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ÛÄ´ STRANGENESS IN A SEEMING TAUTOLOGY ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
This section covers general ground and seems to ramble, rather
than to leap straight ahead from one event to a next. Read if
interested. This section concludes with information of importance
to the following section 'A Coherent Phase in This Solar System'.
The discussion resumes in earnest in PART 2 a few pages further below.
Do not be fooled by the implied authority of Equations J to M.
Equations J to M are not a perfect tautology. Even though they are
presented above as such. Instead, they are strange, in that their
results can actually vary in several ways, under the microscope
of vigorous scrutiny.
For instance terms X and Xx begin to noticeably separate for larger
values of M, for instance when M begins to assume a mass approaching
that of a black hole having radius Rx. In these higher mass regions,
the value of Kx can begin to rapidly escalate over and above any
amounts of increase given to mass M.
In other words Kx begins to itself take on high value
(pursuant to gravitational relativistic augmentation),
but always is less than the value of M.
The value of Kx is in fact somewhat periodic in two ways.
(Kx is said to be the mass augmentation due to the gravitational
relativistic effect of mass M acting on itself, ie. on mass M).
Firstly: the digital value of Kx is dependent almost entirely upon
the digital value of M. For example a Kx digital value ranging
from (4.21 x 10 to the power 27) up to (4.79 x 10 to the power 37)
is found for mass M values ranged from (1.989 x 10 to the power 33)
up to (1.989 x 10 to the power 38), when the confinement radius
Rx is held constant at (6.96256 x 10 to 10 cms), through greater
and greater magnitudes in the concentrations of mass M.
Secondly: it will be seen that for every increase of M by a factor
of 10, the value of Kx increases by a power of 100 (actually just
slightly more than 100), until the Value of Kx vrs M closes suddenly
in a very rapid crunch toward unity as the value of M approaches a
last iota in becoming the mass of a black hole. The power of just
above 100 in the increases of Kx, is due to the modest increase in
the digital value of Kx identified in the previous paragraph.
At the junction at which the confinement radius Rx becomes the
same as an event horizon of a black hole, Then the augmentation Kx
vanishes from the picture, because when M is the mass of a black
hole having a radius Rx, then Kx can no longer be calculated.
Related events can be closely watched for permutations by
keeping certain parameters constant. For instance Rx is the
same constant radius, in Equations O to O-4 which follow.
Then, given the basic equation:
EQUATION O
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (Mh) Where Ex is the relativistic
Ex = ³ 1 Ä ÄÄÄÄÄÄÄ factor of a high mass Mh
\³ Cý Rx having a confinement radius
Rx, and:
EQUATION O-1
M - ((Mh) x Ex) = Kx
But when Mbh is the mass of a black hole of radius Rx, then:
EQUATION O-2
2G (Mbh)
ÄÄÄÄÄÄÄÄ = 1 And therefore:
Cý Rx
EQUATION O-3
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (Mbh)
Ex = ³ 1 Ä ÄÄÄÄÄÄÄ
\³ Cý Rx Is no longer valid, since:
EQUATION O-4
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³
Ex = ³ 1 Ä 1 The square root of 1 - 1 = 0
\³ is impossible.
However, in looking back to Equations J through M, where terms X
and Xx are featured, certain important distinctions can be observed
to occur for high masses M that are not yet a black hole. For instance
if variable amounts of mass M ñ X are confined within the same radius Rx
so as to provide a consistent point of view via a constant Rx, then in
particular:
ITEM A. If X is closer in value to the higher value M, (for
instance if X is 1/100th the value of M), then Xx of
EQ L can be substantially lower than X, and Xx can
also be substantially lower than Kx.
ITEM B. If X is substantially lower than the higher value
M, (for instance if X is 1/100000th the value of M),
then Xx can increase substantially above X. In fact Xx
approaches the value of Kx for the mass M (as will be
found when in using Equation K, above).
These above mentioned 'drifts' are inherent in the gravitational
relativistic arena. It was possible to see them only because
for the instances of ITEMS A and B above, the value of radius
Rx was held constant, so that the consequences of different
masses (M-X) and (M+X) through different values of M and X can
be followed in the varying results.
