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????????????????????????????????????????????????????????????????
? A TEST CASE: ?
??? GOLDEN HARMONIC RATIO IN THE TWO MODES OF RELATIVITY ?
????????????????????????????????????????????????????????????????
Let's look at the critical limit situation in more detail.
An apparent mass aggregate Mk contains an original mass, plus
an augmentation in mass due to gravitational relativity. And
so let the originating mass be Mo, the augmenting mass be Ko,
and the resulting mass be Mk. And therefore:
??????????????????????????
? For Gravity relativity ?
??????????????????????????
EQUATION Z-2
????????????????????????
? 2G (Mo) Mo is an original mass
Eg = ? 1 ? ??????? before augmentation
\? C? R
EQUATION Z-3
(Mo x 1/Eg) - Mo = Ko Ko is the mass augmentation
on Mo, due to effect 1/Eg
EQUATION Z-4
Mo + Ko = Mk Mk is the measured (apparent)
mass, consisting of original
plus augmentive masses
EQUATION Z-5
When Mo = Mc = Mk/GH then: Where Mc is a critical mass
value for original mass Mo
????????????????????????
? 2G Mk
Eg = ? 1 ? ?????? Mk is black hole mass with
? GH horizon radius Rbh, and GH is
? ???????????? the Golden Harmonic Ratio equal
\? C? Rbh to the number 1.61803398875
EQUATION Z-5-1
???????????????????????? Mass Mbh is the same as mass
? 2G Mbh aggregate Mk.
Eg = ? 1 ? ??????
? Ng Ng is ratio Nx when the value
? ???????????? of Nx is GH, which is the
\? C? Rbh Golden Harmonic Ratio
EQUATION Z-6
With digits substituted for GH, then:
????????????????????????
? 2G Mbh
Eg = .61803398875 = ? 1 ? ?????? = 1
? 1.61803398875 ?????????????
? ????????????? 1.61803398875
\? C? Rbh
EQUATION Z-7
because:
????????????????? When and only when Nx = GH.
1 ? 1 The Golden Ratio contains
??? = ? 1 ? ??? this self appreciating
Nx \? Nx mathematical property
and so:
?????????????????
1 ? 1 GH is the Golden Ratio
??? = ? 1 ? ??? 1.61803398875
GH \? GH
??????????????????????????
? For Special relativity ?
??????????????????????????
EQUATION Z-8
??????????????????????????? ??????????????????????
? ?? ??? ? (Vc)?
Es = ? ? C ? = ? 1 ? ??????
? 1 ? ? ???????? ? \? c?
? ? ????? ?
? ? \? Nx ?
? ?? ??
? ??????????????
\? C?
EQUATION Z-9
??????????????????????????? ??????????????????????
? ?? ??? ? (Vc)?
Es = ? ? C ? = ? 1 ? ??????
? 1 ? ? ???????? ? \? c?
? ? ????? ?
? ? \? GH ?
? ?? ??
? ??????????????
\? C?
EQUATION Z-9-A And so:
(Mc x 1/Es) = (Mc x GH) = Mbh, because (Es = 1/GH)
when 1/Es is the special relativitistic effect on
mass Mc which is moving at velocity Vc of EQ Z-9
EQUATION Z-10 As in:
?????????????????????????????????????????
? ?? ???
.61803398875 = ? ? C ?
? 1 ? ? ?????????????????? ?
? ? ??????????????? ?
? ? \? 1.61803398875 ?
? ?? ??
? ????????????????????????
\? C?
????????????????????????????????????????????????????????????????
??? 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:
Vbh = (C / GH) Given as:
EQUATION Z-11
??????????????????????????? ??????????????????????
? ?? ??? ? (Vbh)?
Es = ? ? C ? = ? 1 ? ??????
? 1 ? ? ???? ? \? c?
? ? GH ?
? ?? ??
? ???????????
\? C?
Es turns out to be the reciprocal of the square root
of the Golden Harmonic. That is; Es = (1/?GH).
