Geometry, Geodesics, and the Universe

Robert G. Bill

Description

The story of the development of geometry is told as it emerged from the concepts of the ancient Greeks, familiar from high school, to the four-dimensional space-time that is central to our modern vision of the universe. The reader is first reacquainted with the geometric system compiled by Euclid with its postulates thought to be self-evident truths. A particular focus is on Euclid’s fifth postulate, the Parallel Postulate and the many efforts to improve Euclid’s system over hundreds of years by proving it from the first four postulates. Two thousand years after Euclid, in the process that would reveal the Parallel Postulate as an independent postulate, a new geometry was discovered that changed the understanding of geometry and mathematics, while paving the way for Einstein’s General Relativity.
The mathematics to describe the non-Euclidean geometries and the geometric universe of General Relativity is initiated in the language of mathematics available to a general audience. The story is told as a mathematical narrative, bringing the reader along step by step with all the background needed in analytic geometry, the calculus, vectors, and Newton’s laws to allow the reader to move forward to the revolutionary extension of geometry by Riemann that would supply Einstein with the language needed to overthrow Newton’s universe. Using the mathematics acquired for Riemannian geometry, the principles behind Einstein’s General Relativity are described and their realization in the Field Equations is presented. From the Field Equations, it is shown how they govern the curved paths of light and that of planets along the geodesics formed from the geometry of space-time, and how they provide a picture of the universe’s birth, expansion, and future. Thus, Euclid’s geometry while no longer thought to spring from perceived absolute truths as the ancients believed, ultimately provided the seed for a new understanding of geometry that in its infinite variety became central to the description of the universe, marking mathematics as a one of the great modes of human expression.

Sujets

• Géométrie
• Sciences et techniques
• Geometry
• Mathematics
• Albert Einstein
• Géométrie
• Géométrie

Informations

• Publié par: Xlibris US
• Date de parution: 19 mars 2023
• Nombre de lectures: 0
• EAN13: 9781669853046
• Langue: English
• Poids de l'ouvrage: 3 Mo

Extrait

GEOMETRY, GEODESICS, AND THE UNIVERSE
 
 
 
THE LINES THAT LED FROM EUCLID TO EINSTEIN
 
 
 
 
 
 
 
 
 
 
 
ROBERT G. BILL
 
Copyright © 2023 by Robert G. Bill.
 
Library of Congress Control Number:
2022919890
ISBN:
Hardcover
978-1-6698-5306-0
 
Softcover
978-1-6698-5305-3
 
eBook
978-1-6698-5304-6
 
All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system, without permission in writing from the copyright owner.
 
Cover with photo of the Orion Nebula and interior figures by the author.
 
 
 
 
 
 
Rev. date: 07/11/2023
 
 
 
 
 
Xlibris
844-714-8691
www.Xlibris.com
848109
CONTENTS
Foreword
Prologue
 
PART I
Following the Path of Form
 
1. Lessons From School: Euclid’s Legacy
1.1 Euclid’s self-evident truths
1.2 Consequences of the first four truths
1.3 A not so self-evident truth: The Parallel Postulate and its consequences
 
2. Euclid’s Truths Questioned
2.1 The search for simpler truths
2.2 Saccheri vindicates Euclid – and misses a breakthrough
2.2.1 The Saccheri quadrilateral
2.2.2 The Saccheri Hypotheses of the Acute Angle (HAA), Obtuse Angle (HOA), and the Right Angle (HRA)
2.3 Lambert and intimations of a new geometry on the surface of a sphere
2.3.1 Visions of the HAA on a sphere of imaginary radius
2.3.2 Relating the HOA to a real sphere
 
3. The Discovery of Non-Euclidean Geometry
3.1 Gauss’ insight
3.2 Lobachevsky and Bolyai find a new geometry
3.3 Modeling the hyperbolic plane – a first look
 
PART II
Following the Path of Number
 
4. Geometry Meets Algebra: Descartes’ Insight and Beyond
4.1 Prelude to a revolution – symbolic algebra
4.2 Euclid finds his place on the Cartesian plane – analytic geometry
4.3 Some truths unknown to Euclid and Descartes: the real numbers
4.3.1 The Postulates
4.3.2 Rational numbers as decimals – a starting place for the real numbers
4.3.3 Filling the holes in Euclid’s line with the infinity of real numbers
4.3.4 Counting to infinity
4.4 The real joins the imaginary – the complex plane
 
5. Analyzing Change, Summation, and Direction: the Calculus and Vectors
5.1 Overview
5.2 Changes along the path: differential calculus
5.3 Summing up along the path: integral calculus
5.4 Numbers with direction: vectors
 
6. Gauss Reveals Curvature as the Heart of Geometry
6.1 Curves on the Euclidean Plane
6.2 Surfaces in space
6.3 Gaussian curvature and the demystification of Non-Euclidean geometry
 
7. Models of Non-Euclidean Geometry
7.1 The Hypothesis of the Obtuse Angle (HOA) and the sphere revisited
7.2 Models of surfaces with negative Gaussian curvature
7.2.1 The sphere of imaginary radius
7.2.2 Poincaré’s half-plane and disk models
7.2.3 The pseudosphere – the model that changed mathematics
 
8. Riemann’s New View of Geometry
8.1 The answer to “what are straight lines?” - geodesics
8.2 The foundation of geometry: the metric tensor
8.3 Tensors and a universal geodesic equation for straight lines and the orbits of planets
8.4 Leaving Gauss’ surface – the curvature of space
 
