INTERFACE RELATIVITY Interface between special and general relativity PLEASE NOTE: This document has ascii characters. It reads fine at Dos or in any compatable editor but not in Windows. In Windows the ascii characters are substituted by Ansi surrogates such as capital letters and strange looking symbols. In a Windows Browser you cannot read this document's equations, since the strange codes surround and change the equational content's appearance too drastically. The best view is in any Dos environment such as an editor, or by using the enclosed Viewer.com at the Dos prompt. This file is written in Dos ASCII format. A Dos viewer has been included for your convenience. To use it, shell out to Dos, and at the Dos prompt type: VIEWER RELATIVE.TXT To download the ascii reader utility, on the internet, type: http://cosmicastronomy.com/viewer.com A helpful sonic stereo experiment reports contains possible insights casting more light on Interface Relativity, see the reports here: ABSTRACT: A step toward unifying the forces. The gravitational force, and electro-magnetic forces (strong, electro-weak, and charge) are unified through Interface Relativity equations which join general and special relativitist effects into a fundamental equality. The interface specifies the principles and properties by which the general (gravity) and special relativity (radiation) modes are unified into a single set of of coherent balanced equations. ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± ±±±±±±±±±±±± ±±±±±±±±±±±± ±±±±±±±±±±±± INTRODUCTION TO MASS INCREASES BY ±±±±±±±±±±±± ±±±±±±±±±±±± GRAVITATIONAL RELATIVITY ±±±±±±±±±±±± ±±±±±±±±±±±± ±±±±±±±±±±±± ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± The following proposes that steady state relativistic effects can be understood to occur pursuent to gravitational fields. The wider range of distortions in space embraced by the GENERAL THEORY OF RELATIVITY are put aside and certain specific effects are studied in detail. These specific effects are understood to come under the heading of GRAVITATIONAL RELATIVISTIC EFFECTS. Greydon Moore Ottawa, Canada, June, 1990. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±± GRAVITATIONAL RELATIVITY THEORY ±±±±±±±±±±±± º º CONNECTS CERTAIN SOLAR PLANET MASSES. º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ͹ º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ͹ º ALSO, GRAVITATIONAL AND SPECIAL RELATIVITY THEORIES º º ARE INTRINSICALLY RELATED º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ By assuming a mass and spacial effect in general relativity, a proposed gravitation relativity is evident, in which there is a direct tie-in between effects seen in Special Relativity and in Gravitational Relativity. In fact, properties commonly factored for a star or black hole in Gravitational Relativity, can also be factored in Special Relativity, and visa versa. This suggests not necessarily a unified field theory, but definately a connection betweeen certain properties in gravity, and in electro-magnetism. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ º ABSTRACT º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ Several facets are to be discussed in the following. (Part 1) Arguments demonstrating an increase in mass by the effects of gravitational relativity are shown through events which occur in the solar system. (Part 2) Effects for gravitational and special relativity are shown to be synonymous for a given mass. Critical limits are uncovered in the behaviors of both relativities. In specific situations, mass is locked to a ceiling which is less than, but is determined from, black hole mass equivalents. In this, it is found that the maximum original mass which can be gathered before gravitational relativistic effects are maximized, is that of a black hole's mass divided by a factor of 1.618034 (a number constant known as the Golden Harmonic Ratio). The maximum velocity attainable by this mass when moving in special relativity, is the speed of light divided by the Golden Harmonic Ratio. (Part 3) It is found that for any visible mass, there is a maximum special relativistic limit on the mass. This limit can be known in advance by knowing the maximum velocity the moving mass can attain and still remain visible in the normal sense, when observed by a stationary observer. The maximum effect is a derivative of the speed of light reduced by the relativistic effect of the mass's gravity. This is shown to define an upper limit velocity at which any given mass can appear in the same state of the universe as the stationary observer. Any rest mass reaches this barrier at a plateau that is predictable, and so the mass cannot visibly expand to infinity. (Part 4) Innuendos of a unified field theory are harking loudly, popping out of the framework of relativistic physic. There is a universality in obvious behaviors working directly between the one field's venues (gravity) and the other field's venues (electromagnetism). As to whether these equalities can constitute segments of a full fledged unified field theory is not to be addressed at this time, in the scope of the following disclosures. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º part 1 ±±±±±±±±±±±±±±±± GRAVITATIONAL RELATIVITY ±±±±±±±±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ A little known (entirely unknown) fact is that certain solar planetary masses can be connected as a direct consequence of gravitational relativity. This is shown to be true when it is surmised that relativistic effects of gravity may include an intrinsic increase in the mass comprising the source of the gravity. The relativistic increase for the Sun mass is very small compared to the mass of the Sun itself. Even though the increase in mass is small at roughly 4.23 x 10 to the power 27 grms, the increase is nevertheless nearly 7 times the mass of Mars, and is marginally less than the mass of Venus. Such an increase in the Sun mass, when calculated to advanced accuracy, is found to be exactly equal to the mass difference between Venus and Mars. Another discrete relativistic potential includes 1/2 the mass of Jupiter added to the mass of the Sun. The existence of states makes it possible to infer a more accurate estimate for the existing mass of the Sun. The radius of the Sun is considered to be a constant for various manifestations, shown to correspond to parameters which operate between solar mass equivalents up to the masses of black holes. In this, a link between gravitational and special relativity is shown. The link is the subject of part 2. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º part 2 ±±±±±±±±±±±±±±±±±±±± SPECIAL RELATIVITY ±±±±±±±±±±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ It can be easily demonstrated that a visible mass moving at velocities nearing the speed of light, can never grow to infinite quantities and remain visible in the normal sense, and so can never achieve a velocity equal to the speed of light, in the normal sense. This is because gravitational relativistic effects have to be considered for a moving mass, if it is assumed that gravitational relativity includes an effect that increases the original state of the mass which is the source of the gravity's relativistic effect. It is readily shown that such gravity effect has significance to special relativity. There is a boxed in limit, where the moving mass (bumped in value in special relativity) assumes a value equivalent to the mass of a black hole, when the original rest mass is expanded by the effect of special relativity, in direct accord with the mass's radius contracted by the effect of special relativity. When assuming the mass of a black hole equivalent, the moving mass effectively drops from sight in the normal physical view as seen by a stationary observer. (See Appendix A at the end of this document, for a related discussion involving elementary particles such as the proton). One of the finite limits to which a mass can be accelerated in special relativity, and to which a mass can be accumulated in gravitational relativity, can be explicitly expressed for both modes of relativity as factors of a number constant known as the Golden Harmonic Ratio, 1.61803398875 . In this, the Golden Ratio's significance is to the existence of black holes. Specifically, a black hole's mass includes both an original mass and an augmentive portion from the relativistic effect of gravity, to comprise the total mass involved. The relationship between original, gained, and final black hole mass aggregations, can be expressed in exact terms of the Golden Harmonic ratio. In particular, however, in the dynamic behaviors of both relativities, important boundaries are reached at a certain critical limit whose mathematical significance is the Golden Harmonic Ratio. The parameters here include a black hole's mass aggregate and event horizon. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º part 3 ±±±±±±±±±±±±±±± THE GOLDEN HARMONIC RATIO ±±±±±±±±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ The effects of gravitational relativity can be generally related to the effects of special relativity, to the extent that relativity effects of gravity and of special relativity can be shown to be equated through a single common factor. The maximum velocity attainable by a visible moving mass, is the speed of light reduced by the proportionate effect of the gravitational relativistic effect in the mass being accelerated. The critical limit (maximum velocity) possible, is restricted by bounds achieved in special relativistic effect when the rest mass has increased, and radius has contracted, to a point where the moving entity reaches a state where it forms a black hole and effectively disappears from view, relative to a stationary observer. The barrier limit is easy to calculate and to mathematically confirm, when given the original rest mass and radius. It becomes clear that, generally a visible mass accelerated to relativistic velocities cannot theoretically achieve an infinite mass, and the velocity can never theoretically equal the speed of light. The traditional interpreted statements in special relativity which say any visible mass continues to expand toward infinity, and the velocity continues to the speed of light, are in error about such things. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±± GRAVITATIONAL RELATIVITY THEORY ±±±±±±±±±±±±± º º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±± GENERAL INTRODUCTION for part 1 The Solar System ±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ In the following, the existing orbits of planets are not considered as terms, and all of the events are shown to occur as within a constant confinement radius which is the existing radius of the sun. A general relativistic equation is in common use for gravitational effects. Such an equation has been around in physics since 1916. Variations of the equation are also in common use. Given a known mass for instance, a Schwarzschild radius for that mass confined as a black hole can be immediately calculated. Conversely, given a radius, how much mass would be needed to be confined within that radius as a black hole can also be calculated. Such effects are a steady state system. It is the amount of mass within a specified radius which counts. The effects are constant per given mass and radius, since no outside velocity or acceleration is involved with the masses sitting stationary. The same is true for mass aggregates which are not a black hole, but which have mass sufficiently large, and a radius sufficiently small, for gravitational relativistic effects to be discernible. For stars the size of the Sun, for instance, there are discernible effects, even though they appear to be very slight at first sight. In a closer look, however, the slight effects can reveal many major properties in the fundamental relativistic behavior of gravity. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ GRAVITATIONAL RELATIVISTIC EFFECT ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ In principle, gravitational relativistic effects are calculated via the standard equation, for varying mass and radius, until a meeting point is reached at which the mass and radius correspond to the formal parameters of a black hole. In the standard equation, a term for the relativistic effect results, which has been mainly used to determine the slowing of time in closer vrs more distant proximities to the field generating the effect. The same term can be used to find out how much a gravitational mass's radius can further contract relativistically per given increase in mass, when assuming that gravity relativistically contracts its own confinement radius. The same term can be used to calculate the gravity's relativistic effect on its own mass. This term can be called E (for effect). The value of term E suddenly nose dives toward 0 when the mass is sufficiently large, due to a sudden relativistic upsurge in pull in the greater power of the gravity itself, at which point the existing mass becomes a so called black hole and the existing mass's radius no longer appears to contract, rather, it will begin to increase given further increases in mass. This mass and radius stabilization is considered a physical boundary called the Schwarzschild radius, or event horizon. The stabilization is discussed in 'A Comparison Between Gravitational And Special Relativity' (found directly under the 'General Introduction for Relativity' Part 2', below), and is formally described in Equations 3 to 5 in APPENDIX B at the end of this document. