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R E L A T I V I T Y

BY FORCE

Cover photo:

"Atlas Farnese" bearing the celestial sphere.

This statue of the Titan, preserved in the Naples National Archaeological Museum, is a

Roman copy, dating from the 2nd century, of a Hellenistic sculpture. It invites us to

discover all the interest of the "new force" of Relativity by Force.

Photo credit: Gabriel Seah.

Source: Wikipedia (English).

Creative Commons Attribution-Share Alike 3.0 License

André Laviron

Docteur ès Sciences [PhD]

R E L A T I V I T Y

BY FORCE

RELATIVITY BY FORCE

Copyright © 2017 by André Laviron All rights reserved. Any representation or reproduction in whole or in part of this text made without the written consent of the author or his successors or successors in title is illegal. The same shall apply to translation, adaptation or transformation, arrangement or reproduction by any art or process. The websites indicated are indicative, without any guarantee on their durability.

First printing: 2017

ISBN 978-1-326-96347-7

Legal deposit: 2017 November

Distributor: Lulu Press, Inc.,

627 Davis Drive, Suite 300,

Morrisville, NC 27650, USA www.lulu.com

Author: André Laviron

www.relativity-force.com

CONTENTS

CONTENTS v

Dedication and acknowledgments ix

GLOSSARY AND NOTATIONS xi

FOREWORD xiii

INTRODUCTION 1

1 - INTUITIVE APPROACH 3

1.1 - Classical and relativistic formalisms 3 1.2 - Bases of special relativity. Originality of the RbF 4

1.3 - Bases of general relativity 8

1.4 - Bending mechanism of light rays 9 1.5 - Advance of the perihelion of the planet Mercury 13

1.6 - Relativistic exploration 14

PART I 19

GENERAL RELATIVITY by FORCE

CURVATURE AND DELAY OF LIGHT

2 - FORMULAS OF THE RbF 21

2.1 - Comparison of formalisms 22

2.2 - Force, acceleration and energy in RbF 27 2.3 - Force and electric field in RbF 29

2.4 - Force and gravitational field in RbF 30

3 - GEOMETRY, TIME, GRAVITY 31

3.1 - Accelerated rocket and dimensional equilibrium 31

3.2 - Frequency of local clocks and lengths 35

3.3 - Principle of equivalence. Pound and Rebka's experience 37 3.4 - Geometry in a field of gravity 39

3.5 - Synchronized space H and real space G 42

3.6 - Fundamental relations of gravity 44 3.7 - Law of gravity as a function of distance 46

3.8 - Near a star or in a rocket? 48

vi Relativity by Force ©AL 2017

4 - CURVATURE OF THE TRAJECTORY 50

4.1 - Geometrical and temporal aspects 50

4.2 - Reflections on the straight line 52 4.3 - Speeds in G and in H 55

4.4 - Orders of magnitude at long distances 56

4.5 - Relation between curvatures in G and in H 58

5 - COMMON FORMULAS FOR MASS AND LIGHT IN RbF 60

5.1 - Mass and light trajectories in the rocket 60

5.2 - Same formulas for mass and light in RbF 61

6 - DEVIATION OF A LIGHT RAY 63

6.1 - Curvature of the trajectory of light 63 6.2 - Calculation of angular deviation 63

7 - DELAY OF A RADAR ECHO. SHAPIRO EFFECT 67

7.1 - Speed of light in H 68 7.2 - Delay of an electromagnetic wave 68

7.3 - Reflections on the speed of light 71

PART II 73

GENERAL RELATIVITY by FORCE

ADVANCE OF THE PERIHELION

8 - ACCELERATIONS AND CURVATURES IN G AND IN H 75

8.1 - Gravity force, acceleration, curvature in G 75

8.2 - Perpendicular acceleration, curvature in H 78 8.3 - Longitudinal acceleration in H 79

8.4 - Acceleration in polar coordinates 80

9 - TRAJECTORY OF A PLANET 81

9.1 - Fixed elliptical trajectory 81

9.2 - Elliptical trajectory whose axes turn 83 9.3 - Trajectory "almost" elliptical 85

10 - NUMERICAL SIMULATION IN CLASSICAL MECHANICS 88

10.1 - Simulation and results in classical mechanics 88 10.2 - Precision obtained in classical mechanics 92

11 - RELATIVISTIC SIMULATION FOR MERCURY 95 11.1 - RbF calculation algorithm 95

11.2 - Advance of the perihelion of Mercury 96

11.3 - Areal velocity 97

©AL 2017 Contents vii

12 - CALCULATIONS "BY HAND" IN CLASSICAL MECHANICS 98

12.1 - Law of areas 99 12.2 - Energy 101

12.3 - Differential equation of the trajectory 103

12.4 - Particular quantities and relationships 105

12.5 - Equation of the trajectory in classical mechanics 109 12.6 - Formulas for the trajectory in classical mechanics 111

