Einstein for Everyone

HPS 0410 Einstein for Everyone

HPS 0410

Einstein for Everyone

Spring 2010

LecturesCourse Description ScheduleOld Schedule before snow closing.

Assignments1. Principle of Relativity 2. Adding Velocities Einstein's Way 3. Relativity of Simultaneity 4. Origins of Special Relativity 5. Spacetime 6. Philosophical Significance 7. Non-Euclidean Geometry 8. Curvature 9. General Relativity 10. Relativistic Cosmology 11. Big Bang Cosmology12. Black Holes Not required for submission

Title page, Preface and Table of Contents for Einstein for Everyone Introduction: the questions Special relativity: the basics Special relativity: adding velocities Special relativity: the relativity of simultaneity Is special relativity paradoxical? E=mc 2 Origins of Special Relativity Einstein's Pathway to Special Relativity Spacetime Spacetime and the Relativity of Simultaneity Spacetime, Tachyons, Twins and Clocks What is a four dimensional space like? Philosophical Significance of the Special Theory of Relativity. Euclidean Geometry: The First Great Science Non-Euclidean Geometry: A Sample Construction Spaces of Constant Curvature Spaces of Variable Curvature General Relativity Gravity Near a Massive Body Einstein's Pathway to General Relativity Relativistic Cosmology Big Bang Cosmology Black Holes A Better Picture of Black Holes Atoms and the Quantum

Term paperClockSign in sheet

13. Origins of Quantum Theory 14. Problems of Quantum Theory

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HPS 0410 Einstein for Everyone

Origins of Quantum Theory Quantum Theory of Waves and Particles The Measurement Problem Einstein on the Completeness of Quantum Theory Einstein as the Greatest of the Nineteenth Century PhysicistsFor documents relating to the Fall 2008 offering of this class, click here. For documents relating to the Spring 2008 offering of this class, click here. For documents related to the Spring 2007 offering of this class, click here.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/index.html[28/04/2010 08:17:32 ]

HPS 0410 Course Description

HPS 0410


Spring 2010Back to main course page

Lectures Monday/ Wednesday 1:00 pm - 1:50 pm, CL 232 Recitations

(John D. Norton)

(Register for one.)

Monday 3-3:50 pm, CL 216 (Julia Bursten) Monday 5-5:50 pm, CL 229 (Emi Iwatani) Tuesday 12-12:50 pm, CL 327 (Julia Bursten) Tuesday 1-1:50 pm, CL 327 (Emi Iwatani) Tuesday 3-3:50 pm, CL 129 (Julia Bursten) Tuesday 4-4:50 pm, CL 129 (Emi Iwatani) Instructors John D. Norton, 412-624-1051, [email protected] Room 817 CL. Office hours: Monday 2-3 pm, Wednesday 2-3 pm. Julia Bursten , [email protected] Room 901H CL. Office hours: Tuesday 1-2, Wednesday 12-1. Emi Iwatani, [email protected] Room 901M CL. Office hours: Monday 2-3pm, Tuesday 2-3 pm. Course website Course materials will be posted at the course website http://www.pitt.edu/~jdnorton/teaching/HPS_0410Click here http://www.pitt.edu/~jdnorton/teaching/HPS_0410

We will communicate grades through the Blackboard website at https://courseweb.pitt.edu/ These websites will be the primary means of obtaining course material. To take this course, you must have access the internet. TopicsSpecial relativity: The two postulates and their strange consequences: rods and clocks run amuck. The light barrier. Relativity of simultaneity: the confusion of when and where and the puzzles it solves. Spacetime: time as the fourth dimension. Origins of special relativity: how did Einstein do it?. Puzzles and paradoxes. The most famous equation: E=mc 2. The philosophical dividend.

General relativity: Straightening out Euclid. Acceleration provides the clue: gravitation is just spacetime bent. General relativity passes the tests. Applications of general relativity: Goedel universes and the like: could we take a journey into the past? Cosmology: the biggest picture possible; a beginning and end for time? Black holes: when the fabric of spacetime collapses.http://www.pitt.edu/~jdnorton/teaching/HPS_0410/description.html[28/04/2010 08:17:34 ]


Quantum theory: The puzzle of black body radiation: light comes in lumps. The Bohr atom: where electrons jump. The perversity of matter in the small: both particle and wave. The uncertainty principle. The failure of determinism. The puzzle of Schrdinger's cat: neither alive nor dead.

Assessment Short tests There will be 6 short in-class tests, roughly one each two weeks. (Schedule) The grade is the best 5 of 6. Recitation The grade is divided between assignments (25%) and recitation participation (10%). An assignment is due each week in the recitation. The assignment grade is the best 11 of 14.After cancellation of classes February 8-10, the assignment grade is reset at the best 10 of 13.

35%

35%

Term paper The term paper is by electronic submission to your recitation instructor on the day of the final lecture, Wednesday April 21.

30%

Short Test The short tests will examine material covered roughly in the preceding two weeks. They will be held in the first 15 minutes of class and consist of a series of 3-4 related questions requiring a few sentences each as answers. Policy on Missed Tests and Late Assignments No make up tests will be offered. Since the test grade is the best 5 of 6, one missed test is automatically forgiven. It is strongly recommended that this one forgiven test be used only when illness or emergencies preclude class attendance. Assignments are due each week at the start of the recitation. Late assignments are not accepted. Since the assignment grade is the best 11 of 14, three missed assignments are automatically forgiven. It is strongly recommended that these forgiven assignments be used only when illness or emergencies preclude class attendance.(An exception is made for students who add the course after the start of term. Assignments due prior to the date on which the class was added may be submitted at the next scheduled recitation.)

For added flexibility, a universal makeup assignment is offered to all students. The makeup assignment is a second term paper conforming to the term paper guidelines, but only 500 words in length, due on the day of the last lecture, Wednesday April 21. What do I do if a university break cancels a recitation in which an assignment is due? There will be no recitation held on Martin Luther King Day, Monday, January 18. Assignment 2, due in these cancelled recitations, may be submitted to the recitation instructor at the beginning of the lecture that immediately follows the cancelled recitation

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on Wednesday January 20. Texts The primary text for the class is available on this website as the online text Einstein for Everyone. Supplementary readings are: J. Schwartz and M. McGuinness, Einstein for Beginners. New York: Pantheon. J. P. McEvoy and O. Zarate, Introducing Stephen Hawking. Totem. J. P. McEvoy, Introducing Quantum Theory. Totem. Special Needs If you have a disability for which you are or may be requesting an accommodation, you are encouraged to contact both your instructor and Disability Resources and Services, 216 William Pitt Union, 412-648-7890 or 412-383-7355 (TTY) as early as possible in the term. For more information, see http://www.drs.pitt.edu/The Undergraduate Dean of Arts and Sciences has requested instructors to alert all students to University of Pittsburgh Policy 09-10-01, "E-mail Communications Policy."

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/description.html[28/04/2010 08:17:34 ]

HPS 0410 Schedule

HPS 0410


Spring 2010Back to main course page Clock

ScheduleSchedule as revised after snowstorm closings of February 8-10. Old schedule here. Week 1 2 Lecture Recitation Date Date Wed. Jan. 6 Mon. Jan. 11 Mon. Jan. 11 Tues. Jan. 12 Wed. Jan. 13NO CLASS Mon. Jan. 18. Martin Luther King DayTues. Jan. 19 Add/drop ends

Lecture Topic Introduction: the questions. Special relativity: the basics.

Assignment Due

Test

1. Principle of Relativity Special relativity: adding velocities. Relativity of simultaneity

3

Submitting assignments due on Monday

Tues. Jan. 19 Wed. Jan. 20 4 Mon. Jan 25 Mon. Jan. 25 Tues. Jan. 26

2. Adding Velocities Einstein's Way Is special relativity paradoxical? E=mc 2 3. Relativity of Simultaneity Test 1What

Wed.