It means that when a mass Mc is physically moving at velocity
Vbh relative to a stationary observer, its radius Rbh contracts
by (1/?GH), as its rest mass Mc expands by (?GH), with the result
that a new black hole is formed, having a lesser mass equal to
(Mc x ?GH), and a lesser radius equal to (Rbh x 1/?GH).
As already said, this occurs when velocity Vbh is equal
to the speed of light divided by the Golden Harmonic Ratio.
The new mass can be labelled as Mbh-, which is less than the
gravitational black hole mass Mbh, by a factor of ?GH. As already
indicated, Mbh/Mc = GH, but the special relativistic mass result
Mbh- is not the same as Mbh. There is a series:
EQUATION Z-12
Mc x ?GH = Mbh- x ?GH = Mbh
It means that a visible mass cannot expand to infinity,
because velocities can approach but can never reach the speed
of light, due to built in limiting factors. This statement
is true specifically for visible masses.
For instance, the maximum velocity possible for mass Mc is Vbh
which is C/GH, but this is only when the original mass Mo is at
the critical mass limit Mc which is a black hole mass Mbh divided
by GH. Whereupon the mass becomes a new black hole of mass Mbh-
and disappears from view, relative to a stationary observer.
The ratio C/GH is (C / 1.61803398875)
(The preceding does not take into account any effect that
gravity might have to relativistically reduce the radius of the
mass causing the gravity's relativistic effect. It is realized
that if a reduction in gravitational radius is also needed as a
key term, than the parameters of the critical mass limit Mc regards
the black hole final limit Mbh, will adjust accordingly, as will
the exact factors related to the Golden Harmonic Ratio).
(The question of such possible adjusting is not addressed in
this disclosure, whose prime intention is to simply show that
certain critical limits and equalities do synonymously exist
in the domains of gravitational and special relativity. And
that the Golden Harmonic Ratio is a fundamental primary term).
????????????????????
? 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 Ng for Nx in Equations Z-5 and Z-5-1 ?
A value of (square root of 2) was first tried for Nx, yielding
a mass augmentation result (1/Eg x Mo), which was greater
than mass Mbh, when root 2 for Nx was ratio (Mbh/Mo = Nx).
In intuitional trial and error, an Nx value arbitrarily
selected as 1.8 was next tried. It yielded an (1/Eg x Mo)
value which was slightly less than mass Mbh.
So the two Nx values were averaged as in 1/2(?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 Es result (1/Es x Mo) came to Mbh, using
this familiar number as Nx for a point of reference.
It turned out that 1.61803398875 happened to be the very
term wanted, because the result was perfect. This fast
found number was given the label GH.
When GH was Nx, then (1/Es x Mo) = Mbh.
And so this particular Nx was
labelled Ng (for Golden Ratio).
And Mo was understood to be
the same value as mass Mc.
Equations Z-6 and Z-7 show why Ng is a constant. The
set of Equations Z to Z-10 followed as a consequence
of knowing this.
???????????????????????????????????????????????????????????
? 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 Mc would be travelling to
relativistically increase to mass Mbh-, when radius Rbh
of mass Mc was conjointly contracted to radius Rbh-.
In this thought balloon, Mbh- and Rbh- would be the
parameters forming a new black hole when mass Mo was
travelling at sufficient high velocity.
At this point the rate of joint contraction on mass Mbh
and radius Rbh was not known. And neither was the velocity.
The intention was to find what term Nx is
divided into C to yield the significant velocity.
In a remarkably lucky guess, the first Nx
term tried was GH itself, (in EQ Z-11).
To begin, radius Rbh was modified by (Es x Rbh) as gained
from (EQ Z-11) with Nx equal to GH in the ratio C/GH, to give
contracted radius Rbh-. Then, using EQ 5 of APPENDIX B below
to find the mass of a black hole formed in radius (Es x Rbh-),
a new mass Mbh- was the result. It turned out that the ratios
of masses (Mbh/Mbh-) and (Mbh-/Mc) both equaled the square
root of ratio GH.
It had thus been found that when (C/GH = Vbh), then
EQ Z-11 yielded the square root of GH as the Es value.
