INTERFACE RELATIVITY

A helpful sonic stereo experiment reports contains possible
insights casting more light on Interface Relativity, see the
reports here: http://www.cosmicastronomy.com/appendix.txt

 

ABSTRACT:

A step toward unifying the forces. The gravitational force, and electro-magnetic forces (strong, electro-weak, and charge) are unified through Interface Relativity equations which join general and special relativitist effects into a fundamental equality. The interface specifies the principles and properties by which the general (gravity) and special relativity (radiation) modes are unified into a single set of of coherent balanced equations.

INTRODUCTION TO MASS INCREASES BY

 GRAVITATIONAL RELATIVITY

The following proposes that steady state relativistic effects can be understood to occur pursuent to gravitational fields.

The wider range of distortions in space embraced by the GENERAL
THEORY OF RELATIVITY are put aside and certain specific effects
are studied in detail. These specific effects are understood to
come under the heading of GRAVITATIONAL RELATIVISTIC EFFECTS.

Greydon Moore
Ottawa, Canada, June, 1990.
 

GRAVITATIONAL RELATIVITY THEORY

CONNECTS CERTAIN SOLAR PLANET MASSES

ALSO, GRAVITATIONAL AND SPECIAL RELATIVITY THEORIES ARE INTRINSICALLY RELATED

By assuming a mass and spacial effect in general relativity, a
proposed gravitation relativity is evident, in which there is a
direct tie-in between effects seen in Special Relativity and in
Gravitational Relativity. In fact, properties commonly factored
for a star or black hole in Gravitational Relativity, can also
be factored in Special Relativity, and visa versa. This suggests
not necessarily a unified field theory, but definitely a connection
between certain properties in gravity, and in electro-magnetism.

ABSTRACT

Several facets are to be discussed in the following.

(Part 1) Arguments demonstrating an increase in mass by the
effects of gravitational relativity are shown through events
which occur in the solar system.

(Part 2) Effects for gravitational and special relativity are shown
to be synonymous for a given mass. Critical limits are uncovered in
the behaviors of both relativities. In specific situations, mass is
locked to a ceiling which is less than, but is determined from, black
hole mass equivalents. In this, it is found that the maximum original
mass which can be gathered before gravitational relativistic effects
are maximized, is that of a black hole's mass divided by a factor
of 1.618034 (a number constant known as the Golden Harmonic Ratio).
The maximum velocity attainable by this mass when moving in special
relativity, is the speed of light divided by the Golden Harmonic Ratio.


(Part 3) It is found that for any visible mass, there is a
maximum special relativistic limit on the mass. This limit can be
known in advance by knowing the maximum velocity the moving mass can
attain and still remain visible in the normal sense, when observed by
a stationary observer. The maximum effect is a derivative of the speed
of light reduced by the relativistic effect of the mass's gravity.
This is shown to define an upper limit velocity at which any given
mass can appear in the same state of the universe as the stationary
observer. Any rest mass reaches this barrier at a plateau that is
predictable, and so the mass cannot visibly expand to infinity.


(Part 4) Innuendos of a unified field theory are harking loudly,
popping out of the framework of relativistic physic. There is a
universality in obvious behaviors working directly between the
one field's venues (gravity) and the other field's venues
(electromagnetism). As to whether these equalities can constitute
segments of a full fledged unified field theory is not to be
addressed at this time, in the scope of the following disclosures.

PART 1

GRAVITATIONAL RELATIVITY

A little known (entirely unknown) fact is that certain solar
planetary masses can be connected as a direct consequence of
gravitational relativity. This is shown to be true when it is
surmised that relativistic effects of gravity may include an
intrinsic increase in the mass comprising the source of the
gravity.

The relativistic increase for the Sun mass is very small compared
to the mass of the Sun itself. Even though the increase in mass
is small at roughly 4.23 x 10²⁷ grams, the increase
is nevertheless nearly 7 times the mass of Mars, and is marginally
less than the mass of Venus.

Such an increase in the Sun mass, when calculated to advanced
accuracy, is found to be exactly equal to the mass difference
between Venus and Mars. Another discrete relativistic potential
includes ¹/₂ the mass of Jupiter added to the mass of the Sun.
The existence of states makes it possible to infer a more
accurate estimate for the existing mass of the Sun.

The radius of the Sun is considered to be a constant for various
manifestations, shown to correspond to parameters which operate
between solar mass equivalents up to the masses of black holes.
In this, a link between gravitational and special relativity
is shown. The link is the subject of part 2.

PART 2

SPECIAL RELATIVITY

It can be easily demonstrated that a visible mass moving at
velocities nearing the speed of light, can never grow to infinite
quantities and remain visible in the normal sense, and so can never
achieve a velocity equal to the speed of light, in the normal sense. 

This is because gravitational relativistic effects have to be
considered for a moving mass, if it is assumed that gravitational
relativity includes an effect that increases the original state of
the mass which is the source of the gravity's relativistic effect.
It is readily shown that such gravity effect has significance to
special relativity.

There is a boxed in limit, where the moving mass (bumped in
value in special relativity) assumes a value equivalent to the
mass of a black hole, when the original rest mass is expanded by
the effect of special relativity, in direct accord with the mass's
radius contracted by the effect of special relativity.

When assuming the mass of a black hole equivalent, the
moving mass effectively drops from sight in the normal
physical view as seen by a stationary observer.

(See Appendix A at the end of this document, for a related
discussion involving elementary particles such as the proton).

One of the finite limits to which a mass can be accelerated
in special relativity, and to which a mass can be accumulated
in gravitational relativity, can be explicitly expressed for
both modes of relativity as factors of a number constant known
as the Golden Harmonic Ratio, 1.61803398875 .

In this, the Golden Ratio's significance is to the existence of
black holes. Specifically, a black hole's mass includes both an
original mass and an augmentive portion from the relativistic
effect of gravity, to comprise the total mass involved. The
relationship between original, gained, and final black hole
mass aggregations, can be expressed in exact terms of the
Golden Harmonic ratio.

In particular, however, in the dynamic behaviors of both
relativities, important boundaries are reached at a certain
critical limit whose mathematical significance is the Golden
Harmonic Ratio. The parameters here include a black hole's
mass aggregate and event horizon.
 

PART 3

THE GOLDEN HARMONIC RATIO

The effects of gravitational relativity can be generally related
to the effects of special relativity, to the extent that relativity
effects of gravity and of special relativity can be shown to be
equated through a single common factor.

The maximum velocity attainable by a visible moving mass, is
the speed of light reduced by the proportionate effect of the
gravitational relativistic effect in the mass being accelerated.

The critical limit (maximum velocity) possible, is restricted
by bounds achieved in special relativistic effect when the rest
mass has increased, and radius has contracted, to a point where
the moving entity reaches a state where it forms a black hole and
effectively disappears from view, relative to a stationary observer.

The barrier limit is easy to calculate and to mathematically
confirm, when given the original rest mass and radius.

It becomes clear that, generally a visible mass accelerated to
relativistic velocities cannot theoretically achieve an infinite
mass, and the velocity can never theoretically equal the speed of
light. The traditional interpreted statements in special relativity
which say any visible mass continues to expand toward infinity,
and the velocity continues to the speed of light, are in error
about such things.

GRAVITATIONAL RELATIVITY THEORY

GENERAL INTRODUCTION for part 1 The Solar System

 

In the following, the existing orbits of planets are not
considered as terms, and all of the events are shown to
occur as within a constant confinement radius which is
the existing radius of the sun.

A general relativistic equation is in common use for gravitational
effects. Such an equation has been around in physics since 1916.
Variations of the equation are also in common use. Given a known mass
for instance, a Schwarzschild radius for that mass confined as a black
hole can be immediately calculated.

Conversely, given a radius, how much mass would be needed to be
confined within that radius as a black hole can also be calculated.

Such effects are a steady state system. It is the amount of
mass within a specified radius which counts. The effects are
constant per given mass and radius, since no outside velocity
or acceleration is involved with the masses sitting stationary.

The same is true for mass aggregates which are not a black hole,
but which have mass sufficiently large, and a radius sufficiently
small, for gravitational relativistic effects to be discernible.

For stars the size of the Sun, for instance, there are discernible
effects, even though they appear to be very slight at first sight.
In a closer look, however, the slight effects can reveal many major
properties in the fundamental relativistic behavior of gravity.