Origins of special relativity

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HPS 0410 Schedule

Jan. 27

Einstein's Pathway to Special Relativity

is it on? Grades

5

Mon. Feb. 1 Mon. Feb. 1 Tues. Feb. 2

Spacetime Spacetime and the Relativity of Simultaneity 4. Origins of Special Relativity Spacetime and the Relativity of Simultaneity Spacetime, Tachyons, Twins and ClocksClasses cancelled this week because of snowstorm. This is a revised schedule of classes. Old schedule here. Mon. Feb. 8 Tues. Feb. 9

Wed. Feb. 3

6

Mon. Feb. 8

`Wed. Feb. 10

7


What is a four dimensional space like? Philosophical significance of relativity 5. Spacetime Test 2 Philosophical significance of relativity

Wed. Feb. 17

What is it on? Grades

8

Mon. Feb. 22 Mon. Feb. 22 Tues Feb. 23 Wed. Feb. 24

Euclidean Geometry: The First Great Science Non-Euclidean Geometry: A Sample Construction 6. Philosophical Significance Non-Euclidean Geometry: A Sample Construction Spaces of Constant Curvature Spaces of Constant Curvature Spaces of Variable Curvature

9

Mon. Mar. 1


HPS 0410 Schedule

Mon. Mar. 1 Tues. Mar. 2

7. NonEuclidean Geometry Test 3

Wed. Mar. 3

General relativity


SPRING BREAK 10 Mon. Mar. 15 Mon. Mar. 15 Tues. Mar. 16 Wed. Mar. 17 11 Mon. Mar. 22 Mon. Mar. 22 Tues. Mar. 23 Gravity Near a Massive Body Einstein's Pathway to General Relativity Relativistic cosmology 9. General Relativity Test 4 Wed. Mar. 24 Relativistic cosmology General relativity 8. Curvature


12

Mon. Mar. 29 Mon. Mar. 29 Tues. Mar. 30 Term paper topic submitted Wed. Mar. 31

Big bang cosmology

10. Relativistic Cosmology

Big bang cosmology/ Black holes Black holesOptional: A Better Picture of Black Holes

13

Mon. Apr. 5


HPS 0410 Schedule

Mon. Apr. 5 Tues. Apr. 6

11. Big Bang Cosmology Test 5 Origins of Quantum Theory

Wed. Apr. 7


14

Mon. Apr. 12 Mon. Apr. 12 Tues. Apr. 13 Wed. Apr. 14

Origins of Quantum Theory 13. Origins of Quantum Theory Quantum Theory of Waves and Particles The Measurement Problem Mon. Apr. 19 Tues. Apr. 20 14. Problems of Quantum Theory Einstein on the Completeness of Quantum Theory Test 6What is it on?

15

Mon. Apr. 19

Wed. Apr. 21

Term paper due

Test 1. Wednesday January 27. The test will be in the first 15 minutes of class and will consist of 3-4 questions requiring answers of a few sentences each. The material examinable is the content of the chapters "Special relativity: the basics," "Special relativity: adding velocities," "Relativity of simultaneity," "Is special relativity paradoxical?" and the assignments 1-3. Test 2. Wednesday February 17. The material examinable is the content of the chapters "E=mc 2 ", "Origins of Special Relativity," "Einstein's Pathway to Special Relativity," the three "Spacetime" chapters and the assignments 4 and 5. Test 3. Wednesday March 3. The material examinable is the content of the chapters "Philosophical Significance of Relativity," the chapters on Euclidean and Non-Euclidean Geometry and Spaces of Constant Curvature; and the assignments 6 and 7. Test 4. Wednesday March 24. The material examinable is the content of the chapters "Spaces of Variable Curvature," "General Relativity," "Gravity Near a Massive Body" and "Einstein's Pathway to General Relativity"; and the assignments 8 and 9. Test 5. Wednesday April 7. The material examinable is the content of the chapters "Relativistic Cosmology" and "Big Bang Cosmology" and the assignments 10 and 11.


HPS 0410 Schedule

Test 6. Wednesday April 21. The material examinable is the content of the chapters "Black Holes," "Origins of Quantum Theory," as much as we have covered of "Quantum Theory of Waves and Particles," "The Measurement Problem," "Einstein on the Completeness of Quantum Theory" and the assignments 13 and 14.


HPS 0410 Schedule

HPS 0410


Spring 2010Back to main course page Clock

ScheduleThis is the term's OLD schedule what has been modified as a result of the cancellation of classes on February 8-10 due to snowstorms. The new schedule is here. Week 1 2 Lecture Date Wed. Jan. 6 Mon. Jan. 11 Mon. Jan. 11 Tues. Jan. 12 Wed. Jan. 13NO CLASS Mon. Jan. 18. Martin Luther King DayTues. Jan. 19 Add/drop ends

Recitation Date Lecture Topic Introduction: the questions. Special relativity: the basics.

Assignment Due

Test

1. Principle of Relativity Special relativity: adding velocities. Relativity of simultaneity

3

Submitting assignments due on Monday

Tues. Jan. 19 Wed. Jan. 20 4 Mon. Jan 25 Mon. Jan. 25 Tues. Jan. 26 Wed. Jan. 27 Origins of special relativity Einstein's Pathway to Special Relativity Is special relativity paradoxical? E=mc 2

2. Adding Velocities Einstein's Way

3. Relativity of Simultaneity Test 1What is it on? Grades

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/schedule_old.html[28/04/2010 08:17:38 ]

HPS 0410 Schedule

5


Spacetime Spacetime and the Relativity of Simultaneity 4. Origins of Special Relativity Spacetime and the Relativity of Simultaneity Spacetime, Tachyons, Twins and Clocks What is a four dimensional space like? Philosophical significance of relativity Mon. Feb. 8 Tues. Feb. 9 5. Spacetime Philosophical significance of relativity Euclidean Geometry: The First Great Science Non-Euclidean Geometry: A Sample Construction Mon. Feb. 15 Tues. Feb. 16 Non-Euclidean Geometry: A Sample Construction Spaces of Constant Curvature Spaces of Constant Curvature Spaces of Variable Curvature Mon. Feb. 22 Tues Feb. 23 7. Non-Euclidean Geometry General relativity Test 3What is it on?

Wed. Feb. 3

6

Mon. Feb. 8

Wed. Feb. 10

Test 2What is it on?

7

Mon. Feb. 15

6. Philosophical Significance

Wed. Feb. 17

8

Mon. Feb. 22

Wed. Feb. 24


HPS 0410 Schedule

9

Mon. Mar. 1

General relativity Gravity Near a Massive Body Einstein's Pathway to General Relativity Mon. Mar. 1 Tues. Mar. 2 8. Curvature General relativity

Wed. Mar. 3 SPRING BREAK 10 Mon. Mar. 15 Mon. Mar. 15 Tues. Mar. 16 Wed. Mar. 17 11 Mon. Mar. 22 Mon. Mar. 22 Tues. Mar. 23 Wed. Mar. 24 12 Mon. Mar. 29 Mon. Mar. 29 Tues. Mar. 30 Term paper topic submitted Wed. Mar. 31 13 Mon. Apr. 5 Mon. Apr. 5 Tues. Apr. 6

Relativistic cosmology 9. General Relativity Relativistic cosmology Big bang cosmology 10. Relativistic Cosmology Big bang cosmology/ Black holes Black holes Test 4What is it on?

11. Big Bang Cosmology

A Better Picture of Black Holes A Better Picture of Black Holes 12. Black Holes

Test 5What is it on?


HPS 0410 Schedule

Wed. Apr. 7 14 Mon. Apr. 12 Mon. Apr. 12 Tues. Apr. 13 Wed. Apr. 14 15 Mon. Apr. 19 Mon. Apr. 19 Tues. Apr. 20 Wed. Apr. 21 Term paper due

Origins of Quantum Theory Origins of Quantum Theory 13. Origins of Quantum Theory Problems of Quantum Theory Problems of Quantum Theory 14. Problems of Quantum Theory Problems of Quantum Theory Test 6What is it on?

Test 1. Wednesday January 27. The test will be in the first 15 minutes of class and will consist of 3-4 questions requiring answers of a few sentences each. The material examinable is the content of the chapters "Special relativity: the basics," "Special relativity: adding velocities," "Relativity of simultaneity," "Is special relativity paradoxical?" and the assignments 1-3. Test 2. Wednesday February 10. The material examinable is the content of the chapters "E=mc 2 ", "Origins of Special Relativity," "Einstein's Pathway to Special Relativity," the three "Spacetime" chapters and the assignments 4 and 5. Test 3. Wednesday February 24. The material examinable is the content of the chapters "Philosophical Significance of Relativity" and "Non-Euclidean Geometry" and the assignments 6 and 7. Test 4. Wednesday March 17. The material examinable is the content of the chapters "Spaces of Variable Curvature" and "General Relativity" and the assignments 8 and 9. Test 5. Wednesday March 31. The material examinable is the content of the chapters "Relativistic Cosmology" and "Big Bang Cosmology" and the assignments 10 and 11. Test 6. Wednesday April 21. The material examinable is the content of the chapters "Black Holes," "A Better Picture of Black Holes" and "Origins of Quantum Theory" and the assignments 12 and 13.


