Lecture 2 Relativistic Kinematics II

7/25/2019 Lecture 2 Relativistic Kinematics II

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Lecture 2. Relativistic kinematics, part II

Outline:

Length Contraction Relativistic Velocity Addition

Relativistic Doppler Effect

Red shift in the Universe


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Relativistic effects: length contraction

K

K

Question :how long does the signal tae to co!plete the ro"nd trip#

!ir

ror

An observer in the cars rest RF : 2 x

tc

=

An observer on

the groun :

( )

1 21 2 1 2

1 2 2 2

' ' ' '' ' ' ' '

' ' 1 1 2' ' ' ' '

x V t x V tt t t t t

c c

x x ct t t x x

c V c V c V c V c V

+ = + = =

= = = + = + +

$hese intervals are related %y the ti!e dilation for!"la&2 2

'1 /

ttV c

=

2 2' 1 /x x V c = !"oving ob#ects

are shortene$

2 2

2 2

22 1 / '

x cV c x

c c V

=


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Length %ontraction &cont'

An o%server in the R' K!oving with respect to the R' K0with

the velocity Vdirected parallel to the !eter stic( !eas"res its

length) *n order to do that( he+she finds two points ()and (2in

his+her R' that wo"ld simultaneouslycoincide with the ends

of the !oving stic ,t) *t2-)

0

1x

o%server

V

K.

K

0

2x

( ) ( )2 1 2 10 0 2 12 1

2 2

2 21 1

x x V t t x xx x

V V

c c

= =

0 0

2

21

x Vtx

V

c

=

0

2

21

x Vtx

V

c

+=

Comment *ts easier to write L)$r) for the proper length

interval in the right/hand side&

Of course, the same results stems directly from L.Tr.:

2 1t t=

+roper length L:the length of an o%0ect !eas"red in its rest R' , -)0 0 0

2 1x x x =

/ the end positions are !eas"redsimultaneously in K

2

0 21

VL L

c=

- moving ob#ects

are contracte in

the irection of

their motion

( ) ( )0 02 1 2 1 2 1 2 1x x x x V t t x x = = 0L L=Co!pare&


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Length contraction &cont'

2

0 21

VL L

c=

- moving ob#ects are

contracte in the irection of

their motion

/V c

=

0/L L

1

1

2 Contraction occ"rs only

in the direction of relative

!otion of R's1

V

KK

disc at restthe sa!e disc as seen

%y o%server K

) $o o%serve this effect( the relative speed of the reference

fra!es sho"ld %e large) 'or the fastest spacecraft( the speed

is 23./4c( and the effect is of an order of 3./5&

( )

( )

ct ct x

x x ct

y y

z z

=

=

==


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Recapitulation: eca of cosmic-ra muons

82.994 10 / 0.998 0.998v m s c = = =N06 the n"!%er of !"onsgenerated at high altit"de

N6 the n"!%er of

!"ons !eas"red in

the sea/level la%

6

0 2.2 10t s = *n the !"ons rest fra!e

7y ignoring relativistic effects ,wrong1-( we get the decay length&

( )

6 8

0

0 0

2.2 10 3 10 / 660

20,000exp exp 30

660

L t c s m s m

N N N

= = =

= =

*n fact( the decay length is !"ch greater( the !"ons can %e

detected even at the sea level1

28. !

7eca"se of the ti!e

dilation( in the R' of the

la% o%server the !"ons

lifeti!e is& 6 835 10 3 10 / 10.5L s m s km= =

9"ons are created at high altit"des d"e to collisions of fast cos!ic/ray particles ,!ostly

protons- with ato!s in the Earth at!osphere) ,9ost cos!ic rays are generated in o"r

gala:y( pri!arily in s"pernova e:plosions-

9"on 6 an electrically charged unstableele!entary particle with a rest energy 2 8.; ti!es

greater than the rest energy of an electron) $he !"on has an average half/life of 2.2 )-/s)

60

235 10

1

tt s

=

a

ltit"de

( )0 020,000

exp exp 210,500

N N N

=


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0eca of cosmic-ra muons in the muons RF


7/14

+roblems).$he nearest star to the Earth is =ro:i!a Centa"ri( 4)> light/years away)

/ at what constant speed !"st a spacecraft travel fro! the Earth if it is to reach

the star in 8)? years( as !eas"red %y travelers on the spacecraft#

/ how long does this trip tae according to earth o%servers#

V

2.Consider a disc at rest) years

According to earth o%servers&

/ 4.3

5/ 0.864

L L c yr

t yrV V c= = =


8/14

Relativistic elocit Aition

3pee of light is the largest spee in

nature, no bo nor an signal can

travel 4ith the spee greater than c.

I! K& a particle !oves a

distance d"in a ti!e dt

2 1

2 1

x xv

t t

=

o%server

V

K

K

v

I! K#& a particle !oves a

distance d"#in a ti!e dt#

( )1 1 1'x x Vt=

1 1,x t 2 2,x t 2 1

2 1

' ''' '

x xvt t

=

( )2 2 2'x x Vt=

( )21 1 1' /t t V c x = ( )22 2 2' /t t V c x =

( ) ( )

( ) ( )( )

2 1 2 1

2

2 1 2 1 2

'

/ 1

x x V t t v Vv

vVt t V c x x c

= =

2

'

