Contents
Preamble . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . III
Symbology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XIII
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . XVII
CHAPTER 1
Where is Physics Today? – Synthetic Overview of the State of the Art of Physics Today . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Newton’s Gravitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Electromagnetism/GravitoElectroMagnetism. . . . . . . . . . . . . . . . . . . . 2
1.4 Special Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 General Relativity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.6 Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.7 Gravitational Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.8 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.9 Highlighting the Differences between the Two Pillars of Physics . . . . . 33
1.10 Nature Plays with Our Senses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.11 How to Reconcile the Two Physics . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
CHAPTER 2
First Ask the Right Question . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.1 What is the State of the Art and the Issues that Arise from It? . . . . . 41
2.2 What is the Nature of Space-Time? . . . . . . . . . . . . . . . . . . . . . . . . . . 42
2.3 Can Einstein’s Equation be Reconstructed without Passing
through Newton’s Weak Field Limits? Without Using G? . . . . . . . . . . 42
2.4 What Brings Us Contemporary Data of the Vacuum? . . . . . . . . . . . . . 44
2.5 Space-Time as a Physical Object an Elastic Medium . . . . . . . . . . . . . . 45
CHAPTER 3
A Strange Analogy between S. Timoshenko’s Beam Theory and General Relativity . . . . . . . . . . . . . . . . . . . . 47
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.2 Generalities on General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.3 Analogy between Beam Theory and General Relativity from
the Point of View of the General Principle Curvature = K × Energy
Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.4 Analogy between the Definition of Curvature in Strength of Material
and General Relativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.5 Extension of Curvature to Other Strength of Material Solicitations . . . 57
3.6 Analysis of Einstein’s Equation Applied to the Entire Universe
(Case of Cosmology) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
CHAPTER 4
The Stress Energy Tensor in Theory of General Relativity and the Stress
Tensor in Elasticity Theory are Similar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.1 Definition of the Stress Energy Tensor in General Relativity . . . . . . . . 65
4.2 Definition of Stress Tensor in Elasticity Theory . . . . . . . . . . . . . . . . . . 66
4.3 Demonstration of the Correlation between the Stress Tensor and the Stress Energy Tensor . . . . . . . . . .. . . . . . . . . . . 66
CHAPTER 5
Relationship between the Metric Tensor and the Strain Tensor in Low Gravitational Field . . . . . . . . . . . . . . . . . . . . 71
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.2 Definition of Strain Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
5.3 Determination of the Link between the Metric and the Strain . . . . . . . 73
CHAPTER 6
Relationship between the Stress Tensor and the Strain Tensor in Elasticity
(K) and between the Curvature and the Stress Energy Tensor (κ) in General Relativity in Weak Gravitational Fields . .. . . . . . . . 79
6.1 Reminder of Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2 Some Reminders about the Elasticity Theory . . . . . . . . . . . . . . . . . . . 80
6.3 Highlighting the Parallelism between Elasticity Theory and General Relativity . . . . . . . . . . . . . . . . . . . . . . . 81
6.4 Consequence of Parallelism and Transversalism between the Elasticity Theory and General Relativity. . . . . . . . . . 83
CHAPTER 7
Can Space-be Considered as an Elastic Medium? New Ether? . . . . . . . . . . . . 85
7.1 The Conclusions of Michelson and Morley’s Experiment . . . . . . . . . . . 85
7.2 Einstein’s View of the Ether . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
7.3 Observations Made Demonstrate the Elastic Behaviour of Space-Time . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 86
7.4 Consequence of Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
CHAPTER 8
And if We Reconstructed the Formula of Einstein’s Gravitational Field by no Longer Considering the Temporal Components of the Tensors, but the Spatial Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8.1 Let Us Step Back from Gravitation According to Newton . . . . . . . . . . 93
8.2 The Strengths and Weaknesses of Newton’s Gravitational Approach . . 94
8.3 G a Gravitational Constant of Strange Dimensions as a Combination of Underlying Parameters . . . . . . . . . . . . . . . . . . . . 95
8.4 How to Re-parameterize κ in Einstein’s Gravitational Field Equation . . 96
8.5 The Strengths and Weaknesses of Gravitation According to Einstein . . . 97
8.6 Approach to Reconstructing General Relativity from the Elasticity Theory . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . 98
CHAPTER 9
Re-interpretation of the Results of the Theoretical Calculation of General
Relativity on Gravitational Waves in Weak Field from the Windows of Elasticity Theory . . . . . . . . . . . . . 101
9.2 Re-interpretation of the 2 Gravitational Wave Polarizations in Terms of Space Deformation Tensors in the Sense of Elasticity Theory . . . . . 101
9.3 Consequence in Terms of Oscillating Waves in the Arms of Interferometers . . . . . . . . . . . . . . . . . . . . . . . . 107
9.4 Expression of Einstein’s Linearized Gravitational Equation in the Form of Strains . . . . . . . . . . . . . . . . . . . . . 110
CHAPTER 10
Determination of Poisson’s Ratio of the Elastic Space Material . . . . . . . . . . . 113
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
10.2 First Approach: Analysis of the Movements of Particles Positioned in Space on a Circle Undergoing the Passage of a Gravitational Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3 Second Approach: In the z Direction, the Gravitational Wave is a Transverse Wave and is Not a Compression Wave . . 114