HPS 0410 Term Paper

HPS 0410


Spring 2010

Term PaperAn Amazing Scientific DiscoveryDue by final lecture: Wednesday April 21 Submit in electronic form to recitation instructor 1000 words Topic selection Due in recitation: Mon., Mar. 29/ Tues., Mar. 30

ProjectThis course is a parade of amazing scientific discoveries. They are things that would never occur to us ordinarily: that there may be no fact as to whether two events are simultaneous; that energy and matter are the same thing; that gravity is just funny geometry; that time had a beginning; and more. What makes these all the more amazing is that they are not conjurings of fiction. They are our best attempts to describe how our world really is; and science can tell us a cogent and compelling story as to why we should believe them. For your term paper, you are to identify and describe an amazing idea. Your text should contain: 1. A clear explanation of the amazing scientific discovery. 2. An account of how the discovery was made. Your amazing idea must be drawn from standard science. The goal is not to report on wild speculation that someone, someday thinks might become regular science. You are to seek an amazing discovery that has already become regular science. If you are unsure whether an amazing idea is drawn from standard science, ask if it has experimental or observational evidence in its favor. If it doesn't, it is speculation! Your paper must present material not already covered in lectures and recitations. For this reason you are best advised to write about an amazing idea not already covered in the class. If you do choose one we have covered in class, note that your grade will depend entirely on the extent to which you go beyond class material. Your paper must present novel text written specifically for this class. Because of the breadth of the assignment, you may find you already have something written for another class that suitshttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/paper.html[28/04/2010 08:17:39 ]

HPS 0410 Term Paper

the assignment. You may not "recycle" text written for another class. The point of this assignment is for you to do new research and write new text. Focus on the rational basis of the discovery. Your account of how the discovery was made should focus on what led the scientist or scientists to the discovery and the reasons that they found to believe in its correctness. You need not distract yourself with incidental biographical or other background facts unless they are important to understanding the grounding of the discovery. Keep the discovery narrow. It is easy to tackle too big a topic. Modern cosmology as theory is far too big for this project. One discovery in it--such as the presence of dark matter in galaxies-is already quite a big enough topic for this paper. If in doubt, narrow the topic. The discovery must be in science and not technology. While the achievements of modern technology are amazing, they are not our concern in this paper. You should be looking at things we know, not things we make. Sometimes the latest technology has an amazing scientific discovery behind it; that discovery could be the focus of a paper. If you do decide to pursue a scientific discovery that lies behind some new advance in technology, be careful; very often those discoveries are complicated and can make the paper hard to write.

Selection of TopicA brief statement of the amazing idea selected is due in the recitation, Monday, March 29/ Tuesday, March 30. Submit it as one paragraph, on paper. 1/10th of the term paper grade is assigned for submitting a suitable statement on time. (These are easy points earned just for being on time!) Consult with your recitation instructor if you are uncertain over the idea or need assistance in locating a suitable one.

PresentationThe paper should be headed with your name, the title of the paper and the course to which it is being submitted. The paper should have an introduction and conclusion and be divided into appropriately headed sections. A standard system for footnoting and for referencing your sources must be adopted and used consistently throughout. Consult a guide on writing term papers if you are unsure of such systems. We expect your writing to be clear and simple. That applies both to the thoughts expressed and the words used. The thoughts should develop naturally in small, clear steps. The wording should be plain and direct and the sentences short. There is no gain in a big word, when a little one will do. We expect proper grammar and correct spelling and will penalize major excursions.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/paper.html[28/04/2010 08:17:39 ]

HPS 0410 Term Paper

SubmissionYour paper is to be submitted to us in electronic form via turnitin.com, a plagiarism prevention web resource. Here are the instructions for submitting your paper: 1. Visit http://turnitin.com. 2. Click New Users in the upper right corner. 3. Please contact recitation instructor to obtain the appropriate Turnitin Class ID number and Class Enrollment Password. 4. Finish the registration process. 5. Click on the Einstein for Everyone class link. 6. Click on the Submit icon in the row marked Paper. 7. Upload your paper. Acceptable formats for your paper are MS Word, WordPerfect, PostScript, PDF, HTML, RTF, and plain text. You should also submit your extra credit paper, if you choose to do one, by clicking on the Submit icon in the row marked Extra Credit Paper. All papers (including extra credit papers) must be submitted by midnight of the due date.

Use of SourcesAs is standard in all academic writing, the wording of your paper should be your own; it should not be copied or paraphrased even loosely from another source. If you are uncertain over the correct use of sources, see this Guide.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/paper.html[28/04/2010 08:17:39 ]

Clock

HPS 0410


Spring 2010

Einstein's Time is ...

Main course page Schedule

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HPS 0410 Sign In

HPS 0410


Spring 2010

Name:_______________________________

Major:________________________________

Level:________________________________

Is there anything in particular you would like to cover in this course?

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JOHN D. NORTON

Nullarbor Press 2007revisions 2008, 2010

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/index.html[28/04/2010 08:17:47 ]


Copyright 2007, 2008, 2010 by Nullarbor Press Published by Nullarbor Press, 500 Fifth Avenue, Pittsburgh, Pennsylvania 15260 with offices in Liberty Ave., Pittsburgh, Pennsylvania, 15222 All Rights Reserved

John D. Norton Center for Philosophy of Science Department of History and Philosophy of Science University of Pittsburgh Pittsburgh PA USA 15260

An advanced sequel is planned in this series: Einstein for Almost Everyone

246897531 ePrinted in the United States of America no trees were harmed web*book TM



PrefaceFor over a decade I have taught an introductory, undergraduate class, "Einstein for Everyone," at the University of Pittsburgh to anyone interested enough to walk through door. The course is aimed at people who have a strong sense that what Einstein did changed everything. However they do not know enough physics to understand what he did and why it was so important. The course presents just enough of Einstein's physics to give students an independent sense of what he achieved and what he did not achieve. The latter is almost as important as the former. For almost everyone with some foundational axe to grind finds a way to argue that what Einstein did vindicates their view. They certainly cannot all be right. Some independent understanding of Einstein's physics is needed to separate the real insights from the never -ending hogwash that seems to rain down on us all. With each new offering of the course, I had the chance to find out what content worked and which of my ever so clever pedagogical inventions were failures. By this slow process of trial and error, indulging the indefinitely elastic patience of the students at the University of Pittsburgh, the course has grown to be something that works pretty well--or so it seems from my side of the lectern. At the same time, my lecture notes have evolved. They began as chaotic pencil jottings. Over time they solidified into neater pencil script and overhead transparencies; and then into summaries that I posted on my website; and then finally those summaries were expanded into a full text that can be read independently. That text is presented here. Its content reflects the fact that my interest lies in history and philosophy of science and that I teach in a Department of History and Philosophy of Science. There is a lot of straight exposition of Einstein's physics and the physics it inspired. However there is also a serious interest in the history of Einstein's science. A great deal of my professional life has been spent poring over Einstein's manuscripts, trying to discern how he found what he found. The results of those studies have crept in. In other places I try to show how a professional philosopher approaches deeply intractable foundational issues. The temptation in such cases is let one's standard of rigor drop, since otherwise it seems impossible to arrive at any decision. That is exactly the wrong reaction. When the problems are intractable, we must redouble our commitment to rigor in thought and I have tried to show how we can do this. This texts owes a lot to many. It came about because once Peter Machamer, then chair of the Department of HPS, urged a meandering junior professor to do a course that "did" Einstein and black holes and all that stuff. The text is indebted to the University ofhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/index.html[28/04/2010 08:17:47 ]


Pittsburgh, which has the real wisdom to see that it gets the most from its faculty by letting them do what fascinates them, for they will surely do that best. It owes the greatest debt to the infinite patience of the students who have taken this class, told me what works and what does not, and each year allow me at least indirectly to experience anew that inescapable sense of wonder when one first grasps the beauty of what Einstein did.iii

ContentsPreface 1. Introduction 2. Special Relativity: The Basics 3. Special Relativity: Adding Velocities 4. Special Relativity: Relativity of Simultaneity 5. Is Special Relativity Paradoxical? 6. E=mc2 7. Origins of Special Relativity 8. Einstein's Pathway to Special Relativity 9. Spacetime iii read read read read read read read read read