1

v Vv

vV

c

=

6 anti/parallel

/ / parallel

,v V

, 'v V c v v V


9/14

+roblems

2

1.2' 0.88

1 0.361

v V cv c

vVc

+= =

++

).A person on a rocet traveling at 0.$c ,with respect to the Earth- o%serves a !eteor

passing hi! at a speed he !eas"res as 0.$c) ow fast is the !eteor !oving with respect

to the Earth#

K

Kalilean velocity addition& ' 1.2v v V c= + =

0.6 0.6v c V c= =

Relativistic velocity addition&V

0.6v c=

2.As the o"tlaws escape in their gateway car( which goes >+4c( the police officer fires a%"llet

fro! the p"rs"it car( which only goes 3+8c) $he !"le velocity of the %"llet ,relative to the

g"n- is 3+>c) Does the %"llet reach its target ,a- according to alileo( ,%- according to

Einstein#

1 0.75v c=2 0.5v c=

3 0.33v c=3 2

' 0.83v v V v v c= + = +

K

K

3 23 2

3 2

5 / 6' 0.71

1 / 1 1/ 6

v v cv c

v v c

+= =

+ +


10/14

5ransverse 0oppler 6ffectClassical Doppler effect ,e)g)( Doppler effect in so"nd( the increase in pitch of a so"nd

when its so"rce approaches "s-&

0

1 /

1 /

v c

f f V c

+

=

%6 the speed of an o%server &ith res'ect to air,the

!edi"! where the waves propagate-

V6 the speed of the so"rce of so"nd &ith res'ect to air

0oppler effect in light / a change in the o%served light freF"ency d"e to a relati%e

!otion of the light so"rce and an o%server ,no special R' associated with the !edi"!

where light propagates1-&

). 5ransverse 0oppler effect

lightwave

fronts o%server

$he origin of the transverse Doppler effect is ti!e dilation( this is a 'ure

relati%isticeffect( no co"nterpart in classical !echanics)

2 2

0

0

1 11 1f f

T T = = =

V

0T / the period of oscillationsof the e)/!) field in the

rest R' of the so"rce K

,the proper ti!e-

0 01/f T=

T / the period of oscillations in the R' of the !oving o%server

1 /f T=f is always s!aller thanf.

6 red shift ,shift to lower

freF"encies-

K

K

f ,f0- 6 the freF"ency of so"nd heard %y an o%server ,in the rest fra!e of the so"rce-)


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Longituinal 0oppler 6ffect

V is the velocity of the relative

!otion of an o%server with respect

to the light so"rce)

$he !ost freF"ent enco"nter with Doppler effect in light ,!icrowave-&

police radar speed detectors ,no relativistic effects tho"gh-

1VT V

T T Tc c

= + = +

( )0 0 02

1 1 11

1 11T T T f f

+ = + = = +

V V

light o%server

V

K

K

3.

an e:tra ti!e needed for the ne:tlight front to reach an o%server

( )

0

21 /

TT

V c=

8.

/ the sa!e ti!e dilation as in thecase of the transverse Doppler Effect

/ !re shift$

$he light so"rce and the o%server !ove away fro! each other)

$he light so"rce and the o%server approach each other)

0

1

1f f

+=

/ !blue shift$,shift toward higher freF"encies-


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7ubbles La4 &)828'

$he Universe e:pands& the larger the distance to an o%0ect( the larger the ,relative- speed)

7y !eas"ring the red shift of ,identifia%le- spectral lines( one can calc"late the recessional

speed of the light so"rce with respect to the Earths o%server)

According to 7ubble9s La4( there is a direct proportionality ,at least at not too large

distances- %etween the velocity and the distance to the so"rce&

0V H d=

V/ the o%served velocity of the gala:y away fro! "s

(/ "%%leBs GconstantG ,"nits& s/3-

d / the distance to the gala:y ,3 9egaparsecH>3.Ilight/yrs-

$oday the val"e of (0 is still rather "ncertain( %"t is

generally %elieved to %e 2;.8. !+sec+9pc) "%%les

constant gives "s the age of the )ni%erse 0&

9

0 01/ 13.8 10H yr = 18 1

0

/70 2.3 10

km sH s

Mpc

=

tno&

the horion of

visi%ility H infinite

red shift

c


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6(treme !re shifts$: uasars an %";R

Cosmic *icro&a%e +acground adiation -C*+*n the

standard 7ig 7ang !odel( the radiation is deco"pled fro!

the !atter in the Universe a%o"t >..(... years after the 7ig

7ang( when the te!perat"re dropped to the point where

ne"tral ato!s for! ,T2>...K-) At this !o!ent( the Universe

%eca!e transparent for the pri!ordial photons) $hisradiation is co!ing fro! all directions and its spectr"! is

F"ite distinct fro! the radiation fro! stars and gala:ies-)

$he sub-mm/THzrange contains 2 half of the total l"!inosity of the Universe and

8


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+roblem

*!agine an alien spaceship traveling so fast that it crosses o"r gala:y ,whose rest

dia!eter is 3..(... light/years- in only 3.. years of spaceship ti!e) %servers at rest in

the gala:y wo"ld say that this is possi%le %eca"se the ships speed is so close to 3 that

the proper ti!e it !eas"res %etween its entry into and depart"re fro! the gala:y is !"ch

shorter than the gala:y/fra!e coordinate ti!e ,23..(... ly- %etween those events) 'indthe e:act val"e of the speed that the aliens !"st have to cross the gala:y in 3.. years)

0

21

tt

=

2 3 2 6 2 601 10 1 10 1 10

t

t

= = = =

6

6 101 10 1 0.99999952

= =

( ) ( ) 211 1 .....

2!

n n nn

+ = + + +

ow does it loo to the aliens# $o the!( their clocs are r"nning nor!ally( %"t the gala:y(which !oves %acward relative to the! at speed 3( is Lorent contracted)

Lecture 2 Relativistic Kinematics II

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Transcript of Lecture 2 Relativistic Kinematics II