Once upon a time there was a pendulum

Description

This book is the result of a search that began in 1968, when a very old universal atlas fell in my hands. The first pages caught my attention, for they described the knowledge about the universe at the beginning of the 20th century, how the 19th century idea of the existence of the ether had been discarded by the Michelson-Morley experiment and the theory of relativity. I thought the idea of the ether was beautiful, some fixed spacetime structure to hold, sort of platonic, and I got convinced that there had to be a way to make the existence of the ether compatible with relativity. Somehow, somehow, perhaps making the structure flexible…, its flexibility conspiring to make the speed of light an invariant and to allow relativity. In the following decades, whenever I found a lattice -a honeycomb, the structure of graphite, cubes in playgrounds- I wondered whether that might be the structure of the ether. The answer came as a surprise in 2009, after having spent many months playing with the geometry that appears in Escher´s Waterfall, modifying it so as to turn that impossible fascinating object into something possible. When I realized I had found a special geometry I tried to contrive it to match the ideas described in Sean Carroll´s From Eternity to Here, but the fit was not perfect. Then I turned to the very small, to David Griffiths´ Introduction to Elementary Particles and, to my astonishment, everything seemed to find its right place in the structure, which is not the ether but the structure of the Higgs field, spacetime quantified. Geometry beyond the standard model and relativity: The first part of the book describes how the main questions in particle physics fit in this proposed geometric structure for the ether, built from a modification of that structure that appears in Escher´s Waterfall. There is a geometric representation of quarks, leptons, the four types of gauge bosons, the weak isospin and the geometry and algebra of many interactions and decays. It is explained how to calculate the values of the Weinberg angle and the Cabibbo angle, the parameters in the CKM matrix. Simple formulas for the calculation of the masses of the six quarks are also given. After that a geometric representation of CPT symmetry is described in the structure to give accommodation to: • Special relativity. • The tensors of general relativity. • The geometry of the measurement problem. • The geometry of the uncertainty principle. • The second law of thermodynamics. • Neutrino oscillations and the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix. • The gravitational constant, the dielectric constant, Boltzmann constant and Planck´s constant. • The process of formation of a black hole. • The four fundamental interactions: gravitation, electromagnetism, the strong interaction and the weak interaction. • All the free parameters of the standard model find their precise values in the proposed structure. The fundamental constants -the gravitational constant, Boltzmann constant, Planck´s constant and the vacuum permittivity- are related by the golden ratio. The primes: The link between the distribution of prime numbers and symmetry. I had always thought that there was a hidden rule governing the distribution of the prime numbers among the natural numbers. There is a very simple program that controls that distribution, it is described in the second part of the book. Does this idea and the geometric structure described in the first part of the book mean that we live in a gigantic computing machine? I think so. It is not boring, though. First part: Geometry beyond the standard model: Escher´s Waterfall 13 I. The geometry of the elementary particles 13 1. Introduction [1-11] 13 2. Matrix representation 19 3. Antimatter 21 4. Weak isospin 23 5. Conservation of angular momentum 25 6. Symmetry [15, 16] 25 7. The gauge bosons 26 8. Examples of decays and interactions [6, 7, 8, 9, 10, 11, 17, 18, 19] 30 9. The Weinberg angle, θw 95 10. The Cabibbo angle, θc 96 11. The CKM matrix 97 11. The angles of the three generations 101 II. Geometry of relativity and of some other questions in physics [23-36] 103 1. CPT. Duality of the structure 103 2. Intrinsic parity 106 3. The arrow of time: 106 4. Linking basic units. Velocity 108 5. Frame time and proper time in different inertial frames 111 6. The relativity of simultaneity 112 7. Energy, momentum and mass [38-41] 113 8. The mass of the fermions 118 9. Dark matter and dark energy 121 10. The cosmological constant [42] 122 11. The flatness problem [43] 122 12. Energy levels and the primes 124 13. Lorentz transformations [23] 125 14. Momenergy [23] 126 15. Parametric equations 127 16. Tidal forces 129 17. The stress-energy tensor 130 18. The Weyl tensor 138 19. Gravity and entropy gradients 140 20. The Ricci tensor 141 21. Neighbouring units: Gravity in action 142 22. What happens behind the scenes 143 23. The 8G factor. Spherical objects 145 24. Spherical object collapsing to form a black hole [46] 149 25. Geometric interpretation of the uncertainty principle 151 28. Geometry of the measurement problem 155 29. The infrared cut-off and the ultraviolet cut-off 157 30. Temperature and the Boltzmann constant 159 31. The second law of thermodynamics [32, 47] 160 32. Neutrino oscillation and the PMNS matrix 161 33. The vacuum expectation value, the coupling constants, the mass of the top 165 34. The interactions and the golden ratio 171 35. The hidden transactions within the golden ratio 172 36. The gate 173 37. A possible geometric representation of the space axes for other universes 174 38. Range of the interactions 175 39. Table with the main values obtained from the structure 177 40. The naturalness problem 179 Second part: The symmetry of the prime numbers 183 1. The program of the primes 183 2. The twin primes 194 3. Chained symmetries for the even numbers 196 4. Program of the odd composite numbers 197 Third part: The prime numbers in the structure 201 1. The natural numbers and the energy levels. The primes 202 2. The Goldbach´s conjecture and particle physics 204 3. The primes and the Riemann curvature tensor 209 4. The gluons and the program of the primes 211 References: 213