The result is that with Es equaling the reciprocal of the
square root of the Golden Ratio, when Rbh is multiplied by
Es to yield radius Rbh-, and mass Mc is multiplied by the
reciprocal of Es to yield mass Mbh-, then radius Rbh- and
mass Mbh- are the correct parameters to form a new black
hole from the special relativity effects on both mass Mc
and radius Rbh, when Mc is travelling at a (C/GH) velocity.
???????????????????????????
? How was this verified ? ?
???????????????????????????
The 'dual effect' event was easily
verified by the following:
A. Radius Rbh- was found from radius Rbh,
by using the Es effect of EQ Z-11 in:
Rbh x Es = Rbh-
B. Using radius Rbh- to find mass Mbh- in:
C? Rbh- Finding mass Mbh- needed for a
Mbh- = ??????????????? black hole whose Schwarzschild
2G radius is given as Rbh-
C. Mbh- turned out to be mass Mbh / (1/?GH)
when effect Es (of EQ Z-11) was 1/GH.
D. It meant mass Mbh- and radius Rbh- form a new black hole,
which is less than a black hole of mass Mbh and radius Rbh,
by a factor of the square root of the Golden Ratio for
both Mbh- and Rbh-.
E. This is true when mass Mc is travelling in special
relativity, at a reduced velocity Vbh, as gained
from EQ Z-11.
F. The synonymous special relativistic 'dual effect' event
for a gravitational relativistic event at the critical
mass limit Mc, is gained by using term Nb = GH (as used
in EQ Z-5-1), to find velocity Vbh in EQ Z-11.
?????????????????????????????????????????????????????????????????????????
? ????????????????????????????????????????????????????????????????????? ?
?????????????????????????????????????????????????????????????????????????
? SPECIAL RELATIVISTIC EFFECTS ON ANY MASS AND RADIUS ?
???????????????????????????????????????????????????????
Only certain critical limit cases
(for masses Mo and Mc = black hole mass Mbh/GH)
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 Mbh mass (silent partner mass)
is easily found by:
EQUATION Z-13
C? R Finding mass Mbh needed for a
Mbh = ??????????????? black hole whose Schwarzschild
2G radius is given as R when
R is the radius of the Sun
so that another ratio is found, this being (Mbh/R)
which is the Sun's (black hole mass/radius) ratio.
But actually, term Mbh of EQ Z-13 is worthless. What
we really want to find is what (Mbh-/R-) ratio forms a
black hole out of the original Mo/R parameters, when Mo is
travelling at increasingly faster velocities approaching the
speed of light.
We need a comparative term, to study any differences between
the Sun when standing still, and when moving at a relativistic
velocity. The comparative term we want to know is found as:
EQUATION Z-14
Mbh C? Where ratio C?/2G is a constant,
??? = ???? when C is the speed of light, and
R 2G G is the universal gravitational
constant.
R is the original radius of original mass Mo
Mass Mbh is instantly found from EQ Z-13.
The logical argument formed in advance, was that
any mass result M+, and radius result R-, ensuing
from special relativistic effects on original states
Mo and Ro, should also equal the black hole constant
ratio C?/2G, if mass M+ and R- were relativistically
altered sufficiently to form a new black hole.
Ratio C?/2G can be labeled ratio CR (for 'constant ratio') and
has the value of (6.735275620 x 10 to 27 grs/cm), given a speed
of light whose digital value is 2.99792458, and a gravitational
constant whose digital value is 6.6720 x 10 to -8.
Ratio C?/2G is known as a constant
for the given values of C and G.
