Mary O’Malley

Math G - Dr. Ian Walton

October 9, 2002

Math G. Midterm

Albert Einstein

(1879 –1955)

"Logicalreasoning brings you from a to b, imagination

brings youeverywhere." (Albert Einstein)

I.            Introduction

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

On the other hand, I have always wondered what e=mc² meant.  Here is a phrase I’ve seen orheard all my life.  One of the mostrecognized equations of the 20^th Century and I had no clue what itmeant.  So I decided to take one=mc², and the more I read, the more fascinated I became.  Not just with the formula, but withAlbert Einstein himself.  Einsteinwent 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 askingthe expected questions about observable phenomena like “how come thingsfall” or “why light behaves like it does” and he startedquestioning the basic elements of nature that are unseen and that can only bedreamed about.

So my initial goal of learning about  e=mc²  expanded to include the following:  1) gain a greater understanding of thelife and contributions of Albert Einstein; 2) gain a basic understanding of histwo greatest theories, e=mc² and the Theory ofRelativity; and 3) to share that information with the rest of the class in mypresentation.  I’ll start byintroducing Einstein.

II.          AlbertEinstein – 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 andwhen he was 9 his father commented that he seemed to be a “slow, dreamychild who spoke hesitantly.” His mother hoped he would be able to securea job as a teacher because it would be stable income that would not demand toomuch intelligence.

When Einstein was growing up in Germany, the world outsidehis 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 thisvery structured environment and he did not do well in school.  But he flourished at home where hisUncle Jacob introduced him to algebra, and his mother introduced him to musicand literature.

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

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

What Einstein lacked as a student, he made up for in his social life.  He formed lifelong friendships withsome of the brightest minds of the time. He has been described as a “handsome, irrepressibleromantic” who was popular with the ladies.  During his college years he socialized with the most famousexiled revolutionaries and socialists of the time.  He was introduced to revolutionary socialism by his friendFriedrick 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 determinedsupporter of world government. He insisted that the only way to maintain peaceamong nations in the atomic age was to bring all nations together under onesystem of world law.  One weekbefore his death Einstein sent a letter to Bertrand Russell requesting that hisname go on a manifesto urging all nations to give up nuclear weapons.

He met his first wife Mileva Maric, a Serbian physicsstudent while at the ETH and he married her in 1903.  When Einstein graduated from ETH in 1900 he could not find ajob.  None of his professors wouldrecommend him for positions so Einstein worked at odd jobs as a teacher untilhis friend from school, Marcel Grossman, used his influence to get Einstein ajob as a patent examiner at the Swiss Patent Office where he worked from 1902until 1909. In 1905 Einstein became a Swiss citizen.  Einstein was not particularly career oriented and consideredhis 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.        "AnnusMirabilis" - 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; theidea that light exists as tiny packets or particles that we now callphotons.  Einstein pioneered theidea that we live in a quantum universe built of tiny discrete chunks of energyand matter. In 1921 he was awarded the Nobel Prize for his theory.

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

In June, 1905 he introduced his Special Theory of Relativityin which he demonstrates that measurements of time and distance varysystematically as anything moves relative to anything else. (More on thatlater).

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

All of this work was done by Einstein in his sparetime.  He worked alone at homeafter working an eight hour day at the Swiss Patent Office with no formalsupport from the scientific or mathematic community.  Einstein was 26 years old.

IV.        AlbertEinstein – The Later Years 

Between 1906 and 1927 Einstein continued to work at anamazing pace.  He achieved what noone has been able to equal since: a twenty year career on the cutting edge ofphysics.  His discoveries rangedfrom the most basic (why is the sky blue?) to the most complex questions ofquantum mechanics.  He transformedour understanding of nature on every scale during this 20+ year period. 

In 1911 Einstein was given a fullprofessorship at the German University there. In 1912
he took a position as a professor of Theoretical Physics at the ETH (the schoolwhich once turned him away).  In 1914 he became the director of the Kaiser Wilhelm Institute inBerlin and professor of theoretical physics at the University of Berlin.  In 1932 Einstein wasidentified as a Jew living in Nazi Germany and in 1933 while Einstein wasvisiting England and the United States, Nazi Germany took away his property,his job and his citizenship.  Heand Elsa, his second wife, moved to the United States where Einstein wasoffered a job at the Institute for Advanced Study in Princeton N.J.  He remained in the U.S. for the rest ofhis 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 wassurprised to learn that Einstein really introduced two Theories ofRelativity.  The first one was theSpecial Theory of Relativity published in 1905.  Ten years later, Einstein published his General Theory ofRelativity. 

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

Einstein’s General Theory of Relativity revised his earlier theory andadded gravity and motion to the mix. Confused yet?  You are notalone.  The British astronomer SirArthur 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 understoodgeneral relativity. He is said to have replied, "Who is the third?"[2]  But let’s keep trying.

