旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � A TEST CASE: � 俳� GOLDEN HARMONIC RATIO IN THE TWO MODES OF RELATIVITY � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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: 旼컴컴컴컴컴컴컴컴컴컴컴커 � For Gravity relativity � 읕컴컴컴컴컴컴컴컴컴컴컴켸 EQUATION Z-2 旼컴컴컴컴컴컴컴컴컴컴컴 � 2G (Mo) Mo is an original mass Eg = � 1 � 컴컴컴� 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 旼컴컴컴컴컴컴컴컴컴컴컴 � 2G Mk Eg = � 1 � 컴컴컴 Mk is black hole mass with � GH horizon radius Rbh, and GH is � 컴컴컴컴컴컴 the Golden Harmonic Ratio equal \� C� Rbh to the number 1.61803398875 EQUATION Z-5-1 旼컴컴컴컴컴컴컴컴컴컴컴 Mass Mbh is the same as mass � 2G Mbh aggregate Mk. Eg = � 1 � 컴컴컴 � Ng Ng is ratio Nx when the value � 컴컴컴컴컴컴 of Nx is GH, which is the \� C� Rbh Golden Harmonic Ratio EQUATION Z-6 With digits substituted for GH, then: 旼컴컴컴컴컴컴컴컴컴컴컴 � 2G Mbh Eg = .61803398875 = � 1 � 컴컴컴 = 1 � 1.61803398875 컴컴컴컴컴컴� � 컴컴컴컴컴컴� 1.61803398875 \� C� Rbh EQUATION Z-7 because: 旼컴컴컴컴컴컴컴� When and only when Nx = GH. 1 � 1 The Golden Ratio contains 컴� = � 1 � 컴� this self appreciating Nx \� Nx mathematical property and so: 旼컴컴컴컴컴컴컴� 1 � 1 GH is the Golden Ratio 컴� = � 1 � 컴� 1.61803398875 GH \� GH 旼컴컴컴컴컴컴컴컴컴컴컴커 � For Special relativity � 읕컴컴컴컴컴컴컴컴컴컴컴켸 EQUATION Z-8 旼컴컴컴컴컴컴컴컴컴컴컴컴� 旼컴컴컴컴컴컴컴컴컴컴 � 旼 커� � (Vc)� Es = � � C � = � 1 � 컴컴컴 � 1 � � 컴컴컴컴 � \� c� � � 旼컴� � � � \� Nx � � 읕 켸 � 컴컴컴컴컴컴컴 \� C� EQUATION Z-9 旼컴컴컴컴컴컴컴컴컴컴컴컴� 旼컴컴컴컴컴컴컴컴컴컴 � 旼 커� � (Vc)� Es = � � C � = � 1 � 컴컴컴 � 1 � � 컴컴컴컴 � \� c� � � 旼컴� � � � \� GH � � 읕 켸 � 컴컴컴컴컴컴컴 \� 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: 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 旼 커� .61803398875 = � � C � � 1 � � 컴컴컴컴컴컴컴컴컴 � � � 旼컴컴컴컴컴컴� � � � \� 1.61803398875 � � 읕 켸 � 컴컴컴컴컴컴컴컴컴컴컴컴 \� C� 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 俳� FOR SPECIAL RELATIVITY EFFECT ON BOTH MASS AND RADIUS � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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 旼컴컴컴컴컴컴컴컴컴컴컴컴� 旼컴컴컴컴컴컴컴컴컴컴 � 旼 커� � (Vbh)� Es = � � C � = � 1 � 컴컴컴 � 1 � � 컴컴 � \� c� � � GH � � 읕 켸 � 컴컴컴컴컴� \� 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). 旼컴컴컴컴컴컴컴컴커 � A REMARK � 읕컴컴컴컴컴컴컴컴켸 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'. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 俳� SUPPLEMENTAL REMARKS � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Golden Harmonic Relativistic Effects on Mass Only � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Golden Harmonic Relativistic Effects on Mass and Radius � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴� � How was this verified ? � 읕컴컴컴컴컴컴컴컴컴컴컴컴� 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- = 컴컴컴컴컴컴컴� 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. �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� � 勁袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴箇袴袴袴袴袴袴袴袴� � SPECIAL RELATIVISTIC EFFECTS ON ANY MASS AND RADIUS � 훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� Only certain critical limit cases (for masses Mo and Mc = black hole mass Mbh/GH) have so far been considered. 旼컴컴컴컴컴컴컴컴컴� � QUESTIONS � 읕컴컴컴컴컴컴컴컴컴� 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 ? 旼컴컴컴컴컴컴컴커 � ANSWER � 읕컴컴컴컴컴컴컴켸 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 = 컴컴컴컴컴컴컴� 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, 컴� = 컴컴 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Finding a significant Velocity value, which results in ratio CR � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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' = 컴컴컴컴컴컴컴� 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 = 컴컴컴컴컴컴� 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 旼컴컴컴컴컴컴컴컴컴컴컴컴� 旼컴컴컴컴컴컴컴컴컴컴 � 旼 커� � (Vx)� Es = � � C � = � 1 � 컴컴컴 � 1 � � 컴컴 � \� C� � � Nx � � 읕 켸 � 컴컴컴컴컴� \� 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 旼컴컴컴컴컴컴컴컴컴컴컴컴� � 旼 커� 旼컴컴컴컴컴컴컴컴컴컴 � � C x Egs � � (Vx)� Ess = � � � = � 1 � 컴컴컴 � 1 � 컴컴컴컴컴컴컴� \� 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). 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � Finding that terms M+ and R- are properties of a black hole � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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' = 컴컴컴컴컴컴컴 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' = 컴컴컴컴컴컴컴� 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. 