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Chapter 1: RelativitySpecial Relativity: “all motion is relative”

All motion is described relative to an inertial frame of reference

Inertial frame: frame of reference in which Newton’s law of inertia holds

Inertial frames are (mathematically) simpler in Special Relativity, and are related in a (mathematically simple) manner. Accelerations and forces are accounted for in S.R.

General Relativity = Special Relativity + Gravity (“curved space time” = more complex relations between inertial reference frames)

Postulates of Special Relativity

The laws of physics are the same in all inertial frames of reference. (mathmatical form, conservation laws, etc.)

The speed of light in free space has the same value for all inertial frames of reference. v ~ 3.00E8 m/s

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The First Postulate and “everyday” relativity:

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

x

y

z

x’

y’

z’

v

x’ = x vt

y’ = y

z’ = z

t’ = t

vdx

dtv v

vdy

dtv

vdz

dtv

x x

y y

z z

'

'

'

'

''

''

'

Conservation of momentum, kinetic energy, etc

But waves like sound waves have “special” inertial frame, the frame at rest w.r.t. medium

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Second Postulate: The speed of light in free space has the same value for all inertial frames of reference

need a new relationship between coordinates of different inertial frames

linear, invertible “transformation” = Lorentz Transformations

xx vt

v c

y y

z z

tt v c x

v c

xx vt

v c

y y

z z

tt v c x

v c

'

'

'

'

' '

'

'

' '

1

1

1

1

2

2

2

2

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Time Dialation: the time interval between events depends upon the inertial observer

proper time t0: the time interval between two events in a frame of reference where the events occur at the same place.

timing with a moving mirror

L0

v

L0

v

c t0/2c t/2

v t/2

c t L

c t L v tt

t

v c

0 0

20

2 20

2

2

2 2 1

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Time Dialation and the passage of time

t0 = time interval for clock at rest relative to an observer = proper time

t = time interval for clock in motion relative to an observer, t > t0

v = speed of relative motion

c = speed of light

t

t

v c

0

21

Example 1.1: A spacecraft is moving relative to the earth. An observer finds that, according to her clock, 3601s elapse between 1 PM and 2 PM on the spacecrafts’s clock. What is the spacecraft’s speed relative to the earth?

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Doppler Effect: frequency shifts due to moving source/observer

doppler effect for sound

0 = emitted frequency

= detected frequency

0

1

1

v c

V c

/

/

v = speed of observer

V = speed of source

c = speed of sound

But, light does not require a medium!

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Doppler effect with light: three cases

1- observer moving perpendicular to a line between him and light source

time between ticks is dilated

sour

ce

tt

v c

t

t

v c

0

2

00

0

2

1

1

1

1

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Doppler effect with light: three cases

2- observer moving away from source

time between ticks is dilated

next tick travels farther sour

ce

cvcv

cv

cvcv

cv

t

cv

Tt

cv

tcvt

tT

cv

tt

11

1

11

1

1

1,

1

1

1

0

20

0

2

00

20

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Doppler effect with light: three cases

2- observer moving away towards source

time between ticks is dilated

next tick travels less distance sour

ce

tt

v c

T tvt

ct

v

c

t T

v c

tv

c

v c v c

v c

v c

v c

0

2

00

2

0

0 2

0

1

1

1 1

1

1

1 1

1

1

1

,

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Example 1.2: A driver is caught going through a red light. The driver claims that the color she actually saw was green ( = 5.60x1014 Hz) and not red ( 0= 4.80x1014 Hz) because of the doppler effect. How fast must she have been going?

Example 1.3: A distant galaxy Hydra is receding grom the earth at 6.12x107 m/s. By how much is a green spectral line of wavelength = 500 nm emitted by this galaxy shifted towards the red end of the spectrum?

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Length Contraction: determining the extent of a moving object

L0 =vt

v

L =vt0

v

t t0

1 v c 2

L vt0 vt 1 v c 2 L L0 1 v c 2

Example: Muons created in earth’s atmosphere by cosmic rays have speeds of about 2.994x108 m/s(.998c) At rest, a muon lifetime is about 2.2 s.

-How far would these muons travel in 2.2 s?

-What is the time dialated lifetime? How far do they travel in THIS amount of time?

-What is the length contracted distance of the previous answer, as seen in the muons frame of reference?

