Update on Solar System Radial Distances Recall from previous papers that a "virtual lightspeed" value for each planet can be calculated as:

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1 Angular Momentum by Daniel Browning March 4, 20 Introduction This paper attempts to introduce angular momentum into the sidereal equations of Four Horses. At the same time, a discrepancy found in Four Horses will be reconciled. Recall that a discrepancy was found between the Ζ transformation result for five-halves space and that for three-halves space, for Mercury and Venus. An averaging technique involving radius measures for angular momentum will be used to reconcile this discrepancy. Update on Solar System Radial Distances Recall from previous papers that a "virtual lightspeed" value for each planet can be calculated as: x 4.5 x t 5/3 = n 2/5 = m x. 7 = 36, (Mercury) x 4.5 x t 5/3 = n 2/5 = m x. 7 = 53, (Venus) x 9 x t 5/6 = n 2/3 = 22, (Earth) x 9 x t 5/6 = n 2/3 = 85, (Mars) In Horsepower it was found that an accelerated expansion factor, or aef, could be found for Earth and Mars, that, keyed to the outer horizon of the outer system, is equivalent to the radial distance from the Sun to the respective planet. For example, ~= 92,634, (Earth) ~= 4,74,780.6 (Mars) However, this didn t work for Mercury and Venus. The fix is fairly obvious. Instead of solving Mercury and Venus with the two-fifths transformation, use the two-thirds transformation: x 9 x t 5/6 = n 2/3 = 47, (Mercury) x 9 x t 5/6 = n 2/3 = 88, (Venus) These results lead to the following aef values: ~= 35,849,50.36 (Mercury) ~= 66,974, (Venus) As with Earth and Mars, the values for Mercury and Venus are referenced from the outer horizon of the outer system. All of the solar system radial distances are still high,

2 2 quantitatively, but they are all high by the same amount and so the same scale factor would serve to correct the slight overshoot. The hope was that the scale factor was related to angular momentum. It was thought that the overshoot was due to the lack (or inclusion) of angular momentum in the aef equations. However, no easy or sensible solution could be found that rectified the overshoot. This did, though, lead to the following metric. The Angular Momentum Equation A simple equation for angular momentum, in cubed space, is: J = M 2π 2 v r 2 where M is the Einstein mass for the system in question, v is angular velocity or perhaps an orbital period, and r is a radial distance computed such that angular momentum is conserved, as it must be. The unknown is thus r, since angular momentum J is known. Velocity v is set to reflect the Einstein ratios between event horizons, with the largest value equal to the value of ω in the black string equations, which is 4: v rs = 4 v eh = v tp = where rs, eh, and tp stand for redshift, event horizon, and turning point horizons respectively (also called outer, event, and inner horizons). The default horizon ratios of 3.25, 2. 77, and.25 manifest as: 4 / 3.25 = x.25 =.44 This is all happening in cubed space, as required. The equation can be solved for radius r 2 by plugging in values to the equation: / r 2 = M 2π 2 v / J This is trivial, since all of the right-hand-side values are known. Solving for r To approximate the Mercury and Venus aef, values for mass, angular momentum, and for the event horizons are needed. The mass and angular momentum values are (from Four Horses):

3 3 System M J Compensating inner Canonical Compensating outer As well, horizon values and framework ratios are: Canonical a (outer) b (inner) c (ratio) i 744, , e 773, , o,087, , Compensating i 74, , e 780, , o,089, , The r values and aef approximations for the outer compensating system are: x 4 x 2π2 / = x = n/3 = α rs,089, / α rs = x = outer x.44 x 2π2 / = x = n/3 = α eh 780, / α eh = x = event x.28 x 2π2 / = x = n/3 = α tp 74, / α tp = x = inner The final square root has the effect of unsquaring the radial distance r. Note that the intermediate values are treated as harmonics i.e. they re scaled arbitrarily to achieve the final approximation. The actual magnitudes of these values will be discussed in the final section. The same thing can be done for the inner compensating system. The framework ratios are applied to velocity v x x 2π 2 / = x 902, / α rs = x = outer = n/3 = α rs