13 - CALCULATIONS "BY HAND" IN RbF 112

13.1 - Areal velocity 112

13.2 - Differential equation of the trajectory 116

13.3 - Advance of the perihelion of Mercury 119 13.4 - Harmonic coefficients 123

13.5 - Curvatures of trajectory at perihelion and aphelion 124

13.6 - Abstract and numerical application 126

14 - COMPARISON OF RESULTS

OF RBF WITH THOSE OF EINSTEIN 128

14.1 - RbF and Einsteinian formalism to the first order 128 14.2 - RbF and Einsteinian formalism to the second order 129

14.3 - Areal velocity 130

14.4 - Equation of the trajectory 132 14.5 - Harmonic coefficients 134

14.6 - Curvatures at perihelion and aphelion 135

14.7 - Metric 136 14.8 - Where is the error? 138

14.9 - Reflections 139

14.10 - Confronting with observation 143

PART III 149

SPECIAL RELATIVITY by FORCE

15 - FOUNDATIONS OF THE SPECIAL RELATIVITY 151

15.1 - Lorentz's Relations 151

15.2 - Transformation of speeds 156

15.3 - Accelerations transformation 159

16 - "NEW" FORCE IN RbF 162

16.1 - Force and Energy according to classical mechanics 162 16.2 - Force and Energy according to Einstein 162

16.3 - RbF force 164

16.4 - Energy and momentum 167

viii Relativity by Force ©AL 2017

17 - ELECTRICITY AND GRAVITY IN RbF 169

17.1 - Force of an electric field on a fixed charge 169

17.2 - Electrical force on a mobile charge 171 17.3 - Summary of formulas of special RbF 177

17.4 - Kinetics of a charge in an electric field 178

17.5 - Force and gravitational field 187

EPILOGUE 189

18 - RbF IN BRIEF 191

19 - CONCLUSION 197

ANNEXES 201

20 - STUDY OF AN ACCELERATED LABORATORY 203

20.1 - Dimensional equilibrium 203 20.2 - Function F ensuring dimensional equilibrium 205

20.3 - Abscissa of the "locomotive" and tangent frame of reference

207 20.4 - "Rocket" time and "fixed" time 208

20.5 - Propagation of light in the laboratory 210

20.6 - Spectral shift in an accelerated environment 212

21 - CURVATURE AND ACCELERATION IN H 214

21.1 - Formulas for curvature, polar acceleration 214

21.2 - Curvature of the trajectory in H 221 21.3 - Longitudinal acceleration in H 224

22 - COMPUTER SIMULATION 226 22.1 - Simulation principle 226

22.2 - Test with the Euler method 229

22.3 - Test with the Runge-Kutta method 233 22.4 - Precision and calculation time 235

22.5 - Relativistic results for Mercury 237

REFERENCES AND TOOLS 242

I dedicate this book to my wife Annie.

During the many years of my solitary researches in

relativity, I have benefited from the support of my close

family members, amusing ourselves with the caterpillar, the fly, the explorers of the soft world that you will

discover in your reading. I informed them of my

successes, but also of my errors and failures, which were

much more numerous than the first. In their place, I would have thought that this project was doomed to failure: they

were kind enough never to express it; on the contrary they

were always attentive, interested, encouraging, caring and helping in particular for the final shaping. For all

that, I thank them very affectionately.

At the end of the course, I also benefited from the help of

some very good friends who invested themselves to give

me shaping advices and final readings. If the content is

essential to the author, the wording is also paramount to facilitate reading. By the relevance of their remarks and

their proposals, I am sure that they will have participated

very significantly in the quality of this work. I thank them all very warmly, with special mention to Nicole

Commerçon and Joëlle Gerby in the field of French

writing, to Audrey Bozzetto and Laurent Bruchon for the

cover. I thank my daughter Catherine Cloteaux and my granddaughter Salomé Cloteaux who improved

significantly my translation in English of the original

book "Relativité par la Force". I thank also my son

Raphaël Laviron for his wise assistance in website management and more generally in informatics.

GLOSSARY AND NOTATIONS

The headings below present the main notations used. Some choices allow them to be easily managed with word processing tools (see for example, operators, vector

notation, etc.).

Operators: calculations use the type of notations illustrated in the following examples:

a * b /c /d = a b /(c d) = (a b) /(c d)

4*(a - b /c ) /(d+e) = 4 (a - (b/c))/(d+e) Thus the multiplication sign "*" can be omitted. It is however often added to

clarify some equations.

Vector notation: a vector is written with the sign | followed by the name given to the vector. Examples:

|F is the force vector; F is the intensity of the force.

|E is the electric field vector; E is the intensity of the electric field.

Vector operator:

|a . |b is the scalar product of the vector |a by the vector |b.

Acceleration |a: derivative with respect to time of the speed vector.

|a = d|v /dt

Speed of light c: constant measured in vacuum.

Masse m: constant (independent of speed) characterizing the inertia of a body.

Coefficient of contraction as a function of the speed v1 of a mobile:

αl = √(1- vl2/c

2)

Momentum |p: vector quantity defined for a mass m of speed |vl by

|p = m |vl /αl

Einstein force |Fein: definition of the force used by the Einsteinian formalism.

