Modern Physics
Chapter 2
Companion Website
http://wps.aw.com/aw_harris_mp_2/69/17800/4556992.cw/index.html
• Online Flash demos
Constant Speed of Light
• c = 3 x 108 m/s
• Why did it take so long to figure this out?
Constant Speed of Light
• Think about everyday speeds
Helios
Galileo
Albert Einstein (3/14/1879 – 4/18/1955)
• 1894 (Age 15)
– The Investigation of the State of Aether in Magnetic Fields
– Failed entrance exam to ETH (Swiss Fed. Inst. Tech.)
• 1895
– Chasing light idea, Special Relativity
• 1900
– Finished ETH (physics and math)
– Swiss patent office clerk (EM devices)
• 1905 (Annus Mirabelis)
– 4 papers
•
•
•
•
Photoelectric Effect
Brownian Motion
Special Relativity
Matter and Energy Equivalence
c Constancy
1884 – J. C. Maxwell formulated four equations to describe light, which depend on c being constant
Electric charges cause electric fields
No magnetic monopoles to cause magnetic fields
Changing magnetic fields cause electric fields
Electron currents and changing electric fields cause magnetic fields
Michelson-Morley
• 1887-1907: Find the velocity of the Aether
Michelson-Morley
• 1887-1907: Find the velocity of the Aether
v
ether
v
mirror
Light source
u
c
mirror
screen
Michelson-Morley
• 1887-1907: Find the velocity of the Aether
mirror
mirror
Light source
ether
v
screen
Einstein’s Postulates
1. Physics is the same regardless of inertial
frame
2. Light moves with constant speed in the eyes
of all observers
Postulate 1
Suppose you are sitting in a soundproof, windowless room
aboard a hovercraft moving over flat terrain. Which of the
following can you detect from inside the room? May have
multiple answers.
1. rotation
2. deviation from the horizontal orientation
3. motion at a steady speed
4. acceleration
5. state of rest with respect to ground
Postulate 1
Suppose you are sitting in a soundproof, windowless room
aboard a hovercraft moving over flat terrain. Which of the
following can you detect from inside the room? May have
multiple answers.
1. rotation
2. deviation from the horizontal orientation
3. motion at a steady speed
4. acceleration
5. state of rest with respect to ground
Answer: 1, 2, and 4. There are no
experiments that can detect uniform
motion (or rest); we can sense any motion
that causes acceleration.
Postulate 1
As far as we can tell, we don’t know which frame is actually
doing the moving as long as no acceleration is involved.
Postulate 2
• Sound is longitudinal and has no polarization
• Speed of Light : Speed of Sound as Aether : Air
• c is constant with respect to source or medium?
– Sound depends on both
• vs = vsource + vair
– Aether?
• Can it move radially wrt all sources?
Postulate 2
• The speed of light is constant as found
empirically (Michelson-Morely) and
theoretically (Maxwell)
Consequences
• How do we rectify our conceptual notion of
additive velocities?
– We have to change our idea of space-time
• Space-time is not the same for all inertial frames
Consequences
• How do we rectify our conceptual notion of
additive velocities?
– Simultaneous events in one frame that are at different
locations will not be simultaneous in a different inertial
frame (relative simultaneity)
– Two events occuring at the same location in one
frame will have different temporal separation in
another inertial frame (time dilation)
– The length of an object in one inertial frame will be
different in a different inertial frame (length
contraction)
Relative Simultaneity
The train and platform experiment from the
reference frame of an observer onboard the
train.
The train and platform experiment from
the reference frame of an observer on
the platform.
Space-Time Diagram
Stationary frame (on train)
Moving frame (platform)
ct’
ct’
ct
ct
ct
ct’
t=0
t=1
x’
t=2
x’
x
x’
x
x
Relative Simultaneity
Time Dilation
Events occur in the same location are denoted with the prime frame, S’.
Time Dilation
Events occur in the same location to someone riding along with the timer.
Therefore, on the timer it is the prime frame (t’). Someone off of the timer observes
Events at two different locations. Therefore, a stationary frame is unprimed.
ct'  h
ct’
h
t' 
h
c
ct  h 2  x 2  t '
ct
h
x
x
v
h2  x2
t
c
x  vt
h 2  (vt) 2
t
c
Time Dilation
From previous slide
ct'  h
h
t' 
c
ct  h 2  x 2  t '
h2  x2
t
c
x  vt
h 2  (vt) 2
t
c
t
h 2  (vt ) 2
c
h 2
vt 2



