Fundamental Orbital to Escape Velocity Relationship. Copyright 2009 Joseph A. Rybczyk

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1 Fundamental Orbital to Escape Velocity Relationship Copyright 2009 Joseph A. Rybczyk Abstract Presented herein is the entire orbital velocity to escape velocity relationship recently elevated to the relativistic level by this author. Such relationship transcends its Newtonian/gravitational origins and shows itself to be fundamental to the relativistic principles underlying two inertial frames in motion relative to each other. 1

2 Fundamental Orbital to Escape Velocity Relationship Copyright 2009 Joseph A. Rybczyk 1. Introduction Extensive research by this author has shown in specific detail the underlying relationship between gravitational time dilation and inertial time dilation. This, in turn, led to new discoveries involving the underlying nature of relativistic escape velocity and its relationship to relativistic orbital velocity for circular orbits. That research has now been extended to its fundamental roots involving the relativistic principles between two inertial frames in relative motion to each other. 2. Review of Relativistic Orbital Velocity Derivation In the previous work by this author 1 it was shown how the Newtonian orbital velocity for circular orbits to escape velocity relationship given by 2 1 where 2 2 and 3 could be applied to the previously derived relativistic escape velocity formula to arrive at the relativistic orbital velocity formula 2 5 for circular orbits. In these formulas gravitational potential Φ when given is defined by 2

3 6 where G is the universal gravitational constant, M is the mass of the primary body, R is the radial distance to the center of the primary mass, or the radius of the circular orbit, and c is the speed of light. In the same previously mentioned work 1 also given was 7 where v oa was seen as a possible alternative to the orbital velocity formula given here as equation (5). The relationship between the preferred formula (5) for relativistic circular orbits, and the just given alternative formula (7) has now been fully resolved and will be treated in the remaining sections of this present work. 3. The Relativistic Relationship between Dual Inertial Frames To fully understand the relationship between the preferred formula (5) and the alternative formula (7) it is necessary to review the relativistic relationship between two different inertial frames in relative motion to each other as given next in Figure 1. Referring now to Figure 1 S C 1 S 2 ct ct A vt B Vt FIGURE 1 Dual Inertial Reference Frames assume that a light source gives off a brief pulse of light at point A in the stationary frame of reference as the source travels from left to right to its present location at point B in the stationary frame at a uniform rate of speed v during stationary frame time interval T. Further assume that a point of this light traveling in a perpendicular direction to the source s path of motion travels distance ct from point B to point C in the source s frame of reference in which the source serves as the point of origin. Distance ct then, is the distance light travels at speed c in the source s moving frame of reference during moving frame time interval t that corresponds to stationary frame time interval T. Since, according to the principles of relativity 3, the speed of light is constant at speed c and does not take on the speed of the source, this same point of light travels distance ct from point A to point C in the stationary frame of reference. Also according to the 3

4 principles of relativity, perpendicular distance ct between points B and C is the same in both the moving frame and the stationary frame. In the millennium theory of relativity 4 the spherical distance traveled by a pulse of light in all directions from a point source serves as the inertial reference frame for that pulse of light with the point of propagation at the center of the growing sphere being the point of origin for that particular frame of reference. More specifically, in the moving inertial frame of the source, depicted by the growing light propagation sphere S 2 centered on the source, the source (presently at stationary frame point B) serves as the point of origin, whereas in the stationary frame, the stationary frame point of propagation A at the center of growing stationary frame sphere S 1 serves as the point of origin. Thus, in viewing the relationships given in Figure 1, it can be seen that the moving frame distance ct between points B and C traveled by a point of the propagating light in the perpendicular direction relative to the source s path of travel is also the radius of the propagation sphere S 2. Likewise, this same point of propagating light travels distance ct between points A and C in the stationary frame and since A is the point of origin for stationary frame sphere S 1, distance ct is the radius of that sphere. As has been stated throughout the development of the millennium theory of relativity, the Pythagorean triangle that relates the two propagation spheres shown in Figure 1 is nothing more than a simplified version of the light clock of special relativity. Simplified in the sense that only half of the clock given in special relativity is used because that is all that is needed to define the relationships shown. Now, let us see what these relationships have to do with the relativistic relationship between orbital velocity and escape velocity. 4. Relativistic Speed Transformation By way of preparation for the analysis involving the relativistic orbital to escape velocity relationship we need to first discuss the millennium relativity principle of speed transformation. In accordance with the principles of millennium relativity, real distances are unaffected by relative motion. This simply means that at any instant in time a given distance between two points can be defined in terms of stationary frame values for speed and time, or moving frame values for speed and time. Thus, in terms of stationary frame values, the stationary frame distance traveled by the source between points A and B in Figure 1 can be expressed as distance vt where v is the stationary frame speed in which the distance was traveled by the source during stationary frame time interval T. If this same distance, at the exact same instant, is given in moving frame values for speed and time it could be stated as distance Vt where V is the greater moving frame value for the speed required to traveled that same distance during the dilated time interval t of the moving frame that corresponds to time interval T of the stationary frame. In accordance with the millennium theory mathematical convention 5, upper case letters are used to denote the greater of two corresponding values and lower case letters are used to denote the lesser of two such values. Since, by definition 4

5 8 where the relationship of time interval t to time interval T is given by the Lorentz transformation 9 using the millennium relativity version of the Lorentz transformation factor we can by way of substitution of the right side of equation (9) in place of t in equation (8) arrive at 10 that in turn can be solved for speed V in terms of stationary frame speed v and light speed c. This gives 11 that can alternately be expressed as 12 or 1 13 where in each case V is the speed of the source relative to the stationary frame using the moving frame time interval t in place of the stationary frame time interval T. The importance of these different renditions, (equations (11) to (13)), is in the fact that each gives insights that enable better understanding of the overall relationship between speeds v and V. If we now solve equation (13) for speed v, we obtain

6 giving 20 and 1 21 as two additional important equations involving the relationship between relativistic orbital velocity and escape velocity. In the interest of completeness, from equation (20) we can derive 22 giving 23 as a final version of the speed transformation equation. (Note: The derived transformation factor using transformation speed V in place of v is new to the millennium theory of relativity.) 6

7 5. Gravitational Potential Φ to Transformation Speed V Relationship If we now compare speed v of transformation equation (20) to speed v oa of equation (7), given earlier as an alternative formula for relativistic orbital velocity, we find that they are equal only when or when. Although this is obvious by visual inspection of the two formulas we can nonetheless show it mathematically. That is, if we equate the two equations to each other and solve for V we obtain 24 or giving 30 and 31 as the relationship between the gravitational potential Φ and transformation speed V. In brief, alternative orbital speed v oa equals source speed v of Figure 1 only when the relationship of Φ to V is as given in equations (30) and (31). And, by way of emphasis, that equality occurs only when 0 as will be discussed later in Section 7. Let us now review the 7

8 mathematical derivation of the author s previously derived formula for relativistic escape velocity since it will be needed to arrive at our final conclusion. 6. Relativistic Escape Velocity Derivation In previous works by this author 6 it was shown that the effect that gravity has on time dilation can be expressed mathematically by the gravitational time transformation formula 1 32 where T is the time at the lower gravitational potential (or farthest distance from a primary gravitating body), t is the corresponding dilated time at the higher gravitational potential (or closest distance to a primary gravitating body) and Φ is the difference in gravitational potential between those two points. By way of comparison, inertial time dilation, as given earlier by formula (9) in the present work (repeated below), is defined by 9 where T is the time in the stationary frame, t is the corresponding dilated time in the moving inertial frame, and v is the speed of the moving frame relative to the stationary frame. When equated together and solved for speed v, the corresponding transformation factors from formulas (9) and (32) as given by 1 33 yield

9 ending with 2 4 as given previously, or 2 40 as the formula for relativistic escape velocity v e where the subscript e is added to distinguish it from orbital velocity. This formula was initially derived by direct application of the principles of general relativity 7 and then further validated in a later work through an independent derivation process using relativistic forms of gravitational potential energy and kinetic energy The Relativistic Orbital Velocity to Escape Velocity Relationship In the preceding work 1 it was shown that the relativistic orbital velocity for circular orbits could be derived by simply applying the previously given orbital velocity to escape velocity relationship 2 1 using formula (4) for the relativistic value of escape velocity v e in place of the Newtonian value given by formula (2). This gave 2 5 9

10 as the correct formula for the relativistic orbital speed for circular orbits in place of the Newtonian value given by formula (3). Unfortunately, some confusion resulted with the introduction of the alternative formula 7 given next in that work and for that matter also in this present work. It is now clear, however, from the analysis culminating in Section 5 of this present work that equation (7) gives the correct value for orbital speed v o only when 30 as given previously in equation (30) restated above. Evaluation of the relationship of stationary frame speed v that corresponds to speed V given in formula (30) show that this happens only when or as otherwise stated, only when as is obvious from the equations of (30) and (41). The subsequent correlation of speed v o as given by equation (5) to speed v oa as given by equation (7) was thus indicated in the chart given in Figure 1 of the previous work. That is, their values were equal only at the two points indicated above by the equations of (41) and (42). The underlying nature of speed v oa was not fully understood at that time, however, as has now been accomplished in this present work. In the present work it is seen that the value of v oa as given in equation (7) is actually dependent on the value of transformation speed V as shown in the analysis of Section 5 and not directly on speed v as is the case with the value of v o given by equation (5). 8. Conclusion All of the analysis presented to date, including that just covered in this present work seems to verify that formula (5) given in the present work is the correct formula for relativistic orbital velocity for circular orbits. The relationship of the alternate formula (7) to the inertial reference frames of Figure 1 is nonetheless important in the context of the added understanding it provides regarding the overall gravitational/inertial equivalence principle and its relevance to orbital and escape velocities. There is no doubt in the author s mind that this increased 10

11 understanding will lead to further breakthroughs in the future with regard to the application of relativistic principles to the universe at large. REFFERENCES 1 Joseph A. Rybczyk, Relativistic Orbital Velocity, (2009), 2 Joseph A. Rybczyk, Relativistic Escape Velocity, (2009); Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy, (2009); Relativistic Escape Velocity using Special Relativity, (2009); 3 Albert Einstein, Special Theory of Relativity, (1905), Originally: On the Electrodynamics of Moving Bodies 4 Joseph A. Rybczyk, Millennium Theory of Relativity, (2001), 5 The mathematical convention of the Millennium Theory of Relativity has evolved over the course of all of the works that make up the theory. These changes were necessary to resolve issues arising from the use of microprocessor based computer applications for performing the mathematical research in addition to dealing with the expansion of relativistic principle into areas not previously covered by relativistic theory. 6 Joseph A. Rybczyk, Gravitational Effects on Light Propagation, (2009); Light Based Gravitational and Inertial Equivalence, (2009); Relativistic Escape Velocity, (2009); 7 Joseph A. Rybczyk, Light Based Gravitational and Inertial Equivalence, (2009); Relativistic Escape Velocity, (2009); 8 Joseph A. Rybczyk, Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy, (2009); Fundamental Orbital to Escape Velocity Relationship Copyright 2009 Joseph A. Rybczyk All rights reserved including the right of reproduction in whole or in part in any form without permission. Note: If this document was accessed directly during a search, you can visit the Millennium Relativity web site by clicking on the Home link below: Home 11