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ GENERAL MASS QUANTA EFFECT ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ In variations of the equations, when a quantity of mass is given and the radius containing it is also known, then a simple solution using term E can denote how much of a mass increase may occur in the mass, due to a relativistic augmentation by the mass's gravity. The augmentation can be conjectured to occur in two ways. Either a measured mass is naked (original with no relativistic augmentation), or is augmented (the measured mass includes the augmentation). Hence the augmentation can be conjectured to be in two modes; either a decrease upon the originating mass, or an increase. In keeping with special relativity effects, a mass increase in gravitational relativistic augmentation can be presumed with no difficulties. For instance the Sun (given its mass and radius) is surmised to have a visible radius which is marginally reduced by relativistic augmentation (shrunk), and so the Sun's apparent mass is also surmised to be marginally augmented (expanded) in a mass increase by an equivalent relative proportion. The problem is that such a conjecture (relativistic augment- ation in mass) is hard to prove, since it is not possible to actually separate a given mass from its gravity and so observe any change in the apparent mass, when the mass is compared with vrs without the relativity of the gravity. In which case, any evident mass augmentation will have to be learned by some secondary means. In this solar system such a means is provided mechanically, by the fact that the amount of solar mass augmentation is a meaningful quantity in company with the existing mass of some of the planets. The mass augmentation has a value which is in a quantum correspondence to the existing masses of Venus and Mars. This makes the mass augmentation clearly visible. The fact that the relativistic mass is involved with these planets (in relationship with small particles external from the Sun) is very curious. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ GRAVITATIONAL RELATIVISTIC EFFECTS ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ The standard equation for gravitational relativistic effect is described as follows: EQUATION A ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (Mass) E = ³ 1 Ä ÄÄÄÄÄÄÄ \³ Cý R The square root of ((1 - the product of 2 times the gravitational constant G, times a mass), divided by the radius of that mass times the speed of light squared), yields a gravitational relativistic effect factor, termed E. EQUATION B The radius of the mass times the reciprocal of the E factor, gives the originating radius of the mass, ie., before contraction of the radius by the mass's gravitational relativistic effect. ÚÄ Ä¿ ³ ÚÄ Ä¿ ³ Where Re is the ³ ³ 1 ³ ³ amount of space ³ R x ³ ÄÄÄ ³ ³ - R = Re by which the Sun's ³ ³ E ³ ³ radius is contracted ³ ÀÄ ÄÙ ³ by the relativity ÀÄ ÄÙ in the Sun's mass ÚÄ Ä¿ Ro is the original ÀÄ Ro ÄÙ radius before effect. R is the existing radius (the radius we see) which includes effect (Ro + Re) These (Equations A and B) are well known and nothing new has been so far stated. The relativistic collapse in the Sun's radius is very slight, hardly 1« kilometers. This is learned as the difference between the originating Sun radius Ro, minus the existing (augmented) radius R. The difference seems to be a remarkably close approximation of « the Schwarzschild radius needed for the Sun mass to be a black hole. However this is not surprising, in that the smaller the mass and/or the larger the radius, the closer the radius augmentation is to « the Schwarzschild radius. The 1/2 approximation grows closer, the less the mass aggregate is a black hole. In principle, with little mass and a large radius, there is very little augmentation. Conversely, a very small radius for the small mass is needed as the event horizon for the small mass to become a black hole. The point intended is that as the mass to radius ratio approaches the primes of a black hole, the rates of change due to gravitational relativistic effects climbs up a steepening gradient. At solar quantities, the effects are so slight as to be normally thought of as negligible. But this is not so. If for instance 1/2 the mass of JUPITER is added to that of the Sun, and this enhanced mass sum is regarded as being within the confines of the existing Sun radius, the relativistic mass augmentation effect when applied to the mass of the Sun minus 1/2 the mass of Jupiter, equals the previously noted congress involving Venus and Mars masses, (at the end of 'General Mass Quanta Effect', above). Such state arrays reveal a previously unsuspected property, of relativistic mass quantal arrangements displaced at long distance from the source generating the relativistic mass effect. A first suspicion is that: 'THERE IS AN INCOMPATIBILITY BETWEEN A GRAVITATIONAL FIELD AND THE RELATIVISTIC EFFECT IT GENERATES'. The appearance is that some aspect of the relativistic mass effect generated in a field of gravity, does not stay within the field generating it. In supposition, it appears that some relativistic component is expunged (externalized) from the originating field of gravity. In the case of our solar system's example, the masses of Venus and Mars, along with Jupiter, are external and yet relativistically tied to the Sun mass. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ ESTIMATED ACCURACY OF SOLAR MASSES ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ Masses in the solar system are traditionally published in two ways. A mass for each planet is given as a ratio between it and the mass of the Sun. Since comparative ratios can be inferred to considerable accuracy, the Sun to planet mass ratios for most of the planets are well known. On the other hand estimating the actual mass of a planet or the Sun in terms of (say) gram units, is not so easy, since there is no way of actually sitting a planet on a scale. For that matter, estimating the real mass of the Sun (in say grams) is also difficult since the Sun cannot be weighed on a scale. The problem is compounded in that in order to know a real weight (in grams) requires that the universal gravitational constant (G) be known to high accuracy, which it is not. Whereas determining the mass influences of one body on another, as a ratio, is easier since (G) is not a critical factor for the accuracy. For these reasons the real mass of (for instance) the Sun (in say grams) cannot be stated with great accuracy by ordinary measuring methods. The Sun's mass is currently given as somewhere between 1.989 x 10 to 33 grms, and 1.991 x 10 to 33 grms. Whereas planet masses are currently given in gram figures accurate to between 4 and 5 significant figures. The greater accuracy for planet masses is assisted by the fact that the planets tend to subtlety bounce each other around in orbit, and their bouncing can be closely watched. Whereas the Sun is hardly bounced by the less hardy influence of the planets. The Earth - Moon combination gives the best look at bouncing. But rigorous real weight analysis for the Earth is not so easy when tried, because both the Earth and Moon also subtlety bounce around as a unit. If the gram weight of the Earth (5.976 ñ .004 x 10 to 27 grms) is multiplied by the Sun to Earth mass ratio (332,995.9 ñ .4), then the Sun's gram weight results as (1.9899834 x 10 to 33 grms). This value is actually deemed low to a very minor degree for the equations which follow below. In the following, a Sun mass in the vacinity of (1.990993 x 10 to 33 grms) is explicitly inferred. Another problem in any advanced accuracy is inherent in the weak solar gravitational relativistic effects per se. Because the effect for solar mass quantities is so slight, there is a loss of some accuracy due to inherent truncation in doing the calculations. In the equations which follow, accuracy has been maintained to 13 significant digits, but inherent truncation results at the 7th significant digit of certain of the terms. Such truncation is diminished when dealing with larger masses confined within small radii. The truncation disappears completely when dealing right at the range of black hole masses. Hence, black hole limits can provide a tool for comparing calculations, to determine which calculations produce exactitudes and which produce close approximations only. This is actually more straightforward than it sounds. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ BASIC CONVENTIONS ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ In the following, the existing orbits of planets are not considered as terms. All of the events are shown to occur as within a constant confinement radius, which is the existing radius of the sun. For the sake of convenience, the mass of the Sun is shown as a standard term labeled (MM). In the following, the calculations are accomplished at an accuracy of 10 to the 13 significant digits. Zeros are used to fill gaps between available digits and the 13th significant digit. As already mentioned, some of the terms are accurate only to the 7th significant digit. In fact, some terms cut off at the 7th digit. For this reason, the highest maintained accuracy possible is very important. For the universal gravitational constant G, a recent revision having a digital value of 6.6720 x 10 to -8 is used. The speed of light C of the following value is used: 2.99792458 x 10 to 10 cms/sec. The radius of the Sun is used as a constant R, having the value 6.96265 x 10 to 10 cms. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ MASS CONVENTIONS ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ The following mass aggregates have been adopted as standards for the involved quantities. The high accuracy given them has been by the adjusting of repeated pure math experimental results until a semblance of coherency in the mass standards looked viable. The term 'aggregate mass' is used for denoting a mass (such as the Sun, plus or minus another mass (such as 1/2 the mass of Jupiter). 'Aggregate mass' is also used to denote any apparent mass, since the mass is assumed to include relativistic augmentation due to gravity. Hence, the original mass before augmentation is termed 'original mass', or 'originating mass'. K has been adopted as a term to explicitly denote the relativistic mass augmentation in the Sun's mass due to the Sun's gravity. In determining aggregate mass values, the value of MM for the Sun's apparent mass was first determined, based on an assumed equality that a so called K augmentation factor for the Sun mass is indeed the mass difference between planets Venus and Mars. Without doubt the real values for the mass aggregates (given in grms for instance) will marginally change depending on future adjustments of the universal gravitational constant, and perhaps sharper astronomy techniques. (For that matter, mass MM may not be the true real mass of the Sun. It may turn out that MM is the mass of the Sun ñ something else). It is anticipated that any such changes would nevertheless prove to continue to be coherent within the realms of the gravitational relativistic state equations which involve them. Several tables and basic equations follow. Following these, a discussion begins on how a mass of MM was inferred for the Sun, via gravitational relativistic effects. Table 1 which follows, lists important mass aggregations, and the highest resolved real mass values possible as used to explore their relativistic highlights. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ º INFERRING A GRAVITIONAL RELATIVISTIC º º AUGMENTED MASS VALUE FOR THE SUN º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ TABLE 1 INFERRED VALUES ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ ³ ³ MM = Existing Sun mass, presumed to include ³ ³ original mass plus mass augmentation K ³ ³ ³ ³ = 1.9909930 x 10 to 33 grms ³ ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´ ³ ³ ³ K = Gain in original mass of the Sun, the ³ ³ amount of relativistic augmentation ³ ³ due to the Sun's gravity ³ ³ ³ ³ = 4.226490 x 10 to 27 grms ³ ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´ ³ ³ ³ Mbh = Mass of a black hole having an event ³ ³ horizon equal to the Sun's radius R ³ ³ ³ ³ = 4.689536679 x 10 to 38 grms ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ TABLE 1-A ESTABLISHED VALUES ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ ³ ³ R = Existing Sun radius ³ ³ = 6.96265 x 10 to 10 cms ³ ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´ ³ ³ ³ C = Speed of light ³ ³ = 2.99792458 x 10 to 10 cms/sec ³ ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´ ³ ³ ³ G = Universal gravitational constant ³ ³ = 6.6720 x 10 to -8 cms3/grms secý ³ ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´ ³ ³ ³ CR = A physical constant for Mass/Radius ³ ³ ratio of a black hole ³ ³ = 6.735275620 x 10 to 27 grs/cm ³ ÃÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ´ ³ ³ ³ GH = Golden Harmonic Ratio ³ ³ = 1.61803398875 ³ ³ û = 1.272019649 ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ TABLE 2 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ ³ ³ Planetary masses - Data is from tables found at the ³ ³ back of the following reference: ³ ³ ³ ³ UNIVERSE by Don Dixon, Houghton Mifflin Co., ³ ³ Boston, 1981 ³ ³ ³ ³ Moon = .0735 x 10 to 27 grms ³ ³ ³ ³ Venus = 4.8683 x 10 to 27 grms ³ ³ Earth = 5.976 x 10 to 27 grms ³ ³ Mars = 6.4181 x 10 to 26 grms ³ ³ Jupiter = 1.901 x 10 to 30 grms ³ ³ ³ ³ Sun = 1.9888 x 10 to 33 grms ³ ³ ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ TABLE 3 Certain terms are used to generalize certain types of masses: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ ³ ³ Low mass - Masses in the range of those found ³ ³ in this solar system ³ ³ ³ ³ Enhanced mass - Solar mass aggregates other ³ ³ than the Sun, added or subtracted ³ ³ to the Sun mass ³ ³ ³ ³ - Specifically the mass of the ³ ³ Sun plus 1/2 Jupiter, and mass of ³ ³ the Sun minus 1/2 Jupiter, also mass ³ ³ of the Sun minus mass of Venus ³ ³ ³ ³ Higher mass - Mass of a black hole, and in mass ³ ³ range of a black hole ³ ³ ³ ³ - Specifically the mass for a ³ ³ black hole whose event horizon ³ ³ is the radius of the Sun ³ ³ ³ ³ ³ ³ Originating mass - Original mass accumulation without ³ ³ any relativistic augmentation ³ ³ ³ ³ Augmented mass - Existing mass assumed to include ³ ³ a change from the originating ³ ³ mass due to relativistic effect ³ ³ of gravity ³ ³ ³ ³ Existing mass - As physically measured, with ³ ³ any assumed augmentation present ³ ³ in the measurement ³ ³ ³ ³ Real mass - A real weight, in terms of a ³ ³ physical weight, for instance ³ ³ measured in grms as if weighed ³ ³ on a scale ³ ³ ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ Certain equations are used to generalize mass effects due to gravitational relativity. Certain term conventions are adopted for the sake of convenience in bookkeeping: EQUATION C Determining a relativistic effect factor Em for a mass aggregate, in particular the Sun: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (MM) Where MM is the mass Em = ³ 1 Ä ÄÄÄÄÄÄÄ of the Sun, and R is \³ Cý R the radius of the Sun EQUATION C-1 Determining how much mass augmentation relativistically occurs in the mass aggregate of the Sun: (MM) - ((MM) x Em) = Km Where K is the actual mass augmentation increased on the Sun's original mass due to gravity EQUATION C-2 Determining a relativistic effect factor for a mass aggregate, such as the Sun plus X, where X is anything: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (MM+X) Ex = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄ \³ Cý R EQUATION C-3 Determining how much mass augmentation relativistically occurs in a mass aggregate, such as the combined mass of the Sun + X , when both are confined in radius R : (MM+X) - ((MM+X) x Ex) = K+x EQUATION C-4 For example, determining a relativistic effect factor for such as the Sun plus 1/2 Jupiter combined: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (MM+1/2j) E+1/2j = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄ \³ Cý R EQUATION C-5 Determining how much mass augmentation relativistically occurs in a mass aggregate, such as the combined masses of the Sun and 1/2 Jupiter, when both are confined in radius R : (MM+1/2j) - ((MM+1/2j) x E+1/2j) = K+1/2j ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ º VERIFYING A MASS OF MM FOR THE SUN º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ An aggregate mass MM (being the mass of the Sun) found to have intrinsic relativistic consequences, can be easily verified. If starting with an estimated Sun mass, for instance; (1.989 x 10 to 33 grms); and assuming that the Sun mass is already relativistically augmented, the gravitational relativistic mass increase of a Sun mass of (1.989 x 10 to 33 grms) is found (using Equations C and C-1), to be slightly less than the mass difference between Venus and Mars. That is: Venus mass is 4.8683 x 10 to 27 grms Mars mass is .64181 x 10 to 27 grms Venus - Mars is 4.226490 x 10 to 27 grms whereas the mass augmentation Km of a Sun mass of (1.989 x 10 to 33 grms) is (4.218033 x 10 to 27 grms), which is low. If the Sun's mass is gradually increased, eventually a mass aggregate will be found, in which the relativistic mass augmentation K is precisely (Venus - Mars), that is: K = 4.226490 x 10 to 27 grms. The point of agreement occurs when the mass aggregate for the Sun MM is found to be (1.990993 x 10 to 33 gms). For instance, suppose arbitrary units of Neptune's mass are systematically added to a base mass of (1.989 x 10 to 33 grms). A break point will be reached. At + 18N units of Neptune's mass the relativistic augmentation (Km) of the aggregate mass will be marginally less than (Venus - Mars). And at + 19N units of Neptune's mass, the relativistic augmentation (Km) of the aggregate mass will be marginally more than (Venus - Mars). And so somewhere between (base + 18N) and (base + 19N) is a solar mass component whose resulting augmentation (K) is exactly equal to (Venus - Mars). The search can now be narrowed to (base + X), where (+ X) falls somewhere between (+ 18N and +19N). Fine tune fiddling back and forth using smaller and smaller increments for X, eventually closes in on a result for; (base + 18N + X) in which the relativistic mass augmentation from (base + 18N + X) when using Equation D below, equals (Venus - Mars) exactly. EQUATION D Where b is a base mass ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (1.989 x 10 to 33 grms) ³ 2G (b+X) E = ³ 1 Ä ÄÄÄÄÄÄÄÄ And so (b+X) - ((b+X) x E) = K, \³ Cý R and K = (Venus - Mars) exactly, when (b + X) is exactly (1.990993 x 10 to 33 grms) EQ D can be written so that (b+X) is standardized as MM, so that: EQUATION E Where MM is an inferred Sun ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ mass, so MM - ((MM) x Em) = K ³ 2G MM where K = (Venus - Mars), Em = ³ 1 Ä ÄÄÄÄÄ and Em is the