10. Spacetime and the Relativity of Simultaneity 11. Spacetime, Tachyons, Twins and Clocks 12. What is a Four Dimensional Space Like? 13. Philosophical Significance of the Special Theory of Relativity 14. Euclidean Geometry: The First Great Science 15. Non-Euclidean Geometry: A Sample Construction 16. Spaces of Constant Curvature 17. Spaces of Variable Curvature 18. General Relativity 19. Gravity Near a Massive Body 20. Einstein's Pathway to General Relativity 21. Relativistic Cosmology 22. Big Bang Cosmology 23. Black Holes 24. A Better Picture of Black Holes 25. Atoms and the Quanta 26. Origins of Quantum Theory 27. Quantum Theory of Waves and Particles 28. The Measurement Problem

read read read read read read read read read read read read read read read read read read read



29. Einstein on the Completeness of Quantum Theory 30. Einstein as the Greatest of the Nineteenth Century Physicistsiv

read read


Questions

HPS 0410

Einstein for EveryoneBack to main course page

QuestionsJohn D. Norton Department of History and Philosophy of Science University of PittsburghDo astronauts age more slowly? Can a finite universe have no edge? Can time have a beginning? Is time travel possible? Does the moon change because a mouse looks at it?

Here are the questions that were asked in the description in the course catalog... Answered.

Do astronauts age more slowly?

YESAccording to Einstein's special theory of relativity, all processes slow down when a system moves at high speed. The result applies to astronauts since they are moving rapidly. The amount of slowing is so slight as to be imperceptible for ordinary speeds. It becomes very significant when we get close to the speed of light:An astronaut is really just a quick way of saying "someone who travels away from the earth at high speed and returns."

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

Questions

Rocket at earth's Car at 100 velocity miles per hour (7 miles per second)

escape

Rocket at 100,000 miles per second (53% speed of light)

Rocket at 185,800 miles per second (99% speed of light)

Lose 0.35 (Astronaut is 0.022 seconds seconds in 1,000,000 years younger on returning after a one year trip.) Small effect...

Lose 0.022 seconds in 1 year

Astronaut metabolism slows to Astronaut 4.5% of normal. metabolism slows (One year to 84% of normal. journey aging 16 = days) ...large effect

How can special relativity know that these effects will happen? They arise directly from the basic supposition of the theory: all uniformly moving observers must measure the same speed for light.--186,000 miles per second. At first this seems impossible. Say I send out a light signal from earth. I measure its speed at 186,000 miles per second.

What about another observer that chases after the light signal at, say, half the speed of light. Shouldn't that observer see the light signal slowed to half its speed? All our common sense says yes. Special relativity says no.


Questions

How can that be? Something in our common sense assumptions must be wrong. There is not much room to look for the mistake. We find the speed of the light signal with just two instruments: a measuring rod to determine how far the light signal goes; and a clock to measure how long it takes to go that far. Classically we assume that neither is affected by rapid motion. At least one of these assumptions must be wrong if the speed of light is to remain constant. When we work through the details we find that both are: the rod shrinks in the direction of motion and the clock slows.

So rapidly moving clocks slow. How does that get to a rapidly moving astronaut aging more slowly. An astronaut's metabolism is a clock. You can use yourhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

Questions

pulse to time things if you like. So that metabolism clock must slow too. The legend is that Galileo used his pulse to time the period of a slowly swinging lamp while not attending to a cathedral mass and thereby arrived at the famous result of the isochrony of the pendulum, which just says that the period of a pendulum is fixed by its length. His pulse was the simple clock used to time the pendulum.

Can a finite universe have no edge?

YESWhat is this question asking? It is asking whether we could have a universe with a finite volume. That means if I ask "How many cubic miles of space are there?" the answer is not "infinity" but some definite number. It might be a big number. Say 63 kazillion cubic miles. But it is still a definite number, so that if you started to count off the cubic miles in space, you would eventually come to an end. At the same time it is asking if this finite universe could have no edge. An edge is just what you think. It is a place you get to where you run out of space.http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

Questions

Can both be possible at the same time? Can you run out of space in the sense that you count off all the cubic miles--but you never come to an edge? Both can indeed happen in a more restricted way in a very familiar example. Consider motions on the surface of the earth. If you start in Pittsburgh, choose any direction you like and keep moving straight ahead, you will eventually come back to where you started. There will be no edge for you to fall off. So the surface of the earth has the sort of properties we are looking for. It is finite in area. It just 196,000,000 square miles. But it has no edge.

Of course the example seems strained. While we come back to where we started, we are really not going in a straight line, but in a big circle. While the two dimensional surface of the earth is finite without edge, it gets these properties because it is really curved into a third dimension. Does that fact really make such a difference to the possibility of a surface of finite area but no edge? What

if we were flat beings trapped in the two dimensional surface of the earth , unable to sense

the existence of this third dimension. All we know about the surface of the earth was what can be read off our two dimensional maps. Then all we would know was that we lived in a finite two dimensional space with no edge.


Questions

That a third dimension might have something to do with this, to us would be speculation of little practical importance. We would have no way of accessing this third dimension. Could the analogous thing happen for a three dimensional space ? One of the big discoveries of 19th century geometry was that this is entirely possible. To get us started, imagine that there is a fourth dimension of space into which our three dimensions curve. Then we might end up with a three dimensional space which has finite volume but no edge. No matter which way you voyage in a spaceship, you will eventually come back to where you started, without hitting an edge.

We satisfied ourselves that this is possible by imagining a fourth dimension of space. How seriously should we take this fourth dimension? Our two dimensional surface dwellers could ignore the possibility of a third dimension in doing their geometry. All that mattered to them were the geometrical facts of the earth's surface that they could measure. In the three dimensional case, it is the same. All that matters are the geometrical facts about our three dimensional space that are accessible to us three dimensional beings. In the end, this fourth dimension of space becomes a comfortable fable to help us get used to the idea that a finite three dimensional space without edge is entirely possible. In the 19th century, this sort of space was an interesting mathematical curiosity. In 1917, shortly after Einstein had completed his general theory of relativity, he proposed


Questions

that our cosmic space was really like this. This was the first relativistic cosmology . Whether space has this structure remains one of the most interesting of the open questions of modern cosmology. In Einstein's original universe, space had a finite volume: 1,000,000,000,000,000,000,000,000,000,000 cubic light years That's a one followed by 30 zeros. But there is no edge.

Can time have a beginning?

YESAt first this seems impossible. If time has a beginning, there must be a first event or at least a clustering of events near it. Surely something must have happened before them? Einstein's cosmology of 1917 was the first of many ever stranger cosmologies to be devised on the basis of his general theory of relativity. Einstein's first universe was static in time. The cosmologies that followed, starting in the 1920s, were not. They portrayed space itself as continually expanding . We can think of Einstein's universe as a three dimensional analog of a two dimensional spherical surface, somewhat like a balloon. Then this expansion simply corresponds to the inflation of the balloon.http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

Questions

Here is a picture of this expansion. The universe is represented by a sphere and time advances up the screen. So the small universe of long ago grows up the page to the universe of the present.

As we look further and further back in time, the balloon gets smaller and smaller. In the typical cosmologies considered nowadays, not too long into the past the balloon would have shrivelled to nothing. At that point in our story, space would have ceased to be. One might try to image times before that moment. But it would be futile, since there is no space associated with the time. Indeed there is something highly suspect about the moment at which the balloon shrivels to a point. Then the curvature of the space becomes infinite and the basic equations of Einstein's theory break down. This first moment is not really a moment in time at all. It really amounts to a lower bound on our projections into the past.

imagine this expansion in reverse .Now


Questions

If we think of it as "time 0," then = only moments with a time coordinate greater than 0 have physical meaning. It is the beginning of time and is otherwise known at the "big bang."

To see why it is the "big bang", let us return our imagination to the forward direction and imagine what happens around the beginning of the expansion. Take any moment you like, as close as you like to the big bang. By choosing that moment closer and closer to the big bang, you can make space shrivel up as close to a point in size as you like. From that moment, everything-space and all its matter -- explodes outwards . All this happened not so long ago. It was around 10 billion years ago.

Is time travel possible?