4 x x 2π 2 / = x 635, / α eh = x = event x x 2π 2 / = x 62, / α tp = x = inner = n/3 = α eh = n/3 = α tp Finally, the middle ground in terms of stellar evolution, i.e. the canonical framework, is also involved in this calculation. Since the inner and outer systems of the canonical framework all have the same framework ratios, and since the ratios between event horizons match the velocity formulations, the inner and outer systems and their horizons all have the same value. Only one horizon needs to be solved, as shown below x 4 x 2π2 x 0.85 = x = n/3 = α rs,087, / α rs = x = outer All of these values are aef approximations for Mercury and Venus. The next problem is how to use them. Integration by Parts Recall that the original impetus for using angular momentum values was to correct the five-halves Ζ transformation, which was a little higher than the three-halves transformation. In other words, angular momenta are being used to fine-tune the sidereal periods of the planets. One can think of orbital eccentricity as an offset from the perfect circle. However, the eccentricity may follow the same path over aeons, so it s not a fuzzy boundary. On the other hand, precession may, over the centuries, form a fuzzy boundary since the ellipse of the orbit rotates over itself and smears any coherent line. This freedom to smear over a hard orbital line or position is the essence of angular momentum. A quantum of action, in classical mechanics, is the freedom of motion which involves the product of a distance (or wavelength) and one or more momentum values. The effect is very tiny: it applies to electrons and other fundamental particles. This quantum of action is also called Planck s constant, h. To give credit to Jose Lemos, the author of the black string paper (see reference below), the terms J and M in the black string equations correspond to action, not to planet orbits. And yet, the principle is the same: a wave can t exist independently of motion. Elementary particles can t exist independently of energy and momentum. The motion of the planets must involve energy and momentum, which in turn involve mass, space, and time.

5 5 What s more, the effect of action in one part of a system is not independent of the effect of the action in all of the other parts of the system. Thus, one must think in terms of the integration of many parts. This is reminiscent of the calculus of Newton, which was invented at the same time as his classical mechanics. The more parts that are involved in the final approximation, the better the approximation. One can think of a curve defined by three points, which is a blocky curve, and a curve defined by thirty points. Obviously, the curve with thirty points will be closer to the true curve that the points approximate. The canonical system is essentially the midpoint of the world line of the evolving black hole system; its use as a midpoint is like a one-step integration. If additional intermediate systems were added to the evolving black hole multi-framework, the dynamics of the system could be modeled that much more precisely. Consider the diagram below: compensating outer canonical outer canonical inner compensating inner Each system, being part of the evolutionary curve, has an influence on the action for a particular event horizon or system. Let s say that the horizon being solved for is the outer horizon of the outer compensating system. One might set up a sum of parts across horizons for that system, such that the outer horizon has the greatest influence. A very simple division scheme can be set up by taking the number of influences plus one. For three horizons, this would be four divisions. Then: = x 2 3 = x 2 3 = 3 4 x 2 3 = = This is just a weighting scheme. For three divisions the weights are simply two-thirds and one-third. These are the parts of the integration.