relativistic \³ Cý R effect factor for mass MM In other words the inferred Sun mass MM presents a solar mass factor whose relativistic gravitational augmentation (K) is exactly equal to the mass difference between Venus and Mars. That is: Equation E determines Em and: MM - ((MM) x Em) = K and: K = 4.226490 x 10 to 27 grms which is precisely (Venus - Mars) which also is: 4.226490 x 10 to 27 grms This instantly presents an interesting situation. The inferred mass of the Sun MM appears to involve a relativistic gravitational mass amalgamation which is greater than the mass of the Sun alone. The interesting kink is that the masses of Venus and Mars are found expunged into space, at long distance orbits around the Sun. This orbital existence is not explained at this point and so is noted only as a comment. The other interesting point of view is that although the mass of Mars for instance is very small compared to the mass of the Sun, the mass of Mars is nonetheless highly visible. This is something like the high visibility of the electron's tiny binding energy unit in comparison to the mass of the Proton. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ º SPECIFIC MASS QUANTA EFFECT º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ As described under 'A Comparison Between Gravitational And Special Relativity' (found directly under the 'General Introduction for Part 2', below), gravitational relativity includes at least two variable source terms for its effect. These source terms are the aggregate mass, and the mass's confining radius. It means that different quantities of mass can be said to occupy the same area. In which case there can be (in result) different or identical relativistic mass augmentations, depending on discrete combinations of how much mass is said to be added or subtracted to the initial mass aggregate, confined in the same or in different radii. For instance in mass aggregates which are in the range of the size of the Sun, here, discrete extra mass in the same radius (the Sun's radius) can produce a relativistic factor Ex which when arbitrarily applied to yet another discretely different mass aggregate, can produce a K augmentation which is otherwise gained from yet another different mass aggregate. For instance, the Sun mass MM, plus 1/2 the mass of Jupiter, can provide via EQ C-2 an effect factor (E+1/2j) which when applied to the same mass aggregate, via EQ C-3, results in K+j . But if E+1/2j is applied to a different mass aggregate, for instance to MM-1/2j, a value slightly departed from K+j must result. The resulting slightly lower value in fact once again happens to be K exactly (the mass difference between Venus and Mars). The formal description for this enhanced mass state is: EQUATION E-1 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ (MM+1/2j) is the ³ 2G (MM+1/2j) aggregate of the Sun E+1/2j = ³ 1 Ä ÄÄÄÄÄÄÄÄÄÄÄ mass plus 1/2 the mass of \³ Cý R Jupiter, confined in the existing Sun radius R EQUATION E-2 (MM-1/2j) - ((MM-1/2j) x E+1/2j) = K where K equals the mass of (Venus - Mars), and (E+1/2j) is the relativistic effect of the slightly denser aggregate of the inferred Sun mass MM plus 1/2 the mass of Jupiter, when confined in the Sun's radius R. In keeping with state-like mass aggregates, if EQ E-1 is rewritten so that the initial mass aggregate used in EQ E-1 is now MM-1/2j, and a resulting effect (called E-1/2j) is used in a rewritten form of EQ E-2, then a relativistic mass augmentation equal to K once again results; that is: EQUATION E-3 (MM+1/2j) - ((MM+1/2j) x E-1/2j) = K where K equals the mass of (Venus - Mars). EQUATION E-4 The bifurcation of Jupiter mass around the mass of the Sun to form coherent relativistic states can be generalized as: E+1/2j of mass M+1/2j applied to M-1/2j yields K Em of mass MM applied to MM yields K E-1/2j of mass M-1/2j applied to M+1/2j yields K EQUATION E-5 Such a bifurcation around the mass of the Sun can be generalized as: E+x of mass M+x applied to M-x yields Kx E of mass M applied to M yields Kx E-x of mass M-x applied to M+x yields Kx However, the augmentation quantity Kx only equals known augmentation value K, when M+x and M-x are specifically MM+1/2j, and MM-1/2j. That is, when 1/2 quantas of Jupiter's mass are added, and subtracted, to the inferred mass MM of the Sun. (It should be noted that the bifurcation results of EQ E-4 are not perfect exactitudes. The three resulting values of K happen to look the same for masses in the range of this solar system. For higher mass densities for example MM times 1000, confined in the same radius R, the three K values (shown as Kx in EQ E-5) are noticeably separated). ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ VERIFYING THE COHERENT 1/2j STATES ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ Equations E-1, E-2, E-3, and E-4, were not easily found without a prior insight and a discovery. In question is how come a unit of 1/2 the mass of Jupiter has been arbitrarily used to arrive at a seeming non arbitrary result, this result being where K is twice again calculated, as summarized in Equation E-4. An original intention was to see if the total mass of the solar system could be inferred to be in any way involved in some sort of interphasing between different mass aggregates in this solar system's gravitational relativity. This thought itself came from an original impression that the real mass of the Sun was in the range of base (1.9891 x 10 to 33 grms), and inferred mass MM would be the real Sun mass (base) plus Jupiter's mass, since (MM - base) closes in on an excellent approximation of Jupiter's real mass at (1.901 x 10 to 30 grms), when using EQ D to infer mass MM. For a while it was looking good. It seemed that if MM was the mass of the (Sun + Jupiter), and a mass value just slightly larger than the total mass of the solar system was substituted in EQ C-2, then a mass augmentation of K was again found when the factor Ex of EQ C-2 was substituted in EQ C-3, when Jupiter's mass was subtracted from the solar total mass aggregate and the result of this reduction substituted for MM+X in EQ C-3. In the exploration, a mass term Mt was adopted for the solar mass total, plus some little extra, to give mass term Mtx. And mass term Mtx-j denoted the solar total minus the mass of Jupiter. The value of Mtx could be rigorously inferred, as being exactly the mass aggregate needed in EQ C-2 to result in a mass augmentation effect equal to K in EQ C-3, when mass aggregate Mtx gave augmentation effect