       GRAVITATIONAL RELATIVISTIC EFFECT

In principle, gravitational relativistic effects are calculated via
the standard equation, for varying mass and radius, until a meeting
point is reached at which the mass and radius correspond to the
formal parameters of a black hole.

In the standard equation, a term for the relativistic effect
results, which has been mainly used to determine the slowing
of time in closer vrs more distant proximities to the field
generating the effect.

The same term can be used to find out how much a gravitational
mass's radius can further contract relativistically per given
increase in mass, when assuming that gravity relativistically
contracts its own confinement radius. The same term can be used
to calculate the gravity's relativistic effect on its own mass.

This term can be called E (for effect). The value of term E
suddenly nose dives toward 0 when the mass is sufficiently large,
due to a sudden relativistic upsurge in pull in the greater power
of the gravity itself, at which point the existing mass becomes
a so called black hole and the existing mass's radius no longer
appears to contract, rather, it will begin to increase given
further increases in mass.

This mass and radius stabilization is considered a physical
boundary called the Schwarzschild radius, or event horizon.

The stabilization is discussed in 'A Comparison Between
Gravitational And Special Relativity' (found directly
under the 'General Introduction for Relativity' Part 2',
below), and is formally described in Equations 3 to 5
in APPENDIX B at the end of this document.

        GENERAL MASS QUANTA EFFECT

In variations of the equations, when a quantity of mass is given
and the radius containing it is also known, then a simple solution
using term E can denote how much of a mass increase may occur in
the mass, due to a relativistic augmentation by the mass's gravity.

The augmentation can be conjectured to occur in two ways.
Either a measured mass is naked (original with no relativistic
augmentation), or is augmented (the measured mass includes
the augmentation).

Hence the augmentation can be conjectured to be in two
modes; either a decrease upon the originating mass, or
an increase.

In keeping with special relativity effects, a mass increase
in gravitational relativistic augmentation can be presumed
with no difficulties.

For instance the Sun (given its mass and radius) is surmised
to have a visible radius which is marginally reduced by
relativistic augmentation (shrunk), and so the Sun's apparent
mass is also surmised to be marginally augmented (expanded) in
a mass increase by an equivalent relative proportion.

The problem is that such a conjecture (relativistic augment-
ation in mass) is hard to prove, since it is not possible to
actually separate a given mass from its gravity and so observe
any change in the apparent mass, when the mass is compared with
vrs without the relativity of the gravity.

In which case, any evident mass augmentation will have to
be learned by some secondary means.

In this solar system such a means is provided mechanically,
by the fact that the amount of solar mass augmentation is a
meaningful quantity in company with the existing mass of some
of the planets.

The mass augmentation has a value which is in a quantum
correspondence to the existing masses of Venus and Mars.
This makes the mass augmentation clearly visible. The fact
that the relativistic mass is involved with these planets
(in relationship with small particles external from the Sun)
is very curious.

GRAVITATIONAL RELATIVISTIC EFFECTS

The standard equation for gravitational relativistic effect
is described as follows:

EQUATION A

E = √[ 1 - 2G(mass)/C²R ]                                

 

The square root of [1 - (the product of 2 times the
gravitational constant G, times a mass) /(the radius

of that mass times the speed of light squared) ],
yields a gravitational relativistic effect factor, termed E.

EQUATION B

The radius of the mass times the reciprocal of the
E factor, gives the originating radius of the mass,
ie., before contraction of the radius by the mass's
gravitational relativistic effect.

[ R x (¹/_E) ] - R = Re    Where Re is the amount of space by which the Sun's

                                             radius is contracted by the relativity in the Sun's mass

 

[    Ro   ]                       Ro is the original radius before effect

                                             R is the existing radius (the radius we see) which includes

                                             effect (Ro + Re)                  

These (Equations A and B) are well known and nothing new
has been so far stated.

The relativistic collapse in the Sun's radius
is very slight, hardly 1½ kilometers.

This is learned as the difference between the originating Sun
radius Ro, minus the existing (augmented) radius R. The difference
seems to be a remarkably close approximation of ½ the Schwarzschild radius needed for the Sun mass to be a black hole.

However this is not surprising, in that the smaller the mass and/or

the larger the radius, the closer the radius augmentation is to ½ the Schwarzschild radius. The ½ approximation grows closer, the

less the mass aggregate is a black hole.

In principle, with little mass and a large radius, there is
very little augmentation. Conversely, a very small radius for
the small mass is needed as the event horizon for the small mass
to become a black hole.

The point intended is that as the mass to radius ratio
approaches the primes of a black hole, the rates of
change due to gravitational relativistic effects climbs
up a steepening gradient.

At solar quantities, the effects are so slight as to be
normally thought of as negligible. But this is not so.

If for instance ¹/₂ the mass of JUPITER is added to that of
the Sun, and this enhanced mass sum is regarded as being within
the confines of the existing Sun radius, the relativistic mass
augmentation effect when applied to the mass of the Sun minus
1/2 the mass of Jupiter, equals the previously noted congress
involving Venus and Mars masses, (at the end of 'General Mass
Quanta Effect', above).

 

Such state arrays reveal a previously unsuspected property,
of relativistic mass quantal arrangements displaced at long
distance from the source generating the relativistic mass
effect. A first suspicion is that:

'THERE IS AN INCOMPATIBILITY BETWEEN A GRAVITATIONAL
FIELD AND THE RELATIVISTIC EFFECT IT GENERATES'

The appearance is that some aspect of the relativistic mass
effect generated in a field of gravity, does not stay within
the field generating it.

 

In supposition, it appears that some relativistic component
is expunged (externalized) from the originating field of
gravity. In the case of our solar system's example, the
masses of Venus and Mars, along with Jupiter, are external
and yet relativistically tied to the Sun mass.

         ESTIMATED ACCURACY OF SOLAR MASSES

Masses in the solar system are traditionally published in two
ways. A mass for each planet is given as a ratio between it and
the mass of the Sun. Since comparative ratios can be inferred to
considerable accuracy, the Sun to planet mass ratios for most of
the planets are well known.

On the other hand estimating the actual mass of a planet or
the Sun in terms of (say) gram units, is not so easy, since
there is no way of actually sitting a planet on a scale. For
that matter, estimating the real mass of the Sun (in say grams)
is also difficult since the Sun cannot be weighed on a scale.

The problem is compounded in that in order to know a real
weight (in grams) requires that the universal gravitational
constant (G) be known to high accuracy, which it is not.
Whereas determining the mass influences of one body on another,
as a ratio, is easier since (G) is not a critical factor for
the accuracy.

For these reasons the real mass of (for instance) the
Sun (in say grams) cannot be stated with great accuracy
by ordinary measuring methods.

The Sun's mass is currently given as somewhere between
1.989 x 10³³ grms, and 1.991 x 10³³ grms. Whereas
planet masses are currently given in gram figures accurate
to between 4 and 5 significant figures. The greater accuracy
for planet masses is assisted by the fact that the planets
tend to subtlety bounce each other around in orbit, and their
bouncing can be closely watched. Whereas the Sun is hardly
bounced by the less hardy influence of the planets.

The Earth - Moon combination gives the best look at bouncing.
But rigorous real weight analysis for the Earth is not so easy
when tried, because both the Earth and Moon also subtlety bounce
around as a unit.

If the gram weight of the Earth (5.976 ± .004 x 10²⁷ grms)
is multiplied by the Sun to Earth mass ratio (332,995.9 ± .4),
then the Sun's gram weight results as (1.9899834 x 10³³ grms).

This value is actually deemed low to a very minor degree for the
equations which follow below. In the following, a Sun mass in the
vicinity of (1.990993 x 10³³ grms) is explicitly inferred.

Another problem in any advanced accuracy is inherent in the weak
solar gravitational relativistic effects per se. Because the effect
for solar mass quantities is so slight, there is a loss of some
accuracy due to inherent truncation in doing the calculations.

In the equations which follow, accuracy has been maintained
to 13 significant digits, but inherent truncation results at
the 7th significant digit of certain of the terms.

Such truncation is diminished when dealing with larger
masses confined within small radii. The truncation disappears
completely when dealing right at the range of black hole masses.