6 6 Going back to the problem being solved for, which is the discrepancy between the threehalves and five-halves Ζ transformations, recall that the five-halves transformation for Mercury and Venus uses an aef. The aef for Mercury is: aef merc - = 2π 2 (oo 3 - o 3 ) oo SOL merc Since the angular momenta values are being used to approximate this value (and to improve on it), and since this value uses the outer horizons, then the parts of the integration should be weighted towards the outer horizon. The original formulation in Four Horses could itself be considered an integration by parts, since the left-hand-side is an equation of the inner horizons, and the left-hand-side is balanced by the application of the aef, which uses the outer horizons. The first task is to sum parts over the three horizons for each system. Thus the following sum of parts, 2 (outer) + 3 (event) + 6 (inner) is applied to the three horizons of each system. For the outer compensating system: 2 ( ) + 3 ( ) + 6 ( ) = A Likewise, the inner compensating system is integrated as: 2 ( ) + 3 ( ) + 6 ( ) = B Next, the question arises: should each system (inner and outer) of the canonical framework be counted separately, or just once, since they re all the same values? The fact that the canonical distances are all the same means that the canonical system, evolutionwise, is identical for both the inner and the outer systems. In other words, those systems cohere and don t need to be counted twice. Since all the values are the same, = C The numerical values for A, B, and C are summarized below. System Compensating outer Compensating inner Canonical (A) (B) (C)

7 7 The next question is: how should one further integrate the components A, B, and C? Since the original aef formulation for Mercury and Venus involves both compensating systems, outer and inner, and since the curve of gravitation is better defined with the additional datum of the canonical framework, then all of these values should be integrated. The outer horizons have already been integrated for each system: now the task is to equally integrate the systems. One can start with: 2 (comp in) + 3 (canon) + 6 (comp out) 2 ( ) + 3 ( ) + 6 ( ) = D This can be balanced with: 2 (comp out) + 3 (canon) + 6 (comp in) 2 ( ) + 3 ( ) + 6 ( ) = E The final approximation is the average of D and E: ( ) / 2 = This then replaces the original aef value for Mercury and Venus, Working backwards from the final values of the five-halves Ζ transformations for Mercury and Venus (see Four Horses): Mercury: 97, x / = 97, vs. 97, (goal) Venus: 4, x / = 4, vs. 4, (goal) The goal values are the three-halves Ζ transformation results. Which is Better: Three-Halves or Five-Halves? This question can be answered in a roundabout way. First, it s easy to be deceived by the seeming simplicity of the treatment of the previous section. Mathematically it makes sense, but what does it really mean? The radial component of an angular momentum equation, in unit space, mimics the accelerated expansion factor for Mercury and Venus. The use of this radial component, equivalent to freedom of action, instead of the aef, which is volumetric difference, comes very close to the values achieved without the aef, by using a three-halves formula.

8 8 If the aef approximations are squared they turn out to be lunar values, as shown in the table below. Horizon aef 2 (harmonic) outer : outer outer : event outer : inner inner : outer inner : event inner : inner canonical : any It may be a coincidence that the aef approximations are similar to the lunar period. In real life the lunar period isn t constant. It varies between and 27.6 days: a spread of 0.58 days. The aef spread shown above is 0.42 days, for comparison. The lunar period varies for several reasons. The lunar sidereal period doesn t quite match the lunar axial period. This causes libration, which is the difference in how much of the lunar limb can be seen from Earth. As well, the Moon s orbit is very slightly eccentric. One might think that the five-halves Ζ transformation, which requires an aef correction, is inferior. On the other hand, the three-halves transformation, which hides the aef correction, might be considered inferior. To determine the relative merits of the equations, an analysis can be performed that pinpoints the differences between the five-halves and three-halves Ζ transformations for Mercury and Venus. The differences apply equally to Mercury and Venus, and so only the Venus equations will be analyzed. The five-halves equation (Equation M, Four Horses) is: (oi 5 - i 5 ) 2π 2 T SOL Q 2 v Q e /4 The three-halves equation (Equation S) is: (oi 3 - i 3 ) 2π2 T 4 Q v 720 SOL One difference is: Also: Q e = /2 Q e /4 = = ξ

