Mary O’Malley

Math G - Dr. Ian Walton

October 9, 2002

Math G. Midterm

Albert Einstein

(1879 – 1955)

 

"Logical reasoning brings you from a to b, imagination

brings you everywhere." (Albert Einstein)

 

I.            Introduction

                                            

I will never be a physicist.   The reality is I will never see the inside of a physics classroom.  In fact, until I began Math G, I never thought I would have an interest in any math more demanding than balancing my checkbook (and even that is hit or miss)!  So I approached this midterm project with more than a little hesitation.

 

On the other hand, I have always wondered what e=mc² meant.  Here is a phrase I’ve seen or heard all my life.  One of the most recognized equations of the 20^th Century and I had no clue what it meant.  So I decided to take on e=mc², and the more I read, the more fascinated I became.  Not just with the formula, but with Albert Einstein himself.  Einstein went from being an abstract icon to an amazing human being with amazing ideas, both political and mathematical.

 

Einstein (much like our Math G class) moved away from asking the expected questions about observable phenomena like “how come things fall” or “why light behaves like it does” and he started questioning the basic elements of nature that are unseen and that can only be dreamed about.

 

So my initial goal of learning about  e=mc²  expanded to include the following:  1) gain a greater understanding of the life and contributions of Albert Einstein; 2) gain a basic understanding of his two greatest theories, e=mc² and the Theory of Relativity; and 3) to share that information with the rest of the class in my presentation.  I’ll start by introducing Einstein.

 

II.          Albert Einstein – The Early Years

 

Albert Einstein was born in 1879 in Ulm, Würtemberg, Germany to German-Jewish parents.  His early years did not hint at the genius he would become.  In fact by all appearances he was slow.  He did not speak until he was 3 and when he was 9 his father commented that he seemed to be a “slow, dreamy child who spoke hesitantly.” His mother hoped he would be able to secure a job as a teacher because it would be stable income that would not demand too much intelligence.

 

When Einstein was growing up in Germany, the world outside his home was dominated by military presence. Heads of state, school officials, even taxi drivers wore military type uniforms.  To make matters worse,  Einstein was the only Jew at a traditional Catholic school.  He did not feel comfortable in this very structured environment and he did not do well in school.  But he flourished at home where his Uncle Jacob introduced him to algebra, and his mother introduced him to music and literature.

 

Popular history says that when Einstein was five years old, his father showed him a pocket compass. Einstein was completely fascinated by the mysterious behavior of the compass needle which pointed in the same direction no matter which way he turned it. He later said that looking at that compass at that moment made him feel that "something deeply hidden had to be behind things.[1]"   At age 12, a medical student named Max Talmey, who was a regular visitor to the Einstein home, introduced Einstein to all of the most popular scientific theories available in Germany.  More importantly, he tutored him in geometry and calculus sparking Einstein’s lifelong interest in physics.  At age 12 Einstein found a completely new proof for the Pythagorean Theory.  While Einstein was a voracious reader and an inquisitive student at home, he was bored and disruptive at school. He lacked respect for his instructors and he developed a life-long mistrust of authority.  At 15 Einstein was finally kicked out of school.  He was glad to be out of school and spent a year with his cousins in Genoa.  He renounced his German citizenship and began to think about becoming a theoretical physicist.  After a year off, his father insisted he go back to school to get an engineering degree so that he could support himself.

 

But Einstein had no high school diploma and he failed the entrance examination to the Eidgenossiche Technische Hockschule (the ETH), an elite technical school in Zurich Switzerland. He did however excel at the math portion of the test and was given the opportunity to attend a high school in the Swiss town of Aarau that specialized in math and physics. A year later at the age of 17 he was able to retake and pass the entrance exam at the ETH.  He studied mathematics and physics there from 1896 to 1900.  Even though Einstein passed his classes, he was bored with organized mathematics and he showed a lack of respect for formal education.  He rarely attended classes and alienated all of his instructors.  He was only able to pass his classes because a friend gave him notes from each class so he could pass the tests.  Einstein did however spend a great deal of his time in the school laboratory working on his own experiments, and he managed to graduate in 1900 as a teacher of mathematics and physics.

