«Getting the Measure of Einstein’s Space and Time An Introduction to Special Relativity Len Zane 12/12/2014 The space and time introduced by Albert ...»
UNIVERSITY OF NEVADA, LAS VEGAS
Getting the Measure
of Einstein’s Space
An Introduction to Special Relativity
The space and time introduced by Albert Einstein in 1905 is explained by examining a series of
simple thought or “gedanken” experiments. The development makes extensive use of
spacetime diagrams to help readers appreciate the full extent of these changes.
Table of Contents Introduction
Chapter One: Space and Time Before 1905
Speed and Velocity
The Experimenters are Introduced
The First Set of Experiments: Earth-Based
The Second Set of Experiments: Bus-Based
Bev and Anne Carefully Observe the Experiment Done on the Bus
The Addition of Velocities Formula
A Picture is Worth a Thousand Words
The Galilean Transformation Equations
Summary of Chapter One
The Principle of Relativity and the Isotropy of Space
Space and Time pre-1905
Chapter Two: The Speed of Light
Chapter Three: Space and Time After 1905
A New Set of Experiments are Proposed
New Stop Watches are distributed
Anne and Bev Measure the Speed of Light
The Super Bus Rolls into the Story
Anne and Bev Observe the Light Flash on the Bus: I
Anne and Bev Observe the Light Flash on the Bus: II
The Effect of Motion on Space
The Effect of Motion on Time
Light Moves through Space and Time in a Very Strange Way
Chapter Four: Generalizing the Observations
Defining the New Problem
Finding the Shrinkage Factor
The Ticking Rate of Moving Clocks
Quantifying the Disagreement over Synchronization
Testing the Theory
A Question about Synchronization
Global Positioning System (GPS)
Chapter Five: The Relationship between Bus and Earth Observers
Combining Spacetime Graphs
Bev Generalizes Chuck’s Spacetime Diagram
A Simple Example for Dean
The Revised Equation for the Addition of Velocities
Chapter Six: Chuck Suggests an Experiment and Anne has a Dream
Answers for Experiment I
Answers for Experiment II
Anne has a Dream about a Superluminal Pigeon
Chapter Seven: The Bus and the Garage
The Collision between Theory and Reality
R.I.P. Super Bus
Chapter Eight: The Solar System and Beyond
Synchronizing Watches Separated by Large Distances
Chapter Nine: Chuck and Dean Travel to Alpha Centauri
May 1, Thirteen Years Later
Using the Lorentz Equations
Looking at a Moving Watch – The Doppler Effect
Chapter Ten: Space Travel
The Equivalence Principle
Designing the Space Adventure
The Results of Chuck’s Calculations
How does a Rocket Accelerate?
Bonus Topic: A Peek at General Relativity
Chapter Eleven: Does it Really Happen?
The Far-Away Observer (FAO)
A Dynamic Explanation for the Behavior of Watches and Rulers
A Model for a Hydrogen Atom
Chapter Twelve: E = mc2
Appendix A: Graphing
Appendix B: Scientific Notation
Appendix C: The Light Clock
Appendix D: The Gravitational Clock Effect
Introduction This book was written after many years of teaching special and general relativity to students with varied backgrounds. The presentation emphasizes spacetime diagrams which in my experience helps students visualize space and time. The material is selfcontained, allowing the book to be used as a tutorial for a person with little background in but some familiarity with algebra and a healthy curiosity about special relativity.
There are questions scattered throughout the book to encourage the reader to take some time to review the material presented before moving on to new material. The book can also be used as a textbook for a course in space and time for non-science students, the audience I primarily had in mind when writing it, or as an introductory course for students planning to study science later in their undergraduate careers. Please feel free to contact me at firstname.lastname@example.org if you have any questions or suggestions.
Chapter One establishes a simple methodology for measuring speeds and velocities at non-relativistic values that are encountered in everyday life. In particular, the velocity of an object moving in a bus that is traveling down the road is measured by people on the bus and simultaneously by observers standing on the ground. These measurements are used to probe the rules that govern space and time.
Chapter Two presents a short summary of light’s properties. For us, the most important property is light’s astonishingly large velocity.
Chapter Three replaces the bus in Chapter One with an imaginary one that can move at relativistic speeds. This new “super” bus is used to probe space and time by doing experiments analogous to those done in Chapter One. These new experiments force our experimenters to drastically revise the rules governing space and time that emerged from Chapter One.
Chapter Four and Five generalize the results of the experiments done in Chapter Three culminating in a derivation of the Lorentz transformation equations and the relativistic version of the addition of velocity equation first encountered in Chapter One. Chapter Four ends with a section on how a GPS verifies that moving clocks run slow and that clocks further from the center of Earth run fast.
Chapter Six is primarily a tutorial designed to give readers a chance to review all the earlier material by using spacetime graphs and the Lorentz Transformation equations to analyze two imaginary experiments done with the super bus.
Chapter Seven describes and analyzes the well-known “Pole and Barn” problem and attempts to give a definitive answer to the question, does the pole fit in the barn or not?
