Perspectives On Space and Time

Or, Seeing Space and Time From Both Sides of the Brain

by Kent Heiner

Dedicated to my high school physics, calculus, and art teachers: Dave Trout, Reggie Nelson, and Monte Bianchi.

Contents (under construction)

Book I: Perspectives

  1. Perspective in Art: The Horizon

  2. Perspective in Space, Time and Causality

  3. Perspective in Mathematics

  4. Perspective in Geometry: The Curved Earth

  5. Perspective in Motion: Galilean Relativity and the Doppler Effect

  6. Perspective in Mass, Force, and Energy

  7. Perspective in Art: The Curved Canvas

  8. Motion in Perspective

  9. Perspective and Motion: Electromagnetism

Book II: Beyond Intuition and Experience

  1. A Whole New Perspective: Special Relativity

  2. Special Relativity as Geometry

  3. Relativity and Zeno's Paradoxes

  4. Relativity and the Conic Sections

Book III: Three Shalt Thou Count

  1. Three Facts We Teach About Gravity Which are Almost True (Part One)

  2. Three Facts We Teach About Gravity Which are Almost True (Part Two)

  3. Three Facts We Teach About Gravity Which are Almost True (Part Three)

  4. In Defense of a Three-Dimensional Spacetime (Part One)

  5. In Defense of a Three-Dimensional Spacetime (Part Two)

  6. In Defense of a Three-Dimensional Spacetime (Part Three)


The geographical frontier is a thing of the past. Every place on earth has been visited and mapped out by mankind, and even photographed by satellite. Any place on earth can be seen from any perspective by making a few clicks with a mouse. Before the 21st century is over, “Mapquest,” “Google Earth” and their like are almost certain to be supplemented or superseded by imagery databases that include the surfaces of other planets and their moons.

But at the beginning of the 20th century, inspired by equations specified by James Clerk Maxwell in the later 19th century, Albert Einstein predicted the existence of another frontier no one had ever seen or even thought of. It was not a geographical frontier; it was a mathematical one. Einstein explained that we live in a world where material objects move at very slow speeds compared to the speed of light. He also predicted what would happen in a world where matter approached the speed of light. In the last hundred years, we have only begun to glimpse this incredible place. None of us have ever set foot there.

I remember first reading about relativity and its strange predictions of events being reversed in time order, of clocks slowing down, and objects appearing to shrink. I was disoriented, baffled, perhaps even scandalized, but ultimately fascinated. I tried to picture what it would look like to travel near the speed of light and for a long time I refused to believe that it was impossible to go faster.

The legacy of geographical explorers such as Columbus and Magellan is taught in elementary schools and is fairly well-understood. But the legacy of pioneers of geometry and physics - Descartes, Galileo, Kepler, Newton, Maxwell, and Einstein, to name a few - is much less well-known. It is the journeys of these latter explorers that I intend to celebrate in these pages. In doing so I will also be reliving the significant portion of a lifetime I have spent following (often lamely but almost always enthusiastically) in their footsteps. It is a journey that each of us can make for ourselves; it is a journey of the mind. This is a book about the questions and the search as much as it is about the answers. While I am certainly no expert in physics or mathematics, I have asked many interesting questions and hope I can provide some worthwhile insights to my fellow adventurers. Many mistakes have been made during my journey, but even these were educational. One of my chief aims in this series is to let others benefit from my mistakes.

My formal education in math and physics did not extend beyond my first year of college, and most of the ideas I will be presenting here do not require more than an introductory-level understanding of geometry and physics. I'll be including a little bit of refresher material on these as well, so if you have enough interest in the subject to have picked up this book, you should be able to follow along. I signed up for a course on special relativity and quantum theory during my first year of college, and in my first week of class I found out that the math prerequisites were much greater than what was advertised. I hope to avoid putting you, the reader, in the same situation.

It may surprise the reader to find as much art in this book as mathematics. It turns out that art requires an understanding of perspective that I have found extremely helpful in comprehending relativity. After all, one of Einstein's key lessons is that how you see something depends on where you are and how you are moving. In a coming day, introductory courses on relativity just might include some instruction in perspective drawing.

In many ways, the study of the laws of physics is more than intellectual. It can have spiritual aspects if one is trying, as Einstein was, to understand the mind of God, whatever "God" means to you. I wish you a happy pilgrimage in Einstein's New World.

Preface to the 2015 Edition

Midway through writing the first edition of Perspectives, I picked up a book which I vaguely recalled having some symbolism which would work well for my subject, even though the topic of this other book was quite different. Upon reading, I was astounded to find so many of the subjects I had already chosen for my book to be included in this other. Tremendously pleased by the coincidences, I then consciously aped and shamelessly borrowed to the point where I owed a significant debt to this other author. But in the spirit of fun and discovery in which the other book was written, I leave it to the reader (and the author, should he chance to read this) to puzzle out the source and extent of my inspiration. I leave these clues: I have borrowed the characters Zeno, Achilles and the Tortoise from this author, who borrowed Achilles and the Tortoise from Lewis Carroll, who borrowed them from Zeno.

The first draft of this book was completed in 2008. It was briefly made available as a Kindle e-book in 2011, called "Perspective: Seeing Space and Time From Both Sides of the Brain." Unsatisfied with it (and perhaps more importantly, having seen no sales), I pulled it from Kindle but left it available on my general-purpose web site. After much consideration, I decided to make the book a more interactive experience. There is far too much to be gained by the use of hyperlinks, animated images, embedded videos, and applets. At about the same time, I grew confident enough in my understanding of Einstein's general theory to write about it, which I have been doing in my Wordpress-hosted blog. In late 2014 I decided to launch a new site, Straight Up Physics, to offer guidance to others who are on a similar learning path. This 2015 edition of Perspectives is more than a revised, extended, interactive version of the original; it is an ongoing project which is never to be considered complete. I plan to keep updating it as I think of ways to improve it.