What we can do is follow special relativistic changes upon
both Mo and Ro through successively greater velocities, until
the combined ratios (1/Es x Mo) / (Es x Ro) equals the ratio
C?/2G, as in:
EQUATION Z-14A
((1/Es x Mo) / (Es x Ro)) = (M+/R-) = (C?/2G)
where Es 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 10 to 33 gms
(mass MM from Part 1 above)
1A. Given a Sun radius R of 6.96265 x 10 to 10 cms
1B. Given constant ratio CR = C?/2G
= 6.735275620 x 10 to 27 grms/cms
2. Given the black hole radius parameter
of EQ 4 of APPENDIX B, as:
EQUATION Z-14-1
2G M Finding the Schwarzschild
R' = ??????????????? radius R' of a black hole's
C? event horizon, when given
mass M
3. And given Equation 5 of APPENDIX B, rewritten as:
EQUATION Z-14-2
C? R Finding mass Mbh needed for a
Mbh = ????????????? black hole whose Schwarzschild
2G radius is given as R
Mass Mbh is the black hole silent
partner mass for any given mass M.
4. Given Equation Z-8 above for special relativistic effect
on both an original rest mass and its original radius, based
on a term Nx to determine a velocity, so that:
EQUATION Z-15
??????????????????????????? ??????????????????????
? ?? ??? ? (Vx)?
Es = ? ? C ? = ? 1 ? ??????
? 1 ? ? ???? ? \? C?
? ? Nx ?
? ?? ??
? ???????????
\? C?
5. Given that (1/Es x M) = M+
6. Given that (Es x R) = R-
7. Given that (1/Es x M+) / (Es x R-) = C?/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 = C?/2G
9. A first arbitrary value for Nx was tried, being 1.0001, which
produced results that were too low for the above Item 7 to be
correct.
10. A second arbitrary value for Nx was tried in EQ Z-15, being
1.00001, which was of the right magnitude for a mass M+, but
Item 7 was still not correct.
11. However, it was noticed that 1/1.00001 by itself was in the
magnitude range of gravitational relativistic effect Eg from
the Sun's mass, as determined in EQ C of Part 1 further above.
(MM in EQ C is the same value as Sun mass Mo given in EQ Z-2,
and immediately above in Item 1. And Eg of EQ Z-2 is the same
as Eg used immediately below in Item 12).
12. And so Eg was determined for the Sun's mass M = MM = Mo in
EQ Z-2, and conveniently labelled Egs (for 'effect gravity Sun
mass'), and was substituted as term 1/Nx in EQ Z-15 immediately
above, to give:
EQUATION Z-15-1
???????????????????????????
? ?? ??? ??????????????????????
? ? C x Egs ? ? (Vx)?
Ess = ? ? ? = ? 1 ? ??????
? 1 ? ??????????????? \? C?
\? C?
where velocity Vx is (C x Egs),
and special effect Ess conveniently
means an Es effect related to the
gravitational mass via term Egs.
13. Then; Sun mass M in (M x 1/Ess) = M+
14. And; Sun radius R in (R x Ess) = R-
15. And; ratio (M+/R-) = 6.73527458 x 10 to 27 grms/cms
As found in:
EQUATION Z-15-2
(M x 1/Ess) / (R x Ess) = CR = (M+/R-)
16. Which turned out to be an excellent approximation of ratio
CR (being C?/2G as created in Item 1B immediately above)
Well, this was very good for a first found attempt. How
about for other masses, and how did the ratio result of
Item 15 favorably equate in truth to Item 1B above, in
that the CR result in Item 15 is marginally below the
CR constant in Item 1B ?
17. The mass of the Sun was arbitrarily raised by a factor
of 1000, so that now M = 1.99099305 x 10 to 36 grms
18. A new Egs effect factor was determined using the
larger mass of Item 17, in EQ Z-2 above
19. The new Egs factor was substituted in EQ Z-15-1
to give a new Ess factor
20. The new Ess factor was substituted in the
terms of Items 13, 14, and 15
21. The result M+/R- = 6.735275620 x 10 to 27 gms/cms = CR,
which is exactly the constant of Item 1B
Two things were instantly made clear.
It is clearly evident that Equations Z-15, Z-15-1,
and Z-15-2, are correct for any mass, to yield (M+/R-)
ratios equal to C?/2G.
It is clearly evident that ratio (M+/R-) closes
in on ratio C?/2G, the closer that given original
mass M is to the black hole silent partner mass Mbh
as determined in EQ Z-14-2
(It is also clear from preceding explorations, that
when relativistic effects are to act upon an original
mass, the original mass M can never approach its black
hole silent partner equivalent Mbh any closer than by
Mbh divided by factors of the Golden Ratio).
