Geometry, Geodesics, and the Universe
The Lines that Led from Euclid to Einstein
by
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The subject of my book is the development of geometry from the concepts of the ancient Greeks, familiar from high school, to the fourdimensional 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.i 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 geometry, algebra, and Newton’s laws of motion. These staples of school will be presented sufficiently to reacquaint you with the necessary concepts. Also helpful would be some familiarity with calculus as it would be obtained in a first introductory course; however even in the absence of such a background, concepts from the calculus will be developed as needed. In particular, the basic meaning of the differential and integral calculus will be geometrically motivated by appealing to their relation to the slopes of curves and to the area under a curve. In both these cases, key results will be illustrated using numerical methods that readers could perform for themselves with a spread sheet such as EXCEL. It seems to me that a numerical approach has an advantage over traditional analytic results in that it can make clearer the meaning of the various calculus operations.
A major theme of the book is the expansion of our understanding of geometry beyond that of Euclid. Two thousand years after Euclid, a new geometry was discovered that changed the understanding of geometry and mathematics, while paving the way for Einstein’s General Relativity. In order to follow the revolutionary path to the first new geometry, you will need no other additional skills than those developed in high school to understand Euclid and, unlike many of the followers of Euclid, an open mind. The next steps leading to General Relativity will introduce concepts necessary to characterize the new geometries, including notably the metric tensor and the concept of curvature. These are unquestionable sophisticated mathematical entities, but fortunately you already have at least an intuitive understanding of a non-Euclidean geometry that can be used to introduce the new mathematics, that is, the spherical geometry familiar from the globe found in classrooms and homes. Spherical geometry is therefore used in detailed example calculations so that you can gain a comfort level with all newly introduced mathematical concepts.
As should be clear, my presentation is not meant for specialists in mathematics and physics. With this in mind, I have made use of traditional older tensor component notation rather than its modern abstract coordinate independent counterpart. I believe that results are more easily visualized in the component notation. In this way the presentation should be accessible to all readers from the continuing math enthusiast to those with a science and engineering background who would not have been introduced in their studies to the concepts told here.
With the new mathematical concepts in place, the principles behind Einstein’s General Relativity will be described with the object of providing an introduction to the formulation and the meaning of Einstein’s Field Equations. The solutions to these equations are the source of our understanding of such phenomena as the Big Bang and black holes. Although the full details of the solutions of the Field Equations are beyond the scope of my book, the level of understanding achieved will allow you to see how models of the universe arise from General Relativity, explaining the interconnectedness of the geometry of space-time with the experimentally observed expansion of the universe.
Throughout my book, I have cited references where more details can be obtained. The references have been selected to help readers to continue their journey in mathematics. Thus, they support an equally important goal of my book: opening readers to some of the pleasures of mathematics. In this regard, I dedicate my book to those who long ago put their mathematics aside, but also to those involved in science or engineering who received much training in the practical use of advanced mathematics, but with little regard to its foundations. The concepts addressed here are implicit in much of mathematics taught in high school and beyond, but hidden by an emphasis of math as a tool. My role here is to be a tour guide along a deductive journey pointing out some of the wonders discovered by mathematicians. As such, I have endeavored to include everything needed for your understanding. Thus, I hope you will enjoy the journey and be inspired to continue to explore mathematics as one of humanity’s great expressive and creative achievements.
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About the Book
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.
About the Author
Robert G. Bill was a researcher in fire and fire protection for twenty–five years at FM Global, a major industrial and commercial mutual property insurer which operates the world’s largest full–scale fire research facility. There, he was Assistant Vice President and Director of Research for Fire Hazards and Protection, overseeing research in areas of flammability, fire spread, material reactivity, and fire protection systems. Previous to joining FM Global, he was an Assistant Professor in the Department of Mechanical Engineering at Columbia University conducting research in turbulent combustion. He holds BS, MS and PhD degrees in Mechanical Engineering from Cornell University. During his doctoral program, he minored in theoretical physics and took courses from Nobel Laureates, providing a perspective that, along with his interest in mathematics, has led to his book, Geometry, Geodesics, and the Universe. Dr. Bill's publications include research in the areas of fluid mechanics, micro–meteorology, combustion, and fire protection. In 1994 and 2003, he received the National Fire Protection Association’s (NFPA) Bigglestone Award for communication of scientific concepts in fire protection. From 2002 to 2008 he served on the executive committee of the International Association for Fire Safety Science and in 2009 was elected as a lifetime honorary member equivalent to the grade of Fellow of the Society of Fire Protection Engineering. In 2019, he was a co-winner of NFPA's Philip J. DiNenno Prize for groundbreaking innovations that have had a significant impact in the building, fire and electrical safety fields. Now retired, Dr. Bill enjoys time with family, playing the violin, community volunteering, and walking his fox terrier.