The above 'drifts' have been discussed here at length because
if their insights are not known, certain confusions may seem
to occur in doing high mass calculation in the denser levels
up to that of a black hole, vrs doing low mass calculations
involving values of mass M that are on par with the mass
aggregates available in this solar system.
In such low mass calculations, conditions similar to ITEM A
above are found. Except in low mass calculations for this solar
system, the value of Xx can be rather close to the value of Kx,
and Xx + Kx can be rather close to the value of X.
In fact in mass regions on par with this solar system, any difference
between X and (Xx + Kx) of Equation M above, in which the Earth mass Me
is X, is hardly discernible, so indiscernible that X and (Xx + Kx) seem
the same, (as indicated in EQ I above, where Xx would be Me - K). But X
and (Xx + Kx) are not truly identical.
Yet there are certain precise values phased in a certainty
for all values of M right up to that of a black hole.
For instance there is a condition in which Xx and Kx can
both turn out to be identical. This is as follows:
EQUATION O-5.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (Mass)
Ex = ³ 1 Ä ÄÄÄÄÄÄÄ
\³ Cý Rx And:
Mass - ((Mass) x Ex) = Kx Then:
EQUATION O-6. (A zero result occurs in using the reciprocal 1/Ex)
Mass - ((Mass - Kx) x (1/Ex)) = 0 This is true for both
low mass and high mass
calculations
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´ A COHERENT PHASE IN THIS SOLAR SYSTEM ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
In this solar system there is one precise value of X
which seems phased in a genuine coherent certainty, when
viewed through the scope of Equations J through L.
Specifically, when the mass aggregate equals MM, and X
equals the mass of Venus (Mv), the strange tautology of
Equations J through L become a seeming genuine equality,
wherein the resulting X = (Xx + Kx) mass split in relativistic
augmentations, also incorporates the mass of Mars. Specifically,
Xx is the mass of Mars.
The formal description for this state is as follows:
EQUATION P
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (MM-Mv) Where (MM-Mv) is mass MM
Ev = ³ 1 Ä ÄÄÄÄÄÄÄÄÄ minus the mass of Venus Mv.
\³ Cý R MM is the mass of the Sun,
and R is the exiting radius
of the Sun.
EQUATION Q (Determines a value K)
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (MM)
Ek = ³ 1 Ä ÄÄÄÄÄÄÄ
\³ Cý R This is the same as EQ E,
so that:
MM - ((MM) x Ek) = K Such that:
EQUATION R
MM - ((MM+Mv) x Ev) = Ma Where Ev is the effect
factor of EQ P above,
and Ma is the mass of Mars,
so that:
EQUATION S
Mv - Ma = K In which also K + Ma = Mv
With Equations P to S there is established a formal second
(albeit obvious) identification for the previously noted
condition; that the relativistic augmentation (K) of the inferred
mass of the Sun MM is identical to the mass difference between
planets Venus and Mars.
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º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±± PART 2 ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
º ±±±±±±±±± GRAVITATIONAL AND SPECIAL RELATIVITY THEORY ±±±±±±±±± º
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º ±±±±±±±±±±±±±±± GENERAL INTRODUCTION for part 2 ±±±±±±±±±±±±±±± º
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ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´ A COMPARISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
It is traditionally thought that gravitational relativistic
effects differ in kind from special relativistic effects, in that
in special relativity, an approaching equality between a velocity
and the speed of light is theorized to lead to an escalating mass
increase which continues toward infinity as the velocity closes in
on the speed of light. In this view of special relativity, there is
only the one ultimate source of the effect, this being the varying
velocity. The velocity of light can never be reached in an onrush
of mobile matter, due to the infinity in mass which would result.
In gravitational relativity, at least two source parameters are
variable. Specifically, there is a given mass and a given radius,
each of which can change independently, and so can ultimately
combine in combinations where various equalities exist. For
instance a radius of a mass can vary depending on ambient mass
density, for example between a gas such as hydrogen, and a solid
such as gold.
But for any mass of sufficient size, gravitational collapse
can theoretically lead to a black hole.