10.4 Third Approach: Based on Available Datas . . . . . . . . . . . . . . . . . . . . 115
CHAPTER 11
Dynamic Study of the Elastic Space Strains in an Arm of an Interferometer . . . 117
11.1 Study of an Interferometric Arm Subjected to Gravitational Waves
Causing Compressions and Tractions of the Volume of Empty Space within It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.1.1 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
11.1.2 Determination of Tensorial Equations Associated with Each Arm of the Interferometer . . . . . . . . . . . . . . . . . . 119
CHAPTER 12
Dynamic Study of Simultaneous Elastic Space Strains in the 2 Arms of an Interferometer . . . . .. . . . . . . . . . . 125
12.1 Study of Two Interferometric Arms Subjected to Gravitational Waves Resulting in Compression/Traction of the Volume of Space within Them . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
12.1.1 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
12.1.2 Determination of Tensorial Equation Associated with the Two Arms of the Interferometer . . . . . . . . . . . . . . . 127
CHAPTER 13
Study of an Elastic Space Cylinder Twisted by the Coalescence of Two Black Holes . . . . . . . .. . . . . . . . . . . . . 137
13.1 Study of a Vertical Space Cylinder in Pure Twisting – Use of Shear Speed of the Shear Wave Correlated with the Shear Strains . 138
13.1.1 Assumptions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
13.1.2 Determination of Tensorial Equation Associated with Twisting Space Tube . . . . . . . . . . . 139
CHAPTER 14
New Mechanical Expression of Einstein’s Constant κ . . . . . . . . . . . . . . . . . . 147
14.1 Steps to Obtain the Mechanical Conversion of κ . . . . . . . . . . . . . . . . 148
14.2 Case where We Consider Only One Interferometer Arm . . . . . . . . . . 148
14.3 Cases where the Two Arms of the Interferometer and Poisson’s Ratio are Considered . . . . . . . . . . . . . . . . . . . . . . . . . . 149
14.4 Case of a Pure Torsion of Space Tube . . . . . . . . . . . . . . . . . . . . . . . . 149
CHAPTER 15
Vacuum Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
15.1 Physical Approach or Mathematical Artifact? . . . . . . . . . . . . . . . . . . 151
15.2 The Vacuum Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
15.3 Consistency of Results with Vacuum Data. . . . . . . . . . . . . . . . . . . . . 152
CHAPTER 16
Calibrating the New Mechanical Expression of κ with the Vacuum Data . . . . 155
16.1 Numerical Application to Vacuum Energy – Longitudinal Waves in Interferometric Tubes . . . . . . . . 156
16.1.1 Theoretical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
16.1.2 Intensity Obtained for the New G Parameters Based on Vacuum Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
16.2 Numerical Application to Vacuum Energy – Global Approach
by Twist Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
16.2.1 Theoretical Development . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
16.2.2 Intensities Obtained for New G Parameters Based on Vacuum Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
CHAPTER 17
Let’s Go Back to the Time Components Based on the New Results . . . . . . . 165
17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
17.2 Impact on the Time of this Search . . . . . . . . . . . . . . . . . . . . . . . . . . 166
17.2.1 Time Behaviour as an Elastic Material . . . . . . . . . . . . . . . . . 166
17.2.2 Relating the Time Intervals with the Thickness Fibers of Spatial Space Sheets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
CHAPTER 18
Analogy of Mohr’s Circle with Graviton Spin . . . . . . . . . . . . . . . . . . . . . . . . 175
18.1 Possible Constitution of Space Material. . . . . . . . . . . . . . . . . . . . . . . 175
18.2 Analogy of Mohr’s Circle with Graviton Spin . . . . . . . . . . . . . . . . . . 176
CHAPTER 19
What if We Gave Up the Constant Character of G? . . . . . . . . . . . . . . . . . . . 179
CHAPTER 20
How to Test the New Theory? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
20.1 Experimental Test of Young’s Modulus of the Space Medium . . . . . . 181
20.2 Experimental Test of Pure Space Shear Behavior . . . . . . . . . . . . . . . 182
CHAPTER 21
Other Points in Link with the Strength of Material. . . . . . . . . . . . . . . . . . . . 183
21.1 An Analysis of the Vibrations of the Space Medium at the Time
of the Big Bang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
21.2 The Plastic Behavior of the Space Medium in Strong Fields . . . . . . . 183
CHAPTER 22
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
Appendix A – Chronological Order of Progress of the Author’s Reflection and Related Discoveries. . . . . . . . . 193
Appendix B – Measurements of Space-Time Material Deformations (Strains and Angles) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
Appendix C – History of Physics and Related Formulas . . . . . . . . . . . . . . 205
Appendix D – Calculating the Scalar Curvature R of a Sphere. . . . . . . . . 207
Appendix E – Application of Einstein’s Equation in Cosmology – Demonstration of Friedmann–Lemaitre Equations . . . . . . . 233
Appendix F – Can-We Understand a Black Hole from the Strength of the Materials? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279
Appendix G – Proof of the Relation between Speed c and the Shear Modulus μ of the Elastic Medium in the Case of Gravitational Wave . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289
Appendix H – Proof of Curvature in Beam Theory . . . . . . . . . . . . . . . . . . 297
Appendix I – Proof of Quantum Value of Young’s Modulus of Space Space-Time Obtained in Tables 16.1 and 16.2 . . . . . . . . . . 305
Appendix J – Young’s Modulus of the Space Time from the Energy Density of the Gravitational Wave . . . . . . . . . 309
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
Terms and Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
About the Author . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
