Special relativity:
energy, momentum
and mass
Physics 123
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Lecture IX
1
Outline
• Lorentz transformations
• 4-dimentional
energy-momentum
• Mass is energy
• Doppler shift
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Lecture IX
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Lorentz transformations
System (x’,y’z’,t’) is moving with respect to system (x,y,z,t) with
velocity v
•
•
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•
•
•
•
•
•
•
Galileo
x=x’+vt’
y=y’
z=z’
t=t’
g
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Lorentz
x=g(x’+vt’)
y=y’
z=z’
t=g(t’+vx’/c2)
v
1
0
1 v / c
g  1, g v


c
2
2
Lecture IX
3
Time dilation
• Clocks moving relative to an observer are measured by
the observer to run more slowly ( as compared to clocks
at rest)
 Dt – measured in v=0 frame, Dt0- measured in moving
frame
Dt 
Dt0
1 v2 / c2
Hendrik Antoon Lorentz
Derived time and space
transformations before
Einstein
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Lecture IX
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Twin paradox
• Two twins: Joe and Jane. Joe stays on Earth and Jane goes to Pluto at
v<~c
• Joe observes that Jane's on-board clocks (including her biological
one), which run at Jane's proper time, run slowly on both outbound
and return leg. He therefore concludes that she will be younger than
he will be when she returns.
• On the outward leg Jane observes Joe's clock to run slowly, and she
observes that it ticks slowly on the return run. So will Jane conclude
that Joe will have aged less? And if she does, who is correct?
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Lecture IX
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Length contraction
• No change in directions
perpendicular to velocity
h0
L  L0 1  v 2 / c 2

v
L0
• The length of an object is
measured to be shorter when
it is moving relative to the
observer than when it is at
rest
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h  h0
Lecture IX
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4-dimensional space – time
• Add time to space metric: x1=x, x2=y, x3=z, x4=ict
• 4- dimensional “length”=interval - Lorentz invariant
AB  x 2  y 2  z 2  (ct ) 2
• AB – real – space-like interval, there exists a frame of reference
where the two events happen at the same time (t1=t2 ), but at
different places (r12≠0)
• AB – imaginary – time-like interval, there exists a frame of
reference where the two events happen at the same place (r12=0),
but at different times (t1≠t2)
x A
x
y
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B
y
Lecture IX
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Energy, mass and momentum
m0 – mass at rest
• Relativistic energy:
m0 c
E
2
1 v / c
2
2
 gm0 c
2
• Energy at rest
E=m0c2
• Kinetic energy:
KE  gm0 c 2  m0 c 2
• Relativistic momentum:
p
m0 v
1 v2 / c2
  v/c
 gm0 c • 4-dimensional Energy –
momentum – vector:
• (pxc, pyc, pzc, iE)
• Lorentz invariant interval:
m0 c  E  p c
2
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Lecture IX
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2 2
8
Conservation laws
• Both energy and momentum are conserved in the
 

 
relativistic case:
p1  p2  p3  ...  p1 ' p2 '...
E1  E2  E3  ...  E '1  E '2 ...
• Mass must be considered as an integral component of
energy E=gmc2
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Lecture IX
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Conservation laws
• Energy could be used to create mass
m(e  )c 2  m(e  )c 2  0.5MeV
m( Z )c 2  90GeV  90 103 MeV
e  e  Z
E (e  )  E (e  )  45GeV  45 103 MeV
pc  E 2  (mc2 ) 2  (45 103 ) 2  0.52  45 103 MeV
KE  E  mc2  45 103 MeV  0.5MeV  45 103 MeV
• To conserve momentum electron and positron must
collide head on. Then Z-boson is produced at rest.
 
 
p (e )   p ( e )
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Lecture IX
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Conservation laws
• Mass could be destroyed and converted into energy
m( 0 )c 2  140 MeV
m(g )c 2  0
 0  gg
E (g 1 )  E (g 2 )  70 MeV
pc  E 2  (mc 2 ) 2  (70) 2  0 2  70 MeV
KE  E  mc 2  70MeV  0MeV  70 MeV
• To conserve momentum (zero initially) the photons must be
flying in the opposite direction with the same absolute values of


momenta
p(g )   p(g )
1
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Lecture IX
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Mass and energy
• Mass and energy are interchangeable
• Energy can be used to create mass (matter)
• Mass can be destroyed and energy released
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Lecture IX
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Doppler shift
• Light emitted at f0,l0
• In the source’s r.f.
– the distance between crests is l0
– The time between crests is t0=1/f0= l0/c
• Where are crests in the r.f. moving with speed v wrt
source’s r.f. (chasing the wave)
 l=cDt-vDt=(c-v)Dt
 Dt=gDt0=g l0/c
 l=(c-v) g l0/c
l
cv
c v
f  f0
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Lecture IX
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2
l0  l0
cv
cv
cv
cv
13
Doppler shift
• When the source and the observer move towards each
other the wavelength decrease (redviolet)
cv
cv
l  l0
; f  f0
cv
cv
• When the source and the observer move away from each
other the wavelength increase (violet  red) – Redshift –
used to measure galaxies velocities universe expansion
(Hubble)
l  l0
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cv
cv
; f  f0
cv
cv
Lecture IX
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