Etx, which was used to find the augmenting effect on mass Mtx-j, as in: EQUATION F ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G Mtx Etx = ³ 1 Ä ÄÄÄÄÄÄ \³ Cý R and a mass aggregate of (Mtx - Jupiter) was substituted in EQ C-3, giving: EQUATION G (Mtx-j) - ((Mtx-j) x Etx) = K In other words, the thinking was heading along a line that a sort of formal relativistic interphasing might be occurring, whose boundary was spread between the base mass of the Sun, and the total mass of the solar system. For instance between the Sun, and (Sun + Jupiter), and (Sun + planets + moons), and (Sun + planets + moons - Jupiter). The problem was in that little extra mass bit, (the x of Mtx). What might it represent? It was suddenly and unexpectedly found that the value of Mtx as rigorously inferred, turned out to be exactly (MM + 1/2 Jupiter). This was not a percentage of error type of equality. The figures that suddenly appeared on hand were identical to 8 significant digits. In other words, the rigorously determined value for Mtx, and MM+1/2j, were identical to 8 significant figures. Which dramatically changed the picture. It was now easy to think that MM instead of being a (Sun mass + Jupiter) aggregate, represented the real mass of the Sun itself. In other words, MM could well be the real mass of the Sun. It was also easy to perceive a formal verification for the quanta bifurcation factor involving 1/2 the mass of Jupiter. By using Equations F and G to find a result equal to K, a mass quanta increment of (+X) added upon MM eventuates in an interphase involving (MM-X) for the K result, only when X is exactly 1/2 Jupiter, when using the same inferencing technique as was used to infer MM in the first place, as described above under 'Verifying a Mass of MM For The Sun'. A slightly more accurate inferencing for MM itself was thus made possible. In order for Equations E-1 to E-4 to yield results definitely equal to K, the value of MM is adjusted to the greater accuracy of (1.99099305 x 10 to the 33 grms). It made the explorations involving solar mass total aggregates Mt and Mtx not important. This avenue of reasoning was dropped, and is mentioned above only to reveal how a quantal value of ñ 1/2 Jupiter as displayed in Equations E-1 to E-4 came to be an issue. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ OTHER MASS AGGREGATE STATES ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ In applying such interphasing logic to the solar system, the study is narrowed to include only mass quantities which currently exist; these being the Sun, and certain planets. In the case of a bifurcated Jupiter mass, a theoretical attribute is identified. This is where mass aggregates and resulting gravitational relativistic effects can phase in and out (in a continuation of certain coherent effects), through a range of mass densities confined within a single constant radius. A form of harmonic interphasing through a realm of masses is definitely sensed. In gist; a higher relativistic effect from an enhanced mass aggregate is applied to a lower mass aggregate, such that the resulting augmentation is lower or different than would be expected for either the originating enhanced mass, or the reduced mass. This type of reasoning should only be speculative, except that the mass augmentation which actually results when +1/2 Jupiter and -1/2 Jupiter are involved, is already a recognized quantity, this being mass term K, already independently seen for a mass aggregate which is other than an effect that is expected straight across for an enhanced or diminished sum of the Sun plus or minus 1/2 Jupiter. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ OTHER MASS EFFECT COHERENCIES ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ Other mass effect coherencies seem to occur. One involves the mass of the Earth (Me), which, when subtracted from mass MM, yields an aggregate mass whose relativistic effect factor (herein called Ee), which when applied to mass aggregate MM, results in a discrete mass split which is precisely equal to the mass of the Earth Me minus K. This formula (as exemplified in EQ H and I below), might at first seem tautological until further studies show that a relativistic factor Ex for any mass aggregate (M + X) or (M - X) does not phase in perfectly to an exact result for (MM - (MM x Ex)) = X - Kx for any value assumed for mass X. Only certain precise values of ñ X are seemingly phased in a coherency. For instance when: 1. X equals the mass of Earth 2. X equals the mass of Venus 3. X equals ñ 1/2 the mass of Jupiter The case of X being equal to ñ 1/2 the mass of Jupiter has already been demonstrated in Equations E-1 to E-4. When X equals the mass of Venus, then a mass split resulting in a discrete relativistic augmentation, also incorporates the mass of Mars. This is shown further below in Equations Q to S. A formal description for the interphasing state involving the Earth is as follows: EQUATION H ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (MM-Me) Where (MM-Me) is mass MM Ee = ³ 1 Ä ÄÄÄÄÄÄÄÄÄ minus the mass of the Earth Me. \³ Cý R MM is the mass of the Sun EQUATION I MM - ((MM + Me) x Ee) = Me - K Where Me is the mass of Earth, and K is (Venus - Mars) This formula (as exemplified in EQ I), might at first seem exciting until it is recognized that it is rather a sort of strange tautology. That is, further exploration shows that a relativistic factor Ex for any low mass aggregates in the range available for this solar system, for instance (MM + X) or (MM - X), phases in to a seeming predictable result where: when Ex is determined as the relativistic effect factor for mass MM-X (for instance using EQ H), then: MM - ((MM+X) x Ex) = Xx = (X - K) where Xx = (X - K) results for any reasonable value assumed for mass X. But for higher masses (much beyond MM), the equality actually breaks down, demonstrating that there was no tautological equality to begin with. A formal description for showing the breakdown is: EQUATION J ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (M-X) Where (M-X) is mass M minus Ex = ³ 1 Ä ÄÄÄÄÄÄÄ any other mass X, and radius \³ Cý Rx Rx is the same for any values of (M-X), then: EQUATION K M - ((M) x Ex) = Kx And: EQUATION L M - ((M+X) x Ex) = Xx And: EQUATION M Xx - X = Kx Where: Xx + Kx = X And: Xx = X - Kx Where X is the original arbitrary mass that was subtracted from M in EQ J, and was then added to M in EQ L ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ 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 effect 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. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±± PART 2 ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º º ±±±±±±±±± GRAVITATIONAL AND SPECIAL RELATIVITY THEORY ±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±± GENERAL INTRODUCTION for part 2 ±±±±±±±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ 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. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ³ A UNISON BETWEEN GRAVITATIONAL AND SPECIAL RELATIVITY ³ ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ 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 ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±± GENERAL INTRODUCTION for part 3 ±±±±±±±±±±±±±±± º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ 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 . ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ A TEST CASE: ³ ÛÄ´ GOLDEN HARMONIC RATIO IN THE TWO MODES OF RELATIVITY ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ For Gravity relativity ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ EQUATION Z-2 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G (Mo) Mo is an original mass Eg = ³ 1 Ä ÄÄÄÄÄÄÄ before augmentation \³ Cý 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 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G Mk Eg = ³ 1 Ä ÄÄÄÄÄÄ Mk is black hole mass with ³ GH horizon radius Rbh, and GH is ³ ÄÄÄÄÄÄÄÄÄÄÄÄ the Golden Harmonic Ratio equal \³ Cý Rbh to the number 1.61803398875 EQUATION Z-5-1 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ Mass Mbh is the same as mass ³ 2G Mbh aggregate Mk. Eg = ³ 1 Ä ÄÄÄÄÄÄ ³ Ng Ng is ratio Nx when the value ³ ÄÄÄÄÄÄÄÄÄÄÄÄ of Nx is GH, which is the \³ Cý Rbh Golden Harmonic Ratio EQUATION Z-6 With digits substituted for GH, then: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ 2G Mbh Eg = .61803398875 = ³ 1 Ä ÄÄÄÄÄÄ = 1 ³ 1.61803398875 ÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄ 1.61803398875 \³ Cý Rbh EQUATION Z-7 because: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ When and only when Nx = GH. 1 ³ 1 The Golden Ratio contains ÄÄÄ = ³ 1 Ä ÄÄÄ this self appreciating Nx \³ Nx mathematical property and so: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 1 ³ 1 GH is the Golden Ratio ÄÄÄ = ³ 1 Ä ÄÄÄ 1.61803398875 GH \³ GH ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ For Special relativity ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ EQUATION Z-8 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ÚÄ Ä¿ý ³ (Vc)ý Es = ³ ³ C ³ = ³ 1 Ä ÄÄÄÄÄÄ ³ 1 Ä ³ ÄÄÄÄÄÄÄÄ ³ \³ cý ³ ³ ÚÄÄÄÄ ³ ³ ³ \³ Nx ³ ³ ÀÄ ÄÙ ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ \³ Cý EQUATION Z-9 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ÚÄ Ä¿ý ³ (Vc)ý Es = ³ ³ C ³ = ³ 1 Ä ÄÄÄÄÄÄ ³ 1 Ä ³ ÄÄÄÄÄÄÄÄ ³ \³ cý ³ ³ ÚÄÄÄÄ ³ ³ ³ \³ GH ³ ³ ÀÄ ÄÙ ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄ \³ Cý 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: ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ÚÄ Ä¿ý .61803398875 = ³ ³ C ³ ³ 1 Ä ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ³ ³ \³ 1.61803398875 ³ ³ ÀÄ ÄÙ ³ ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ \³ Cý ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ FOR SPECIAL RELATIVITY EFFECT ON BOTH MASS AND RADIUS ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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 ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ ³ ÚÄ Ä¿ý ³ (Vbh)ý Es = ³ ³ C ³ = ³ 1 Ä ÄÄÄÄÄÄ ³ 1 Ä ³ ÄÄÄÄ ³ \³ cý ³ ³ GH ³ ³ ÀÄ ÄÙ ³ ÄÄÄÄÄÄÄÄÄÄÄ \³ Cý Es turns out to be the reciprocal of the square root of the Golden Harmonic. That is; Es = (1/ûGH). It means that when a mass Mc is physically moving at velocity Vbh relative to a stationary observer, its radius Rbh contracts by (1/ûGH), as its rest mass Mc expands by (ûGH), with the result that a new black hole is formed, having a lesser mass equal to (Mc x ûGH), and a lesser radius equal to (Rbh x 1/ûGH). 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 ûGH. 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 ûGH = Mbh- x ûGH = 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). ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ A REMARK ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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'. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ÛÄ´ SUPPLEMENTAL REMARKS ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ Golden Harmonic Relativistic Effects on Mass Only ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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(û2 + 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. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ Golden Harmonic Relativistic Effects on Mass and Radius ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ How was this verified ? ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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: Cý Rbh- Finding mass Mbh- needed for a Mbh- = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ black hole whose Schwarzschild 2G radius is given as Rbh- C. Mbh- turned out to be mass Mbh / (1/ûGH) 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. ÉÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍ» º ±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±±± º ÌÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍËÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ º SPECIAL RELATIVISTIC EFFECTS ON ANY MASS AND RADIUS º ÈÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍÍͼ Only certain critical limit cases (for masses Mo and Mc = black hole mass Mbh/GH) have so far been considered. ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ QUESTIONS ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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 ? ÚÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ¿ ³ ANSWER ³ ÀÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄÙ 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 Cý R Finding mass Mbh needed for a Mbh = ÄÄÄÄÄÄÄÄÄÄÄÄÄÄÄ 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 Cý Where ratio Cý/2G is a constant, ÄÄÄ = ÄÄÄÄ 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 Cý/2G, if mass M+ and R- were relativistically altered sufficiently to form a new black hole. Ratio Cý/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 Cý/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