???????????????????????????????????????????????????????????????
? 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 Ess value in Item 19, which as applied
to Item 13, was:
3.055623494 x 10 to 27 grms
23. The value of R- from the same Ess in Item 19, applied
to Item 14, was:
4.536746031 x 10 to 9 cms
24. Looking to Equations Z-14-1 and Z-14-2, it was found in
EQ Z-14-2 (given mass M+ of Item 22), and found in EQ Z-14-1
(given radius R- of Item 23), that (M+/R-) = CR. This is shown
in the following three equations:
EQUATION Z-15-3
2G M+ Finding the Schwarzschild
R' = ?????????????? radius R' of a black hole's
C? event horizon, when given
mass M+
R' was 4.536746031 x 10 to 9 cms,
exactly the same as R- in Item 23
EQUATION Z-15-4
C? R- Finding mass M' needed for a
M' = ??????????????? black hole whose Schwarzschild
2G radius is given as R-
M' was 3.055623493 x 10 to 27 grms,
exactly the same as M+ in Item 22
EQUATION Z-15-5
And so: M' of EQ Z-15-4, divided by R' of EQ Z-15-3, = CR
as in: (M'/R') = CR
where: CR is the constant of Item 1B
proving: that M+ of Item 22 and R- of Item 23 are the
correct parameters of a new black hole created
by relativistic effect Ess of Item 19, on higher
mass M of Item 17, using EQ Z-15-1 to determine
Ess, after using EQ Z-16 to determine Egs.
?????????????????????????????
??? 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
????????????????????
? 2G M M is any mass, R is its
Egs = ? 1 ? ?????? radius, and Egs is the
\? C? R gravitational relativistic
effect of mass M
EQUATION Z-16-1
???????????????????????????
? ?? ??? ??????????????????????
? ? C x Egs ? ? (Vx)?
Ess = ? ?? ?? = ? 1 ? ??????
? 1 ? ??????????????? \? C?
\? C?
Ess is the special relativistic effect ensuing from
velocity Vx, determined as the direct consequence of
the speed of light reduced by the mass's gravitational
relativistic effect Egs.
EQUATION Z-16-2
(M x 1/Ess) = M+
EQUATION Z-16-3
(R x Ess) = R-
EQUATION Z-16-4
(M+/R-) = C? = CR
????
2G
and mass M+ and radius R- are a relativistic transfiguration of
M and R into the parameters of a black hole, when ratio (M+/R-) = CR.
CR is a physical constant in black holes,
whose value is given as the speed of light squared
divided by twice the gravitational constant, and
whose value is 6.735275620 x 10 to 27 gms/cms.
EQUATION Z-16-5
And ultimately, Ess can be determined directly
from Egs, by:
Ess? = 1 - (Egs)?
Ess is not the same value as Egs. Ess can be higher
or lower than Egs. The exact relationship between the
value of Egs and Ess is known by:
EQUATION Z-16-6
????????????????
Ess = \? 1 - (Egs)?
????????????????
Egs = \? 1 - (Ess)?
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.
?????????????????????????????????????????????????????????????
?????????????????????????????????????????????????????????????????????????
? ????????????????????????????????????????????????????????????????????? ?
? UNIFIED EFFECTS IN FIELD BEHAVIOR ?
? ????????????????????????????????????????????????????????????????????? ?
?????????????????????????????????????????????????????????????????????????
?????????????????????????????????????????????????????????????
?????????????????????????????????????????????????????????????????????????
? ??????? 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 Mbh, and plateaus below this, articulated as functions
of the Golden Harmonic Ratio 1.61803398875.
For the test cases, it is desirable to see what happens
mathematically for events which are right at the brink of
a black hole mass, compared to masses well below the brink.
The phenomenology is thus most easily watched in detail.