Hence, black hole limits can provide a tool for comparing
calculations, to determine which calculations produce
exactitudes and which produce close approximations only.
This is actually more straightforward than it sounds.

         BASIC CONVENTIONS

In the following, the existing orbits of planets are not
considered as terms. All of the events are shown to occur
as within a constant confinement radius, which is the
existing radius of the sun.

For the sake of convenience, the mass of the
Sun is shown as a standard term labeled (M_M).

In the following, the calculations are accomplished at
an accuracy of 10 to the 13 significant digits. Zeros are
used to fill gaps between available digits and the 13th
significant digit. As already mentioned, some of the terms
are accurate only to the 7th significant digit. In fact,
some terms cut off at the 7th digit. For this reason, the
highest maintained accuracy possible is very important.
 

For the universal gravitational constant G, a recent
revision having a digital value of 6.6720 x 10^-8 is used.

The speed of light C of the following value is used:
2.99792458 x 10¹⁰ cm/sec.

The radius of the Sun is used as a constant R, having
the value 6.96265 x 10¹⁰ cm.

MASS CONVENTIONS

The following mass aggregates have been adopted as standards for
the involved quantities. The high accuracy given them has been
by the adjusting of repeated pure math experimental results until
a semblance of coherency in the mass standards looked viable.

The term 'aggregate mass' is used for denoting a mass (such as
the Sun, plus or minus another mass (such as ½ the mass of
Jupiter).

'Aggregate mass' is also used to denote any apparent
mass, since the mass is assumed to include relativistic
augmentation due to gravity.

Hence, the original mass before augmentation is

termed 'original mass', or 'originating mass'.

K has been adopted as a term to explicitly denote the
relativistic mass augmentation in the Sun's mass due
to the Sun's gravity.

In determining aggregate mass values, the value of M_M for the
Sun's apparent mass was first determined, based on an assumed
equality that a so called K augmentation factor for the Sun mass
is indeed the mass difference between planets Venus and Mars.

Without doubt the real values for the mass aggregates (given
in grms for instance) will marginally change depending on
future adjustments of the universal gravitational constant,
and perhaps sharper astronomy techniques.

(For that matter, mass M_M may not be the true real
mass of the Sun. It may turn out that M_M is the mass
of the Sun ± something else).

It is anticipated that any such changes would nevertheless
prove to continue to be coherent within the realms of the
gravitational relativistic state equations which involve them.

Several tables and basic equations follow. Following these,
a discussion begins on how a mass of M_M was inferred for the
Sun, via gravitational relativistic effects.

Table 1 which follows, lists important mass aggregations,
and the highest resolved real mass values possible as used
to explore their relativistic highlights.

                                   TABLE 1:  INFERRED VALUES

M_M 

= Existing Sun mass, presumed to include
   original mass plus mass augmentation K

=1.9909930 x 10³³ grms

K 

= Gain in original mass of the Sun, the
   amount of relativistic augmentation 
   due to the Sun's gravity

= 4.226490 x 10²⁷ grms

M_bh

= Mass of a black hole having an event 
   horizon equal to the Sun's radius R

= 4.689536679 x 10³⁸ grms

                             TABLE 1-A ESTABLISHED VALUES

R 

= Existing Sun radius

=6.96265 x 10¹⁰ cm

C

=  Speed of light

= 2.99792458 x 10¹⁰ cm/sec

G

= Universal gravitational constant

= 6.6720 x 10^-8 cm³/grms sec²

C_R

= A physical constant for Mass/Radius
   ratio of a black hole

= 6.735275620 x 10²⁷ grms/cm

G_H

= Golden Harmonic Ratio

= 1.61803398875

√G_H = 1.272019649

                            TABLE 2

Planetary masses - Data is from tables found at the
                     back of the following reference:          

   UNIVERSE by Don Dixon, Houghton Mifflin Co.,  Boston, 1981

Moon     = 0.0735 x 10²⁷ grms
Venus    = 4.8683 x 10²⁷ grms
Earth     = 5.976   x 10²⁷ grms
Mars      = 6.4181 x 10²⁶ grms
Jupiter  = 1.901   x 10³⁰ grms

Sun       = 1.9888 x 10³³ grms

                            TABLE 3

                            Certain terms are used to generalize  certain types of masses:

Certain equations are used to generalize mass effects
due to gravitational relativity. Certain term conventions
are adopted for the sake of convenience in bookkeeping:

EQUATION C

Determining a relativistic effect factor E_m
for a mass aggregate, in particular the Sun:

[using Equation A above]

E_m=√[1 –  ^2G·MM/_C2_·R]              where M_M is the mass of the Sun

                                                         and R the radius of the Sun

EQUATION C-1

Determining how much mass augmentation relativistically
occurs in the mass aggregate of the Sun:

M_M – (M_M x E_m) = K_m       Where K is the actual mass
                                         augmentation increased on
                                         the Sun's original mass
                                         due to gravity

EQUATION C-2

Determining a relativistic effect factor for a mass
aggregate, such as the Sun plus X, where X is anything:

E_x=√[1 –  ^2G·(MM^+X)/_C2_·R]      

EQUATION C-3

Determining how much mass augmentation relativistically
occurs in a mass aggregate, such as the combined mass
of the Sun + X , when both are confined in radius R :

(M_M+X) – [(M_M+X) × E_x] = K_x   [originally K+x]

EQUATION C-4

For example, determining a relativistic effect factor
for such as the Sun plus ½ Jupiter combined:

E_+½j=√[1 –  ^2G·(MM^{+ ½j)}/_C2_·R]       [ E_+½j originally E+½j ] 

EQUATION C-5

Determining how much mass augmentation relativistically
occurs in a mass aggregate, such as the combined masses
of the Sun and ½ Jupiter, when both are confined in
radius R :

(M_M+½j) – [(M_M+½j) × E_+½j] = K_+½j  

                                              [ E_+½J originally E+½j  , K_+½J originally K+½j ] 

 

An aggregate mass M_M (being the mass of the Sun) found to have
intrinsic relativistic consequences, can be easily verified.

If starting with an estimated Sun mass, for instance;
(1.989 x 10³³ grms); and assuming that the Sun mass is
already relativistically augmented, the gravitational relativistic
mass increase of a Sun mass of (1.989 x 10³³ grms) is found
(using Equations C and C-1), to be slightly less than the mass
difference between Venus and Mars.

That is:    Venus mass       is      4.8683     x 10²⁷ grms
                Mars mass         is        .64181   x 10²⁷ grms
                Venus - Mars     is      4.226490 x 10²⁷ grms

whereas the mass augmentation K_m of a
Sun mass of (1.989 x 10³³ grms) is
(4.218033 x 10²⁷ grms), which is low.

If the Sun's mass is gradually increased, eventually a
mass aggregate will be found, in which the relativistic
mass augmentation K is precisely (Venus - Mars), that is:

K = 4.226490 x 10²⁷ grms.

                       

The point of agreement occurs when the mass aggregate
for the Sun MM is found to be (1.990993 x 10³³ gms).

For instance, suppose arbitrary units of Neptune's mass are
systematically added to a base mass of (1.989 x 10³³ grms).
A break point will be reached. At + 18N units of Neptune's mass
the relativistic augmentation (K_m) of the aggregate mass will be
marginally less than (Venus - Mars). And at + 19N units of
Neptune's mass, the relativistic augmentation (K_m) of the
aggregate mass will be marginally more than (Venus - Mars).

And so somewhere between (base + 18N) and (base + 19N) is a solar
mass component whose resulting augmentation (K) is exactly equal
to (Venus - Mars). The search can now be narrowed to (base + X),
where (+ X) falls somewhere between (+ 18N and +19N).

Fine tune fiddling back and forth using smaller and smaller
increments for X, eventually closes in on a result for;

(base + 18N + X)

 

in which the relativistic mass augmentation
from (base + 18N + X) when using Equation D
below, equals (Venus - Mars) exactly.