VI.        TheSpecial Theory of Relativity

In this theory, published in 1905, Einstein demonstrated that measurements oftime and distance vary systematically as anything moves relative to anythingelse.  It stated that relative tothe 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 examplegiven by Albert Einstein.

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

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

 that's relativity." (Albert Einstein)

The following rules of relativity and examples will helpmake 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 tapeanalogy.  You ask a friend to videotape your wedding and he agrees but says he can only tape for one hour.  He decides to call the tape “Onelast hour of bachelorhood”. He starts by zooming in on your watch which shows it is 12:00.  He continues to film the wedding butevery fifteen minutes or so he zooms back in on the watch to show the hourpassing by.  Later when you areshowing the tape of your wedding to friends, you decide to fast forward throughthe first 15 minutes of the tape to get to the kiss at the end of theceremony.   As the tape beginsyou see your watch on the screen and it is 12:00.  Three minutes later, you have fast forwarded through thetape to the kiss and you see your watch on the screen says 12:15.  You have passed through 15 minutes oftime in 3 minutes.  This is not anactual example of relativity, but it is an analogy that makes relativity easierto understand.  Motion affectsmeasurement.

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

Let’s say that you normally drive to work at 50 milesper hour and it takes you 10 minutes to get there.  One day you oversleep and are late for an important meetingso you jump in your car  and driveto work at 100 miles per hour. Along the way, you notice that the passing scenery looks different whenyou pass it at twice the speed. You notice that the local McDonald’s appears to be somewhere nearhalf its usual size.  If you werewalking past that same building you would notice that the building appears tobe much bigger than it looked when you drove by at 50 miles per hour.  Although the building looks differentat 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 anexploding star.  You say“Look… it’s happening right now!”   But the reality is that the staractually exploded several thousand years ago and is long gone by the time yousee it.  The only reason you see itwhen you do is that it took this long for the star’s final light waves toreach the earth.  Given the greatdistance between you and the star it is easy to see that what looked like asimultaneous event was actually separated by thousands of years.  You can test this out on smallerdifferences in time as well. Imagine that you are talking to your friend on your cell phone at 12:00o’clock and your friend is about 2 miles away in front of a fire stationwhen the alarm goes off.  While youare talking to your friend you hear the sound of the alarm over the phone and aminute or so later you hear the same alarm again as the sound actually reachesyou.  The alarm went off at 12:00but it took a minute or so for the sound to reach you two miles away.  There will always be some differentialin time no matter how close or far you are to something.

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

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

VII.      TheGeneral Theory of Relativity

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

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

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

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

Einstein’s Theory of General Relativity led to advances in physicsthat led us to the nuclear era; it made it possible to understand themicroworld of elementary particles and their interactions; and revolutionizedour understanding of astronomical phenomena such as the big bang, neutronstars, black holes and gravitational waves. 

VIII.    e=mc²

Still with me?  Good.  Now we cantackle  e=mc² (or as oneof my reference books called it e=(mass x confusion)[4]).   The formula e=mc² meansthat the energy content of a body is equal to the mass of the body times thespeed 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 speedof light.

This formula is really a part of the Special Theory ofRelativity which was explained above. Einstein was looking for an explanation of relativity when he came upwith e=mc².  It wasnot until later when he discovered his General Theory of Relativity that herealized that e=mc² was a special case of a more general law.  Put simply, the theory means mass willchange according to speed and who’s looking and from where.

IX.        Summary

This paper could not begin to cover the genius of AlbertEinstein.   It introduced usto 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 muchmore to the man who has been called the greatest genius of all time. Theproblems that Einstein could not solve remain ones that today’sphysicists, mathematicians and scientist find most challenging.  So I will leave you with some of themore interesting quotes by Albert Einstein and hope that you are as interestedas 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 becalled research,

 wouldit?”

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

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

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

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

 direction.”

References:

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

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

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

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

© 1990 ISBN 0-385-24228-X

Clarice Swisher, “Relativity: Great Mysteries OpposingViewpoints” Greenhaven Press, Inc. P.O. Box 289009, San Diego, CA92128-9009 © 1933 ISBN 0-89908-076-6

Jeremy Roberts, “How Do We Know the Laws ofMotion” The Rosen Publishing Group, Inc. New York © 2001 ISBN0-8239-3383-0

Websites:

http://www.bartleby.com/173/  Bartleby.com – Great Books Online:  Relativity – The Special andGeneral 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 accessed9/15/02)

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 Canyou please explain Einstein's theory of relativity… (Firstaccessed 9/15/02)

http://www.geocities.com/einstein_library/index.htm  Einstein Library (First accessed9/15/02)

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

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

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