For this, such masses are arbitrarily created, and assumed to exist
in violation of the statements in Part 2 above (which delineate that
a mass of black hole equivalent includes an original mass Mo, a mass
augmentation unit Ko, and resultant mass aggregate which is that of a
black hole or less. If the mass is that of a black hole, the original
mass is at a critical mass limit Mc, and the ratio Mbh/Mc = Ng is a
function of the Golden Ratio. For masses other than than Mc, ratio Ng
is given the general label Nx).
In the following, the cases for Mc and Ng parameters are ignored by
conveniently looking the other way. In the test cases which follow,
the existence of discrete portions denoted by terms such as Mo, Mc,
and Ng, are expeditiously put aside, and a mass value is assumed which
can be anything less than Mbh, even if less than Mbh by a few parts in
a thousand. This is called a HIGH mass, for convenience.
?????????????????????????????????????????????????????????????????????????
? ????????????????????????????????????????????????????????????????????? ?
?????????????????????????????????????????????????????????????????????????
? TEST CASE ?
?????????????
In a test case, a HIGH mass value is studied which hangs right
below the mass of a black hole Mbh. This is in a deliberately
selected HIGH mass range which as already said ignores properties
such as a critical mass factor (Mc) outlined in Part 2 above.
The intention this time is to follow test case examples in
excruciating digital detail, so that the effects and their
changes are unmistakable.
The sole intention of the following, is to observe how certain
properties are universally united in a general way through various
transformations between gravity and electromagnetic field behaviors.
And so a new study model is created, based on the arbitrary
criteria that any job needed to do a certain job is good enough
for the purpose intended.
A HIGH mass gravitational event and a LOW mass event are thus
arbitrarily created from the same Mbh term, which is the mass
of a black hole confined in the Sun's radius. Mbh for the Sun's
radius is (4.689536679 x 10 to 38 grms).
The Sun's radius (6.96265 x 10 to 10 cms) has been chosen as an
easily recognized radius for use as a constant to investigate
the effects of different mass densities confined in a fixed
(unchanged) area. Otherwise, the Sun's radius has no physical
significance when tied to the following arbitrary mass aggregates.
To supply the study, a small ratio Nx has been selected for a
control in the study. Nx is meaningless other than its value
is the charge to mass ratio of the hydrogen atom, ie.:
((Proton + electron) / electron) = 1.000544617 = Nx.
(The interpretation is that the negative electron charge
of the lightweight electron influences the heavy proton
by only 1.000544617 of the effect the proton has on the
electron, since both particles have the same quantity of
charge (opposite) despite widely divergent rest masses.
This is mentioned only to satisfy curious minds. As said,
the real value for the above ratio Nx has no intrinsic
significance in the following).
MASS1 In our study model, Mbh is arbitrarily reduced by the
small ratio Nx to give a HIGH Mass1 term, which is very
slightly below Mbh.
MASS2 Mass1 is then arbitrarily reduced by a factor of 100,000 to
give a LOW Mass2 term having the same digits but much lower
magnitude then Mass1.
The intention is to be able to follow certain relativistic field
effects in detail by following the digital results of both the
HIGH mass term (Mass1), and LOW mass term (Mass2), to more openly
follow the unifying effects between the two fields (being gravity
and electromagnetism).
In the study model, as already said, the value of Nx has no
significance except that it provides a convenient low value
Nx ratio to arrive at a HIGH mass term for the study model.
Nx is given to 13 significant digits as gained from the
ratio (P 938.2796 mev + E .5110034 mev) / (P 9382796 mev)
= 1.000544617404
TABLE 4-A
????????????????????????????????????????????????????????????
? ARBITRARY STUDY MODEL DATA ?
????????????????????????????????????????????????????????????
? ?
? Nx = 1.000544617404 = (P + E) / E ?
? Mbh = 4.689536679 x 10 to 38 grms ?
????????????????????????????????????????????????????????????
? ?
? HIGH mass1 = Mbh / Nx ?
? = 4.686984066 x 10 to 38 grms ?
? Nx = 1.000544617404 ?
? ?
? LOW mass2 = Mass1 / 100,000 ?
? = 4.686984066 x 10 to 33 grms ?