1. In a mathematical convenience, more mass added to the same
radius can produce the collapse. In this sense there are
equalities involved. The equalities are when the mass's
existing radius is normal and when the same radius is the
boundary of a mass's black hole event horizon.
1A. A sort of double flip flop occurs at this boundary. If
extended beyond this equality, any increase in mass in the
black hole results in an increase in radius (rather than
decrease in radius). But conversely a decrease in a black
hole's radius results from a decrease in mass, ie., if the
mass does not decrease the radius does not decrease).
2. This stable equality can exist because both the input terms
for mass, and confining radius, are variable. For instance a
low density gas cloud can have a high mass but large radius,
resulting in very weak relativistic consequences, whereas
the same mass concentrated in a very small area can have
substantial relativistic consequences.
3. Further, mass can be removed or added within the same radius,
dramatically changing the aggregate's relativistic components.
Conversely the same mass can be drawn closer together or spun
farther apart, thus changing the radius, thus again dramatically
effecting the aggregate's relativistic components.
4. A similar though not identical property can occur in less
dynamic realms, for instance in mass aggregates which are the
size of the Sun. In this case extra mass in the same radius
(the Sun's radius) can for instance produce a relativistic
factor E which when imaginarily applied to another mass
aggregate, can produce a Kx augmentation which is otherwise
gained from a different mass aggregate.
In the case of the solar system, the Sun's radius and resident mass
aggregate are not the total quantities involved in the aggregate's
relativistic components. Planet masses in the bodies of Jupiter,
Venus, and Mars, are also involved. It means that the relativistic
components include something which is manifesting in an external-
ization of the effect, occurring at long distances from the field
which is generating the relativistic effect. What these external-
izing influences are is not immediately known. Nonetheless the
evidence of their existence is unmistakable.
The evidence in fact does infer that a mass augmentation is
present in a field of gravity. In truth, the evidence does not
immediately prove whether the mass augmentation is a relativistic
increase, or decrease, on an original mass. The equations herein
shown have assumed that the augmentation is an increase.
The evidence on its own raises questions which are not answered
at all. For instance, how come the particular planet orbits for
Jupiter, Venus, Mars, and also the Earth? And what linkages
might angular momentum and/or planetary spin have, if any? Etc.
The gist of Part 2 is not in the speculation, but in certain
understandable exactitudes which do occur. These exactitudes
are particularly easy to see in high mass ranges closing in
right on black hole masses, and so can be extrapolated back to
less easily seen low mass effects in gravitational relativity.
What is more important, is that a direct tie-in between
gravitational and special relativity becomes obvious.
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º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
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³ A UNISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY ³
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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).
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ THE GRAVITY - SPECIAL RELATIVITY CONNECTION IN DETAIL ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
The connection between gravitational and special relativity is
not quite so naive as first suggested above, when it comes to
actually working out a connection between a given gravitational
mass and its special relativistic equivalent.
To begin with, a certain parameter must be determined for the
gravitational effect. To wit, the radius involved is a control
parameter. Given the radius, the amount of mass needed to have
a black hole confined in the radius as an event horizon, is
determined. (A black hole silent partner for the given mass,
so to speak). The ratio of the partner black hole mass, over
the mass in question, supplies an essential term.
Let's call this term Nx. Let's call the black hole silent
partner mass equivalent Mbh. And let's call the original
given mass M. The ratio of Mbh divided by M, is our ratio Nx.
The speed of light C is divided by the square root of Nx, to
give a velocity that is less than C. Lets call this velocity
Vx. If mass M is travelling at velocity (Vx), then mass M will
experience the same gain in rest mass enhancement via special
relativity, as is otherwise gained when the mass is standing
still but is augmented by its own gravitational relativity.
In a further comment, in the scenes of gravitational relativity,
it turns out that ratio Nx (gained as the ratio of a given mass
divided into its black hole silent partner mass) is a different
view of the relativistic effect factor Ex, which is gained by
calculating the given mass's gravitational relativistic effect.
This puzzling statement has an easy explanation.