 

What Einstein lacked as a student, he made up for in his social life.  He formed lifelong friendships with some of the brightest minds of the time.  He has been described as a “handsome, irrepressible romantic” who was popular with the ladies.  During his college years he socialized with the most famous exiled revolutionaries and socialists of the time.  He was introduced to revolutionary socialism by his friend Friedrick Adler and as a result he supported socialism his entire life.  Until the rise of Nazism in the 1930's, Einstein was an ardent pacifist. After the war, he became an equally determined supporter of world government. He insisted that the only way to maintain peace among nations in the atomic age was to bring all nations together under one system of world law.  One week before his death Einstein sent a letter to Bertrand Russell requesting that his name go on a manifesto urging all nations to give up nuclear weapons.

He met his first wife Mileva Maric, a Serbian physics student while at the ETH and he married her in 1903.  When Einstein graduated from ETH in 1900 he could not find a job.  None of his professors would recommend him for positions so Einstein worked at odd jobs as a teacher until his friend from school, Marcel Grossman, used his influence to get Einstein a job as a patent examiner at the Swiss Patent Office where he worked from 1902 until 1909. In 1905 Einstein became a Swiss citizen.  Einstein was not particularly career oriented and considered his job a blessing because it allowed him to “…leave [work behind] at the end of the day free to go home and work on his science….”

 

III.        "Annus Mirabilis" - Einstein's Miracle Year  

 

1905 is known as Einstein’s "Annus Mirabilis" or Einstein's "Miracle Year."

 

In March 1905 he developed the quantum theory of light; the idea that light exists as tiny packets or particles that we now call photons.  Einstein pioneered the idea that we live in a quantum universe built of tiny discrete chunks of energy and matter. In 1921 he was awarded the Nobel Prize for his theory.

 

In April and May he published two articles that together proved the existence of the atom.

 

In June, 1905 he introduced his Special Theory of Relativity in which he demonstrates that measurements of time and distance vary systematically as anything moves relative to anything else. (More on that later).

 

Finally, he added his famous formula e=mc² to his Theory of Special Relativity.

 

All of this work was done by Einstein in his spare time.  He worked alone at home after working an eight hour day at the Swiss Patent Office with no formal support from the scientific or mathematic community.  Einstein was 26 years old.

 

IV.        Albert Einstein – The Later Years 

 

Between 1906 and 1927 Einstein continued to work at an amazing pace.  He achieved what no one has been able to equal since: a twenty year career on the cutting edge of physics.  His discoveries ranged from the most basic (why is the sky blue?) to the most complex questions of quantum mechanics.  He transformed our understanding of nature on every scale during this 20+ year period. 

 

In 1911 Einstein was given a full professorship at the German University there. In 1912
he took a position as a professor of Theoretical Physics at the ETH (the school which once turned him away).  In 1914 he became the director of the Kaiser Wilhelm Institute in Berlin and professor of theoretical physics at the University of Berlin.  In 1932 Einstein was identified as a Jew living in Nazi Germany and in 1933 while Einstein was visiting England and the United States, Nazi Germany took away his property, his job and his citizenship.  He and Elsa, his second wife, moved to the United States where Einstein was offered a job at the Institute for Advanced Study in Princeton N.J.  He remained in the U.S. for the rest of his life and in 1940 he became an American citizen.  On April 16, 1955 Einstein died of heart failure.

 

V.                Einstein’s Theories of Relativity

We often associate Einstein with his Theory of Relativity, but I was surprised to learn that Einstein really introduced two Theories of Relativity.  The first one was the Special Theory of Relativity published in 1905.  Ten years later, Einstein published his General Theory of Relativity. 

Einstein’s Special Theory of Relativity stated that length and time, previously believed to be fixed and unchanging, were not. Empty space could contract or expand depending on how close you were to an object, and the rate at which time passes could change as well. Space and time could even change depending on who was measuring them.

Einstein’s General Theory of Relativity revised his earlier theory and added gravity and motion to the mix.  Confused yet?  You are not alone.  The British astronomer Sir Arthur Eddington, one of the first to fully understand the theory in detail, was once asked if it were true that only three people in the world understood general relativity. He is said to have replied, "Who is the third?"[2]  But let’s keep trying.

VI.        The Special Theory of Relativity

In this theory, published in 1905, Einstein demonstrated that measurements of time and distance vary systematically as anything moves relative to anything else.  It stated that relative to the observer, both space and time are altered near the speed of light -- distances appear to stretch and clocks tick more slowly.