? Nx = 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 Egs ratio is substituted for ?
? the algebraic term Egs. ?
????????????????????????????????????????????????????????????
EQUATION Z-17 HIGH gravitational Mass1 results:
??????????????????????????????????????????????
? 2G (4.686984066 x 10 to 38 grms)
Egs = ? 1 ? ???????????????????????????????????
\? C? R
Mass1 has been given in
terms of a real weight.
Radius R is the radius of the Sun.
Egs is the gravitational relativistic effect of Mass1
??????????????
HIGH gravity field effect Egs = ? .023330687 ?
Egs is closing toward 0 ??????????????
EQUATION Z-17-1 Electromagnetic field effect results
(Ess is special relativistic effect)
?????????????????????????????????????
? ?? ???
? ? C x .023330687 ? V?
Ess = ? ? ? = ????
? 1 ? ?????????????????????? C?
\? C?
.023330687 is effect Egs
of EQ Z-17
Ess = 1 - (Egs)?
As in: 1 - (.023330687)? = .999727802
??????????????
LOW special field effect Ess = ? .999727802 ?
Ess is closing toward 1 ??????????????
V velocity is starting to
close toward 0
EQUATION Z-17-2 LOW gravitational Mass2 results:
??????????????????????????????????????????????
? 2G (4.686984066 x 10 to 33 grms)
Egs = ? 1 ? ???????????????????????????????????
\? C? R
Mass2 has been given in
terms of a real weight.
??????????????
LOW gravity field effect Egs = ? .999995002 ?
Egs is closing toward 1 ??????????????
EQUATION Z-17-3 Electromagnetic field effect results
(Ess is special relativistic effect)
?????????????????????????????????????
? ?? ???
? ? C x .999995002 ? V?
Ess = ? ? ? = ????
? 1 ? ?????????????????????? C?
\? C?
.999995002 is effect Egs
of EQ Z-17-2
Ess = 1 - (Egs)?
As in: 1 - (.999995002)? = .003161416
??????????????
HIGH special field effect Ess = ? .003161416 ?
Ess is closing toward 0 ??????????????
V velocity is closing toward 1
???????????????????????????????????????????????????????????????
??? 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 Ess from Equations Z-17-1, and Z-17-3), when applied
to the HIGH mass of EQ Z-17, and LOW mass of EQ Z-17-2, will yield
appropriate M+ and R- terms for each of the masses. These are
listed in the following:
TABLE 5
????????????????????????????????????????????????????????????
? ?
? HIGH MASS GRAVITY ?
? ?
? MASS1 = (4.686984066 x 10 to 38 grms) ?
? ?
? RADIUS R = 6.96265 x 10 to 10 cms ?
? ?
? Ess EFFECT = .999727802 ; from EQ Z-17-1 ?
? ?
? M+ = (Mass1 x 1/Ess) ?
? = 4.688260199 x 10 to 38 grms ?
? ?
? R- = (radius R x Ess) ?
? = 6.9607547839 x 10 to 10 cms ?
? ?
? CR = ratio (M+/R-) ?
? = 6.735275620 x 10 to 27 grms/cm ?
? ?
????????????????????????????????????????????????????????????
TABLE 6
????????????????????????????????????????????????????????????
? ?
? LOW MASS GRAVITY ?
? ?
? MASS2 = (4.686984066 x 10 to 33 grms) ?
? ?
? RADIUS R = 6.96265 x 10 to 10 cms ?
? ?
? Ess EFFECT = .003161416 ; from EQ Z-17-3 ?
? ?
? M+ = (Mass1 x 1/Ess) ?
? = 1.482558107 x 10 to 36 grms ?
? ?
? R- = (radius R x Ess) ?
? = 2.201183848 x 10 to 8 cms ?
? ?
? CR = ratio (M+/R-) ?
? = 6.735276152 x 10 to 27 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 = C?/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). ?
????????????????????????????????????????????????????????????
?????????????????????????????????????????????????????????????????????????
? ????????????????????????????????????????????????????????????????????? ?
?????????????????????????????????????????????????????????????????????????
