1. 1 PROJECTIVE GEOMETRICAL SPACE, DUALITY, HARMONICITY AND THE INVERSE SQUARE LAW Dionysios G. Raftopoulos www.dgraftopoulos.eu UNIFIED FIELD MECHANICS II Preliminary Formulations and Empirical Tests 10th International Symposium Honouring Mathematical Physicist Jean-Pierre Vigier PortoNovo (Ancona) Italy, 2016 ATHENS - GREECE

2. 2 The Choice of Geometrical Space After the fundamental distinction between Perceptible Space and Geometrical Space given by my late professor, at the National Technical University of Athens, Panagiotis Ladopoulos, and since all the logically consistent geometries are equally valid, we select the Projective Space as the Geometrical Space of choice to formulate a natural theory named: “The Theory of the Harmonicity of the Field of Light” The Projective Geometry was established in the beginning of the 19 th century mostly by the French mechanical engineer and mathematician Jean-Victor Poncelet. Although Poncelet, and his predecessors (Pappus of Alexandria, Johannes Keppler, Gerard Desargues) applied the synthetic method, today Projective Geometry is mostly developed by following the analytical one. In this work, however, we shall adhere to the synthetic geometric spirit. Projective Space is established with eight (8) Axioms and its Geometry with nine (9).

3. 3 The Axioms of Projective Geometry A. The Positional Axioms I. Two points define a straight line, on which they lie. IV. Two planes define a straight line, which lies on them. II. Three points, not on the same straight line, define a plane on on which they lie. V. Three planes, not intersecting at the same straight line, define a point, which lies on them. VI. A plane and a straight line not lying on it, define a point which lies on them. III. A point and a straight line, not passing through it, define a plane on which they lie.

4. 4 B. The axioms of order and of the projective character of the direction of movement VII. If an element O is defined on a first degree geometric formation, the remaining of its elements can be ordered in such a way so that element O precedes all the other elements. In this order, there is always an element preceding every other element and between two elements A and B of the formation, if A precedes B, there is always an element that comes after A and before B. VIII. In a first degree geometric formation there are two specific directions of movement opposite each other. If a first degree formation results from another via a finite number of projections and sections, a specific direction of movement on the first, corresponds to a specific direction of movement on the second.

5. 5 C. The Dedekind’s Axiom of Continuity IX. If AB is a first degree geometric formation segment on which a direction of movement has been defined and if this segment is divided into two parts so that: a. Every element of the segment AB belongs to one of the two parts. b. Element A belongs to the first part and B to the second. c. A random element of the first part precedes a random element of the second, Then there is an element C of the segment AB (which may belong to one of the parts), so that every element of AB preceding C belongs to the first part and every element after C belongs to the second part.

6. 6 Some Remarks on the Axioms of Projective Geometry 1. The axioms of Projective Geometry introduce automatically to the Projective Space also its elements “at infinity”, which are considered totally equivalent to those “not at infinity” and consequently undistinguishable. As a result of this introduction and equivalence, the fifth Euclidean postulate, (parallel straight lines do not intersect), has absolutely no meaning in Projective Space. 2. The VII Axiom of order “closes” the straight line. The projective straight line is a closed line via its point at infinite, as opposed to the Euclidean which is an open one. It follows that the projective plane is also a closed surface via its straight line at infinity (ideal line). 3. Projective Space was established with the first eight axioms. Dedekind's continuity axiom does not contribute to the establishment of Projective Space. It was introduced, however, to Projective Geometry by the Italian Mathematician Federico Enriques to offer Projective Geometry autonomy. Dedekind's continuity axiom, as formulated, constitutes the geometrical expression of the continuity principle of real numbers in Analysis.

7. 7 The Principle of Duality in Projective Geometrical Space The Dual Fundamental Concepts in Space: Point - Plane The Dual Fundamental Concepts on the Plane: Point - Straight Line The Dual Fundamental Concepts in Central Beam: Straight Line - Plane The Principle of Duality: Every true statement of Projective Geometry is transformed to an equally true statement, if the dual fundamental concepts swap their roles and the third fundamental concept remains unchanged.