Einstein said that it is all about your frame of reference.  Let’s look at a simple example given by Albert Einstein.

"When you sit with a nice girl for two hours, it seems like two minutes.

 When you sit on a hot stove for two minutes, it seems like two hours

 that's relativity." (Albert Einstein)

 

The following rules of relativity and examples will help make relativity easier to understand[3]

 

Relativity Rule #1    Time is not constant.  The faster you go, the slower it goes and vice versa.

 

One way to demonstrate this is to use a video tape analogy.  You ask a friend to video tape your wedding and he agrees but says he can only tape for one hour.  He decides to call the tape “One last hour of bachelorhood”.  He starts by zooming in on your watch which shows it is 12:00.  He continues to film the wedding but every fifteen minutes or so he zooms back in on the watch to show the hour passing by.  Later when you are showing the tape of your wedding to friends, you decide to fast forward through the first 15 minutes of the tape to get to the kiss at the end of the ceremony.   As the tape begins you see your watch on the screen and it is 12:00.  Three minutes later, you have fast forwarded through the tape to the kiss and you see your watch on the screen says 12:15.  You have passed through 15 minutes of time in 3 minutes.  This is not an actual example of relativity, but it is an analogy that makes relativity easier to understand.  Motion affects measurement.

 

Relativity Rule #2      Moving objects appear to be shortened in the direction in which they are moving.

 

Let’s say that you normally drive to work at 50 miles per hour and it takes you 10 minutes to get there.  One day you oversleep and are late for an important meeting so you jump in your car  and drive to work at 100 miles per hour.  Along the way, you notice that the passing scenery looks different when you pass it at twice the speed.  You notice that the local McDonald’s appears to be somewhere near half its usual size.  If you were walking past that same building you would notice that the building appears to be much bigger than it looked when you drove by at 50 miles per hour.  Although the building looks different at each of these speeds, it has not changed.  What has changed is time. 

 

              

Relativity Rule #3    There is no such thing as simultaneous events.

 

Imagine that you look up into the sky and you see an exploding star.  You say “Look… it’s happening right now!”   But the reality is that the star actually exploded several thousand years ago and is long gone by the time you see it.  The only reason you see it when you do is that it took this long for the star’s final light waves to reach the earth.  Given the great distance between you and the star it is easy to see that what looked like a simultaneous event was actually separated by thousands of years.  You can test this out on smaller differences in time as well.  Imagine that you are talking to your friend on your cell phone at 12:00 o’clock and your friend is about 2 miles away in front of a fire station when the alarm goes off.  While you are talking to your friend you hear the sound of the alarm over the phone and a minute or so later you hear the same alarm again as the sound actually reaches you.  The alarm went off at 12:00 but it took a minute or so for the sound to reach you two miles away.  There will always be some differential in time no matter how close or far you are to something.

 

Relativity Rule #4    The speed of light is the fastest rate at which any object may travel.

 

This one is easy.  Nothing can travel faster than the speed of light (at least we haven’t discovered it yet).  At 186,000 miles per second, it is nature’s fundamental speed limit.  It is also the one constant in the theory e=mc².

 

VII.      The General Theory of Relativity

 

In 1907, Einstein began revising his Special Theory of Relativity and in 1915 his General Theory of Relativity was published.  This new theory proposed that gravity, as well as motion, can also affect the intervals of time and of space. The key idea of General Relativity, called the Equivalence Principle, is that gravity pulling in one direction is completely equivalent to acceleration in the opposite direction.

 

For example:  when your car accelerates forward, you feel gravity pushing you back against your seat, or when you are in an elevator accelerating upwards, you feel gravity pushing you into the floor.

 

Einstein’s General Theory of Relativity says that the gravity of any mass, such as our sun, will have the effect of warping space and time around it. For example, the angles of a triangle no longer add up to 180 degrees and clocks tick more slowly the closer they are to a gravitational mass like the sun.

With this theory, Einstein showed that we reside not in the flat, Euclidean space and uniform, absolute time of everyday experience, but in another environment: curved space-time.  This is the basis for the explanation of why the shortest distance between two points is not always a straight line.  As we discussed in class, the globe (like space) is curved.

