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

2. 2 Acknowledgments

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

4. 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 (July 17th, 2010)  The three elements theory (Dec 18th, 2010).  A specific measurement of G (March 10th, 2012)

5. 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. 6  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. The gravitational model (reminder) (1)

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

8. 8  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  Linearity of gravitational forces is no longer valid in any cases and must be whatched carefully. The gravitational model (reminder) (3)

9. 9  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). The gravitational model (reminder) (4)

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

11. 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. 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. 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. 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. 15 Lorentz transform (2)  YES. The rule is applying postulate 1 figure in the normal map of the 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. and then apply the base transformation in the laboratory frame:

16. 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. 17 Postulate 3 (slide 2)  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.  YES

18. 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. 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. 20 Geodesics (3) **: 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.  In Minkowskian pseudo-Riemannian metric yields:  And in this Riemannian metric it yields:

21. 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. 22  There are no black holes (as predicted by the model).  Schwarzschild Minkowskian metric in the Gravitational model of the three elements theory: Black holes (1)

23. 23 Black holes (2)  This time coefficient cannot be equal to 0 but only tend to 0 when speed tend to c: 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.  And therefore the Swarzschild ray is equal to 0

24. 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. 25 Conclusion

26. 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. 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. 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: Mathematical Explanations," Journal of Modern Physics, Vol. 4 No. 7, 2013, pp. 1027-1035. doi: 10.4236/jmp.2013.47138. *: 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.

29. 29 QUESTIONS

30. 30 APPENDIX 1 VersionEquationEquation with the slope angle Minkowski (correct one) Old one ”mystmass.doc” Riemannian time-line geodesic Not interesting 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. 31 APPENDIX 2 VersionLimited development Minkowski (correct one) Old version”mystmass.doc” Riemannian time-line geodesic 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. 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. 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. 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. 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. 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. 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 [ 2 ]R. Francisco, F.; Bertolami, O.; Gil, P. J. S.; Páramos, J., “Modelling the reflective thermal contribution to the acceleration of the Pioneer spacecraft”, Physics Letters B, Volume 711, Issue 5, p. 337-346.Francisco, F.Bertolami, O.Gil, P. J. S.Páramos, J.