In this chapter, the super bus replaces the pole and a newly constructed garage takes the place of the barn.
Chapter Eight introduces the astronomical distance unit, the light year, and gives a brief description of our home galaxy, the Milky Way.
Chapter Nine analyzes in some detail the famous “Twin Paradox.” This iconic paradox of special relativity arises when one of the twins travels away from Earth in a space ship and returns years later. Upon returning, it is discovered that the twin that remained on Earth aged more than the traveling twin.
Chapter Ten uses the laws governing space and time to study the motion of a rocket that moves with constant acceleration. The accelerating rocket is used to examine the possibilities of human travel to other galaxies. The last section of Chapter Ten explains why a clock further from the center of Earth runs faster than an identical closer to the center.
Chapter Eleven gives two separate arguments designed to show that rulers really do shrink and watches actually run slow and these effects are not just illusory or “theoretical.” Chapter Twelve gives a simple “derivation” of Einstein’s famous equation, E = mc2.
Though not directly related to the primary theme of the book, it is difficult to write a book that purports to cover special relativity and not include that equation.
Chapter One: Space and Time Before 1905 This chapter examines some simple experiments that establish the space and time of Galileo and Newton. This view prevailed until 1905 when Albert Einstein introduced the world to Special Relativity. It is also the space and time of our everyday lives where rulers and watches are well behaved.
Speed and Velocity The concepts of speed and velocity are colloquially understood to be a distance traveled divided by the elapsed time. In this chapter, velocities and speeds will be measured in feet/second or ft/s.
To physicists, speed and velocity are related but not interchangeable ideas. Velocity includes a sense of direction in its definition. Throughout this book, anything or anybody moving from left to right will have a positive velocity, for example +10 ft/s. Note that a velocity of +10 ft/s is the same as 10 ft/s. While any object or person moving from right to left will have a negative velocity, for example -10 ft/sec,. The speed of an object is the size, or magnitude, of its velocity.
In the above examples, both objects would have the same speed, 10 ft/sec, although those speeds would be in opposite directions.
Later we will look very carefully at how speed is measured. But for now, it is clear that to find the velocity or speed of an object, it is necessary to measure both a distance covered and the time it took to traverse that distance. Rulers and stop watches are the usual instruments used to measure distance and time. Imagine a warehouse full of excellently fabricated rulers and stop watches. All the rulers are identical to one another and the same is true for the stop watches.
These instruments will be handed out to specially-trained observers who will collect the data used to determine the speed or velocity of test objects in a variety of circumstances. A key point to keep in mind is that the experiments described are all perfectly reasonable and doable, at least in principle, though some may be technologically too challenging to be done with currently available rulers, stop watches, and observers.
The Experimenters are Introduced Anne, Bev, Chuck, and Dean are good friends, astute observers, and curious by nature.
One evening, after watching an episode of Star Trek, they begin talking about space and time and Special Relativity, subjects none of them knows very much about. Finally Bev suggests that instead of speculating about the meaning of Special Relativity, they ought to do some careful experiments to get first-hand knowledge about space and time. Her three friends quickly agree, thinking it could be enlightening to get an experimentally based understanding of the nature of space and time.
Anne, Bev, Chuck, and Dean are the main characters in this book. Anne and Bev are stationed on Earth during all the experiments with Bev always being to the right of Anne. Conversely, Chuck and Dean are the traveling pair of experimenters. They ride in buses and rocket cruisers with Dean always situated to the right of Chuck. A & B and C & D are shorthand for Anne and Bev and Chuck and Dean.
The sections in italics represent the author interjecting himself into the narrative. The hope is that these asides will add to and not disrupt the main story, the experimental probing of space and time.
The next morning, bright and early, our friends meet to map out a set of experiments.
They recall that Special Relativity has something to do with light and its velocity and the way it is perceived by different observers moving with respect to one another.
Though they are not sure what it is about light that is so peculiar, they do know that the speed of light is very, very large. Anne suggests that they do some simple experiments with something that moves at a pedestrian speed; for example, one of the trained pigeons that she has seen in the park. These pigeons all fly at exactly 20 ft/s.
At the time scientists were grappling with the concepts of space and time, there was general agreement that light was a wave. One of the principle characteristics of a wave is that it travels through some medium. For example; sound travels through air and other material substances.
The speed of a wave is the speed at which it moves through that medium. Scientists imagined space being permeated by an ethereal substance, the “lumeniferous ether,” through which light moved. The speed of a pigeon is the speed that it moves through air. This is analogous to the way a wave moves through a medium, or more specifically the way scientists pictured light moving through the ether. Thus Anne’s choice of a pigeon as the object to study was reasoned and not fortuitous.
Dean volunteers to head to the park to sign up a pair of pigeons for their experiments.
After he leaves, his friends decide to break themselves into two teams--Anne and Bev and Chuck and Dean. Anne and Bev will be the team that measures things from the perspective of Earth while C & D will ride in a laboratory that moves with respect to Earth. Chuck agrees with this plan and immediately heads down to the used bus lot to find a vehicle that could be appropriately modified to become the rolling laboratory for him and Dean.