Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy. Copyright (2009) Joseph A. Rybczyk

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1 Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy Copyright (2009) Joseph A. Rybczyk Abstract Presented herein is the ultimate theoretical proof of the correctness of the relativistic escape velocity formula previously introduced by this author. The proof is in the form of a new and independent derivation of the relativistic escape velocity formula using relativistic gravitational potential energy in combination with relativistic kinetic energy. 1

2 Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy Copyright (2009) Joseph A. Rybczyk 1. Introduction The previously introduced relativistic form of the escape velocity formula 1 based on the integration of the principles of general relativity 2 with the principles of millennium relativity 3 is further validated through an independent derivation using a new relativistic form of gravitational potential energy in combination with the earlier derived relativistic form of the kinetic energy formula 4. The resultant derivation provides independent theoretical verification of the validity of the newly introduced relativistic escape velocity formula. 2. Review of the Newtonian Principles The standard Newtonian derivation of the escape velocity formula is as follows: First we are given the formula for the potential energy of a mass m at any radius R from the center of a celestial body of mass M as stated by 1 where Φ R is the gravitational potential energy. Then we are given the Newtonian kinetic energy formula in the form where K is kinetic energy of a mass m moving at speed v. Next we are given the relationship that gives the minimum energy required for mass m to escape the gravity of the celestial body of mass M. Solving this equation for v then gives

3 2 6 where speed v of the final equation is the Newtonian escape velocity. This is considered the minimum speed in a radial direction that an unpowered object would have to be moving at when it reaches radial distance R from the center of mass in order to escape the gravity of a celestial body of mass M. Since it is given that 7 where Φ is the gravitational potential, by way of substitution into equation (6) we obtain 2 8 as the final form of the Newtonian formula for escape velocity. (Only now, as will be shown next in Section 3, is it known for certain that equation (1) gives the Newtonian value for gravitational potential energy Φ R and not the relativistic value.) 3. Relativistic Escape Velocity To begin the relativistic derivation process, first we need to convert the formula for gravitational potential energy Φ R to its relativistic form. We do this by simply factoring it with the millennium relativity version of the Lorentz transformation factor 5. This gives 9 where the term to the far right is an exact version of the Lorentz factor commonly referred to as gamma, or by the Greek symbol γ. Simplification of equation (9) then gives 10 where Φ R is the relativistic potential energy for the relationships defined. This conversion is justified by the evidence supported observations of the asymptotic increase in energy needed to accelerate an object to speed v. An object at that potential then by definition contains that amount of potential energy and not the amount given by equation (1). For our next step in the derivation process we employ the millennium relativity formula for relativistic kinetic energy 4 3

4 11 where K now gives the correct value for kinetic energy at all speeds of v from 0 to c. This formula has previously been acknowledged as a correct formula for relativistic kinetic energy by scientific authorities 6. To continue the derivation process we now simply follow the same procedure used in the previous Newtonian derivation beginning with 0 12 where now we show the minimum relativistic escape energy and not the minimum Newtonian escape energy needed to overcome the gravitational force of celestial body M. This then gives 13 that when solved for v using a computer mathematical application gives 2 14 that simplifies to and since 7 we obtain 4

5 2 18 for the relativistic version of the escape velocity formula. This is the formula previously arrived at by integrating the principles of general relativity with the principles of millennium relativity in the two earlier works 1 by this author. It gives virtually the same results for v as the Newtonian version of the formula, equation (8), except for gravitational potential Φ values that approach c 2. At that potential the escape velocity v takes on the value of v = c as a quick visual inspection of equation (18) will indicated. At that same potential the Newtonian equation (8) will give a value of v = (square root of 2)c, or v = 1.414c as a quick visual inspection of that formula will show. 4. Conclusion In consideration of the fact that the independent mathematical derivation presented in this work gives an identical result to that of the previous derivations that used completely different principles of general relativity and millennium relativity, it seems more than reasonable to conclude that the result must be valid. Otherwise one must question the validity of the underlying theories themselves and that includes the general theory of relativity, at least in the domain of applicability covered by these latest works of the present author. REFERENCES 1 Joseph A. Rybczyk, Light Based Gravitational and Inertial Equivalence, (2009), Equation (59); Relativistic Escape Velocity, (2009), Equation (13) 2 Albert Einstein, On the Influence of Gravitation on the Propagation of Light, Annalen der Physik, 35, (1911), as presented in, The Principle of Relativity, Dover Publications, Inc., 31 East 2 nd Street, Mineola, N.Y Joseph A. Rybczyk, A compilation of 22 main works beginning with the Millennium Theory of Relativity, (2001), all available from the home page of the Millennium Relativity web site at 4 Joseph A. Rybczyk, Time and Energy, (2001); Time and Energy, Inertia and Gravity, (2002), (Re. 3 above) 5 The mathematical factor used by Dutch physicist H. A. Lorentz in 1895 to explain the null results of the Michelson and Morley interferometer experiments of The use of the factor was expanded upon by Albert Einstein in his Special Theory of Relativity, (1905) 6 The validity of equation (11) for relativistic kinetic energy has been verified by a theoretical physicist at SLAC and also by scientists at the U.S. National Science Foundation in connection with NSF Proposal, Theoretical Relativistic Physics, PHY , Submitted August 28, 2003, Principal Investigator, Joseph A. Rybczyk 5

6 Relativistic Escape Velocity using Relativistic Forms of Potential and Kinetic Energy 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 6