EQUATION D

E =√[1 –  ^2G·(b+X)/_C2_·R]     Where b is a base mass

                                                   =(1.989 x 10³³ grms)

                                                     And so  (b+X) – [ (b+X) × E ] = K,

                                                     and K=(Venus-Mars) exactly,

                                                     when (b+X) is exactly 1.990993 x 10³³ grms

EQ D can be written so that (b+X) is standardized as MM, so that:

EQUATION E

E_m=√[1 –  ^2G·MM/_C2_·R]       Where M_M is an inferred Sun mass,

                                                                so  M_M – (M_M× E_m) = K

                                                                                     where K =(Venus-Mars),

                                                                and E_m is the relativistic

                                                                effect factor for mass M_M

In other words the inferred Sun mass M_M presents a solar
mass factor whose relativistic gravitational augmentation (K)
is exactly equal to the mass difference between Venus and Mars.

 

That is:     Equation E determines Em
     and:     M_M – (M_M × E_m) = K
     and:     K = 4.226490 x 10²⁷ grms

 

                                            which is precisely (Venus - Mars)
  which also is:            4.226490 x 10²⁷ grms

 

This instantly presents an interesting situation. The inferred
mass of the Sun M_M appears to involve a relativistic gravitational
mass amalgamation which is greater than the mass of the Sun alone.

The interesting kink is that the masses of Venus and Mars
are found expunged into space, at long distance orbits around
the Sun. This orbital existence is not explained at this point
and so is noted only as a comment.

The other interesting point of view is that although the mass
of Mars for instance is very small compared to the mass of the
Sun, the mass of Mars is nonetheless highly visible. This is
something like the high visibility of the electron's tiny
binding energy unit in comparison to the mass of the Proton.

 

 

As described under 'A Comparison Between Gravitational And Special
Relativity' (found directly under the 'General Introduction for
Part 2', below), gravitational relativity includes at least two
variable source terms for its effect. These source terms are the
aggregate mass, and the mass's confining radius. It means that
different quantities of mass can be said to occupy the same area.
In which case there can be (in result) different or identical
relativistic mass augmentations, depending on discrete combinations
of how much mass is said to be added or subtracted to the initial
mass aggregate, confined in the same or in different radii.

For instance in mass aggregates which are in the range
of the size of the Sun, here, discrete extra mass in the
same radius (the Sun's radius) can produce a relativistic
factor E_x which when arbitrarily applied to yet another
discretely different mass aggregate, can produce a K
augmentation which is otherwise gained from yet another
different mass aggregate.

For instance, the Sun mass MM, plus ½ the mass of Jupiter,
can provide via EQ C-2 an effect factor E_+½j which when
applied to the same mass aggregate, via EQ C-3, results in
K_+½j .     [ K_+½j originally K+j ]

  

But if E_+½j is applied to a different mass aggregate, for
instance to M_M-½j, a value slightly departed from K_+½j must
result. The resulting slightly lower value in fact once again
happens to be K exactly (the mass difference between Venus
and Mars).

The formal description for this enhanced mass state is:

EQUATION E-1

E_+½j=√[1 –  ^2G·(MM⁺ ½j)/_C2_·R]      M_M+½j is the

                                                                          aggregate of the Sun

                                                                          mass plus ½ the mass of

                                                                         Jupiter, confined in the

                                                                         existing Sun radius R

EQUATION E-2
 

( MM-½j ) – [ ( M_M –½j ) x E_+1/2j] = K

where K equals the mass of (Venus - Mars), and
E_+1/2j is the relativistic effect of the slightly
denser aggregate of the inferred Sun mass M_M plus ½
the mass of Jupiter, when confined in the Sun's radius R.

In keeping with state-like mass aggregates, if EQ E-1 is
rewritten so that the initial mass aggregate used in EQ E-1
is now M_M-½j, and a resulting effect (called E-½j) is
used in a rewritten form of EQ E-2, then a relativistic mass
augmentation equal to K once again results; that is:

 

EQUATION E-3

( MM+½j ) – [ ( M_M+½j ) x E_–1/2j] = K
where K equals the mass of (Venus - Mars).
 

EQUATION E-4

The bifurcation of Jupiter mass around the mass of the Sun
to form coherent relativistic states can be generalized as:

E_+1/2j   of mass   M+½j   applied to   M–½j   yields K
E_m        of mass   M_M        applied to   M_M       yields K
E_-1/2j    of mass   M–½j    applied to   M+½j  yields K

 

EQUATION E-5

Such a bifurcation around the mass of the Sun
can be generalized as:

E_+x  of mass  M+x   applied to   M–x    yields K_x
E     of mass  M       applied to   M       yields K_x
E_-x   of mass  M–x    applied to   M+x   yields K_x

However, the augmentation quantity K_x only equals known
augmentation value K, when M+x and M-x are specifically
M_M+½j, and M_M–½j. That is, when ½ quantas of Jupiter's
mass are added, and subtracted, to the inferred mass M_M of
the Sun.

 

(It should be noted that the bifurcation results of EQ E-4
are not perfect exactitudes. The three resulting values of
K happen to look the same for masses in the range of this
solar system. For higher mass densities for example M_M
times 1000, confined in the same radius R, the three K
values (shown as K_x in EQ E-5) are noticeably separated).

         VERIFYING THE COHERENT ½j STATES

Equations E-1, E-2, E-3, and E-4, were not easily found without a
prior insight and a discovery. In question is how come a unit of 1/2
the mass of Jupiter has been arbitrarily used to arrive at a seeming
non arbitrary result, this result being where K is twice again
calculated, as summarized in Equation E-4.

An original intention was to see if the total mass of the solar
system could be inferred to be in any way involved in some sort
of interphasing between different mass aggregates in this solar
system's gravitational relativity. This thought itself came from
an original impression that the real mass of the Sun was in the
range of base (1.9891 x 10³³ grms), and inferred mass M_M
would be the real Sun mass (base) plus Jupiter's mass, since
(M_M - base) closes in on an excellent approximation of Jupiter's
real mass at (1.901 x 10³⁰ grms), when using EQ D to infer
mass M_M.

For a while it was looking good. It seemed that if M_M was the
mass of the (Sun + Jupiter), and a mass value just slightly
larger than the total mass of the solar system was substituted
in EQ C-2, then a mass augmentation of K was again found when
the factor Ex of EQ C-2 was substituted in EQ C-3, when
Jupiter's mass was subtracted from the solar total mass
aggregate and the result of this reduction substituted for
M_M+X in EQ C-3.

In the exploration, a mass term M_t was adopted for the solar
mass total, plus some little extra, to give mass term M_tx.
And mass term M_tx–j denoted the solar total minus the mass
of Jupiter.

The value of M_tx could be rigorously inferred, as being
exactly the mass aggregate needed in EQ C-2 to result in
a mass augmentation effect equal to K in EQ C-3, when mass
aggregate M_tx gave augmentation effect E_tx, which was used
to find the augmenting effect on mass M_tx-j, as in:

 

EQUATION F

E_tx=√[1 –  ^2G·Etx/_C2_·R]       

and a mass aggregate of (M_tx - Jupiter) was substituted
in EQ C-3, giving:

 

EQUATION G

(M_tx–j) – ((M_tx–j) x E_tx) = K

In other words, the thinking was heading along a line that a
sort of formal relativistic interphasing might be occurring,
whose boundary was spread between the base mass of the Sun,
and the total mass of the solar system. For instance between
the Sun, and (Sun + Jupiter), and (Sun + planets + moons),
and (Sun + planets + moons - Jupiter). The problem was in that
little extra mass bit, (the x of M_tx). What might it represent?

It was suddenly and unexpectedly found that the value
of M_tx as rigorously inferred, turned out to be exactly
(M_M + ½ Jupiter). This was not a percentage of error
type of equality. The figures that suddenly appeared on
hand were identical to 8 significant digits.

In other words, the rigorously determined value for M_tx,
and M_M + ½j, were identical to 8 significant figures.

Which dramatically changed the picture.

It was now easy to think that M_M instead of being
a (Sun mass + Jupiter) aggregate, represented the
real mass of the Sun itself. In other words, M_M
could well be the real mass of the Sun.

It was also easy to perceive a formal verification for the
quanta bifurcation factor involving ½ the mass of Jupiter.

By using Equations F and G to find a result equal to K,
a mass quanta increment of (+X) added upon M_M eventuates in
an interphase involving (M_M-X) for the K result, only when
X is exactly ½Jupiter, when using the same inferencing
technique as was used to infer M_M in the first place, as
described above under 'Verifying a Mass of M_M For The Sun'.

