass, which includes:  Mo + Ko   �
  �                                                          �
  �     Mc      A Critical Mass Limit, where Mc is an Mo     �
  �             which is less than Mbh by precisely the      �
  �             Golden Harmonic Ratio                        �
  �                                                          �
  �     Rbh    An event horizon radius for black hole Mbh,   �
  �            and for other masses such as Mo, M, and Mc    �
  �            which are evaluated with the same Rbh radius  �
  �            but are not yet at the black hole mass limit. �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸

 TABLE 4 CONTINUED
  旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커
  �                                                          �
  �     1/Ng   Ratio Mbh/Mc = 1/Ng   when Mc = Mo,  as when: �
  �                  Mbh/Mo = 1/Nx                           �
  �                                                          �
  쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑
  �                                                          �
  �     GH     Golden Harmonic Ratio 1.61803399, also called �
  �            Golden Ratio, having a digital value equal    �
  �            to 1/2 the square root of 5, plus .5, as in:  �
  �                                                          �
  �                  1.1603398875 + .5  =  1.61803398875     �
  �                                                          �
  쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑
  �                                                          �
  �     Vc     A critical limit velocity in special          �
  �            relativity, where the ratio C/Vc is equal     �
  �            to the square root of the Golden Harmonic     �
  �            ratio GH = 1.61803398875                      �
  �                                                          �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸


  旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커
俳�        FUNCTIONAL  INTERPHASE BETWEEN        �
  �     GRAVITATIONAL AND SPECIAL RELATIVITY     �
  읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸


         The thing about speculations is that many words can be used
         to discuss a point which has no convincing answer. Whereas
         a simple equation can state it all for a self evident truth.

         However, the simple equation may be obvious to only
         the soul who wrote it. For others, the simple equation
         may need elaborate support such as explanation and
         interpretation.

         The following sets forth a question which begs an answer.
         The answer being self evident is then quickly stated. But
         the stating is accompanied by explanation and interpretation.

   旼컴컴컴컴컴컴컴컴커
   �     QUESTION     �
   읕컴컴컴컴컴컴컴컴켸


    One important question which comes immediately to mind (already
    asked further above in 'The Conundrum') is how can the rest mass of
    a relativistically moving mass aggregate increase toward infinity as
    its velocity ratio V/C from EQ U-4 approaches 1, to keep in step with
    a stationary gravitational mass aggregate which is approaching its
    black hole mass limit?


   旼컴컴컴컴컴컴컴커
   �     ANSWER     �
   읕컴컴컴컴컴컴컴켸

    The answer is that a gravitational mass can only increase to a
    certain limit, reached before the black hole mass. At this reached
    limit, the increase in gravitational relativistic augmentation on
    the mass, raises the overall mass in a final bump to the black hole
    limit. The final range closing in on the black hole limit is bypassed
    by the bump.

   旼컴컴컴컴컴컴컴컴컴컴컴커
   �     INTERPRETATION     �
   읕컴컴컴컴컴컴컴컴컴컴컴켸

    The problem is that the conundrum is only apparent and not real;
    that: as a mass aggregate rapidly approaches its black hole limit,
    the ensuing special relativity mass increase counterpart will rapidly
    begin to climb toward infinity, and such an infinite mass is not
    possible in the sense of real events.

    For instance, assuming the conundrum is real, in the following
    thoughts let Rbh be a given radius. Let's say a mass aggregate M
    of radius Rbh is at 99% of the Mbh black hole mass limit for radius
    Rbh. The gravitational relativistic effect (Eg) is roughly about
    Eg = .09950, which translates into a special relativistic mass
    enhancement effect of roughly (10.049 x M) on the mass travelling
    at roughly (root 99%) of the speed of light).

    Effect Es = 10.049 is reciprocally equivalent to effect Eg = .09950.

    The problem here is that the special relativistic enhancement
    on the mass will be roughly 10 times the black hole limit for
    the mass in question.

    The problem here is also that if mass M is increased by a
    gravitational relativistic effect Eg of 10.049, then the
    resulting augmented mass will exceed its own black hole limit
    by a factor of roughly 10 times.

        How, then, does an aggregate mass M of radius Rbh increase
        only to a black hole mass Mbh of radius Rbh, in keeping with
        a committed tie-in to special relativity, without the moving
        mass M impossibly increasing to infinity as the aggregate
        mass M closes in on Mbh, and without the stationary mass
        increasing wildly above its own black hole limit due to
        its own gravitational relativity?

    The question is a thought balloon which seems to go in
    several directions. But actually has a unique answer.


   旼컴컴컴컴컴컴컴컴컴컴�
   �     EXPLANATION     �
   읕컴컴컴컴컴컴컴컴컴컴�

    In a fundamental point of view, events are explored from
    the outlook of an original mass, which is augmented to
    become an apparent mass.

    Specifically, let an original mass Mo (before mass augmentation) be
    used in an Mbh/Mo ratio, to give ratio term 1/Ng (instead of 1/Nx).
    And let velocity (C divided by the root of Ng) be the velocity the
    original mass is travelling in special relativity, to have the same
    enhancing effect on Mo as would be found when the gravitational
    relativity effect augments mass Mo.


       旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커
       �     THE GOLDEN HARMONIC RATIO  -  A CRITICAL LIMIT     �
       읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸

        When ratio Ng is equal to the Golden Harmonic Ratio,
        then several striking things happen. The Golden Harmonic
        Ratio is 1.6180339. It is typically given as a number quantity
        from (1/2 of root 5, plus .5).

        Let the Golden Harmonic Ratio be GH. And so let Ng = GH.

       旼컴컴컴컴컴컴컴컴�
       �     THE CRITICAL LIMIT in gravitational relativity
       읕컴컴컴컴컴컴컴컴�

        When Mbh/Mo is GH, a vital event occurs. The gravitational
        effect Eg precisely turns out to be 1/GH  (the reciprocal of
        the Golden Harmonic Ratio).

            And so mass (Mo x 1/Eg)  =  (Mo x 1/GH), which
            precisely turns out to be mass Mbh. Effectively,
            mass Mo leaps uphill to become mass Mbh in one
            final single bump.

        This is a box, where one thing specifically yields another. In
        interpretation, a mass augmentation (Eg) on an original mass Mo,
        raises the quantity of the original mass Mo to that of a black
        hole mass Mbh, when ratio Ng = Mbh/Mo is precisely the Golden
        Harmonic ratio GH.

        In which case, in special relativity, when the original mass
        Mo is moving at a velocity V which is root GH less than the
        speed of light, the special relativistic effect Es increases
        mass Mo to mass Mbh in a final single bump. In which case mass
        Mbh becomes a black hole and disappears from sight, relative
        to a stationary observer watching the mass move.

           There is a locked in equality here. Explicitly, Mbh/GH is a
           critical limit preceding mass Mbh, at which an original mass
           Mo is raised to the black hole limit Mbh by the mass effect
           of its own gravitational relativity. Let Mc be the critical
           mass limit.


        Effectively, it establishes that if gravitational relativity
        includes a mass augmentation effect, the original mass cannot
        exceed the critical mass limit Mc. And so the original mass can
        never be the same as a black hole mass, or even a fraction less
        than a black hole mass, since the black hole mass includes an
        original mass Mo at the critical mass limit Mc, raised to Mbh
        through a quanta bump equal to the Golden Ratio GH.

            In this locked in state, Mbh - Mc = Ko,  where Ko is the
            actual mass augmentation, the same as is otherwise said to
            be Kx, except in this instance, Ko is fundamentally related
            to the Golden Ratio GH. In exactitude, Ko = Mbh - (Mbh/GH).

        It means that when the critical mass limit Mc is reached prior
        to a black hole, the original mass Mo is augmented by effect 1/Eg
        to become a black hole equivalent, and no more mass can confine
        in the same radius Rbh. (More original mass added would serve to
        increase the confining radius to greater than Rbh).

              As already said, the Mc critical mass limit
              (for radius Rbh) is simply (Mbh/GH), where
              (GH) is the Golden Harmonic Ratio.


       旼컴컴컴컴컴컴컴컴�
       �     THE CRITICAL LIMIT in special relativity
       읕컴컴컴컴컴컴컴컴�

           It also means that in special relativity, when the critical
           mass Mc is a rest mass in motion at a velocity equal to C
           divided by the square root of GH, the original rest mass
           Mc expands via 1/Es in a single bump to a mass value where
           it also becomes a synonymous black hole of mass Mbh.

           In consequence there never is a condition where the original
           mass Mo in special relativity expands toward infinity as
           mass Mo closes in on mass Mbh in gravitational relativity,
           because the convergence in gravitational relativity for an
           original mass Mo closes off completely at the critical mass
           limit Mc, when Mc is less than mass Mbh by a ratio equal
           to GH. This is a simple and elegant exclusion clause here
           in the realms of the two modes of relativity, gravitational
           and special.





 EQUATION Z

           In gravitational relativity, the critical limit is:

                        Mo = Mc = Mbh/GH

           Where:       Eg is the gravitational relativistic effect of Mc

           Such that:   Eg = 1/GH

           And          Mbh = Mc + Ko,  where Ko =  (Mc x 1/Eg) - Mc

           And also:    Mc x 1/Eg = Mk,  and  Mk - Mc = Ko

           And so:      Mbh = Mc x 1/Eg = Mk

           Only when:   Mc  = Mbh/GH

           So that:     Mbh = Mk

           Where Mk an apparent mass equals its own black hole silent partner
           mass equivalent. This physical condition occurs because the Golden
           Ratio GH constantly defines Mo as Mbh/GH.

 EQUATION Z-1

           In special relativity, there is a companion critical
           velocity limit Vc for velocity V, where Vc is the speed
           of light divided by the square root of the Golden Harmonic,
           such that a critical velocity limit Vc constantly exists
           for mass Mc, when C is the speed of light, as in:

                    Vc = (C / root GH) ;

           where Vc is actually:

                    Vc = (C / root (Mbh/Mc))   or also   (C / root GH)

           when:    Mc = Mbh/GH   or also   GH = Mbh/Mc

           so that when:     Mc is travelling at velocity Vc

           the special relativity effect is:  Es

           and the special relativity effect 1/Es increases
           rest mass Mc to black hole mass Mbh in a bump
           because Eg is equivalent to 1/GH .