The Geometry of Special Relativity

Tevian Dray

• Publisher: Chapman and Hall/CRC
• Year: 2021
• ISBN:
  9781351663205
• Edition: 2
• Topic: Matematica
• Subtopic: Matematica generale

This unique book presents a particularly beautiful way of looking at special relativity. The author encourages students to see beyond the formulas to the deeper structure.The unification of space and time introduced by Einstein's special theory of relativity is one of the cornerstones of the modern scientific description of the universe. Yet the unification is counterintuitive because we perceive time very differently from space. Even in relativity, time is not just another dimension, it is one with different propertiesThe book treats the geometry of hyperbolas as the key to understanding special relativity. The author simplifies the formulas and emphasizes their geometric content. Many important relations, including the famous relativistic addition formula for velocities, then follow directly from the appropriate (hyperbolic) trigonometric addition formulas.Prior mastery of (ordinary) trigonometry is sufficient for most of the material presented, although occasional use is made of elementary differential calculus, and the chapter on electromagnetism assumes some more advanced knowledge.
Changes to the Second Edition

• 
  
  

• The treatment of Minkowski space and spacetime diagrams has been expanded.

• 
  
  

• Several new topics have been added, including a geometric derivation of Lorentz transformations, a discussion of three-dimensional spacetime diagrams, and a brief geometric description of "area" and how it can be used to measure time and distance.

• 
  
  

• Minor notational changes were made to avoid conflict with existing usage
  in the literature.

Table of Contents

Preface
1. Introduction.
2. The Physics of Special Relativity.
3. Circle Geometry.
4. Hyperbola Geometry.
5. The Geometry of Special Relativity.
6. Applications.
7. Problems III.
8. Paradoxes.
9. Relativistic Mechanics.
10. Problems II.
11. Relativistic Electromagnetism.
12. Problems III.
13. Beyond Special Relativity.
14. Three-Dimensional Spacetime Diagrams.
15. Minkowski Area via Light Boxes.
16. Hyperbolic Geometry.
17. Calculus.
Bibliography.

Author Biography

Tevian Dray is a Professor of Mathematics at Oregon State University. His research lies at the interface between mathematics and physics, involving differential geometry and general relativity, as well as nonassociative algebra and particle physics; he also studies student understanding of "middle-division" mathematics and physics content. Educated at MIT and Berkeley, he held postdoctoral positions in both mathematics and physics in several countries prior to coming to OSU in 1988. Professor Dray is a Fellow of the American Physical Society for his work in relativity, and an award-winning teacher.

1

Introduction

1.1 Newton's Relativity

Our daily experience leads us to believe in Newton's laws. When you drop a ball, it falls straight down. When you throw a ball, it travels in a uniform (compass) direction—and falls down. We appear to be in a constant gravitational field, but apart from that no forces act on the ball. This isn't the full story, of course, because we are ignoring such things as air resistance and the spin of the ball. Nevertheless, Newton's laws seem to give a pretty good description of what we observe, and so we base our intuitive understanding of physics on them.

But Newton's laws are wrong.

Yes, gravity is more complicated than this simple picture. The earth's gravitational field isn't really constant. Furthermore, other nearby objects act gravitationally on us, notably the moon. As discussed in the final chapter of this book, this action causes tides.

A bigger problem is that the earth is round. Anyone who flies from San Francisco to New York is aware that due east is not a straight line, defined in this case as the shortest distance between two points. In fact, if you travel in a straight line that (initially) points due east from my home in Oregon, you will eventually pass to the south of the southern tip of Africa!¹

So east is not east.

But the real problem is that the earth is rotating. Try playing catch on a merry-go-round! Balls certainly don't seem to travel in a straight line! Newton's laws don't work here, and, strictly speaking, they don't work on (that is, in the reference frame of) the earth's surface. The motion of a Foucault pendulum can be thought of as a Coriolis effect caused by an external pseudoforce. Also, a plumb bob doesn't actually point toward the center of the earth!

 

¹You can check this by stretching a string on a globe so that it goes all the way around, is as tight as you can make it, and goes through Oregon in an east/west direction.

So down is not down.

1.2 Einstein's Relativity

All of the problems above come from the fact that, even without worrying about gravity, the earth's surface is not an inertial frame. An inertial frame is, roughly speaking, one in which Newton's laws do hold. Playing catch on a train is little different from on the ground—at least in principle, as long as the train is not speeding up or slowing down. Furthermore, an observer on the ground would see nothing out of the ordinary, it merely being necessary to combine the train's velocity with that of the ball.

However, shining a flashlight on a moving train—especially the description of this from the ground—turns out to be another story, which we study in more detail in Chapters 2 and 6. Light doesn't behave the way balls do, and this difference forces a profound change in our description of the world around us. As we will see, this forces moving objects to change in unexpected ways: Their clocks slow down, they change size, and, in a certain sense, they get heavier.

So time is not time.

Of course, these effects are not very noticeable in our daily lives, any more than Coriolis forces affect a game of catch. But some modern conveniences, notably global positioning technology, are affected by relativistic corrections.

The bottom line is that the reality is quite different from what our intuition says it ought to be. The world is neither Euclidean nor Newtonian. Special relativity isn't just some bizarre theory; it is a correct description of nature (ignoring gravity). It is also a beautiful theory, as I hope you will agree. Let's begin.

2

The Physics of Special Relativity

In which it is shown that time is not the same for all observers.

2.1 Observers and Measurement

This chapter provides a very quick introduction to the physics of special relativity, intended as a review for those who have seen it before, and as an overview for those who have not. It is not a prerequisite for the presentations in subsequent chapters.

Special relativity involves comparing what different observers see. But we need to be careful about what these words mean.

A reference frame is a way of labeling each event with its location in space and the time at which it occurs. Making a measurement corresponds to recording these labels for a particular event. When we say that an observer “sees” something, what we really mean is that a particular event is recorded in the reference frame associated with the observer. This has nothing to do with actually seeing anything—a much more complicated process that would involve keeping track of the light reflected into the observer's eyes. Rather, an “observer” is really an entire army of observers who record any interesting events and an “observation” consists of reconstructing from their journals what took place.

2.2 The Postulates of Special Relativity

Postulate I is the most fundamental postulate of relativity.

Postulate I: The laws of physics apply in all inertial reference frames.

The first ingredient here is a class of preferred reference fram...