YESThe "yes" is intriguing, but there is a catch. The questionhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

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did not ask if there really is time travel; it asked

only if it is possible. Something can be possible without actually happening. It is possible for our earth to have two moons. In fact it has only one. While we have no evidence that time travel actually occurs, all our latest work in theories of space and time tell us that it is entirely possible . Broadly speaking, there are two senses of time travel, both possible.

first sense is the the H. G. Wells sense. This one is named after the author of the most famous story about time travel in which a voyager hops into a machine and travels about in time. Special relativity has room for something close. If we had things that traveled faster than light, then, for some observers, they would travel backwards in time. These faster than light objects are "tachyons." For some observers, they would leave today and arrive yesterday. Of course how we could get ourselves to travel faster than light is an unsolved problem! We cannot accelerate through the speed of light. But is there some wayhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

1. The

The effects that bring this about are closely related to those that lead to the slower aging of rapidly moving astronauts. That effect depended on rapidly moving clocks not behaving as we expected. The time travel effect arises from anomalies in how observers in rapid motion set their clocks at different places in space

Questions

to recreate ourselves traveling faster than light? If so, some observers would judge us to be traveling backwards in time.

"There was a young lady named Bright, Whose speed was far faster than light. She set out one day In a relative way, And returned home the previous night."--Arthur Henry Reginald Buller.

called " Goedelian " (by John Earman) in honor of the great logician Kurt, Goedel, who was a friend of Einstein's and did pioneering work on spacetimes that admit time travel.

2. The second sense is more topological and has been


Questions

We can imagine space and time as forming a huge sheet of paper. the vertical line is the complete history through time of a person, experiencing the years ..., 1980, 1981, ... etc.

What Einstein did in 1917 was to get us to wrap up the sheet of paper in the spatial direction so travel in the direction "left" is wrapped around to meet travel in the direction "right". That way we always end up where we started.

What Einstein's theory also allows is that travel into the future of time can be wrapped around to connect with the past, so that if we persist long enough in time we end up back at the present.


Questions

This is less a type of time travel that we create with a machine. The best advice to someone who wants to travel in time this way is that they should be sure to be born into the right universe! However there are special circumstances that might bring it about. It might happen near black holes generated by gravitational collapse. It also may happen if we get very dense, very rapidly rotating matter.

Does the moon change because a mouse looks at it?


Questions

YESThis "yes" depends upon quantum mechanics, in whose founding Einstein played a major role. It is our best theory of matter and is usually applied to deal with matter in the very small, that is, little particles like electrons. It tells us that matter in the very small has properties quite unlike the ones we are used to with ordinary objects. We are used to the idea that ordinary objects are either particles or waves. It turns out that in the small, particles are both particles and waves . They have a dual character that is quite preplexing when you first learn of it and, as far as I can tell, that perplexity never really goes away, even if you know a lot about them.


Questions

Take electrons, for example. They are familiar to us from old-fashioned television tubes. The electrons are fired from a glowing element at the back of the tube. They are formed into a beam by deflecting magnetic fields. When the electron is in flight in the beam, it behaves just like a wave. It is spreads out in space, has a wavelength and frequency and can produce all sorts of wavelike phenomena, like interference patterns. These are just like the rippled patterns that water waves make on the surface of a pond when pebbles are dropped in. We can only get them because the waves are spread out in space. When these electrons strike the screen of the TV tube, they behave very differently. According to the standard text book accounts of quantum mechanics, they instantly cease to be wave. They collapse to a point, so they are now behaving like a particle . We see that localization through the emitting of a brief flash of light from just one point on the screen. (Many of those flashes combine to make the


Questions

images we watch.)

So sometimes an electron behaves like a wave; and sometimes like a particle. So what? The odd part is what decides whether the electron behaves like a wave or a particle. In the standard text book treatments, we decide by the act of observing the electron. An electron left to itself behaves like a wave. The moment we observe it --for example by having it smash into the screen of a TV tube so that we can see where it is from the flash of light produced --then it behaves like a particle. That is the odd part. Standard, text book quantum mechanics tells us that the act of our observing the electron has caused it to collapse to a point. This astonishing idea troubled Einstein very greatly and he could never accept it. What difference does it make to the electron if we observe it or not? What Einstein also saw was that the difficulty could not be confined to minute objects like electrons. If individual particles have this dual wave-particle, then so do collections of particles. Our observing of them will also cause them to collapse. Big objects like steam locomotives, moons and planets are just many, many particles all in one place. They will also have a slight wave character, too small for us to notice, but there nonetheless. And when we observe them, they will collapse!

His collaborator and biographer Abraham Pais reports "...during one walk, Einstein suddenly stopped, turned to me, and asked whether I really believed that the moon exists only when I look at it." The famous physicist (and inventor of the name "blackhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Questions/index.html[28/04/2010 08:17:54 ]

Questions

hole") John Wheeler also reported of Einstein "...No one can forget how he expressed his discomfort about the role of the observer, 'When a mouse observes, does that change the state of the universe?'" The question above is a combination of these two remarks and the answer of yes is just standard text book physics.

Copyright John D. Norton. February, 2002; July 2006; January 3, 2007.


Special Relativity Basics

HPS 0410


Special Theory of Relativity: The BasicsJohn D. Norton Department of History and Philosophy of Science University of PittsburghInertial and Accelerated Motion Absolute versus Relative Motion I. The Principle of Relativity II. The Light Postulate A Light Clock Light Clocks are Slowed by Motion All Moving Clocks Are Slowed by Motion Moving Rods Shrink in the Direction of Their Motion What you need to know:

Background reading: J. Schwartz and M. McGuinness, Einstein for Beginners. New York: Pantheon.. pp. 66 - 151.

"On the Electrodynamics of Moving Bodies" In June 1905, when Albert Einstein was still a patent examiner in Bern, Switzerland, he sent a paper with this title to the journal Annalen der Physik. It contained his special theory of relativity. He argued that altering our understanding of the behavior of space and time could resolve certain problems in electrodynamics. (See page one in German or English.) To understand what these alterations were, we need some preliminary notions.

Inertial and Accelerated MotionThere is a preferred motion in space known an inertial motion. Any body left to

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itself in space will default to an inertial motion, which is just motion at uniform speed in a straight line. The easiest example to visualize is a huge spaceship with the engines turned off, gliding through space. At any point in space, many inertial motions are possible. They will be pointed in different directions and will be at different speeds. Any other motion is accelerated. This includes motion at uniform speed in a circle. While the speed stays the same, the direction does not. So the motion is accelerated. Sometimes we will talk of an observer," which is just an moving inertially.

"inertialobserver

Such an observer might set up an elaborate system of measuing rods and other physical devices to fix the positions of events; and an elaborate system of clocks to fix their timing. Such a system is an inertial frame of reference.

Absolute versus Relative Motion

respect to another. For example, our spaceship might move relatively to a nearby planet.

Relative motion arises when one body moves with



Correspondingly the spaceship.

planet

moves

relative

to

the

Prior to Einstein, it was generally thought that there was another sense of motion, absolute motion . According to this sense, there is a fact of the matter as to whether the spaceship is moving, without regard to whether it moves relative to another object, such as a planet. There is an absolute state of rest in space, according to this earlier view. Either the spaceship is in this state and at rest; or it is not and it is moving.

Einstein found it most convenient to base his theory of relativity on two postulates ; once they were assumed it became an exercise in logic to develop the whole theory. The two postulates are I. The Principle of Relativity and II. The Light Postulate.

I. The Principle of RelativityAll inertial physics. observers find the same laws of

moving inertially in space but with different velocities. If we conduct experiments on either ship aimed at determining a law of physics, we will end up with the same law no matter which spaceship we are on. Or, more simply , the laws of physics simply tell us which

What this says is just this: imagine two spaceships, each



physical process can happen and which cannot. So if all inertial observers find the same laws, that just means that any process that can happen for one inertial observer can happen for any other. Here are some important consequences of the principle: No experiment aimed at detecting a law of nature can reveal the inertial motion of the observer. Absolute velocity has no place in any law of nature. No experiment can reveal absolute motion. Notice that the principle of relativity is limited to inertial motions. In special relativity, this relativity of motion does not extend to accelerated motion . If something accelerates, then it does so absolutely; there is no need to say that it "accelerates with respect to..." A traditional indicator of accelertion is inertial forces. If you are in an airplane that flies uniformly in a straight line, you have no sense of motion. If the airplane hits turbulence and accelerates, you sense immediately the acceleration as inertial forces throw things around in the cabin.