? FIRST INTERPRETATION ?
????????????????????????
Thus M+ can approach but never equal or exceed Mbh. As the Egs
effect approaches 0 (greatest power in gravity field strength),
the Ess effect approaches 1 (the least power, no effect), in
velocity related relativistics.
At the point where the gravity effect has its greatest value;
at Egs = 0 ; the special relativistic effect ceases to exist
(comes to a standstill), because there is no velocity, as when:
EQUATION Z-17-4
(C/0) / C = 0/C = 0 .
This closes right in on a clear insight regards the question
of how maximum potential relativistic gravity effect can
contain light - effectively cancel the velocity of light.
The velocity of light is not cancelled. The ability to have
a velocity related to any special relativistic effect is
cancelled. It appears this amounts to the same thing as a
counteracting of the velocity of light.
?????????????????????????????????????????????????????????????????????????
? ????????????????????????????????????????????????????????????????????? ?
?????????????????????????????????????????????????????????????????????????
? DIRECT INTERPRETATION ?
?????????????????????????
A first interpretation of the consequences of Equations Z-17 to
Z-17-3, is that a HIGH gravitational mass density results in a
LOW special relativistic synonymity. And a LOW gravitational
mass density results in a HIGH special relativistic synonymity.
It has the immediate interpretation that things run faster in
LOW gravitational events, and slower in HIGH gravitational events.
It adds another picture to the experimentally
confirmed property that proximity to gravity,
relativistically causes time to slow.
Intuitively, it answers a question as to how gravity at
its highest can confine light. A see saw (or yin yang)
characteristic in the works is summarized in the following:
TABLE 7
????????????????????????????????????????????????????????????
? HIGH mass gravity Effect Egs approaches 1 ?
? Effect Ess approaches 0 ?
? ?
? LOW mass gravity Effect Egs approaches 0 ?
? Effect Ess approaches 1 ?
????????????????????????????????????????????????????????????
? You can see at a glance how gravity can confine ?
? light. As gravity effect Egs closes in on 1, ?
? special effect Ess closes down toward 0 velocity. ?
? When Egs is right at 1, Ess is closed down right ?
? to 0 and the velocity of light C in a V/C ratio ?
? is vanished when 0/C = 0 . ?
? ?
? Conversely, when Egs is low and closing down to 0, ?
? effect Ess intensifies with a velocity approaching ?
? 1, which is equivalent to approaching the full ?
? speed of light. ?
????????????????????????????????????????????????????????????
? In another sense, it is clearly seen that events ?
? are free to move more rapidly in activities of a ?
? HIGH velocity, in a LOW gravity field density. ?
? ?
? And in a HIGH gravity field density, events are ?
? constrained to low velocity activity approaching ?
? 0 velocity, when the gravity field approaches the ?
? density of a black hole, re: special relativity. ?
????????????????????????????????????????????????????????????
? ?
? Notes: ?
? ?
? In real events, as summarized above in Part 2, ?
? if a mass augmentation is assumed for gravity ?
? effect Egs, then when a mass's density (without ?
? augmentation) reaches a critical mass factor Mc, ?
? the mass augmentation amount Ko is sufficient to ?
? jump the mass amalgamation in one whole bump to a ?
? black hole quantity Mbh, such that effect Egs = 1. ?
? And thus effect Ess = 0; which is the equivalent ?
? of a 0 velocity for light. ?
? ?
? The proportionate bump of mass Mc to Mbh is a ?
? function of the Golden Ratio 1.61803398875. ?
? ?
? It means there never is a situation where effects ?
? Egs and Ess slowly converge to 1 and 0, as is ?
? fictitiously indicated in Equations Z-17 and ?
? Z-17-1. As show in Part 2 further above, effects ?
? Egs and Ess will jump in a final leap to 1 and 0 ?
? in a single bump via Golden Ratio functions, when ?
? the gravity mass density reaches Mc before ?
? reaching black hole mass Mbh. ?
? ?
????????????????????????????????????????????????????????????
-- Continued in RELATIVE.4 --
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