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Chapter 02Special Relativity
General Bibliography1) Various wikipedia, as specified
2) Thornton-Rex, Modern Physics for Scientists & Eng, as indicated
Version 110906, 110907, 110908, 110913
Outline
• Galilean Transformations• Names & Reference Frames• The Ether River• Michelson-Morley Experiments• Einstein Postulates• Lorentz Transformations
– Position– Velocity
• Space-Time Diagrams• Relativistic Forces & Momentum• Relativistic Mass• Relativistic Energy
CLASSICAL / GALILEAN / NEWTONIANTRANSFORMATIONS
Galilean Transformations
t
z
y
x
z
y
x
v
v
v
v
K’ frame moving with speed v
K frame fixed
K & K’ coincided at t=0. Sketch shown at time t later.
How do the position, velocity, acceleration, & time between the 2 frames compare?
'
'
'
'
t
z
y
x
'
'
'
z
y
x
v
v
v
z
y
x
a
a
a
'
'
'
z
y
x
a
a
a
Fam Fam
'
Galilean Transformations
v
K’ frame moving with speed v
K frame fixed
K & K’ coincided at t=0. Sketch shown at time t later.
How do the position, velocity, acceleration, & time between the 2 frames compare?
K K’
x = x’ + vt
K’ K
x = x’ - vt
Newtonian Principle of Relativity
• If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system.
• This is referred to as the Newtonian principle of relativity or Galilean invariance.
• K is at rest and K’ is moving with velocity
• Axes are parallel
• K and K’ are said to be INERTIAL COORDINATE SYSTEMS
Inertial Frames K and K’
The Galilean Transformation
For a point P In system K: P = (x, y, z, t) In system K’: P = (x’, y’, z’, t’)
x
K
P
K’ x’-axis
x-axis
Conditions of the Galilean Transformation
• Parallel axes (for convenience)
• K’ has a constant relative velocity in the x-direction with respect to K
• Time (t) for all observers is a Fundamental invariant, i.e., the same for all inertial observers
speed of frameNOT speed of object
x’ = x – v t
y’ = y
z’ = z
t’ = t
Galilean Transformation Inverse Relations
Step 1. Replace with .
Step 2. Replace “primed” quantities with
“unprimed” and “unprimed” with “primed.”
speed of frameNOT speed of object
x = x’ + v t
y = y’
z = z’
t = t’
General Galilean Transformations
'
'
'
tt
yy
vtxx
11'
''
''
'
dt
dt
dt
dt
vvdt
dy
dt
dy
vvvvdt
dx
dt
dx
samethearetandt
yy
xx
yyyy
xxxx
aadt
dv
dt
dv
aadt
dv
dt
dv
samethearetandt
''
'0'
'
inertial reference frame
FamFam '
11'
''
''
'
dt
dt
dt
dt
ttdt
dy
dt
dy
vuuvdt
dx
dt
dx
samethearetandt
yy
xx
frame K frame K’
Newton’s Eqn of Motion the same at face-value in both reference frames
Pos
ition
Vel
ocity
Acc
eler
atio
n
Classical Reference Frames
• Inertial Reference Frame– Non-accelerating– Newton’s Laws apply at face-value
• Non-Inertial Reference Frame– Examples:
• Rocket during acceleration phase• Rotating merry-go-round• Rotating Earth
Youtube clips (part 1)
• Galilean/Classical Relativity Part 1 – The Cassiopeia Project http://www.youtube.com/watch?v=6rl3Z9yCTn8
The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.