旼컴컴컴컴컴컴컴컴컴컴컴컴컴� 俳� SUMMARY EQUATIONS � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 旼컴컴컴컴컴컴컴컴컴 � 2G M M is any mass, R is its Egs = � 1 � 컴컴컴 radius, and Egs is the \� C� R gravitational relativistic effect of mass M EQUATION Z-16-1 旼컴컴컴컴컴컴컴컴컴컴컴컴� � 旼 커� 旼컴컴컴컴컴컴컴컴컴컴 � � C x Egs � � (Vx)� Ess = � 읕 켸 = � 1 � 컴컴컴 � 1 � 컴컴컴컴컴컴컴� \� 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 컴컴 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 旼컴컴컴컴컴컴컴 Ess = \� 1 - (Egs)� 旼컴컴컴컴컴컴컴 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. �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� � � UNIFIED EFFECTS IN FIELD BEHAVIOR � � 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� � 훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� 훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � 굇굇굇� GENERAL INTRODUCTION for part 4 Unified Fields 굇굇굇� � 훤袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� '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. �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� � 勁袴袴袴袴袴箇袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� � TEST CASE � 훤袴袴袴袴袴� 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 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � ARBITRARY STUDY MODEL DATA � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � � � Nx = 1.000544617404 = (P + E) / E � � Mbh = 4.689536679 x 10 to 38 grms � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � � � 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 � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � � � 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. � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 EQUATION Z-17 HIGH gravitational Mass1 results: 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 � 2G (4.686984066 x 10 to 38 grms) Egs = � 1 � 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� \� 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 旼컴컴컴컴컴커 HIGH gravity field effect Egs = � .023330687 � Egs is closing toward 0 읕컴컴컴컴컴켸 EQUATION Z-17-1 Electromagnetic field effect results (Ess is special relativistic effect) 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 旼 커� � � C x .023330687 � V� Ess = � � � = 컴컴 � 1 � 컴컴컴컴컴컴컴컴컴컴컴 C� \� C� .023330687 is effect Egs of EQ Z-17 Ess = 1 - (Egs)� As in: 1 - (.023330687)� = .999727802 旼컴컴컴컴컴커 LOW special field effect Ess = � .999727802 � Ess is closing toward 1 읕컴컴컴컴컴켸 V velocity is starting to close toward 0 EQUATION Z-17-2 LOW gravitational Mass2 results: 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴 � 2G (4.686984066 x 10 to 33 grms) Egs = � 1 � 컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� \� C� R Mass2 has been given in terms of a real weight. 旼컴컴컴컴컴커 LOW gravity field effect Egs = � .999995002 � Egs is closing toward 1 읕컴컴컴컴컴켸 EQUATION Z-17-3 Electromagnetic field effect results (Ess is special relativistic effect) 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� � 旼 커� � � C x .999995002 � V� Ess = � � � = 컴컴 � 1 � 컴컴컴컴컴컴컴컴컴컴컴 C� \� C� .999995002 is effect Egs of EQ Z-17-2 Ess = 1 - (Egs)� As in: 1 - (.999995002)� = .003161416 旼컴컴컴컴컴커 HIGH special field effect Ess = � .003161416 � Ess is closing toward 0 읕컴컴컴컴컴켸 V velocity is closing toward 1 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 俳� COMPARING M+ AND R- RESULTS FOR HIGH AND LOW MASSES � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴� 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 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � � � 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 � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 TABLE 6 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � � � 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 � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � 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. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � 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. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � 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. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � 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. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � (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). � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� � 勁袴袴袴袴袴袴袴袴袴袴藁袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� � FIRST INTERPRETATION � 훤袴袴袴袴袴袴袴袴袴袴暠 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. �袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴敲 � 굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇굇� � 勁袴袴袴袴袴袴袴袴袴袴袴箇袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴袴� � DIRECT INTERPRETATION � 훤袴袴袴袴袴袴袴袴袴袴袴� 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 旼컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴커 � HIGH mass gravity Effect Egs approaches 1 � � Effect Ess approaches 0 � � � � LOW mass gravity Effect Egs approaches 0 � � Effect Ess approaches 1 � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � 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. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � 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. � 쳐컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴캑 � � � 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. � � � 읕컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴컴켸 -- Continued in RELATIVE.4 -- Item D if you are using the HELP MENU