A slightly more accurate inferencing for M_M itself was thus
made possible. In order for Equations E-1 to E-4 to yield
results definitely equal to K, the value of M_M is adjusted
to the greater accuracy of (1.99099305 x 10³³ grms).

It made the explorations involving solar mass total aggregates
M_t and M_tx not important. This avenue of reasoning was dropped,
and is mentioned above only to reveal how a quantal value of
± ½ Jupiter as displayed in Equations E-1 to E-4 came to be
an issue.

         OTHER MASS AGGREGATE STATES

In applying such interphasing logic to the solar system, the
study is narrowed to include only mass quantities which currently
exist; these being the Sun, and certain planets.

In the case of a bifurcated Jupiter mass, a theoretical attribute
is identified. This is where mass aggregates and resulting
gravitational relativistic effects can phase in and out (in a
continuation of certain coherent effects), through a range of
mass densities confined within a single constant radius.

A form of harmonic interphasing through a realm of masses
is definitely sensed.

In gist; a higher relativistic effect from an enhanced mass
aggregate is applied to a lower mass aggregate, such that
the resulting augmentation is lower or different than would
be expected for either the originating enhanced mass, or the
reduced mass.

This type of reasoning should only be speculative, except that
the mass augmentation which actually results when +1/2 Jupiter
and -1/2 Jupiter are involved, is already a recognized quantity,
this being mass term K, already independently seen for a mass
aggregate which is other than an effect that is expected
straight across for an enhanced or diminished sum of the Sun
plus or minus 1/2 Jupiter.

         OTHER MASS EFFECT COHERENCIES

Other mass effect coherencies seem to occur. One involves the
mass of the Earth (M_e), which, when subtracted from mass M_M,
yields an aggregate mass whose relativistic effect factor
(herein called E_e), which when applied to mass aggregate M_M,
results in a discrete mass split which is precisely equal to
the mass of the Earth M_e minus K.

This formula (as exemplified in EQ H and I below), might at first
seem tautological until further studies show that a relativistic
factor E_x for any mass aggregate (M + X) or (M - X) does not phase
in perfectly to an exact result for (M_M - (M_M x E_x)) = X - K_x for
any value assumed for mass X. Only certain precise values of ± X
are seemingly phased in a coherency. For instance when:

1. X equals the mass of Earth
2. X equals the mass of Venus
3. X equals ± ½ the mass of Jupiter

The case of X being equal to ± ½  the mass of Jupiter
has already been demonstrated in Equations E-1 to E-4.

When X equals the mass of Venus, then a mass split resulting
in a discrete relativistic augmentation, also incorporates the
mass of Mars. This is shown further below in Equations Q to S.

A formal description for the interphasing state involving
the Earth is as follows:

EQUATION H

E_e=√[1 –  ^2G·(MM^{– M}e⁾/_C2_·R]      Where (M_M-M_e) is mass MM

                                                                    minus the mass of the Earth M_e.

                                                                    M_M is the mass of the Sun

 EQUATION I

M_M - [(M_M + M_e) x E_e] = M_e - K     Where Me is the mass of Earth,
                                                                   and K is (Venus - Mars)

This formula (as exemplified in EQ I), might at first seem
exciting until it is recognized that it is rather a sort of
strange tautology.

That is, further exploration shows that a relativistic factor E_x
for any low mass aggregates in the range available for this solar
system, for instance (M_M + X) or (M_M - X), phases in to a seeming
predictable result where:

when E_x is determined as the relativistic effect factor
for mass M_M-X (for instance using EQ H), then:

M_M – [ (M_M+X) x E_x ] = X_x = (X - K)

where X_x = (X - K) results for any
reasonable value assumed for mass X.

But for higher masses (much beyond MM), the equality actually
breaks down, demonstrating that there was no tautological
equality to begin with.

A formal description for showing the breakdown is:

EQUATION J

E_e=√[1 –  ^2G·(MM^–Me⁾/_C2_·Rx]      Where (M_M-X) is mass M minus

                                                                    any other mass X, and radius

                                                                    R_x is the same for any values

                                                                    of (M-X), then:

EQUATION K

M - [(M) x E_x] = K_x                          And:

 

EQUATION L

M - [(M+X) x E_x] = X_x                     And:

 

EQUATION M

X_x - X = K_x                                         Where:

X_x + K_x = X                                And:

X_x = X - K_x                                 Where X is the original arbitrary
                                                  mass that was subtracted from M in
                                                  EQ J, and was then added to M in EQ L

 

         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 X_x 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 R_x. In these higher mass regions,
the value of K_x can begin to rapidly escalate over and above any
amounts of increase given to mass M.

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

The value of K_x is in fact somewhat periodic in two ways.
(K_x 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 K_x is dependent almost entirely upon
the digital value of M. For example a K_x digital value ranging
from (4.21 x 10²⁷) up to (4.79 x 10³⁷) is found for mass M values

ranged from (1.989 x 10³³) up to (1.989 x 10³⁸), when the

confinement radius R_x is held constant at (6.96256 x 10¹⁰ cm),

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 K_x increases by a power of 100 (actually just
slightly more than 100), until the Value of K_x 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 K_x identified in the previous paragraph.

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

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

Then, given the basic equation:

EQUATION O

E_x=√[1 –  ^2G·Mh/_C2_·Rx]      Where E_x is the relativistic
                                                            effect factor of a high mass M_h

                                                            having a confinement radius R_x,

                                                            and:

EQUATION O-1

M – [(M_h x E_x) = K_x 
 

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

EQUATION O-2

^2GMbh/_C2Rx =1                   And therefore:

EQUATION O-3

E_x=√[1 –  ^2G·Mbh/_C2_·Rx]      Is no longer valid, since:
 

EQUATION O-4

E_x=√[1 – 1]                                 The square root of 1-1=0

                                                             is impossible.

However, in looking back to Equations J through M, where terms X
and X_x 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 R_x
so as to provide a consistent point of view via a constant R_x, 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 X_x of
                    EQ L can be substantially lower than X, and X_x can
                    also be substantially lower than K_x.

ITEM B.       If X is substantially lower than the higher value
                   M, (for instance if X is ¹/₁₀₀₀₀₀th the value of M),
                   then X_x can increase substantially above X. In fact X_x
                   approaches the value of K_x 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 X_x can be rather close to the value of K_x,
and X_x + K_x 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 (X_x + K_x) of Equation M above, in which the Earth mass Me
is X, is hardly discernible, so indiscernible that X and (X_x + K_x) seem
the same, (as indicated in EQ I above, where X_x would be Me - K). But X
and (X_x + K_x) 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 X_x and K_x can
both turn out to be identical. This is as follows:

EQUATION O-5.

E_x=√[1 –  ^2G·(Mass)/_C2_·Rx]      And:
 

Mass - [(Mass) × E_x] = K_x             Then:

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

Mass - [ (Mass - K_x) × (¹/_Ex) ] = 0    This is true for both
                                                        low mass and high mass calculations

         A COHERENT PHASE IN THIS SOLAR SYSTEM

 

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 M_M, and X
equals the mass of Venus (M_v), the strange tautology of
Equations J through L become a seeming genuine equality,
wherein the resulting X = (X_x + K_x) mass split in relativistic
augmentations, also incorporates the mass of Mars. Specifically,
X_x is the mass of Mars.

 

 The formal description for this state is as follows:

EQUATION P

E_x=√[1 –  ^2G·(MM-Mv)/_C2_·R ]     Where (M_M-M_v) is mass M_M of the mass

                                                                    of the Sun minus the mass of Venus M_v.

                                                                    R is the exiting radius of the Sun.

EQUATION Q (Determines a value K)

E_k=√[1 –  ^2G·(MM)/_C2_·R ]            This is the same as EQ E, so that:

                                                                  

M_M - [(M_M) x E_k] = K                      Such that:

EQUATION R

M_M - [(M_M+M_v) x E_v]= M_a              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 M_M is identical to the mass difference between
planets Venus and Mars.

PART

GENERAL INTRODUCTION for part 2 

         A COMPARISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY

 

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 K_x 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.

 

 

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).

THE GRAVITY - SPECIAL RELATIVITY CONNECTION IN DETAIL

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 (see equation T below), supplies an essential term.