9 9 (oi 3 - i 3 ) /2 = 409,683,523.5 (oi 5 - i 5 ) /4 = 9,045, Factor: = α Another difference is: /2 = /4 = π 2 T SOL 2π 2 T SOL Factor: = β Putting it all together: α / β = = = µ µ ξ = ( ) - vs (average) This explains the difference in a numerical sense. In a philosophical sense, however, the numbers are meaningless. There is absolutely no indication that the three-halves transform is better than the five-halves, or vice versa. The Q values hold the key. They are defined as velocity values. The only difference between the five-halves equations for Earth and Mars and for Mercury and Venus are the Q ratios. As well, the equations for Earth and Mars don t use an accelerated expansion factor. The Q ratio for Mars (with the fourth-root applied) is simply: Q mars /4 The Q ratio for Earth is: Now: Q 2 /4 e Q mars Q 2 e = = n/4 = This is close to the angular momentum value for the inner compensating system: For Mars: Q mars = = n /8 =

10 0 This is close to the angular momentum value for the outer compensating system: This is an exciting discovery, because the Q values go back to the axial spin equations of Three Sheets to the Wind, and a general framework that covers both sidereal periods and axial periods is now given new life through angular momentum J. It also raises many questions. For example, why is the angular momentum for Mars and the outer system squared? A related question is: why does the equation for Earth use a ratio of angular momenta for the inner and outer compensating systems? Since the Earth Q is used in the five-halves equations for Mercury and Venus, a third question might be: what is the influence of the inner compensating system on Mercury and Venus? Since the five-halves equations expose angular momentum they are to be preferred. For Earth and Mars they are mandatory, since no equivalent three-halves Ζ equations exist for those planets. Magnitudes The next task is to try to resolve the magnitudes that were glossed over in the derivation of the approximations of the aef values for the various systems and horizons. Working with the canonical system for simplicity, and given: Then: v = 4, M = , J = 0.85 r 2 = = x = n/3 = To get the required ratio of = x = one really wants a magnitude of 394, , which can be called a pseudo horizon. In which case the related lunar value is:,087, / 394, = In cubed space,

11 394, = n 3 = x = x 0-7 = r 2 so that: r 2 x J / M / 2π2 = 4 x 0 8 = 2.5 x 0-9 = v Velocity v is very small. The magnitude of 0-5 is about the orbital period (in seconds) of an electron in a Bohr atom. The same magnitude applies (in meters) to the wavelength of a particle like a neutron or proton. This doesn t mean that we live inside of an atom. Rather, the magnitudes scale inversely between solar horizons and classical atoms. Conclusion An angular momentum equation for circular motion has been proposed. Its native environment is cubic space, where it complements mass M, velocity ω, and the horizon ratios. When solved to preserve angular momentum, a unique radius can be found for each horizon. When reduced to unit space through a cube root, this radius value, in conjunction with the precipitating horizon radius, approximates the accelerated expansion factor for Mercury and Venus, about.66. Thus, the angular momentum values for Mercury and Venus are significant in cubed space. However, the Q values used for Earth and Mars are applied in a fifth-power space, and the cubic values for J are only revealed when the fifth-power environment is reduced by a fourth-rooting the whole thing being the five-fourths Ζ transformation. Therefore the use of angular momentum for Earth and Mars, and for Mercury and Venus, differs. The bright side of things is that angular momentum does indeed fit into the solar equations, since the Q values for Earth and Mars are in fact angular momentum values. Angular momentum provides a measure of freedom of action such that orbits can have some amount of smearing or overlapping, or in other words, fuzziness. This concept is usually applied to atoms, using the Planck constant. But the black string equations show that the concept applies also to the solar system. The effect is very small, both at the atomic and at the cosmic level. Which is better? Three halves or five halves? The answer seems to be five-halves space, because the equations leave room for angular momentum. However, without the threehalves transformation there is no longer a cross-relation between the Ζ and Ζ2 transformations. (See Four Horses.) However, a cross-relation may turn up elsewhere. Symmetry, though pleasing, is not the be all and end all of mathematical endeavour.

12 2 References Ridley, B. K. Time, Space and Things, Peregrine / Penguin Books, 976 Lemos, Jose. Cylindrical Black Hole in General Relativity, Brazil Observatory, CERN preprint gr-qc/940404