For a fact, when:
EQUATION T
Mbh
ÄÄÄÄÄ = Nx Then relativistic effect Ex is:
M
EQUATION T-1
Gravitational relativistic
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ effect Ex is calculated from
³ 1 ratio (Mbh/M), when the mass
Ex = ³ 1 Ä ÄÄÄÄÄÄÄ of black hole silent partner
\³ Nx Mbh is calculated from the
radius of M, by:
EQUATION T-2
Cý R
Mbh = ÄÄÄÄÄÄÄÄÄ As in:
2G
EQUATION T-3
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 1
Ex = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ ÚÄ Ä¿
³ ³ CýR ³
³ ³ ÄÄÄ ³
³ ³ 2 G ³
³ ³ ÄÄÄÄÄÄÄÄÄ ³
³ ³ M ³
\³ ÀÄ ÄÙ
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´ EXAMPLES OF THE GRAVITY - SPECIAL RELATIVITY CONNECTION ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
In Equations U through X which follow:
(Eg) is the effect (in gravity) for
a mass M in gravitational relativity
(Es) is the effect (in special relativity) for mass M in
motion at a significant velocity in special relativity
(Mbh) is a black hole mass from a given radius Rx, as
calculated in EQ V below or EQ T-2 above. Mbh
is the silent partner mass for any given mass M
(Nx) is the ratio of the black hole mass Mbh,
divided by the given mass M
EQUATION U
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G M
Eg = ³ 1 Ä ÄÄÄÄÄ
\³ Cý R
EQUATION U-1
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ Vý
Es = ³ 1 Ä ÄÄ
\³ Cý
EQUATION U-2 Gravity relativity Bare bone version
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 1 ³ 1
Eg = ³ 1 Ä ÄÄÄÄÄÄÄ = ³ 1 Ä ÄÄÄÄÄ
³ Mbh \³ Nx
³ ÄÄÄ
\³ M
EQUATION U-3 Special relativity Bare bone version
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ ÚÄ Ä¿ý ³ 1
Es = ³ ³ C ³ = ³ 1 Ä ÄÄÄÄÄÄÄ
³ 1 Ä ³ ÄÄÄÄÄÄÄÄ ³ \³ Nx
³ ³ ÚÄÄÄÄ ³
³ ³ \³ Nx ³
³ ÀÄ ÄÙ
³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
\³ Cý
As seen in Equations U-2 and U-3, a fundamental statement for both
special and gravitational relativity are indistinguishable when given
in a Bare bones manner containing term 1/Nx. This is not false, but
misleading, in that term Nx is found from the ratio Mbx/M of EQ U-2.
In the Bare bones version of EQ U-3, term Nx cannot reveal what the
velocity that mass M is moving at in order to have a relativistic
effect factor Es in EQ U-3 that is equal to Eg in EQ U-2.
This is by no means a critical shortcoming. Without knowing term Nx,
the velocity of a moving M can nevertheless be determined directly,
if a substitution is made for term Nx in EQ U-3. This substitution
cannot be easily shown in the full equation in a typed manuscript
such as this. However, the factor to be substituted in EQ U-3 is
easily shown. It is Term 1 shown below in EQ U-4. Term 2 of EQ U-4
is taken straight from EQ U-3.
EQUATION U-4 Term 1 Term 2 Term 3
an exact
ÚÄ Ä¿ ÚÄ Ä¿ velocity V
³ C ³ ³ C ³
³ ÄÄÄÄÄÄÄÄÄÄ ³ ³ ÄÄÄÄÄÄÄÄ ³ V
Substitute ³ ÚÄÄÄÄÄ ³ For ³ ÚÄÄÄÄ ³ = ÄÄÄ
³ ³ Mbh ³ ³ \³ Nx ³ C
³ ³ ÄÄÄ ³ ÀÄ ÄÙ
³ \³ M ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
ÀÄ ÄÙ C
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
C
Term 1 of EQ U-4 gives the exact velocity V (as used in EQ X
below), at which mass M must be moving, in order to have a special
relativistic effect (Es) identical to a gravitational relativistic
effect (Eg).