-How far does the earth move relative to the muons during the 2.2 slifetime?

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The Twin “Paradox”: a counter-intuitive result

Two twins on earth are separated temporarily when one twin takes a trip at .80c to a star twenty light years away.

Earth bound twin: round trip takes 2x(20 ly / .8ly/y) = 50 years

Traveling twin: Earth-Star distance is length contracted to

L L0 1 v c 220ly 1 .8 2

12ly

so round trip takes 2x(12 ly / .8ly/y) = 30 years

Twins age differently! Travelling twin does not stay in one inertial reference frame, situation is not symetric.

Example 1.4: Each twin emits a radio signal once a year for the duration of the voyage. How many signals does each twin recieve?

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Electricity and MagnetismMaxwell’s Equations

Integral Form Differential Form

E dAQ

B dA

B dl Id

dt

E dld

dt

enc

o

o c oE

M

0

( )

E

B

B JdE

dt

EdB

dt

o

o c o

4

0

( )

Lorentz Force Law F q(

E

v B )

Mathematically invariant formulation

electric charge is a relativistic invariant quantity

Maxwell’s equations Wave Equation2

2

22

21

t

E

cx

E yy

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Relativistic Inertia (“relativistic mass”)

consider elastic collision between two identical (when at rest) masses

x

y

zx’

y’

z’v

S

S’

Y

V’B

VA

Ball A moves vertically only in frame S with speed VA , Ball B moves vertically only in frame S’ with speed VB ’= VA . Ball A rebounds in S with speed VA , Ball B rebounds in S’ with speed VB’ .

Y/2

Y/2

Collision in S Collision in S’

VA T0 2Y 2 VB 'T0 Y

momentum conservation: mAVA mBVB

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momentum conservation in S :

mAV A mBVB

VB Y T

T T0 1 v c 2

mA Y T0 mB Y T0 1 v c 2

mB mA 1 v c 2

massrest

1

Inertia) tic(Relativis Mass" icRelativist"

0

20

m

cvmm

Example 1.5: Find the mass of an electron (m0 = 9.1E-31 kg)whose speed is .99c.

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Relativistic Momentum

p m

v

m0

v

1 v c 2

Newton's Second Law F

dp

dt

dmv

dtm

dv

dt

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Relativistic Kinetic Energy

KE Fds0

s

d mv

dtds

0

s

d mv v0

v

mv 2 mv dv0

v

m0v 2

1 v c 2

m0v

1 v c 2dv

0

v

m0v 2

1 v c 2 m0 c2 1 v c 2

0

v

m0v 2

1 v c 2 m0 c2 1 v c 2

m0c2

KE m0 c2

1 v c 2 m0c2 mc2 m0c2

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KE mc2 m0c2

mc2 KE m0c 2 E KE E0

E mc2 total energym0c2

1 v c 2

E0 m0c 2 rest energy

at low speeds (v c)

KE m0 c2

1 v c 2 m0c2 m0c2 1 v c 2 1 2

1

KE m0c2 11

2v c 2

1

1

2m0v 2

Correspondence Principle!

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Example 1.6: A stationary body explodes into two fragments each of mass 1kg that move apart at speeds 0.6c relative to the original body. Find the rest mass of the original body.

Example 1.7: Solar energy reaches the earth at a rate of about 1.4kW/m2. Given that the average radius of the earth’s orbit is 1.5E11 m, how much mass does the sun lose per second?

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General momentum-energy relations

E m0c 2

1 v c 2p

m0v

1 v c 2

E2 p2c2 m0

2c4

1 v c 2

m0

2v 2c2

1 v c 2

m0

2c4

1 v c 21

v 2

c2

m0

2c4

E2 p2c2 m0

2c4

for a massless particle (photon, neutrino, ...)

E pc

(and v c always)

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Electronvolts: typical energy scale for atomic physics

1eV = 1.6E-19 Coulomb x 1 Volt = 1.6E-19 J

X-Ray energies ~ keV

Nuclear Physics ~ MeV

Particle Physics ~ GeV

Example : What are the momenta and speeds of a 5 MeV

-electron? (m0 = .511 MeV/c2,

-proton? (m0 = 938 MeV/c2,

-photon? (m0 = 0 MeV/c2).

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Chapter 1 problems: 1,2,4,5,7,9,10,11,12,16,19,20,22,32,33,34,35,36,41,47,49