t c  c
ct 2
vt 2
t  c  c 
2
ct 2
vt 2
t  c  c 
2
2
vt 2

t  t'  c 
2
2
vt 2
t'  t   c 
2
t '2  t 2 1   vc  
2
t '  t 1   vc 
Bring c into square
root by squaring it
Replace h with ct’
Square both sides to
get rid of square root
Cancel c in first term
on right
Solve for t by bringing
it to left hand side by
itself.
Simplify by pulling t
out of both terms on
the right hand side by.
Square root both sides
Time Dilation
The Relativistic Correction, g
t '  t
t ' 
t
g
v 2

1 c 
Plot of gn
An Example of Time Dilation
• 1971
– U. S. Naval Observatory flew jets around the
world
• Some East, with Earth’s rotation
• Some West, against Earth’s rotation
The Twin Paradox
Events occur in the same location to the spaceship. Therefore, it is the prime frame.
Planet X
Earth
t=0s
t=?
20 ly
t’ = 0 s
t’ = ?
v = 0.80 c
The Twin Paradox
• How do we reconcile that the relative
motion between Earth and spaceship
should make it impossible to determine
which one sees time dilation?
The Relativistic Correction, g
Time for those moving at high speed appears
to go slower than when stationary.
t '  t
t ' 
t
g
v 2

1 c 
Length Contraction
• An object in motion will appear shorter than an object
at rest.
v
x'  vt '
t1
t2
t=t2-t1
x  vt
Muon Decay
Evidence of Special Relativity
• Muons (μ-) decay to mu neutrino (vμ),
electron anti-neutrino (ve), and an electron
(e-) in 2.2 μs (2.2 x 10-6 s) in the muon’s
rest frame
 n  n e  e


N  No e
Feynman diagram
t /
Muon Relativity
Muon speed
v = 0.994 c
2 km
At the top of a 2 km high mountain we detect 560 muons/hr.
How many should reach sea level (2 km below)?
Muon Relativity
Classical Galilean Relativity
v
Lorentzian Relativity
v
Lorentzian Relativity
S’
S
Light flash occurs on a
moving train
x
x’
vt
x
x’
Lorentzian Relativity
S’
S
Light flash occurs on a
moving train
x
x’
vt
x
x’
Lorentzian Relativity
Velocity Transformation
Velocity Transformation
Velocity Transformation
Velocity Transformation
S frame
v=0
S’ frame
v = 0.8c
meteor
u = - 0.6c
Velocity Transformation
S frame
v=0
S’ frame
v = 0.8c
u = -c
Mechanics
• Requirements
– Valid in all inertial frames
• Physics does not change with relative velocity
– Reduce to classical expectations at low speed
– Agreement with experiment/observation
• Expectations
–


p  mu


pi  p f
Lorentzian Momentum
y’
y
2
2
2
2
2
2
1
1
1
1
1
x’
x
S
S’
v
Momentum
Momentum
Momentum
Mechanics
• Requirements
– Valid in all inertial frames
• Physics does not change with relative velocity
– Reduce to classical expectations at low speed
– Agreement with experiment/observation
• Expectations
–
E  mv
1
2
2
 Ei   E f
Energy
E  g u mc
2
Energy
Energy
Energy
Higgs Boson - LHC
Circumference is 27 km
Higgs Boson - LHC
proton
mp = 1.6726217 x 10-27 kg
c = 2.99792458 x 108 m/s
Each Beam
450 GeV  vp = 0.999998 c
3.5 TeV  vp = 0.999999991 c
proton
The Neutrino – Neutron Decay
e
• Wolfgang Pauli (1930)
n
p
– Neutron decays don’t conserve energy or
momentum
– Hypothesized the neutrino
• SuperKamiokande
– 1998-present
– Measure the mass of the neutrino
• MINOS – Main Injector Neutrino
Oscillation Search
– 2005-present
v
The Neutrino – Neutron Decay
e
n
p
v
Energy
• Accelerator vs. Collider
Energy
• Massless Particles
Length Contraction
Muon Relativity
Muon speed
v = 0.98 c
3 km
t = ? μs
? km
t’ = 2.2 μs