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Physics IClass 26

The Special Theory ofRelativity*

Rev. 21-Apr-04 GB

*This material is optional.It will not be on any exam.

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Newton’s Laws of Motion

1. Newton’s First Law: No net force, no change in motion.

2. Newton’s Second Law: amFnet

3. Newton’s Third Law: All forces come in pairs.(equal magnitudes and opposite directions)

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Are Newton’s Laws True?

It’s been over 300 years since Newton published Principia Mathematica.How have his laws done since then?

The First Law is still doing fine. In modern times, many types of verylow-friction motion (space travel, magnetic bearings, air hockey tables,etc.) make this notion more intuitively appealing than in the past.

The Third Law is also doing fine. All forces currently known to physicsobey this law. Any force not obeying this law would cause big problemsin physics, like getting free mechanical energy from nothing.

However, the Second Law in the form we learn it in Physics I is notexactly correct. Where did Newton go wrong?

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Where Did Newton Go Wrong?

Newton defined time and space as follows:

“Absolute, true, and mathematical time, of itself and from its own nature,flows equably without relation to anything external…”

“Absolute space, in its own nature, without relation to anything external,remains always similar and immovable.”

As the 19th Century drew to a close, it became evident that there wassomething wrong with these assumptions.

Isaac Newton, 1642-1727

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Maxwell’s Electromagnetic Theory - 1873M a x w e l l d e v e l o p e d a t h e o r y o f e l e c t r o m a g n e t i s m t h a te x p l a i n e d a l l t h e p h e n o m e n a o f e l e c t r i c i t y a n dm a g n e t i s m k n o w n t h e n a n d p r e d i c t e d s o m e t h i n g n e w :e l e c t r o m a g n e t i c w a v e s . T h i s p r e d i c t i o n w a s c o n f i r m e db y H e r t z i n 1 8 8 6 a n d l i g h t w a s s o o n s h o w n t o b e a t y p eo f e l e c t r o m a g n e t i c w a v e .

t

BE

D

t

DJH

0B

B u t a q u e s t i o n r e m a i n e d : I f l i g h t i s a w a v e , w h a t i s i t sm e d i u m o f p r o p a g a t i o n ? M o s t p h y s i c i s t s a s s u m e d t h a tt h e r e m u s t b e o n e a n d c a l l e d i t t h e e t h e r . E t h e r w a sa s s u m e d t o d e f i n e a f i x e d r e f e r e n c e f r a m e f o r t h eu n i v e r s e ( N e w t o n ’ s “ a b s o l u t e s p a c e ” ) t h r o u g h w h i c he l e c t r o m a g n e t i c w a v e s t r a v e l a t s p e e d c .

James Clerk Maxwell(1831-1879)

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The Michelson-Morley Experiment - 1887

Michelson’s interferometer was designed to measure slight differences in phasebetween two light beams that travel in orthogonal directions. By measuring thephase differences at various places in earth’s orbit, Michelson and Morley hopedto measure the speed of earth with respect to the ether.However, no effect was found. The speed of light is constant for all observers.

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Einstein’s Postulates of theSpecial Theory of Relativity

Albert Einstein (1879–1955)

Studying Maxwell’s equations and noting aremarkable symmetry in them between space andtime, Einstein replaced Newton’s definitions ofspace and time with two new postulates that leddirectly to the Special Theory of Relativity.

Einstein’s Two Postulates of Special Relativity (1905):

1. The laws of physics are the same in all inertial frames.

2. The speed of light, c, is constant in all inertial frames.

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What is an Inertial Frame?

An inertial frame (of reference) is a real or imaginary set ofdevices for measuring position and time that are in motion togetheraccording to Newton’s First Law; in other words, these devicesare not accelerating (or rotating).

Neglecting gravity and the small acceleration of Earth (those arecovered in General Relativity), the track and motion detector thatwe use in our activities, along with the clock in your PC when yourun LoggerPro, comprise an inertial reference frame with a specialname: the laboratory reference frame (because this is the framewe use to make measurements).

If we were to set the same equipment up in the Ferris Wheel westudied earlier, that would not be an inertial reference frame.Instead, we call that an accelerated frame.

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Where Did Einstein’s Postulates Come From?