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

The speed of light C is divided by the square root of N_x, to
give a velocity that is less than C. Lets call this velocity
V_x. If mass M is traveling at velocity (V_x), 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 N_x (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 E_x, 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/_M = N_x                        Then relativistic effect E_x is:

EQUATION T-1

E_x=√[1 –  ¹/_Nx ]              Gravitational relativistic effect E_x

                                                     is calculated from ratio (^Mbh/_M), when

                                                      the mass of black hole silent partner M_bh

                                                     is calculated from the radius of M, by:

EQUATION T-2

M_bh = C²R/_{2G                  As in:}

 

EQUATION T-3

         A COMPARISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY

In Equations U through X which follow:

(E_g) :

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

(E_s) :

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

(M_bh) :

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

(Nx) :

 is the ratio of the black hole mass M_bh,
divided by the given mass M

EQUATION U

E_g=√[1 –  ^2G·M/_C2_·R ]            

EQUATION U-1

E_g=√[1 –  V²/_C2_·R ]            

 

EQUATION U-2

Gravity relativity                            Bare bone version

E_g=√[1 –  ¹/_Mbh/M ]           =√[1 –  ¹/_Nx ]          

EQUATION U-3 

Special relativity                               Bare bone version

E_s=√[1 –  [^C/√Nx]²/_C2 ]           =√[1 –  ¹/_Nx ]          

 

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 N_x is found from the ratio ^Mbx/_M of EQ U-2.
In the Bare bones version of EQ U-3, term N_x cannot reveal what the
velocity that mass M is moving at in order to have a relativistic
effect factor E_s in EQ U-3 that is equal to E_g in EQ U-2.

This is by no means a critical shortcoming. Without knowing term N_x,
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-3 

                     Term 1                             Term 2              Term 3

                                                                                      an exact velocity V

Substitute    [^C/√(Mbh/_M)]_/C  For  [^C/√Nx]_/C       = ^V/_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 (E_s) identical to a gravitational relativistic
effect (E_g).

In this connective equality between relativities, identical augmenting
effects on the moving rest mass (Mass)(¹/E_s) of special relativity, and
aggregate mass (Mass)(¹/E_g) 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.

         A Support equation for gravitational relativity follows next

EQUATION V

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

M_bh = M × [ ¹/_{( 1-(Eg)}2 ₎ ]

 

EQUATION V-1   However:

 [ ¹/_(1-(Eg)2₎ ]                 also equals   √[1 –  ¹/_Nx ]

 

EQUATION V-2   So that EQ V simplifies to:

M ×  ^Mbh/_M =  M × N_x     So that         N_x = ^Mbh/_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 N_x,
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 N_x 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 N_x.

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.

         Summary equations for the two modes of relativity follow next

EQUATION W     Basic Gravitational relativity equation

E_g=√[1 –  ^2G·(Mass)/_C2_·R]      EQ W is the  

                                                   same as EQ C further above
 

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

 

EQUATION W-1

(Mass) - ((Mass) x E_g) = K_x              Where K_x is an augmentation
                                                       of (Mass) by gravitational
                                                       relativistic effect E_g

EQUATION X         Basic special relativity equation

E_x=√[1 –  V²/_C2]      Many text books cite a greek letter for effect

                                             E_s, and for ratio V²/_C2

Effect ¹/E_s increases the mass. E_s 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

(M_bh) 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:

M_bh = C²R/_2G                 Finding the mass (M_bh) needed for

                                         a black hole whose Schwarzschild

                                         radius is given as R. EQ X-1 is

                                         the same as EQ 5 of APPENDIX B below

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 (K_x) included. The augmentation (K_x)
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 K_x 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/N_x of the black hole
mass M_bh 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 N_x (as in ^C/√N_x) 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 Velocity

C

C

For

√1

=

C

=

1

√

Mbh

C

Mbh

C

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/N_x 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
M_bh. 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 N_x factor directly from the ratio of ^Mbh/_M .

 

 

THE CONUNDRUM

 

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 K_x 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

E_g=√[1 –  ^{2G·(Mass)(1/E}g⁾/_C2_·R(Eg)]      Mass is increased by 1/E_g,   

                                                                   Radius is decreased by E_g

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

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

EQUATION Y-1

E_g=√[1 –  ^2G·(Eg^)(Mass)/_C2_·(1/Eg)·R]      Gravity is decreased by Eg,

                                                                                    is increased by ¹/E_g

then the ratio portion (2G)(Eg) / C²(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

E_g=√[1 –  ^2G(Eg^)(Mass)(1/Eg⁾/_C2_{·(1/Eg)·R(Eg)}]     

and the net internal effect is again simply
2G (Mass) / C²R, 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.

GENERAL INTRODUCTION for part 3 

                        TABLE 4

KEY TERMS

                      TABLE 4 Continued

          FUNCTIONAL INTERPHASE BETWEEN
          GRAVITATIONAL AND SPECIAL RELATIVITY

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.

QUESTION

 

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?

 

ANSWER

 

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.

 

 

INTERPRETATION

 

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 R_bh be a given radius. Let's say a mass aggregate M
of radius R_bh is at 99% of the M_bh black hole mass limit for radius
R_bh. The gravitational relativistic effect (E_g) is roughly about
E_g = .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 E_g 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 R_bh increase
only to a black hole mass M_bh of radius R_bh, 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 M_bh, 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.

 

EXPLANATION

 

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.
 

 

THE GOLDEN HARMONIC RATIO - A CRITICAL LIMIT

 

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 G_H. And so let Ng = G_H.

 

 

THE CRITICAL LIMIT in gravitational relativity

 

When ^Mbh/_Mo is G_H, a vital event occurs. The gravitational
effect E_g precisely turns out to be ¹/G_H (the reciprocal of
the Golden Harmonic Ratio).

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

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

In which case, in special relativity, when the original mass
Mo is moving at a velocity V which is root G_H less than the
speed of light, the special relativistic effect Es increases
mass Mo to mass M_bh in a final single bump. In which case mass
M_bh 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/G_H is a
critical limit preceding mass M_bh, at which an original mass
Mo is raised to the black hole limit M_bh by the mass effect
of its own gravitational relativity. Let M_c 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 M_c. 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 M_c, raised to M_bh
through a quanta bump equal to the Golden Ratio G_H.

In this locked in state, M_bh - M_c = K_o, where K_o is the
actual mass augmentation, the same as is otherwise said to
be K_x, except in this instance, K_o is fundamentally related
to the Golden Ratio G_H. In exactitude, K_o = M_bh - (^Mbh/G_H).

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

As already said, the Mc critical mass limit
(for radius R_bh) is simply (M_bh/G_H), where
(G_H) is the Golden Harmonic Ratio.

 

THE CRITICAL LIMIT in special relativity

 

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

In consequence there never is a condition where the original
mass M_o in special relativity expands toward infinity as
mass M_o closes in on mass M_bh in gravitational relativity,
because the convergence in gravitational relativity for an
original mass M_o closes off completely at the critical mass
limit M_c, when M_c is less than mass M_bh by a ratio equal
to G_H. 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/G_H

Where: E_g is the gravitational relativistic effect of M_c

Such that: E_g = 1/G_H

And M_bh = M_c + K_o, where K_o = (M_c x 1/E_g) - M_c

And also: M_c x 1/E_g = M_k, and M_k - M_c = K_o

And so: M_bh = M_c x 1/E_g = M_k

Only when: M_c = ^Mbh/G_H

So that: M_bh = M_k

 

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

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/√G_H) ;

where Vc is actually:

Vc = [^C/√(Mbh/_Mc)] or also (^C/√G_H)

when: M_c = ^Mbh/G_H or also G_H = ^Mbh/M_c
so that when: M_c is traveling at velocity V_c
the special relativity effect is: E_s
and the special relativity effect ¹/E_s increases
rest mass M_c to black hole mass M_bh in a bump
because E_g is equivalent to ¹/G_H.

