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) $he !"on has an average half/life of 2.2 )-/s)
60
235 10
1
tt s
=
a
ltit"de
( )0 020,000
exp exp 210,500
N N N
=
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0eca of cosmic-ra muons in the muons RF
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+roblems).$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= = =
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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
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+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
+= =
+ +
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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)
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