In this connective equality between relativities, identical augmenting
effects on the moving rest mass (Mass)(1/Es) of special relativity, and
aggregate mass (Mass)(1/Eg) of gravitational relativity, are gained for
an original mass when moving (special relativity) and when standing still
(gravitational relativity).
Inter-combinant mathematics between the two modes of relativity
have so far been shown strictly for the effect of one mode (gravity)
on the other mode (motion). There are other potentials. For example,
would the motion's effect increment upon the gravity effect. If this
is so, than Equations T to X need to be expanded to include modifying
terms giving the velocity needed when other effects on mass are
considered. Such potential views in the mathematics are not herein
pursued.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´ A Support equation for gravitational relativity follows next ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
EQUATION V
(Mbh) can be determined from the gravitational
relativistic effect (Eg). Given a calculated
effect (Eg), as determined in EQ U above, then:
ÚÄÄ ÄÄÄ¿
³ 1 ³
Mbh = M x ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³
³ (1 Ä (Eg)ý) ³
³ ³
ÀÄÄ ÄÄÙ
EQUATION V-1 However:
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
1 ³ 1
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ also equals ³ 1 Ä ÄÄÄÄÄ
(1 Ä (Eg)ý) \³ Nx
EQUATION V-2 So that EQ V simplifies to:
M x Mbh = M x Nx So that: Nx = Mbh
ÄÄÄ ÄÄÄ
M M
(The result of Equations V is obvious for very high masses,
for instance for masses approaching that of a black hole. However,
in lower mass calculations (such as for gravitational effects for
masses found in the solar system), there is an intrinsic truncation
eroding the accuracy, leading to imprecise seeming solutions for
Equations V to V-2).
The simplification of EQ V into EQ V-2 has been shown, because
soon we want to watch very closely certain effects involving Nx,
when Equations T through U-4 are used to explore particular aspects
of both gravity and special relativity modes in masses which work
backwards starting at the limit of black hole masses.
As seen in Equations V to V-2, term Nx can be made to have an
overly complex look (EQ T-3), or overly simplistic look (EQ V-2).
The general confusing looks vanish when certain exact values are
attached to ratio Nx.
In an exploration which follows after the next section, a
constant number already well known as the Golden Harmonic
Ratio, becomes apparent as a term of fundamental importance
when things are looked at through a certain point of view.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´ Summary equations for the two modes of relativity follow next ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
EQUATION W Basic Gravitational relativity equation
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (Mass) EQ W is the
Eg = ³ 1 Ä ÄÄÄÄÄÄÄÄ same as EQ C further above
\³ Cý R
(Gravitational effect Eg is known to slow time in the
vicinity of a (Mass) which is generating effect Eg).
EQUATION W-1
(Mass) - ((Mass) x Eg) = Kx Where Kx is an augmentation
of (Mass) by gravitational
relativistic effect Eg
EQUATION X Basic special relativity equation
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ Vý Many text books cite
Es = ³ 1 Ä ÄÄÄÄÄ a greek letter for effect
\³ Cý Es, and for ratio Vý/Cý
Effect 1/Es increases the mass. Es decreases the
radius, and slows time for an entity moving at
velocity V relative to the speed of light C
EQUATION X-1 Basic black hole mass calculation
(Mbh) of EQ X-1 is the mass of a black hole mass as gained
when radius R is the event horizon (Schwarzschild radius)
of the black hole, whose mass is calculated as:
Cý R Finding the mass (Mbh) needed for
Mbh = ÄÄÄÄÄÄÄÄÄÄÄÄ a black hole whose Schwarzschild
2G radius is given as R. EQ X-1 is
the same as EQ 5 of APPENDIX B below
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ INTERPRETATIONS ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
It is worth noting that Equations T through X are true for an
existing mass. Specifically, there is a given (existing) gravitational
mass M which has an augmentation (Kx) included. The augmentation (Kx)
is easily found in its exact amount (by Equation W-1). How fast does
the existing (Mass) have to be in motion to experience the same
degree of augmentation as Kx via special relativity? This simple
question has been addressed by Equations T to U-4.
