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PROJECTIVE GEOMETRICAL SPACE, DUALITY,HARMONICITY AND THE INVERSE SQUARE LAW
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
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The Choice of Geometrical Space
After the fundamental distinction between Perceptible Space and GeometricalSpace given by my late professor, at the National Technical University of Athens,Panagiotis Ladopoulos, and since all the logically consistent geometries areequally valid, we select the Projective Space as the Geometrical Space ofchoice 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 19th centurymostly by the French mechanical engineer and mathematician Jean-VictorPoncelet. Although Poncelet, and his predecessors (Pappus of Alexandria,Johannes Keppler, Gerard Desargues) applied the synthetic method, todayProjective Geometry is mostly developed by following the analytical one. Inthis work, however, we shall adhere to the synthetic geometric spirit.
Projective Space is established with eight (8)
Axioms and its Geometry with nine (9).
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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 thesame straight line, define a point, whichlies on them.
VI. A plane and a straight line not lyingon 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.
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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, theremaining of its elements can be ordered in such a way so that element Oprecedes all the other elements. In this order, there is always an elementpreceding every other element and between two elements A and B of theformation, if A precedes B, there is always an element that comes after Aand before B.
VIII. In a first degree geometric formation there are two specific directionsof movement opposite each other. If a first degree formation results fromanother via a finite number of projections and sections, a specific directionof movement on the first, corresponds to a specific direction of movementon the second.
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C. The Dedekind’s Axiom of Continuity
IX. If AB is a first degree geometric formation segment on which a directionof movement has been defined and if this segment is divided into two partsso 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 ofthe parts), so that every element of AB preceding C belongs to the first partand every element after C belongs to the second part.
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Some Remarks on the Axioms of Projective Geometry
1. The axioms of Projective Geometry introduce automatically to the Projective Space alsoits elements “at infinity”, which are considered totally equivalent to those “not at infinity”and consequently undistinguishable. As a result of this introduction and equivalence, thefifth Euclidean postulate, (parallel straight lines do not intersect), has absolutely nomeaning in Projective Space.
2. The VII Axiom of order “closes” the straight line. The projective straight line is a closedline via its point at infinite, as opposed to the Euclidean which is an open one. It followsthat 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 continuityaxiom does not contribute to the establishment of Projective Space. It was introduced,however, to Projective Geometry by the Italian Mathematician Federico Enriques to offerProjective Geometry autonomy. Dedekind's continuity axiom, as formulated, constitutesthe geometrical expression of the continuity principle of real numbers in Analysis.
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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 equallytrue statement , if the dual fundamental concepts swap their roles andthe third fundamental concept remains unchanged.
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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 theirlocality. We adhere to the philosophical thought of Weizsacker and Heisenberg, as itis presented by the latter in his well-known book, “Physics and Philosophy”, and canbe summarized in the following statement by the former: “Nature is earlier thanman, but man is earlier than the natural science”. We therefore believe that everynatural science ought definitely not to ignore the existence of the Observer, whoobserves and measures the events. Of course the “Observer” concept is ratherelastic and able to refer even to the instrument through which the registration andmeasurement of events takes place.
The foundation of this theory is briefly presented below:
1. The Philosophy of the Theory
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The second hypothesis of the Special Relativity Theory refers to theindependence of the speed of light from the speed of its source. Weshall extend it to cover all interactions of matter but refer it to theGeometrical 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 isessentially constant and independent of the relative speed of theinteracting elements of matter. More specifically, matter interactionsconducted through light, occur in Geometrical Space at a speedessentially constant in magnitude, independent of the relative speed ofinteracting elements of matter and equal to the speed of light which, Imeasure in my Perceptible Space at the place where I am located”
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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 Linear Array of Synchronized Clocks (LASC) :
The Definition of the Measurement
of the Speed of a Material Point
(Classical Mechanics) :
Α
Β
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The Harmonic Bundle (Plato’s “Shadows”)
Fig.1 The two conjugate positions for a given position
(2)
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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 ConjugatePositions is, in reality, thesimultaneous arrival to a
distant localized Observer,
(or to a recording-measuringinstrument) of images (shadows)of past events. Those eventsnever actually occurredsimultaneously.
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Figure 3. The Geometrical Locus of Observation Position for given Conjugate Positions (υ<c)
(5)
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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΄΄
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(10)
Figure 4. The Gravitational Interaction between the point-mass O and the linearly expanded mass Α΄Α΄΄
(11)
(12)
(13)
(14)
(15)
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(16)
(17)
(18)
(19)
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(20)
The Briefest Computation Principle (BCP)
This conditionality via which we arrived at the InverseSquare Law I name: Briefest Computation Principle (BCP).
(21)
(6a)
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Figure 5. The Total Gravitational Interaction Eo lies on the Bisector of the angle Α΄OΑ΄΄where A lies (see Harmonicity).
(22)
(23)
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(24)
(26)
(25)
(27)
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Relativistic Kinematics in Projective Space
Figure 6. The Apollonian Circumference at theFoot of the Perpendicular (υ<c).
,
,
R2 = AS.OS
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The Electrostatic Field outside
a grounded conductive sphere
with a charge q at O
,
,
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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)
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Summary
The Theory of the Harmonicity of the Field of Light is based on two fundamentalacceptances:
1. The adoption of the natural philosophy of Werner Heisenberg and the schoolof Copenhagen, according to which a consistent natural description of the Cosmosshould not ignore the existence of the Observer or at least the instrument ofobservation and measurement and
2. The choice of the Projective Space as the Geometrical Space of its naturaldescription. This choice is validated following the fundamental separation of thePerceptible Space, which is objective, and the Geometrical Space, that exists onlyin our minds. As all logically consistent Geometries are accepted in MathematicalScience, the adoption of a Geometrical Space by a Theory of Physics is free.
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Further on, this theory adopts as its first fundamental hypothesis the secondhypothesis of the Special Relativity Theory, properly modified. Then, during thestudy of the kinematics of the material point, the harmonic cross-ratio emergespractically automatically.
However, as the Principle of Duality is a fundamental property of the ProjectiveSpace, this principle governs the development of the whole theory and leads tosome very important conclusions in both the Relativistic and the QuantumMechanics.
One application of the Principle of Duality in the research of the GravitationalField, guides to the creation of the Inverse Square Law, during which the BriefestComputation Principle (BCP) also emerges practically automatically. Thisprinciple, that is probably related to the Principle of Least Action, with its wellknown important consequences on Classical as well as Quantum Mechanics.
Moreover, this work establishes that there is an internal relation between theElectrostatics and the Relativistic Kinematics of the Material Point in the ProjectiveSpace, which might offer future researchers, alternative routes of approachleading towards the final unification of the four known dynamic Fields.
Thank you all
for your kind patience
and attention.
Dionysios G. Raftopoulos
www.dgraftopoulos.eu
ATHENS - GREECE