This is the time derivative of the momentum vector.

|Fein = d|p /dt

Force |F: definition of the force used by Relativity by Force.

|F = m |a /αl3

xii Glossary and notations ©AL 2017

Relativity by Force (RbF): original formalism presented in this book, based on a

new definition of force, used in the context of relativity (special and generalized)

Einsteinian Relativity: formalism followed by Einstein to develop his theory of

relativity (special and generalized)

Special relativity: part of the theory of relativity concerning non-accelerated

frames of reference, without gravity

General relativity: part of the theory of relativity concerning gravitational fields and accelerated frames of reference

Longitudinal frame of reference: frame associated with the movement of a mobile. It includes:

- a longitudinal axis oriented in the direction of the speed vector of the mobile,

to which the index l and a unit vector |ul are associated

- an axis perpendicular to the preceding one to which the index p and a unit vector |up are associated

Radial frame of reference: frame associated with the polar coordinates (R,Θ), with:

- a radial axis oriented in the direction of increasing distances R,

to which the index r and a unitary vector |ur are associated - an axis orthogonal to the preceding one, called orthoradial,

to which the index o and a unit vector |uo are associated

Unitary vectors: the notations of these vectors are generally indicated in the following way:

- |ux and |uy in Cartesian coordinates

- |ur and |uo are respectively called "radial" and "orthoradial", in polar coordinates

- |ul and |up are respectively called "longitudinal" and "perpendicular",

in longitudinal coordinates. |ul is collinear with the speed vector.

RbF: Relativity by Force

Unless otherwise indicated, the studies are carried out in a two-dimensional space. The three-dimensional generalization is left to the initiative of any interested

reader.

The figures are identified in the paragraphs as follows: {25}.

References are listed on the last pages and identified in the text by a statement

such as: [7].

FOREWORD

This book is the result of an ambitious dream: to better understand the foundations

of relativity (special and general) with mathematics as "simple" as possible,

starting from a minimum number of laws known experimentally. Initially, the ambition and hopes for success were very limited. The magic of dreams has

operated beyond all hope, culminating in the new formalism of Relativity by

Force (RbF), the subject of these pages.

From this moment on, everyone has the right to ask whether all this is very

serious, inasmuch as the traditional formalism developed by Einstein's genius is

only affordable if one has a very advanced mathematical training in very particular domains. Be aware that for a long time I have asked myself the same question. I

have gradually gained confidence over the years in this new approach, observing

that I was able to find all the known results of the special relativity and then those of general relativity. This confidence-building was very slow, with many moments

of discouragement, but also many other moments full of passion. Even if all this

now appears almost as a "flowing" story, do not think that the development of the corresponding formalism has been a "quiet long river".

The main characteristic of the RbF calculations is that they are modeled on those

of classical mechanics, thanks to a new definition of force, collinear with the acceleration it produces on a mass, which is not the case for the Einsteinian

formalism. For the latter, the force is used very little in favor of the notions of

momentum and energy. Thus, in RbF, with this "new force", we will calculate "simply" the deviation of a light ray in the gravitational field of the Sun, the

spectral shift in gravitational medium, the delay of a wave in a gravity field, the

advance of the perihelion of the planet Mercury. As you will see, the RbF formalism works perfectly while remaining very attached to the simple concepts

of classical mechanics, with formulas adapted to account for relativity.

The aim was thus achieved. It has even been largely surpassed since you will discover, at the very end of the study of general relativity, results that surprised me

enormously and that should surprise you just as much.

xiv Foreword ©AL 2017

The introduction and conclusion of this book is accessible to a wide audience.

Calculations and demonstrations require a minimum of scientific studies. To give

an idea of the required mathematical level, we can quote the following words: "derivative, integral, limited developments, vector, rotation in a plane ...". The

words "quadrivector, tensor, space-time, geodesic ..." are totally excluded, not

because they would be "dirty words", but because they correspond to specialized mathematics with which each one is not necessarily familiar. This linguistic

exclusion does not in any way call into question the interest of the corresponding

formalisms, of which everyone knows both historical and practical importance.

In calculations and reasoning, a balance has been sought between a certain

conciseness and explanations sufficiently detailed for a relatively easy reading,

even by a reader who wants to reactivate old knowledge.

I invite you to gradually enter the process of the RbF with the introduction

devoted to a first intuitive approach. Unlike traditional presentations, the study of

general relativity will be dealt with directly in Part I from a very brief description of the formulas of special relativity in RbF. We will continue with the study of the

curvature and the delay of electromagnetic waves in gravity. Part II is, in the field

of general relativity, devoted to the trajectory of the planets, leading in particular to the advance of the perihelion. The results are compared with those obtained by

the Einsteinian formalism. Part III presents special relativity by detailing how the

RbF formulas presented in Part I were obtained. The epilogue summarizes the approach of this research and compares it with the results of the Einstein method,

and concludes. The appendices present various calculations and additions that

enrich the previous Parts.

Depending on your own sensitivity, you can begin by reading Part III, dealing

with special relativity, before Parts I and II dedicated to general relativity without

any difficulty.

I hope you will find in this new approach to relativity as much passion to discover

it as the one I have set to elaborate it.

André Laviron

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