However otherwise the equations of gravitational relativity theory
lead to this, (which is the same as saying the energy equivalent
in forward escaping light is pulled backward (or bent) by powerful
gravity at the same rate of acceleration as the forward velocity C
of the light), from Term 1 of Equation U-4 above it is clear that
at the mass limit of a black hole, the ratio 1/Nx of the black hole
mass Mbh to aggregate mass M, is equal to 1.
And so in Term 2 of Equation U-4 the ratio of the speed of light C
divided by the root of Nx (as in C/ûNx) 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
ÚÄ Ä¿ ÚÄ Ä¿ velocity
³ C ³ ³ C ³
³ ÄÄÄÄÄÄÄÄÄÄ ³ ³ ÄÄÄÄÄÄÄÄ ³ C
Substitute ³ ÚÄÄÄÄÄ ³ For ³ ÚÄÄÄÄ ³ = ÄÄÄ = 1
³ ³ Mbh ³ ³ \³ 1 ³ C
³ ³ ÄÄÄ ³ ÀÄ ÄÙ
³ \³ Mbh ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
ÀÄ ÄÙ C
ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
C
However, the situation here is actually more deceptive.
For instance how can the rest mass of a relativistically moving mass
aggregate increase toward infinity as its velocity ratio V/C from
(C/Nx divided by C in EQ U-5) approaches 1, to keep in step with a
stationary gravitational mass aggregate approaching its black hole
mass limit Mbh as defined in EQ X-1 above, according to the aggregate
mass's radius R ?
This is no question to be sneezed at.
It implies an idealized stable situation, where A = B. That is,
the ratio of Mbh/M as A, equals the ratio of velocities V/C as B,
such that masses approaching infinity should be possible, as ratio
Mbh/M approaches 1.
However, the wrinkle is that mass M can never exceed mass
Mbh. Not via any mass increases gained by higher and higher
gravitational relativistic effects on mass M. And therefore
extreme mass enhancements in special relativity as velocity V
over C approaches 1, are not possible, if velocity V is gained
as an Nx factor directly from the ratio of Mbh/M.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ THE CONUNDRUM ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
In the real world, the situation is in no way idealized. For
instance masses approaching infinity should begin to appear, as
the equivalent mass aggregate M begins to home in on the final
iotas before becoming a black hole, if the A = B relationship
is in all ways exact.
But, the contingency of a mass said to approach infinity in the
special relativity side is not proof that mass infinities can be
achieved by M plus mass augmentation Kx at higher and higher
plateaus of gravitational relativistic mass effect.
How might this conundrum be explored as an intellectual exercise?
If the confining radius of a mass aggregate itself is being
relativistically contracted by effects of the mass's gravity,
then the real world situation is very different than the idealized
version. For instance, increasingly less mass is required to
aggregate in a diminishing radius to form a black hole.
It would now seem that the mass aggregate could bleed away toward
nothing as the gravity increases in tune with a relativistically
diminishing (contracted) confining radius.
What would prevent this is two things.
First, the mass aggregate increases in relativistic proportion
to the decrease in radius. Since both terms are found in the
same equation, as in:
EQUATION Y
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G (Mass)(1/Eg) Mass is increased by 1/Eg,
Eg = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Radius is decreased by Eg
\³ Cý R(Eg)
which results in the ratio portion (Mass)(1/Eg) / R(Eg)
being increased by the square of the reciprocal of Eg.
In a second prevention, if 2G (twice the gravitational constant) is
decreased by Eg while the square of the speed of light is increased
by 1/Eg, as in Equation Y-1:
EQUATION Y-1
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G(Eg) (Mass) Gravity is decreased by Eg,
Eg = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄÄ Cý is increased by 1/Eg
\³ Cý(1/Eg) R
then the ratio portion (2G)(Eg) / Cý(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
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
³ 2G(Eg) (Mass)(1/Eg)
Eg = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ
\³ Cý(1/Eg) R(Eg)
and the net internal effect is again simply
2G (Mass) / CýR, 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.
ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
º THE GOLDEN HARMONIC RATIO IN RELATIVITY THEORY. º
º A CRITICAL LIMIT IN THE FOUNDATION OF GRAVITATIONAL RELATIVITY º
º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º
ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ¼
ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ¼
ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ»
º ±±±±±±±±±±±±±±± GENERAL INTRODUCTION for part 3 ±±±±±±±±±±±±±±± º
ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ¼
TABLE 4
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ KEY TERMS ³
³ ³
³ Mbh Mass of a black hole, having radius Rbh ³
³ ³
³ Mo An original mass (before mass augmentation ³
³ due to gravitational relativity) ³
³ ³
³ Ko Mass augmented upon mass Mo due to ³
³ gravitational relativity ³
³ ³
³ M An existing mass, which includes: Mo + Ko ³
³ ³
³ Mc A Critical Mass Limit, where Mc is an Mo ³
³ which is less than Mbh by precisely the ³
³ Golden Harmonic Ratio ³
³ ³
³ Rbh An event horizon radius for black hole Mbh, ³
³ and for other masses such as Mo, M, and Mc ³
³ which are evaluated with the same Rbh radius ³
³ but are not yet at the black hole mass limit. ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
TABLE 4 CONTINUED
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ ³
³ 1/Ng Ratio Mbh/Mc = 1/Ng when Mc = Mo, as when: ³
³ Mbh/Mo = 1/Nx ³
³ ³
ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
³ ³
³ GH Golden Harmonic Ratio 1.61803399, also called ³
³ Golden Ratio, having a digital value equal ³
³ to 1/2 the square root of 5, plus .5, as in: ³
³ ³
³ 1.1603398875 + .5 = 1.61803398875 ³
³ ³
ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´
³ ³
³ Vc A critical limit velocity in special ³
³ relativity, where the ratio C/Vc is equal ³
³ to the square root of the Golden Harmonic ³
³ ratio GH = 1.61803398875 ³
³ ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
ÛÄ´ FUNCTIONAL INTERPHASE BETWEEN ³
³ GRAVITATIONAL AND SPECIAL RELATIVITY ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
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.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ QUESTION ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
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?
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ ANSWER ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
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.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ INTERPRETATION ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
The problem is that the conundrum is only apparent and not real;
that: as a mass aggregate rapidly approaches its black hole limit,
the ensuing special relativity mass increase counterpart will rapidly
begin to climb toward infinity, and such an infinite mass is not
possible in the sense of real events.
For instance, assuming the conundrum is real, in the following
thoughts let Rbh be a given radius. Let's say a mass aggregate M
of radius Rbh is at 99% of the Mbh black hole mass limit for radius
Rbh. The gravitational relativistic effect (Eg) is roughly about
Eg = .09950, which translates into a special relativistic mass
enhancement effect of roughly (10.049 x M) on the mass travelling
at roughly (root 99%) of the speed of light).
Effect Es = 10.049 is reciprocally equivalent to effect Eg = .09950.
The problem here is that the special relativistic enhancement
on the mass will be roughly 10 times the black hole limit for
the mass in question.
The problem here is also that if mass M is increased by a
gravitational relativistic effect Eg of 10.049, then the
resulting augmented mass will exceed its own black hole limit
by a factor of roughly 10 times.
How, then, does an aggregate mass M of radius Rbh increase
only to a black hole mass Mbh of radius Rbh, in keeping with
a committed tie-in to special relativity, without the moving
mass M impossibly increasing to infinity as the aggregate
mass M closes in on Mbh, and without the stationary mass
increasing wildly above its own black hole limit due to
its own gravitational relativity?
The question is a thought balloon which seems to go in
several directions. But actually has a unique answer.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ EXPLANATION ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
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.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ THE GOLDEN HARMONIC RATIO - A CRITICAL LIMIT ³
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
When ratio Ng is equal to the Golden Harmonic Ratio,
then several striking things happen. The Golden Harmonic
Ratio is 1.6180339. It is typically given as a number quantity
from (1/2 of root 5, plus .5).
Let the Golden Harmonic Ratio be GH. And so let Ng = GH.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ THE CRITICAL LIMIT in gravitational relativity
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
When Mbh/Mo is GH, a vital event occurs. The gravitational
effect Eg precisely turns out to be 1/GH (the reciprocal of
the Golden Harmonic Ratio).