 

                                           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 M_k contains an original mass, plus
an augmentation in mass due to gravitational relativity. And
so let the originating mass be M_o, the augmenting mass be K_o,
and the resulting mass be M_k. And therefore:

For Gravity relativity

EQUATION Z-2

E_g=√[1 –  ^2G·Mo)/_C2_·R]      Mo is an original mass

                                                           before augmentation,

EQUATION Z-3

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

EQUATION Z-4

M_o + K_o = M_k                        M_k is the measured (apparent)
                                                         mass, consisting of original
                                                         plus augmentive masses

EQUATION Z-5
When

Mo = Mc = ^Mk/G_H                          Where Mc is a critical mass
                                                                    value for original mass Mo

then:

E_g=√[1 –  ^(2G·Mk^/GH⁾/_C2_·R]      M_k is black hole mass with

                                                                    horizon radius R_bh, and G_H is

                                                                    the Golden Harmonic Ratio equal

                                                                    R_bh to the number 1.61803398875

EQUATION Z-5-1

E_g=√[1 –  ^(2G·Mbh^/Ng⁾/_C2_{·R bh}]  Mass M_bh is the same as mass

                                                                    aggregate M_k.

                                                                    Ng is ratio Nx when the value

                                                                    of Nx is GH, which is the

                                                                    Golden Harmonic Ratio.

EQUATION Z-6

With digits substituted for G_H, then:

E_g=0.61803398875=√[1 –  ^(2G·Mbh^{/1.61803398875)}/_C2_{·R bh}] = ¹/_{1.61803398875}  

EQUATION Z-7
because:

¹/N_x=√[1 –  ¹/N_x]             When and only when N_x = G_H.

                                                      The Golden Ratio contains

                                                      this self appreciating

                                                      mathematical property.

and so:

¹/G_H=√[1 –  ¹/G_H]             G_H is the Golden Ratio

                                                       1.61803398875

 

 

For Special relativity

EQUATION Z-8

E_s=_√[1 –  (C/√Nx)²/_C2] =_√[1 –  (V_C)²/_C2] 

EQUATION Z-9

E_s=_√[1 –  (C/√GH)²/_C2] =_√[1 –  (V_C)²/_C2] 

EQUATION Z-9-A   And so:
 

(M_c x 1/E_s) = (M_c x G_H) = M_bh, because (E_s = 1/G_H)

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

EQUATION Z-10      As in:

0.61803398875=√[1 –  (C/√1.61803398875)²/_C2]

       FOR SPECIAL RELATIVITY EFFECT ON BOTH MASS AND RADIUS

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:

V_bh = (^C/G_H)                    Given as:

EQUATION Z-11

E_s=_√[1 –  (C/GH)²/_C2] =_√[1 –  (V_bh)²/_C2] 

 

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

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

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

The new mass can be labelled as M_bh–, which is less than the
gravitational black hole mass M_bh, by a factor of √G_H. As already
indicated, ^Mbh/M_c = G_H, but the special relativistic mass result
M_bh– is not the same as M_bh. There is a series:

EQUATION Z-12

M_c x √G_H = M_bh–  x  √G_H = M_bh

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 M_c is V_bh
which is ^C/G_H, but this is only when the original mass M_o is at
the critical mass limit Mc which is a black hole mass M_bh divided
by G_H. Whereupon the mass becomes a new black hole of mass M_bh–
and disappears from view, relative to a stationary observer.

The ratio ^C/G_H 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 M_c regards
the black hole final limit M_bh, 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).

A REMARK

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'.

       SUPPLEMENTAL REMARKS

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.

Golden Harmonic Relativistic Effects on Mass Only

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

A value of (square root of 2) was first tried for N_x, yielding
a mass augmentation result (¹/E_g x M_o), which was greater
than mass M_bh, when root 2 for N_x was ratio (^Mbh/M_o = N_x).

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

So the two N_x values were averaged as in ¹/₂(√2 + 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 E_s result (¹/E_s  x M_o) came to M_bh, using
this familiar number as N_x 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 G_H.

When G_H was N_x, then (¹/E_s x M_o) = M_bh.

And so this particular N_x was
labeled N_g (for Golden Ratio).

And M_o was understood to be
the same value as mass M_c.

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

Golden Harmonic Relativistic Effects on Mass and Radius

 

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 M_c would be travelling to
relativistically increase to mass M_bh-, when radius R_bh
of mass Mc was conjointly contracted to radius R_bh-.
In this thought balloon, M_bh- and R_bh- 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 M_bh
and radius R_bh was not known. And neither was the velocity.

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

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

To begin, radius R_bh was modified by (E_s × R_bh) as gained
from (EQ Z-11) with N_x equal to G_H in the ratio C/G_H, to give
contracted radius R_bh-. Then, using EQ 5 of APPENDIX B below
to find the mass of a black hole formed in radius (E_s × R_bh-),
a new mass M_bh- was the result. It turned out that the ratios
of masses (^Mbh/M_bh-) and (^M_bh-/M_c) both equaled the square
root of ratio G_H.

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

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

How was this verified ?

 

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

 

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

R_bh x E_s = R_bh-

B.     Using radius R_bh- to find mass M_bh- in:

 

M_bh- = C²R_bh-/_2G                  Finding mass M_bh- needed for a

                                               black hole whose Schwarzschild

                                               radius is given as R_bh-

C.   M_bh- turned out to be mass  M_bh/(1/_√GH)
      when effect E_s (of EQ Z-11) was 1/G_H.

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

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

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

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

QUESTIONS

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 ?

 

ANSWER

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 M_bh mass (silent partner mass)
is easily found by:

EQUATION Z-13

M_bh = C²R/_2G                  Finding mass M_bh needed for a

                                          black hole whose Schwarzschild

                                          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 M_bh 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/_R= C²/_{2G                 Where ratio} C²/_{2G is a constant,}

                                          when C is the speed of light, and

                                          G is the universal gravitational

                                          constant.

R is the original radius of original mass Mo

Mass M_bh 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  C²/_2G, if mass M+ and R- were relativistically
altered sufficiently to form a new black hole.

Ratio C²/_2G can be labeled ratio CR (for 'constant ratio') and
has the value of (6.735275620 x 10²⁷ 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^-8.

Ratio C²/_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/E_s x M_o) / (E_s x R_o) equals the ratio
C²/_2G, as in:

EQUATION Z-14A

[ (¹/E_s x M_o)/(E_s x R_o) ] = (^M+/_R-) = C²/_2G

where E_s is the special relativistic effect.

Finding a significant Velocity value, which results in ratio CR

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 10³³ gms
        (mass M_M from Part 1 above)

1A.   Given a Sun radius R of 6.96265 x 10¹⁰cm

1B.   Given constant ratio CR =  C²/_2G
                                               = 6.735275620 x 10²⁷ grms/cm

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

       EQUATION Z-14-1

       R'= 2G·M/_C2         _{Finding the Schwarzschild}

                                      _{radius R' of a black hole's}

                                      _{event horizon, when given}

                                      _{mass M}

 

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

       EQUATION Z-14-2

       M_bh= C²R/_2G         Finding mass M_bh needed for a

                                      black hole whose Schwarzschild

                                      radius is given as R_.

                             Mass M_bh 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 N_x to determine a velocity, so that:

       EQUATION Z-15

       E_s=_√[1 –   (C/Nx)²/_C2  ] =_√[1 –   (Vx)²/_C2 ] 

5.   Given that (¹/E_s × M) = M+

6.   Given that (E_s × R) = R-

7.   Given that (¹/E_s × M+) / (E_s × R-)   =  C²/_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 = C²/_2G

9.   A first arbitrary value for N_x 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 N_x 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 E_g from
       the Sun's mass, as determined in EQ C of Part 1 further above.
       (M_M in EQ C is the same value as Sun mass M_o given in EQ Z-2,
       and immediately above in Item 1. And E_g of EQ Z-2 is the same
       as E_g used immediately below in Item 12).

12.  And so E_g was determined for the Sun's mass M = M_M = M_o in
       EQ Z-2, and conveniently labelled E_gs (for 'effect gravity Sun
       mass'), and was substituted as term 1/N_x in EQ Z-15 immediately
       above, to give:

       EQUATION Z-15-1

       E_s=_√[1 –   (C×Egs)²/_C2  ] = _√[1 –   (Vx)²/_C2 ] 

              where velocity V_x is (C × E_gs),
              and special effect E_ss conveniently
              means an E_s effect related to the
              gravitational mass via term E_gs.