II. The Light PostulateAll inertial observers find the same speed for light. That speed is 186,000 miles per second or 300,000 kilometers per second. Because this speed crops up so often in relativity theory, it is represented by the letter "c". That Einstein should believe the principle of relativity should not come as such a surprise. We are moving rapidly on planet earth through space. But our motion is virtually invisible to us, as the principle of relativity requires. Why Einstein should believe the light postulate is a little harder to see. We would expect that a light signal would slow down relative to us if we chased after it. The light postulate says no. No matter how fast an inertial observer is traveling in pursuit of the light signal, that observer will always see the light signal traveling at the same speed, c.http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_basics/index.html[28/04/2010 08:18:00 ]


The principal reason for his acceptance of the light postulate was his lengthy study of electrodynamics, the theory of electric and magnetic fields. The theory was the most advanced physics of the time. Some 50 years before, Maxwell had shown that light was merely a ripple propagating in an electromagnetic field. Maxwell's theory predicted that the speed of the ripple was a quite definite number: c. The speed of a light signal was quite unlike the speed of a pebble, say. The pebble could move at any speed, depending on how hard it was thrown. It was different with light in Maxwell's theory. No matter how the light signal was made and projected, its speed always came out the same. The principle of relativity assured Einstein that the laws of nature were the same for all inertial observers. That light always propagated at the same speed was a law within Maxwell's theory. If the principle of relativity was applied to it, the light postulate resulted immediately.

A Light ClockOne cannot have both of Einstein's postulates and leave everything else unchanged. We can only retain both without contradiction if we make systematic changes throughout our physics . Let us begin investigating these changes, which include our basic, classical presumptions about space and time. One of them is that we learn that a moving clock runs slower. To see how this comes about, we could undertake a detailed analysis of a real clock, like a wristwatch or a pendulum clock. That would be difficult and complicated--and unnecessarily so. All we need is to demonstrate the effect for just one clock and that will be enough, as we shall see shortly, to give it to us for all clocks. So let us pick the simplest design of clock imaginable, one specifically chosen to make our analysis easy. A light clock is an idealized clock that consists of a rod of length 186,000 mileshttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_basics/index.html[28/04/2010 08:18:00 ]


with a mirror at each end. A light signal is reflected back and forth between the mirrors. Each arrival of the light signal at a mirror is a "tick" of the clock. Since light moves at 186,000 miles per second, it ticks once per second.

Light Clocks are Slowed by MotionTo see the effect of motion on this light clock, imagine that it has been set into rapid motion. To begin, we will assume that the motion is perpendicular to the rod and that it is very fast--99.5% the speed of light. (We'll write this compactly as "0.995c.") An observer traveling with the clock will still see the light signal bounce backwards and forwards between the mirrors as before. Let us view this process from the perspective of an observer who stays behind and does not move with the clock.

That observer sees a light signal leave one end of the rod and arrive at the other end. But that end is now rushing away fromhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_basics/index.html[28/04/2010 08:18:00 ]


the light signal at 99.5% the speed of light. A quick calculation shows that that the signal will now take 10 seconds to reach the other end of the rod.

To see this, note that in ten seconds the rod will move 1,850,700 miles, as shown in the figure above. So to get to the end of the rod, the light signal must traverse the diagonal path shown. A little geometry tells us that a right angle triangle with sides 186,000 miles and 1,850,700 miles will have a diagonal of 1,860,000 miles. Pythagoras' theorem tells us the diagonal 1,860,000 miles since 1,860,000 miles2 = 1,850,700 miles2 + 186,000 miles2 is

Since light moves at 186,000 miles per second, it will need ten seconds to traverse the diagonal. light signal must travel so much farther to traverse the rod of a moving clock, it takes much longer to do it. So a moving light clock ticks slower. In this case, for a clock moving at 99.5% the speed of light, it ticks once each ten seconds instead of once each second.

Setting the arithmetic aside , the result is simple. Since the

All Moving Clocks Are Slowed by MotionA simple application of the principle of relativity shows that all clocks must be slowed by motion, not just light clocks. We set a clock of any construction next to a light clock at rest in an inertial laboratory.



We notice that they both tick at the same rate. That must remain true when we set the laboratory into a different state of inertial motion. But since the light clock has slowed with the motion, the other clock must also slow if it is to keep ticking at the same rate as the light clock. You might be tempted to say that the other clock would not keep pace with the light clock. But then you would have devised a device that detects absolute motion , in contradiction with the principle relativity. That device would pick out absolute rest as the only state in which the two clocks run at the same rate.

Moving Rods Shrink in the Direction of Their MotionSo far, we have considered a light clock whose rod is perpendicular to the direction of its motion. If we now consider a light clock whose rod is oriented parallel to the direction of motion , we will end up concluding that its rod must shrink in the direction of its motion. To get to this result, we need two steps:

First Step: Light clocks oriented perpendicular to one another run at the same speed. Take the light clock considered above. Image a second, identical light clock with its rod oriented parallel to the direction of the motion. Once again the principle of relativity requires that both clocks run at the same speed. We could just leave it at that--an application of the earlier result. However it is reassuring to go through it from scratch. To begin, we don't need the principle of relativty to see that the clocks at rest run



at the same rate. They will run at the same rate simply because they are the same clocks oriented in different directions. That just follows from the isotropy of space. All its directions are equivalent. So the orientation of the clock cannot affect its speed. Now imagine that we take the entire system of the two clocks and set it into rapid motion at, say, 99.5% the speed of light, in the direction of one of the light clocks.

An observer moving with the two light clocks must see them continue to run at the same rate. We now do need the principle of relativity to establish this. Our earlier symmetry argument doesn't work anymore, since the two directions of the clocks are intrinsically different. One is perpendicular to the direciton of motion; the other is parallel to it. The principle of relativity requires that they run at the same rate. For, if they ran at different rates, the device would be an experiment that could detect absolute motion. Second Step: The rod oriented in the direction of motion must shrink.

We could detect absolute motion just by taking two light clocks perpendicular to each other and checking if they run at the same rate. Only when we are rest would they run at the same rate. If they do not run at the same rate we would know we are moving absolutely. The principle of relativity prohibits an experiment that can do this. So the two clocks must run at the same rate.

We know from the earlier analysis that a light clock (indeed any clock) moving at 99.5% the speed of light is slowed so that it ticks only once in ten seconds. So now we know that the light clock oriented parallel to the direction of motion must tick



once each ten seconds. But that cannot happen if everything is just as we describe it. Imagine the outward bound journey of the light signal.

How do I get this? If you have to know, here are the details. The light signal chases at 100% c after the leading end of the rod. That end is initially 186,000 miles away and moving at 99.5% c. So the light signal approaches the end of the rod at 0.5% c, which is 930 miles per second. The distance to cover is 186,000 miles, so it takes 186,000/930 = 200 seconds.

The light signal has to go from one end to the other of a 186,000 mile rod. The light moves at 186,000 miles per second. But the rod is also moving in the same direction at 99.5% the speed of light. So the light has to chase after a rapidly fleeing end and will need much more than a second to catch it. With a little arithmetic it turns out that the light will need 200 seconds to make the trip.

But the light clock has to tick once every ten seconds! Something has gone badly wrong. What has gone wrong is our assumption that the rod parallel to the direction of motion retains its length. That is incorrect. That rod actually shrinks to 10% of original length, so the moving pair of clocks really looks more like:

Now the light signal has time to get from one end of the rod to the other and keep the clock ticking at once each ten seconds as expected. The signal just has far less distance to travel so now it can maintain the rate of ticking expected.

There are more details in this last calculation that I don't want to bother you with. But since some of you will ask, here they are--but only for those who really want them. Overall it will turn out that the light signal now needs 20 seconds to complete the journey from the trailing end of the rod to the front and then back. That is what we expect. The round trip journal is "two ticks" and should take 2x10=20 seconds. The



catch is that virtually all of the 20 seconds will be spent in the forward trip and virtually none of it in the rearward trip. This effect actually figures in the relativity of simultaneity which we will discuss at some length later. If you want to see this for yourself you should redo the calculations. If you do, you'll need to undo my rounding off. The rod is not contracted exactly 10%--I rounded things off to keep life simple. It is 9.987%. The ticks are not exactly 10 seconds apart, but 10.0125 seconds. The forward trip will take 19.9750 seconds. The rearward trip will take 0.05 seconds. That gives a total round trip of 20.025 seconds = 2x10.0125 as expected.