8. 8 The Theory of the Harmonicity of the Field of Light The Science of Physics, contrary to Mathematical Science, is basically empirical, practiced by human beings, who, being of a material nature, are restricted by their locality. We adhere to the philosophical thought of Weizsacker and Heisenberg, as it is presented by the latter in his well-known book, “Physics and Philosophy”, and can be summarized in the following statement by the former: “Nature is earlier than man, but man is earlier than the natural science”. We therefore believe that every natural science ought definitely not to ignore the existence of the Observer, who observes and measures the events. Of course the “Observer” concept is rather elastic and able to refer even to the instrument through which the registration and measurement of events takes place. The foundation of this theory is briefly presented below: 1. The Philosophy of the Theory

9. 9 The second hypothesis of the Special Relativity Theory refers to the independence of the speed of light from the speed of its source. We shall extend it to cover all interactions of matter but refer it to the Geometrical Space only. Thus, our first hypothesis is as follows: 2. The First Fundamental Hypothesis of the Theory “Matter interactions occur in the Geometrical Space at a speed that is essentially constant and independent of the relative speed of the interacting elements of matter. More specifically, matter interactions conducted through light, occur in Geometrical Space at a speed essentially constant in magnitude, independent of the relative speed of interacting elements of matter and equal to the speed of light which, I measure in my Perceptible Space at the place where I am located”

10. 10 The Concept of the Linear Array of Synchronized Clocks (LASC) The Definition of Synchronized Clocks (A. Einstein) : The Definition of the Measurement of the Speed of Light (A. Einstein) : The Definition of the L inear A rray of S ynchronized C locks (LASC) : The Definition of the Measurement of the Speed of a Material Point (Classical Mechanics) : ΑΒ

11. 11 The Harmonic Bundle (Plato’s “Shadows”) Fig.1 The two conjugate positions for a given position (2)

12. 12 Kinematics of the Material Point moving with superluminal speed (υ>c) measured by the LASC (3) Fig.2. The two conjugate positions for superluminal speed The co-perception of Conjugate Positions is, in reality, the simultaneous arrival to a distant localized Observer, (or to a recording-measuring instrument) of images (shadows) of past events. Those events never actually occurred simultaneously.

13. 13 Figure 3. The Geometrical Locus of Observation Position for given Conjugate Positions (υ<c) (5)

14. 14 From the Projective Principle of Duality to the Inverse Square Law (6)(8) (7)(9) Figure 4. The Gravitational Interaction between the point-mass O and the linearly expanded mass Α΄Α΄΄

15. 15 (10) Figure 4. The Gravitational Interaction between the point-mass O and the linearly expanded mass Α΄Α΄΄ (11) (12) (13) (14) (15)

16. 16 (16) (17) (18) (19)

17. 17 (20) The Briefest Computation Principle (BCP) This conditionality via which we arrived at the Inverse Square Law I name: Briefest Computation Principle (BCP). (21) (6a)

18. 18 Figure 5. The Total Gravitational Interaction Eo lies on the Bisector of the angle Α΄OΑ΄΄where A lies (see Harmonicity). (22) (23)

19. 19 (24) (26) (25) (27)

20. 20 Relativistic Kinematics in Projective Space Figure 6. The Apollonian Circumference at the Foot of the Perpendicular (υ<c).,, R 2 = AS.OS

21. 21 The Electrostatic Field outside a grounded conductive sphere with a charge q at O,,

22. 22 Figure 8. The Gravitational Field when the Conjugate position Α΄ is at the Foot of the Perpendicular. (36) What is the Electric Field ? (35) (37)

23. 23 Summary The Theory of the Harmonicity of the Field of Light is based on two fundamental acceptances: 1. The adoption of the natural philosophy of Werner Heisenberg and the school of Copenhagen, according to which a consistent natural description of the Cosmos should not ignore the existence of the Observer or at least the instrument of observation and measurement and 2. The choice of the Projective Space as the Geometrical Space of its natural description. This choice is validated following the fundamental separation of the Perceptible Space, which is objective, and the Geometrical Space, that exists only in our minds. As all logically consistent Geometries are accepted in Mathematical Science, the adoption of a Geometrical Space by a Theory of Physics is free.

24. 24 Further on, this theory adopts as its first fundamental hypothesis the second hypothesis of the Special Relativity Theory, properly modified. Then, during the study of the kinematics of the material point, the harmonic cross-ratio emerges practically automatically. However, as the Principle of Duality is a fundamental property of the Projective Space, this principle governs the development of the whole theory and leads to some very important conclusions in both the Relativistic and the Quantum Mechanics. One application of the Principle of Duality in the research of the Gravitational Field, guides to the creation of the Inverse Square Law, during which the Briefest Computation Principle (BCP) also emerges practically automatically. This principle, that is probably related to the Principle of Least Action, with its well known important consequences on Classical as well as Quantum Mechanics. Moreover, this work establishes that there is an internal relation between the Electrostatics and the Relativistic Kinematics of the Material Point in the Projective Space, which might offer future researchers, alternative routes of approach leading towards the final unification of the four known dynamic Fields.

25. Thank you all for your kind patience and attention. Dionysios G. Raftopoulos www.dgraftopoulos.eu ATHENS - GREECE