13.  Then; Sun mass M in (M × ¹/E_ss) = M+

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

15.  And; ratio (^M+/_R-) = 6.73527458 × 10²⁷ grms/cm
       As found in:

EQUATION Z-15-2

 

(M x ¹/E_ss) / (R x E_ss) = CR = (^M+/_R-)

16.  Which turned out to be an excellent approximation of ratio
       CR (being C²/_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 × 10³⁶ grms

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

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

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

21.  The result ^M+/_R- = 6.735275620 × 10²⁷ gms/cm = 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 C²/_2G.

It is clearly evident that ratio (^M+/_R-) closes
in on ratio C²/_2G, the closer that given original
mass M is to the black hole silent partner mass M_bh
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 M_bh any closer than by
M_bh divided by factors of the Golden Ratio).

Finding that terms M+ and R- are properties of a black hole

 

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 E_ss value in Item 19, which as applied
        to Item 13, was:
                                 3.055623494 × 10²⁷grms

23.   The value of R- from the same E_ss in Item 19, applied
        to Item 14, was:
                                4.536746031 × 10⁹cm

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

        R'=  2G·M+/_C2        _{Finding the Schwarzschild}

                                         radius R' of a black hole's

                                         event horizon, when given

                                         mass M+

R' was 4.536746031×10⁹ cm,
    exactly the same as R- in Item 23

EQUATION Z-15-4

        M'=  C²R-/_{2G           Finding mass M' needed for a}

                                         _{black hole whose Schwarzschild}

                                         _{radius is given as R-}

                                        

M'  was 3.055623493 × 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 E_ss of Item 19, on higher
                mass M of Item 17, using EQ Z-15-1 to determine
                E_ss, after using EQ Z-16 to determine E_gs.

       SUMMARY EQUATIONS

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

E_gs=_√[1 –   2G·M/_C2_·R  ]                     M is any mass, R is its

                                                            radius, and E_gs is the

                                                            gravitational relativistic

                                                            effect of mass M.

EQUATION Z-16-1

E_ss=_√[1 –   (C×Egs)²/_C2  ] = _√[1 –   (Vx)²/_C2 ]

 

E_ss is the special relativistic effect ensuing from
velocity V_x, determined as the direct consequence of
the speed of light reduced by the mass's gravitational
relativistic effect E_gs.

EQUATION Z-16-2

(M x ¹/E_ss) = M+

 

EQUATION Z-16-3

(R x E_ss) = R-
 

EQUATION Z-16-4

^M+/_R-  = C²/_2G  = CR

 

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 10²⁷ gms/cm.

EQUATION Z-16-5

And ultimately, E_ss can be determined directly
from E_gs, by:

 

E_ss² = 1 - E_gs²
 

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

EQUATION Z-16-6

E_ss =√[1 - E_gs²]
 

E_gs =√[1 - E_ss²]
 

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.

GENERAL INTRODUCTION for part 4   Unified Fields 

'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 M_bh, 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 K_o, 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 M_bh/M_c = N_g is a
function of the Golden Ratio. For masses other than than Mc, ratio N_g
is given the general label N_x).

In the following, the cases for M_c and N_g 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 M_o, M_c,
and N_g, are expeditiously put aside, and a mass value is assumed which
can be anything less than M_bh, even if less than M_bh by a few parts in
a thousand. This is called a HIGH mass, for convenience.

TEST CASE

In a test case, a HIGH mass value is studied which hangs right
below the mass of a black hole M_bh. This is in a deliberately
selected HIGH mass range which as already said ignores properties
such as a critical mass factor (M_c) 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 M_bh term, which is the mass
of a black hole confined in the Sun's radius. M_bh for the Sun's
radius is (4.689536679 × 10³⁸ grms).

The Sun's radius (6.96265 × 10¹⁰cm) 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 N_x has been selected for a
control in the study. N_x is meaningless other than its value
is the charge to mass ratio of the hydrogen atom, ie.:

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

(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 N_x has no intrinsic
significance in the following).

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


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 N_x has no
significance except that it provides a convenient low value
N_x ratio to arrive at a HIGH mass term for the study model.


N_x      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

   ARBITRARY STUDY MODEL DATA

N_x      =   1.000544617404 = (P + E) / E
M_bh   =   4.689536679 × 10³⁸ grms

HIGH mass1     =   M_bh / N_x
                        =   4.686984066 × 10³⁸ grms
                 N_x   =   1.000544617404
 
LOW mass2      =   Mass1 / 100,000
                        =   4.686984066 × 10³⁸ grms
                 N_x   =   100054.4617404 

In the following, Equations Z-17-1 and Z-17-3
are the same as EQ Z-15-1 above, except, the real
digit value of each E_gs ratio is substituted for
the algebraic term E_gs.

EQUATION Z-17 HIGH gravitational Mass1 results:

E_gs =√[1 - 2G (4.686984066 × 10³⁸ grms)/_C2_R]

Mass1 has been given in
terms of a real weight.

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

HIGH gravity field effect Egs is closing toward 0

Egs =

0 .023330687

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

E_ss=_√[1 –   (C × 0.023330687)²/_C2  ] =  (Vx)²/_C2

0.023330687 is effect Egs
of EQ Z-17

E_ss =   [1 - E_gs²]
As in:  [1 - 0.023330687²]= 0.999727802
 

LOW special field effect Ess is closing toward 1 V velocity is starting to close toward 0
	Ess =	0.999727802

EQUATION Z-17-2            LOW gravitational Mass2 results:

E_ss=_√[1 –   (2G × 4.686984066 ×10³³ grms)²/_C2_R  ]

                                                                             Mass2 has been given in
                                                                             terms of a real weight.

LOW gravity field effect Egs is closing toward 1
	Egs =	0..999995002

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

E_ss=_√[1 –   (C × 0.999995002)²/_C2  ] =  (Vx)²/_C2

                                                                            0.999995002 is effect E_gs
                                                                                                  of EQ Z-17-2

E_ss =   [1 - E_gs²]
As in:  [1 - 0.999995002²]= 0.003161416
 

HIGH  special field effect Ess     is closing toward V        velocity is closing toward 1
	Ess =	0.003161416

       COMPARING M+ AND R- RESULTS FOR HIGH AND LOW MASSES

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 E_ss 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

HIGH MASS GRAVITY                    MASS1   =   (4.686984066 x 1038 grm)                RADIUS R  =   6.96265 x 1010 cm             Ess EFFECT  =   0.999727802 ; from EQ Z-17-1                       M+ =    (Mass1 x 1/Ess)                           =    4.688260199 x 1038 grms                        R- =    (radius R x Ess)                           =    6.9607547839 x 1010 cm                       CR =    ratio (M+/R-)                           =    6.735275620 x 1027 grms/cm

 

TABLE 6

LOW MASS GRAVITY                    MASS2   =   (4.686984066 x 1033 grm)                RADIUS R  =   6.96265 x 1010 cm             Ess EFFECT  =   0.003161416 ; from EQ Z-17-3                       M+ =    (Mass1 x 1/Ess)                           =    1.482558107 x 1036 grms                        R- =    (radius R x Ess)                           =    2.201183848 x 108 cm                       CR =    ratio (M+/R-)                           =    6.735276152 x 1027 grms/cm

 

It is seen that results M+ , though higher than an       originating mass, are lower than the ceiling mass Mbh       in LOW mass results, and close in on ceiling mass Mbh       in HIGH mass results. (Ceiling mass means a black       hole mass equivalent Mbh formed in radius R.

In HIGH mass situations, M+ can look like the high       mass itself, but in low mass situations, M+ is far       removed from the low mass itself.

Also, it is obvious that M+ of LOW mass results can       gain substantially over the LOW mass itself, yet still       remain substantially below the final mass Mbh, whereas       M+ hardly gains over its originating HIGH mass, and       can also look very much like final mass Mbh, when       the HIGH mass itself looks closely like Mbh.

In real situations, the HIGH mass will be fixed at a       maximum ceiling of critical limit Mc. In this current       test case situation M+ looks neither like Mc, or Mbh.       Yet M+ will be explicitly Mc x √GH, and Mbh/√GH, when       GH the Golden Ratio 1.618034 is term Nx.

(Ratio CR in the LOW mass situation, is seen to be       marginally more than CR = C2/2G . This shift might       be due to intrinsic truncations in the digital       accuracy of the equations for lower mass densities.       It is hard to tell, in the scope of a digital       accuracy limited to 13 significant figures).