PART III
The Geometric Universe
 
9. Newton
9.1 On the shoulders of giants
9.2 The stage of Newton’s universe: time and space
 
10. Beyond Newton: Conservation of Energy
 
11. Einstein
11.1 Special Relativity: the merging of time and space
11.1.1 The Lorentz transformation
11.1.2 Four vector momentum and E = mc 2
11.2 General Relativity: a universe driven by geometry
11.2.1 Einstein’s insight: the Equivalence Principle
11.2.2 Newton’s Law of Gravity revisited as an equation of gravitational potential
11.2.3 Einstein’s Field Equations
 
12. Consequences of General Relativity
12.1 The first three predictions: the spectral redshift, Mercury’s orbit, and the bending of light
12.2 Black holes and the expanding universe
 
Epilogue : Philosophic Thoughts
Appendix A : Proof that √2 is irrational
Appendix B : Euclid’s Book I Propositions: 1 to 28
Appendix C : An Alternative Approach to Covariant Derivatives and the Christoffel Symbols
Bibliography
Notes
About the Author
LIST OF FIGURES
1-1 Proposition IE1
1-2 Proposition IE16
1-3 Proposition IE27
1-4 Proposition IE28
1-5 Proposition IE29
1-6 Proposition IE32
1-7 Proposition IE37
1-8 Proposition IE47 –The Pythagorean Theorem
1-9 Similar triangles
1-10 Similar right triangles: △ABC & △ A´ B´ C
2-1 Proof of Playfair’s Postulate from the Parallel Postulate
2-2 Saccheri Proposition 1
2-3 Saccheri Proposition 2
2-4 Saccheri Proposition 3
2-5 Asymptotic parallel line DE
2-6 Congruence of similar triangles under the HOA or HAA
2-7 Additivity of triangle deficit, 𝛿d, or excess, 𝛿e
2-8 Saccheri-Legendre Theorem
2-9 Triangles on a sphere
3-1 The Hyperbolic Postulate
3-2 Lobachevsky Angle Theorem 21
3-3 Lobachevsky Right Triangle Theorem 22
3-4 Proof of the HAA from the Hyperbolic Postulate
3-5 Poincaré half-plane hyperbolic geometry model
3-6 Poincaré disk model with parallel lines
4-1 Graph of y = mx + b
4-2 Graph of a circle
4-3 Law of Cosines
4-4 Cantor counts the rational numbers
4-5 Cantor attempts to count the real numbers
4-6 The complex plane
5-1 Graphical estimates of the slope of a parabola at x = 1
5-2 Tangent line and radius of the unit circle at the point ( x c, y c)
5-3 Finding the area under y = x
5-4 Finding the length of a curve
5-5 Addition of vectors
5-6 Angle between vectors
5-7 Vector cross product
5-8 Volume of a parallelepiped
6-1 Spherical coordinates
6-2 Surface r( x 1(u, v), x 2(u, v), x 3(u, v))
6-3 Double Integration
6-4 Curvature vectors at a circle of constant colatitude
7-1 Intersecting lines in the elliptic plane
7-2 Boundless and finite lines in the elliptic plane
7-3 A spherical triangle
7-4 Great circle route from Boston to Rome
7-5 Proof for angle of parallelism Π
7-6 Triangle with one side a limiting parallel
7-7 Half-plane model
7-8 Half-plane triangle
7-9 Half-plane angles of parallelism
7-10 The pseudosphere
8-1 Vector representation in rotated Cartesian axes
8-2 Parallel displacement of a vector around a closed curve
9-1 Calculation of the descent of a falling body under gravity
11-1 A rotating space station in deep space
11-2 Flux of a gravitational field
11-3 Divergence of the vector A
C- 1 Orthogonal bases on a spherical surface
LIST OF TABLES
Table 3-1 Comparison of the HAA, HRA, and HOA
Table 4-1 Comparisons of Algebraic Notations
Table 4-2 Approximating 2 1/2 with rational numbers
Table 4-3 Matching infinite sets of integers with the natural numbers
Table 4-4 Matching the natural and positive rational numbers
Table 5-1 Estimates of the Slope of a Parabola at x = 1
Table 5-2 Derivatives and integrals (anti-derivatives) of common functions
Table 5-3 Numerical approximation of π from area of a quarter circle
Table 7-1 Half-plane calculations of Π
Foreword
The subject of my book is the development of geometry from the concepts of the ancient Greeks, familiar from high school, to the four-dimensional space-time that is central to our modern vision of the universe. For the specialist, there are many technical books that will delve into the subjects of modern geometry and physics at an advanced mathematical level. However, for many who are not specialists, this approach will be impossible to follow. Alternatively, there are books without any, or virtually without any, mathematics whatsoever. Such books can qualitatively describe geometrical concepts including those leading to Special and General Relativity, but I believe that many readers could obtain a more exact understanding of these subjects from a mathematical presentation that takes advantage of their typical math background.
Mathematics is, of course, the natural language to express the concepts of geometry and Relativity and brings with it a precision in meaning which I believe is otherwise impossible. Moreover, the experience of following a mathematical description rather than a qualitative verbal description is akin to the difference between playing music and just listening to it. The experiences are intellectually and aesthetically quite different. 1 Along the way you will encounter concepts that are crucial to mathematics, science, the meaning of truth, space and time, the infinite, and the origin of the universe - not a bad list!
My ideal readers would be those who enjoyed encountering new concepts in math and physics in high school or introductory college courses and puzzling out their distinctive problems, although they did not pursue them further in their vocation. I only assume in my presentation familiarity with high school