The analysis is now complete. We have learned that a clock moving at 99.5% the speed of light, slows by a factor of ten. It ticks once each ten seconds instead of once each second. A rod, oriented in the direction of motion, shrinks to 10% of its length. Rods perpendicular to the direction of motion are unaffected. The two effects are not noticeable as long as our speeds are far from that of light. They become marked when we get close to the speed of light . The closer we get the the speed of light, the closer clocks come to stopping completely and rods come to shrkinking to no length in the direction of motion. For more details of how the effects depend on speed, see What Happens at High Speeds.

What you need to know:Inertial and accelerated motion. Absolute versus relative motion. Einstein's two postulates and how to apply them. What a light clock is and how it is affected by motion. Moving rods are shrunk in the direction of their motion.Copyright John D. Norton. January 2001, August 30, 2002, July 20, 2006; January 8 2007, January 3, August 21, 27, 2008.


Special Relativity Adding Velocities

HPS 0410


Special Theory of Relativity: Adding VelocitiesJohn D. Norton Department of History and Philosophy of Science University of PittsburghNothing Can Be Accelerated Through the Speed of Light Setting up the Challenge Prohibited by the Principle of Relativity Adding Velocities Einstein's Way Light? What you need to know

Nothing Can Be Accelerated Through the Speed of LightThe speed of light clearly has a special place in this theory. If something is traveling at the speed of light c, then all observers will find it to be traveling at exactly same speed. A similar thing happens to things traveling at less than the speed of light. If one observer finds an object to be traveling at less than light, say, then so must every other. There is no way that observers can change their states of motion so as to find the object traveling at faster than the speed of light. And there is a similar result for objects traveling at faster than the speed of light-if such things exist. If one observer finds them traveling at faster than the speed of light, then so must all. One of light's most important roles as a limiting velocity follows from this: no matter how hard we try, it is impossible to accelerate something through the speed of light . More generally, the speeds of things are divided into three groups: --things that travel slower than light, --things that travel at exactly the speed of light, --and things that travel faster than the speed of light.

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We cannot slow down or speed up anything so that it crosses the barrier of the speed of light. Yet it looks like it would be pretty easy to violate the limiting character of the speed light by accelerating something through the speed of light. We might have a gun that can fire particles at, say, 2,000 miles per second. That is well below the speed of light. We put the gun on a spaceship that we accelerate up to 185,000 miles per second--a mere 1,000 miles per second short of the speed of light. If we fire the gun in the direction of motion, would it not accelerate the particle through the speed of light? The limiting character of the speed of light is sufficiently striking for it to be worth seeing how it follows from the principle of relativity.

Setting up the Challenge

To see it, let us set up the challenge quite solidly. Imagine that a machine that can fire particles at 100,000 miles per second, which is more than half the speed of light, 186,000 miles per second.

Now we will try to push things past the speed of light. Imagine that the machine is placed on a spaceship that also moves at 100,000 miles per second in the direction that the machine fires the particles; that is, it moves at this speed with respect to a second observer on the earth.

So, let us ask the



obvious question. What will the earth bound observer find for the speed of the particle? The calculation seems irresistible . The spaceship moves at 100,000 miles per second with respect to the earthbound observer; and the particle moves at 100,000 miles per second with respect to the spaceship. So...

100,000 + 200,000 ??

100,000 =

But that would be faster than the speed of light , 186,000 miles per second.

Prohibited by the Principle of RelativityTo see that the principle of relativity prohibits this faster than light outcome, imagine that a light signal passes the particle emitting machine at the moment that the particle is emitted. The observer moving with the machine would (obviously) judge that the light signal overtakes the particle. Now imaginehttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_adding/index.html[28/04/2010 08:18:06 ]


this same process viewed by the the Earthbound observer. That observer

particle.

must also see the light signal overtake the

It is just the one experiment, so both observers must judge the same outcome. What else would you expect ? Might it be that the light signal would overtake a particle emitted by the machine, when the machine is on earth. But when the machine emits on a rapidly moving spaceship, then the particle overtakes the light? That is

exactly what the principle of relativity prohibits! For then we have an

experiment that can detect absolute motion. The resting machine emits particles that don't overtake light; the rapidly moving machine emits particle that do overtake light. The principle of relativity demands that the experiment must proceed in the same way when carried out on earth or a rapidly moving spaceship.

(For experts) Those who have read ahead might worry that each

observer might find a different outcome, perhaps as an artefact of the relativity of simultaneity (below). That won't happen. Whether light overtakes the particle or not can be reduced to local facts independent of judgments of simultaneity. Imagine that the light signal and the particle are to traverse the same interval in space AB. Both depart A at the same moment--judged locally. If light outstrips the particle, it will arrive at B before the particle. That earlier arrival is once again a local fact that obtains just at point B.

Adding Velocities Einstein's WayWhat this shows is that the principle of relativity prohibits us adding velocities in the usual way. We cannot add velocities by the ordinary rule 100,000 + 100,00 = 200,000. More generally,http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_adding/index.html[28/04/2010 08:18:06 ]


the classical rule for the composition of velocities fails: Velocity of B Velocity of A Velocity of A + = with respect to C with respect to B with respect to C In its place we need a new rule for the composition of velocities. It ought to look like the ordinary rule as long as velocities are small--we do know that the ordinary rule works for slow moving things like cars on freeways and trains. But it must look very different at high speeds. If we use it to add two velocities close to light, we must get a resultant that is still less than the velocity of light. Einstein found that the principle of relativity forces a particular rule. For the case of velocities oriented in the same direction in space, the relativistic rule for composition of velocities is: Velocity of A with respect to B + Velocity of B with respect to C

Velocity of A = __________________________________ with respect to C reduction factor All the work is done in this new rule by the reduction factor . When the velocities are small, this factor is close to 1. So it is as if it isn't really there and Einstein's rule just behaves like the classical rule. But when the velocities get to be close to that of light, the factor starts to get larger and larger and in just the right way to prevent any composition of velocities less than light exceeding that of light. If we use the rule to add 100 mph to 100 mph, the reduction factor is almost exactly one, so the ordinary rule works: 100 + 100 = 200. If we use the rule for adding 100,000 miles per second to 100 miles per second, we are now dealing with velocities that are 100,000/186,000 = 0.54 the speed of light. For that sum, the reduction factor is 1.29, so the composition yields: (100,000 + 100,000)/1.29 = 200,000/1.29 = 155,000 which is still less than the speed of light. What is most instructive is to see what happens if we start with a velocity of 100,000 miles; and add 100,000 miles per second to it; and add it again; and again; and again. To picture physically what we are doing, imagine that we start with our base machine "I" that happens already to be moving at 100,000 miles per second. From it we shoot out a second smaller version of the same machine --call it "II" --at 100,000http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_adding/index.html[28/04/2010 08:18:06 ]

here to see the complete formula.

(For experts only) Click


miles per second with respect to "I."

Now let's repeat the operation. From the smaller machine "II," we'll shoot out a yet smaller version of the same machine at 100,000 miles per second with respect to "II." Call it "III."

Then machine "III" will shoot out machine "IV"; and so on; and so on. As we pass through the series of machines "I," "II,", "III," "IV," etc., we are boosting each with a speed of 100,000 miles per second with respect to the one before. The cumulative effect of the repeating boosting by 100,000 miles per second is shown below. The total speed of the last boosted machine increases as we proceed along the sequence "I," "II," etc. But the increases become smaller and smaller.

No matter how often we add 100,000 miles per second, we never get past the speed of light--here set at exactly 186,000 miles per second. We get closer and closer to it. But never past it. One way to think of it is as an "Einstein tax ," that copies the way a very severe progressive taxation might increase the amount of tax paid as we get more income. We keep adding 100,000 miles per second to the speed, but the Einstein tax-implemented through the reduction factor --precludes our total speed ever exceeding that of light. That the ordinary addition rule fails follows from the principle ofhttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_adding/index.html[28/04/2010 08:18:06 ]


relativity. Why should the ordinary rule fail? Here's way to get comfortable with the the failure. In the original example, the spaceship observer uses rods and clocks that move with the spaceship to measure the speed of the emitted particle as 100,000 miles per second. The earthbound observer now wants to find the speed of the emitted particle. That observer, however, cannot directly use measurements made with the spaceship rods and clocks, for the earthbound observer thinks that they have shrunk and slowed. The earthbound observer must correct the spaceship observer's measurements for effects such as these. The result of the these corrections is Einstein's formula!

Light?This special role for the speed of light sometimes arouses special wonder. What is so special about light , we may be drawn to ask, that everything else takes such special note of it? Once one starts along this path, all sorts of confusions may arise. Is it that light is used for communication and finding things out? Does everything somehow respond to how we find things out? Does special relativity still work in the dark? Well--you can forget all this mystical mumbo-jumbo, if ever it attracted you. There is nothing special about light. It's space and time that is special. They have properties we don't expect. Space and time are such that rapidly moving objects shrink and their processes slow down. For a long time, we didn't notice these effects because we did not have a thorough account of a probe of space and time that moves very fast. That changed in the



nineteenth century when we developed good theories of light. It is the probe that moves very fast and, for the first time, begins to reveal to us that space and time are not quite what we thought. There is one further fact about space and time. It harbors a special velocity, one that is the same for all inertial observers. It is an invariant (="unchanging") velocity. Light is just something that happens to go as fast as it possibly can and thereby ends up going at that speed. There's nothing special about light. What is special is the speed at which it goes.

What you need to know:Nothing can be accelerated through the speed of light. Adding velocities Einstein's way.Copyright John D. Norton. January 2001, August 30, 2002, July 20, 2006; January 8 2007, January 3, August 21, 27, 2008, January 13, 2010..



HPS 0410


Special Theory of Relativity

Relativity of SimultaneityJohn D. Norton Department of History and Philosophy of Science University of PittsburghUsing Light Signals to Judge the Time Order of Events What the Relativity of Simultaneity is NOT What you need to know:

When Einstein first hit upon special relativity, he thought one effect of special importance, so much so that it fills the first section of his "On the Electrodynamics of Moving Bodies." It is the relativity of simultaneity. According to it, inertial observers in relative

motion disagree on the timing of events at different places.If one observer thinks that two events are simultaneous, another might not. At first this will seem like just another of the many novel effects relativity brings. However, as we explore more deeply, you will see that this is the central adjustment Einstein made to our understanding of space and time in special relativity . Once you grasp it, everything else makes sense. (And until you do, nothing quite makes sense!)

Using Light Signals to Judge the Time Order of EventsThere is a quick way to see how this comes about. Imagine a long platform with an observer located at its midpoint. At either end, at the places marked A and B, there are two momentary flashes of light. The light propagates from these events to the observer. Let us imagine that they arrive at the same moment, as they do in the animation below. Noticing that they arrive at the same moment and that they come from places equal distances away, the observer will decide that the two events happened simultaneous. Another outcome is closely related. Imagine also that there arehttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_rel_sim/index.html[28/04/2010 08:18:16 ]


clocks located at A and B. If both clocks show the same reading at the events of the two flashes, then we would judge the two clocks to be properly synchronized . That is what the platform observer judges since, as the animation shows, both clocks read "0" when the flashes occur at each location.

Here's a version that isn't animated.

So far, nothing remarkable has happened. That is about to change. Now consider this process from the point of view of an observer who moves relative to the platform along its length. For that new observer, the platform moves rapidly and, in the animation, in the direction from A towards B. Once again there will be two flashes and light from them will propagate towards the observer at the midpoint of the platform. However the midpoint is in motion. It is rushing away from light coming from A; and rushing toward the light coming from B. Nonetheless, the two signals arrive at the midpoint at the same moment.

http://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_rel_sim/index.html[28/04/2010 08:18:16 ]


Here's a version that isn't animated.

What is the new observer to make of this? For the new observer, the light from A must cover a greater distance to catch up with the receding midpoint; and the light from B must cover a lesser distance to arrive at the midpoint rushng towards it. So if the two arrive at the same moment, the light from A must have left earlier than the light from B to give it greater time to cover the greater distance to get to the midpoint. That is, the flash at A happened earlier than the flash at B. The two events were not simultaneous, according to the new observer.

Notice that the reasoning

requires the light postulate: both light

flashes must move at the same speed; that is, each must require the same time to cover the same distance.

The reasoning extends to the clocks. The clocks at A and B show the same time when the flash events happen at each.These two events are not simultaneous for the new observer. Therefore the new observer will judge the clocks at A and B not be properly synchronized. In fact clock A is set ahead of clock B.

happened simultaneously and that the two clocks are properly

In short, the platform observer will say that the two flashes



synchronized; the new observer will say the A -flash happened first and that the A -clock is set ahead of the B -clock. It is not a matter that one or other of them is somehow misinformed. They are both using the same information. Rather it is that judgements of the simultaneity of spatially separated events depend on the observer, just as the rate of clocks and lengths of bodies depends of the observer in special relativity. That a moving clock slows and moving rod shrinks is something most of us get used to with a little thought. The same is not true of the relativity of simultaneity. It is harder to get used to it, since it amounts to a more fundamental breakdown. It tells us that that there is no absolute fact about the relative timing of events at distant places. Imagine that you have candles on a birthday cake in Pittsburgh and on one in far -away Sydney. You plan to have them blown out at exactly the same moment. The relativity of simultaneity tells you that there is no absolute fact to whether you succeed. Relative to an earth-bound observer, you may succeed. But that can mean that relative to an observer on the moon, who moves relative to the earth, you did not succeed. The relativity of simultaneity adds to the repertoire of quantities that are relative and not absolute. There is no absolute fact to whether a spaceship is moving uniformly or is at rest. It can only be said to be at rest relative to another body. There is no absolute fact as to whether a rod is foot long or a process lasts for one minute. They can only true with respect an observer with a definite state of motion. To this list we add that there is no absolute fact to whether two spatially separated events are simultaneous; or whether two spatially separated clocks are synchronous. These can only be true relative to an observer with a definite state of motion.

What the Relativity of Simultaneity is NOTThere is a quite benign way in which observers can disagree on the simultaneity of events. It is not the effect at issue. To see the benign way, imagine that a flash of lightning strikes the tree you are standing under. Let us say the strike comprises two events: the flash of the light and the boom of the thunder. For you standing under the tree, if you survive, the two events are simultaneous. It would not appear so for someone standing on a distant hill top watching the lightning strike. That observer would see the flash and then, several seconds later, hear the boom of thehttp://www.pitt.edu/~jdnorton/teaching/HPS_0410/chapters/Special_relativity_rel_sim/index.html[28/04/2010 08:18:16 ]


thunder. For you the flash and the boom are simultaneous. For the distant observer they are not simultanteous; or, more precisely, they do not appear simultaneous.

This same effect can arise in more abstruse settings. When we look at a distant galaxy 10 million light years away, we are seeing it as it appeared 10 million years ago. So if we see some event occuring now, such as a star in the galaxy exploding, that event really happened 10 millions years ago. It will appear to us that it happened now, at the same time as the events of the present day. In fact it did not. We know that and we correct for the time the starlight took to reach us in judging the timing of the event.

sense, merely if our sensations of them happen at the same moment. Or they fail to be simultaneous in this sense if our sensations of them happen at different times.

"appearance simultaneity." Events are simultaneous in this

These two examples

illustrate the oddities of what we can call

That sort of simultaneity is not the sort that is at issue in the relativity of simultaneity. The idea is that we correct for differences in appearance simultaneity. For example, when we hear the boom of the thunder coming after we see the flash of the lightning, we routinely allow for the fact that light travels very rapidly, but sound travels slowly--roughly one mile in five seconds. So even though we sense the flash and boom at different times, we judge the two originating events to be simultaneous. Here's another case. Two lightning bolts strike at points D and E, where D is farther away from the observer. Let's say that the strikes are timed so that light signals from the bolts arrive at the same moment at the observer. The observer would see both flashes at the same time. The bolts would appear simultaneous. But the observer would then correct for the greater distance that the light signal from D must travel. So that the observer sees the flashes at the same time means the observer judges the D bolt to have struck



earlier.

The relativity of simultaneity of relativity theory arises after we have corrected for the oddities of appearance simultaneity. Even after those corrections have been made, it turns out that observers in relative motion will not agree on the timing of spatially separated events. In the thought experiment above with the A and B clocks, it turns out that no corrections for appearance simultaneity are needed. Since the observer is located at the midpoint of the platform, the flashes of light at A and B are delayed equally. That is why the observer was placed there.

What you need to know:What the relativity of simultaneity is. What the relativity of simultaneity is not.Copyright John D. Norton. January 2001, August 30, 2002, July 20, 2006; January 8 2007, January 3, August 21, 27, 2008; January 13, 2010..


Problem of Reciprocity

HPS 0410

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