Einstein’s Theory of General Relativity led to advances in physics that led us to the nuclear era; it made it possible to understand the microworld of elementary particles and their interactions; and revolutionized our understanding of astronomical phenomena such as the big bang, neutron stars, black holes and gravitational waves. 

 

VIII.    e=mc²

 

Still with me?   Good.  Now we can tackle  e=mc² (or as one of my reference books called it e=(mass x confusion)[4]).   The formula e=mc² means that the energy content of a body is equal to the mass of the body times the speed of light squared.

 

               E  = energy

               M = mass

               C  = speed of light  (not infinity! In physics, the symbol "c" denotes the speed of

light.)

C² = the square of the speed of light.

 

 

This formula is really a part of the Special Theory of Relativity which was explained above.  Einstein was looking for an explanation of relativity when he came up with e=mc².  It was not until later when he discovered his General Theory of Relativity that he realized that e=mc² was a special case of a more general law.  Put simply, the theory means mass will change according to speed and who’s looking and from where.

 

IX.        Summary

 

This paper could not begin to cover the genius of Albert Einstein.   It introduced us to Einstein, and it gave us a basic overview of two of his best known theories, e=mc² and the Theory of Relativity, but there is so much more to the man who has been called the greatest genius of all time. The problems that Einstein could not solve remain ones that today’s physicists, mathematicians and scientist find most challenging.  So I will leave you with some of the more interesting quotes by Albert Einstein and hope that you are as interested as I am to keep on reading about the life and contributions of Albert Einstein.

 

 

“Imagination is more important than knowledge.”

 

 

“If we knew what it was we were doing, it would not be called research,

 would it?”

 

 

“I know not with what weapons World War III will be fought, but

World War IV will be fought with sticks and stones.”

 

 

“Any intelligent fool can make things bigger, more complex, and more violent.

 It takes a touch of genius -- and a lot of courage -- to move in the opposite

 direction.”

 

 

 

References:

 

Joseph Schwartz and Michael McGuinness, “Einstein for Beginners” Pantheon Books, New York, ©1979 ISBN 0-679-72510-5

 

David Bodanis, “E=mc²; A Biography of the World’s Most Famous Equation” Walter & Company, New York © 2000 ISBN 0-8027-1352-1

 

Daniel Orange, Ph.D., and Gregg Stebben, “The Pocket Professor:  Everything You Need to Know About Physics” Simon & Schuster Inc. 1230 Avenue of the Americas, New York, NY 10020 © 1999 ISBN 0-671-53490-4

 

Ira M. Freeman, “Physics Made Simple” Revised by William J. Durden, Doubleday

© 1990 ISBN 0-385-24228-X

 

Clarice Swisher, “Relativity: Great Mysteries Opposing Viewpoints” Greenhaven Press, Inc. P.O. Box 289009, San Diego, CA 92128-9009 © 1933 ISBN 0-89908-076-6

 

Jeremy Roberts, “How Do We Know the Laws of Motion” The Rosen Publishing Group, Inc. New York © 2001 ISBN 0-8239-3383-0

 

Websites:

 

http://www.bartleby.com/173/  Bartleby.com – Great Books Online:  Relativity – The Special and General Theory, Albert Einstein (First accessed 9/15/02)

 

http://www.time.com/time/time100/poc/magazine/albert_einstein5a.html  Time.com: Person of the Century – Albert Einstein (First accessed 9/15/02)

 

http://www.pbs.org/wgbh/nova/einstein/ Nova Online:  Einstein Revealed (First accessed 9/15/02)

 

http://whyfiles.org/052einstein/ Everything’s Relative –Einstein: Still Right After All These Years (First accessed 9/15/02)

 

http://www.aip.org/history/einstein/  A. Einstein:  Image and Impact (First accessed 9/15/02)

 

http://ask.yahoo.com/ask/20011211.html  Ask Yahoo Can you please explain Einstein's theory of relativity… (First accessed 9/15/02)

 

http://www.geocities.com/einstein_library/index.htm  Einstein Library (First accessed 9/15/02)

 

http://www2.slac.stanford.edu/vvc/theory/relativity.html

 

http://pratt.edu/~arch543p/help/theory_of_relativity.html

 

http://curious.astro.cornell.edu/relativity.php

 

Footnotes: