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Gravitational Model of the Three Elements Theory : Mathematical Details

Gravitational Model of the Three Elements Theory : Mathematical Details. Acknowledgments. Table of content. Introduction The gravitational model (reminder) Used mathematical model Lorentz transformation (postulate 1) Postulate 3 Geodesics Black holes Conclusion. Introduction (1).

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Gravitational Model of the Three Elements Theory : Mathematical Details

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  1. Gravitational Model of the Three Elements Theory : Mathematical Details

  2. Acknowledgments

  3. Table of content Introduction The gravitational model (reminder) Used mathematical model Lorentz transformation (postulate 1) Postulate 3 Geodesics Black holes Conclusion

  4. Introduction (1) • This work • The three elements theory (1983 first idea, 1999 first version) • The gravitational model of the three elements theory (since 2007) • Last NPA visio-conferences • The gravitational model of the three elements theory (July17th, 2010) • The three elements theory (Dec18th, 2010). • A specific measurement of G (March10th, 2012)

  5. Introduction (2) • The aim of this visio-conference • Describing (more in depth) the mathematical basis of this gravitational model. • « Reminding » that the Riemannian (locally euclidean) metric gives another interesting view of relativity.

  6. The gravitational model (reminder) (1) • The idea • Gravitational mysteries might comes from the fact that relativity could not be coherent enough : Lorentz transform acts like an algebraic postulate in front of GR beautiful principles. • The idea is therefore to explain Lorentz transform with the help of GR “space-time deformation by energy” principle. • This idea yields a local space-time deformation postulate (postulate 1), and then a global space-time deformation postulate (postulate 3). In between, postulate 2 must be added for coherence (mattter is made of indivisible particles, allways travelling at c speed). • Calculations with this global space-time deformation yields a modification of Newton’s law. • This modification of Newton’s law is completely compatible with relativity, by construction.

  7. The gravitational model (reminder) (2) • A modification of Newton’s law

  8. The gravitational model (reminder) (3) • Main theoretical result : • G is no longer a constant, but a variable which value is a function of matter distribution. • The role of this distribtution must be taken in account locally, but also globally in the universe. • Linearity of gravitational forces is no longer valid in any cases and must be whatched carefully.

  9. The gravitational model (reminder) (4) • Experimental results : • Explanation of the mysterious galaxy speed profiles. • Explanation of the anomaly in the speed of the galaxies and in the deviation of light beams. • Explanation of the Pioneer anomaly to be compared with reference[2], • Explanations for miscellaneous physics mysteries: • sideral gravity, • Impossibility of an accurate measurement of G • “spurious forces” with asymmetric objects (linearity violation) , • non-Newtonian role of surrounding matter (linearity violation and variable G). • “Missing asteroids” in the main belt, • Sagnac effect (see reference [1] at the end of this presentation).

  10. Used Mathematical model (1) • The idea • Using a Riemannian model for understanding relativity: • In place of the pseudo-Riemannian one:

  11. Used Mathematical model (2) • Why using this unusual representation ? • Euclidean representation is the used representation, from the beginning of the construction of the three elements theory model : • Postulate 1: Lorentz transform, local space-time deformation. • Postulate 3: global space-time deformation. • Description and understanding luminous points trajectories.

  12. Used Mathematical model (3) • Differentiated version : • This Riemannian metric • with all positives, • must be linked to the usual pseudo-Riemannian one: • with negatives for

  13. Used Mathematical model (4) • Link between classical Minkowskian pseudo-Riemannian metric and used Riemannian metric: • which leads to this equation: , Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  14. Lorentz transform (1) • Postulate 1 figure Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054.

  15. Lorentz transform (2) • YES. The rule is applying postulate 1 figure • in the normal map of the Riemannian metric: and then apply the base transformation in the laboratory frame: Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  16. Postulate 3 (slide 1) • Postulate 3 figure Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  17. Postulate 3 (slide 2) • YES • This equation comes from the parallel transport of time vectors in this Riemannian representation of space-time: Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  18. Geodesics (1) • Is the following geodesics principle still valid in the context of this Riemannian metric ? ? Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  19. Geodesics (2) • YES the « following geodesics » principle is still valid in the Schwarzschild metric for this Riemannian version: • as a first order approximation for the law deformations • in the reference frame attached to the attracting object. Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  20. Geodesics (3) • In Minkowskian pseudo-Riemannian metric yields: • And in this Riemannian metric it yields: **: F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  21. Geodesics (4) • BUT this « following geodesics » principle is fundamentally only valid in the Minkowskian metric. • This proeminent role of the Minkowskian metric is explained by a rule which comes from the three elements theory model: • The maximisation of mass energy, therefore the minimisation of motion energy, is the rule when determining free falling particle trajectories: • is maximised because • is maximised and is minimized. Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  22. Black holes (1) • There are no black holes (as predicted by the model). • Schwarzschild Minkowskian metric in the Gravitational model of the three elements theory:

  23. Black holes (2) • This time coefficient cannot be equal to 0 but only tend to 0 when speed tend to c: • And therefore the Swarzschild ray is equal to 0 Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  24. Black holes (3) • Remark Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  25. Conclusion

  26. CONCLUSION (1) • This Riemannian metric allows to explain the gravitational model of the three elements theory in a coherent manner. • Applying postulate 1 in the Riemannian metric normal map allows to retrieve Lorentz transform. • This describe the local space-time deformations. • Applying postulate 3 in this Riemannian metric using the parallel transport of time vectors allows to calculate the relativistic coefficient. • This describe the global space-time deformations. • It will yield the modification of Newton’s law. • Geodesics • The « following geodesics » principle is still valid in the Riemannian metric in the case of the law deformations and in the referential frame which is attached to the attracting object. • The Physical explanation of this principle is explained in a straightforward manner. • This Riemannian metric allows to understand relativity in a more human sensitive manner.

  27. CONCLUSION (2) • Next to come • This “In depth” understanding of the Riemannian metric validates once more the gravitational model of the three elements theory. ? • Help!

  28. ACKNOWLEDGMENTS • NPA and WorldSci organisations • Journal of Modern Physics from which some material where used in this presentation (*, **). *: F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054. **: F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  29. QUESTIONS

  30. APPENDIX 1 Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054.

  31. APPENDIX 2 Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054.

  32. APPENDIX 3 Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054.

  33. APPENDIX 4 Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054.

  34. APPENDIX 5 Extracted from F. Lassiaille, "Gravitational Model of the Three Elements Theory," Journal of Modern Physics, Vol. 3 No. 5, 2012, pp. 388-397. doi: 10.4236/jmp.2012.35054.

  35. APPENDIX 6 **: F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  36. APPENDIX 7 **: F. Lassiaille, "Gravitational Model of the Three Elements Theory: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138.

  37. REFERENCES [ 1 ] R. Wang, Y.Zheng, A.Yao, D.Langley, “Modified Sagnac experiment for measuring travel-time difference between counter-propagating light beams in a uniformly moving fiber”, Physics Letters A 312 (2003) 7-10. DOI:10.1016/S0375-9601(03)00575-9

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