1. The laws of physics are the same in all inertial frames.

This idea goes back to Galileo. Imagine an experiment like our carttrack and hanging weight being performed in an airliner movinguniformly in one direction at a constant speed and altitude. (Assumeno turbulence and the altitude is low enough so that the force ofgravity is about the same as on the ground.) Our measurements thatwe take with LoggerPro should be the same as what we did in classif we set things up carefully.

2. The speed of light, c, is constant in all inertial frames.

Few wanted to believe this prior to the Michelson-Morleyexperiment in 1887. There are now many measurements confirmingthis postulate and none contradicting it. However, do a web searchand you will see that many people still refuse to believe it!

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Postulate #1

The first postulate states that anyexperiment performed in anyinertial reference frame (like thestarship Enterprise moving at 0.5 c)will exhibit the same laws ofphysics as in any other inertialframe (like space station DeepSpace 9 at rest). In fact, there is noway to know which inertial frameis moving and which is at rest.

0.5 c

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Postulate #2

The second postulate states that nomatter what the state of motion ofany source of electromagneticwaves (like Enterprise sending asensor beam), the speed of thewaves as measured in any inertialframe (like Deep Space 9 or theEnterprise) will be exactly c.

0.5 c

c

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How Is This Possible?

Kirk: “Scotty, measure the speed of oursensor beam relative to Enterprise.”Scott: “Exactly 299,792,458 m/s, sir!”

0.5 c

c

Sisko: “O’Brien, measure the speed of Enterprise’ssensor beam relative to us in Deep Space 9.”O’Brien: “Exactly 299,792,458 m/s, sir!”

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Consequence #1Time Dilation (Part 1)

Engineer Scott on Enterprise constructs a special kind of “clock” thatemits a pulse of light, bounces it off a mirror, and detects the returningpulse. Scott makes the distance to the mirror exactly 14.9896229 cm, sothe round trip takes exactly one nanosecond. One billion “ticks” of thisclock = one second.

Scott synchronizes all of the clocks on Enterprise to this standard clock.

mirror

pulsed lasersource & detector

d0

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Consequence #1Time Dilation (Part 2)

How it looks to Chief O’Brien on Deep Space 9:The path of light is longer because the device is moving, but light still travels at thesame speed of light. Therefore, O’Brien measures a longer time for each “tick”.If t0 is the time of a tick as measured on Enterprise, t is the time on Deep Space 9:

22

0

cv1

tt

Because of postulate #1, this time dilation effect applies not just to Scott’s speciallaser clock, but to all clocks on Enterprise, even the biological clocks of the crew!

A little algebra (not rocket science): Path length:

22

120 tvd2s

Speed of light: tcs Solve for t and use t0 = 2 d0 / c

d

vt/2

0

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Consequence #2Length Contraction (Part 1)

S c o t t h a s a s e c o n d c l o c k t h a t i s o r i e n t e d b o w - s t e r n i n s t e a d o f u p - d o w n .T h i s c l o c k i s o t h e r w i s e i d e n t i c a l t o t h e f i r s t a n d k e e p s t h e s a m e t i m e .

H o w i t l o o k s t o C h i e f O ’ B r i e n o n D e e p S p a c e 9 :O n t h e f i r s t p a r t o f t h e p a t h , l i g h t i s “ c h a s i n g ” t h e m i r r o r . O n t h e w a y b a c k , t h ed e t e c t o r i s m o v i n g t o w a r d t h e l i g h t . T h e t o t a l t i m e o f t h e “ t i c k ” i s

2221 cv1

cd2

vc

d

vc

dttt

T h i s t i m e m u s t b e t h e s a m e a s O ’ B r i e n m e a s u r e d f o r t h e f i r s t c l o c k o r P o s t u l a t e # 1w o u l d b e v i o l a t e d . D o i n g t h e a l g e b r a , t h e o n l y p o s s i b l e c o n c l u s i o n i s t h a t d a sm e a s u r e d b y O ’ B r i e n m u s t b e d i f f e r e n t t h a n d 0 a s m e a s u r e d b y S c o t t :

220 cv1dd

lightvlight

d

v

d

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Consequence #2Length Contraction (Part 2)

The length contraction effectapplies only in the direction ofrelative motion of an object asobserved from another inertialframe of reference.