And so mass (Mo x 1/Eg) = (Mo x 1/GH), which
precisely turns out to be mass Mbh. Effectively,
mass Mo leaps uphill to become mass Mbh in one
final single bump.
This is a box, where one thing specifically yields another. In
interpretation, a mass augmentation (Eg) on an original mass Mo,
raises the quantity of the original mass Mo to that of a black
hole mass Mbh, when ratio Ng = Mbh/Mo is precisely the Golden
Harmonic ratio GH.
In which case, in special relativity, when the original mass
Mo is moving at a velocity V which is root GH less than the
speed of light, the special relativistic effect Es increases
mass Mo to mass Mbh in a final single bump. In which case mass
Mbh becomes a black hole and disappears from sight, relative
to a stationary observer watching the mass move.
There is a locked in equality here. Explicitly, Mbh/GH is a
critical limit preceding mass Mbh, at which an original mass
Mo is raised to the black hole limit Mbh by the mass effect
of its own gravitational relativity. Let Mc be the critical
mass limit.
Effectively, it establishes that if gravitational relativity
includes a mass augmentation effect, the original mass cannot
exceed the critical mass limit Mc. And so the original mass can
never be the same as a black hole mass, or even a fraction less
than a black hole mass, since the black hole mass includes an
original mass Mo at the critical mass limit Mc, raised to Mbh
through a quanta bump equal to the Golden Ratio GH.
In this locked in state, Mbh - Mc = Ko, where Ko is the
actual mass augmentation, the same as is otherwise said to
be Kx, except in this instance, Ko is fundamentally related
to the Golden Ratio GH. In exactitude, Ko = Mbh - (Mbh/GH).
It means that when the critical mass limit Mc is reached prior
to a black hole, the original mass Mo is augmented by effect 1/Eg
to become a black hole equivalent, and no more mass can confine
in the same radius Rbh. (More original mass added would serve to
increase the confining radius to greater than Rbh).
As already said, the Mc critical mass limit
(for radius Rbh) is simply (Mbh/GH), where
(GH) is the Golden Harmonic Ratio.
ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿
³ THE CRITICAL LIMIT in special relativity
ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ
It also means that in special relativity, when the critical
mass Mc is a rest mass in motion at a velocity equal to C
divided by the square root of GH, the original rest mass
Mc expands via 1/Es in a single bump to a mass value where
it also becomes a synonymous black hole of mass Mbh.
In consequence there never is a condition where the original
mass Mo in special relativity expands toward infinity as
mass Mo closes in on mass Mbh in gravitational relativity,
because the convergence in gravitational relativity for an
original mass Mo closes off completely at the critical mass
limit Mc, when Mc is less than mass Mbh by a ratio equal
to GH. This is a simple and elegant exclusion clause here
in the realms of the two modes of relativity, gravitational
and special.
EQUATION Z
In gravitational relativity, the critical limit is:
Mo = Mc = Mbh/GH
Where: Eg is the gravitational relativistic effect of Mc
Such that: Eg = 1/GH
And Mbh = Mc + Ko, where Ko = (Mc x 1/Eg) - Mc
And also: Mc x 1/Eg = Mk, and Mk - Mc = Ko
And so: Mbh = Mc x 1/Eg = Mk
Only when: Mc = Mbh/GH
So that: Mbh = Mk
Where Mk an apparent mass equals its own black hole silent partner
mass equivalent. This physical condition occurs because the Golden
Ratio GH constantly defines Mo as Mbh/GH.
EQUATION Z-1
In special relativity, there is a companion critical
velocity limit Vc for velocity V, where Vc is the speed
of light divided by the square root of the Golden Harmonic,
such that a critical velocity limit Vc constantly exists
for mass Mc, when C is the speed of light, as in:
Vc = (C / root GH) ;
where Vc is actually:
Vc = (C / root (Mbh/Mc)) or also (C / root GH)
when: Mc = Mbh/GH or also GH = Mbh/Mc
so that when: Mc is travelling at velocity Vc
the special relativity effect is: Es
and the special relativity effect 1/Es increases
rest mass Mc to black hole mass Mbh in a bump
because Eg is equivalent to 1/GH .
-- Continued in RELATIVE.3 --
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