To the equipment and peopleinside Enterprise, nothingunusual is noticed. As far asthey are concerned, they are atrest while everything aroundthem is moving.0.95 c

0.8 c

0.5 c

v = 0

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Are There Real Values for Time and Length?

Evidently, time and length can vary depending on which inertialframe of reference we use, but can we find the real values of timeand length, as opposed to apparent values? All values that wemeasure in a repeatable way are real, but physicists have two specialways of measuring time and length so that everyone can agree:

Proper Time:The proper time interval between two events is the time measured inthe inertial frame in which the events happen at the same location.

Proper Length:The proper length of an object is the length measured in the inertialframe in which the object is at rest.

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Special Relativity Applied toElectric and Magnetic Fields

An electron is moving in the +X direction at 1000 km/s in an electricfield of 1000 V/m in the +Y direction and a magnetic field of 0.001 Tin the +Z direction. What is the net force on the electron?

Take a few minutes to determine the answer before we continue.

Y

B = 0.001 T Xelectron

E = 1000 V/mv = 1000 km/s

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Special Relativity Applied toElectric and Magnetic Fields

You should have gotten exactly 0 because the electric force cancelsthe magnetic force. The electron follows Newton’s First Law.

Now look at the situation from the electron’s inertial frame.

Since the electron is not moving in this frame, the magnetic forcemust be zero no matter what the magnetic field. But the total forcemust still be zero since the electron stays at rest in this inertial frame.

What happened to the electric field or electric force?

Y

B' = ? X'electron

E' = ?v = 0

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Special Relativity Applied toElectric and Magnetic Fields

The electric and magnetic fields are not separate fields! They arereally components of the same field and these components transforminto each other when we switch reference frames. In this case, thetransformation equations are

22

zyy

cv1

BvEE

22

2yz

zcv1

cEvBB

You can verify that E' = 0 for this case. The reason there is no electricforce in the electron’s frame is that the electric field is zero.

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Einstein’s Corrections toNewtonian Mechanics

T h e fa m ilia r q u a n titie s o f N e w to n ia n m e c h a n ic s h a v e n e wd e fin it io n s in S p e c ia l R e la tiv ity :

M o m e n tu m : vcv1

mp

22

T o ta l E n e rg y : 22

22

cv1

cmKcmE

K in e tic E n e rg y :

1

cv1

1cmK

22

2

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Einstein’s Corrections toNewtonian Mechanics

R e l a t i o n o f p a n d E : 42222 cmcpE

R e l a t i o n o f p a n d K : 2222 cmK2Kcp

N e w t o n ’ s 2 n d L a w ( N e w F o r m ) :

22netcv1

vm

td

d

td

pdF

F r o m t h e l a b o r a t o r y i n e r t i a l f r a m e o f r e f e r e n c e , a p a r t i c l e ’ s m a s sa p p e a r s t o i n c r e a s e a s t h e p a r t i c l e m o v e s f a s t e r . T h i s e f f e c t i ss i g n i f i c a n t o n l y w h e n t h e s p e e d o f t h e p a r t i c l e a p p r o a c h e s c .

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If “F = m a” isn’t true,why do we still use it?

The original form of Newton’s Second Law is true to a very goodapproximation when dealing with velocities much less than the speed oflight. For most calculations involving ordinary objects, it is close enoughfor practical purposes. For some calculations, we need relativity.

“Disintegration of the Persistence ofMemory” by Salvador Dalí, 1931

Art inspired by theTheory of Relativity?

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What’s so “special” aboutSpecial Relativity?

The “special” part means that it deals only with inertial frames ofreference. Einstein worked another decade to generalize his ideas toaccelerating frames of reference. In 1915, his General Theory ofRelativity revolutionized our ideas about gravity and the nature ofspace and time. Unfortunately, we can’t go into the details today.

To find out more, check the website on the next slide.

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http://www.amnh.org/exhibitions/einstein/?src=h_h

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Activity #26Special Relativity

Objectives of the Activity:

1. Think about and resolve the “paradoxes” of Special Relativity.2. Try some simple calculations using Special Relativity.3. Have fun. (Well, for some people this will be fun.)