http://www.cassiopeiaproject.com/
To read more about the Theory of Special Relativity, you can start here:
http://www.phys.unsw.edu.au/einsteinlight/
http://www.einstein-online.info/en/elementary/index.html
http://en.wikipedia.org/wiki/Special_relativity
THE ETHER RIVERHISTORY OF ETHER
MICHELSON-MORLEY EXPTS
The Ether River
A
C
Dv
Maximum speed of the boat is ‘c’ meters/sec
The Ether River
Time t1 from A to C and back:
Time t2 from A to D and back:
So that the difference in trip times is:
2
2
2
22
22
1
122
cvcvc
t
2
21
2
2
212
11
2
cv
cvc
ttt
down river
Timeline of luminiferous aether(http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether)
• 4th cent BC – Light propagates in air – Aristole• 1704 – Aetheral medium for light & heat – Newton• 1818 – aether – Fresnel wave theory• 1830 – problems emerge, explained by “aether drag”, Fresnel &
Stokes
• 1830 – ~1955 – mixed experimental conclusions
Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg
Timeline of luminiferous aether(http://en.wikipedia.org/wiki/Timeline_of_luminiferous_aether)
• 1830 – ~1955 – mixed experimental conclusions
• 1887 – 1st Michelson-Morley expt doesn’t find aether• 1889(1895) – Fitzgerald hypothesis (Lorentz)• 1902-1904 – Refined Michelson-Morley measurements• 1905 – Trouton-Rankine expt doesn’t support Fitz-Loentz hypothesis• 1958- nearly all measurements do not find evidence for aether
Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg
Michelson-Morley Expt “the most famous failed experiment”
Cronholm144, http://en.wikipedia.org/wiki/File:AetherWind.svg
Michelson-Morley: Ether River - Revisited
A
C
Dv
Measure two orientations because don’t know direction of aether river
A C
D
v
Ether River - Revisited
2
21
2
2
2121
11
2
cv
cvc
tttorient
Orientation 1
Orientation 2
2
2
1
2
22
122
11
2
cv
cvc
tttorient
Difference in Orientations
2
2
21
2
2
21
2
221
21
1
11
2
c
v
cc
vc
vctt orientorient
down river
down river
Michelson-Morley Measurements
l1+l2
(m)
to1-to2
(sec)
c[to1-to2]
1887 2*1.5 1E-16 30 nm
~1903 2*30 2E-15 600 nm
Earth-Moon 3.8E8 1.3E-8 3.8 m
v=30 km/s c=3E8 m/s
http://en.wikipedia.org/wiki/Lunar_Laser_Ranging_Experiment
Apollo 11 Apollo 15
~2002 accuracy ~1 mm
Crises with Reference Frame Xformations
• Can’t find the Ether• Maxwell’s Equations not Galilean Invariant
Speed of Light fixed by EM constants
1
c
Fitzgerald-Lorentz Hypothesis1889 (1895)
{only a partial explanation}
2
21
1
cv
LL
POSTULATE: the null results are due to changes in length in the direction of travel.
2
21
2
2
2
22
2121
11
22
cv
cvcvc
tttorient
2
2
1
2
22
22
2122
11
22
cv
cvcvc
tttorient
021 orientorient tt
EINSTEIN’s 1905 POSTULATES
• All laws for physics have the same functional form in any inertial reference frame
• Speed of Light (in vacuum) is same in any inertial reference frame.
LORENTZ POSITION-TIME
TRANSFORMATIONS
Lorentz Transformations
c
xtt
zz
yy
vtxx
'
'
'
'
21
1
x
K
P
K’
x’-axis
x-axis
x’
v
c
v
K’ K
Lorentz Transformations
c
xtt
zz
yy
vtxx
''
'
'
''
21
1
x
K
P
K’
x’-axis
x-axis
x’
v
c
v
K K’
ExampleAs observed from a large asteroid, an explosion occurs at x=3000, y=500, z=-500 and t=5 us.
P
v
A spaceship approaches at a high speed v=0.6c .The reference frames coincided at t=0, t’=0
At what position does the spaceship observe the explosion to occur?
K: 3km, 5usK’: 2.6km, -1.25us
ExampleThe reference frames coincide at x=0, x’=0 & t=0, t’=0
A spaceship has indicator lights which are flashed at the same time. At t’=0 the lights flash. The locations of the lights are x’rear=4km & x’front=+4km.
The spaceship is observed from the spacestation.The spaceship is observed to move at v=0.6c .
At what position does the spacestation observe the lights to flash?
K rear -5km, -10 us front +5km, +10 us
KK’
x’-axis
x-axis
v
ExampleThe reference frames coincide at x=0, x’=0 & t=0, t’=0
As viewed from the Earth, a meteorite impacts the lunar surface at 3E8m and 2.5s .
The impact is observed from 2 passing spaceships, one traveling to the right at 40% c and the other to the left at 40% c.
Where do the 2 spaceships observe the impact to occur ?
0m, 2.3 s6.5E8, 3.2 s
Length Contraction(Lorentz-Fitzgerald)
A meter stick, lying parallel to the x-axis, is moving with speed v
v
How long does the stick appear to be to a stationary observer who makes the observation of the length at t=0?
xleft & tleft=0 xright & tright=0
Moving objects appear shorter
Time Dihilation(distinct from the L-F)
A clock, located at x’=0, makes ticks at t’1, t’2, …
v
What is the interval between ticks to a stationary observer, who observes the clock to move at speed v?
x’1=0 & t’1 x’2=0 & t’2
Moving clocks
run slow
Distorted Picturesstationary moving to the right
Our brain records photographs (frames in a movie) – light rays arriving at the same time.
“Jump to Light Speed”
Distorted Pictures
Lorentz Transformation - DerivationLight propagates with speed c in all inertial reference frames
Spherical wavefronts in K:
Spherical wavefronts in K’:
K K’
ct ct’
1) Let x’ = (x – vt) so that x = (x’ + vt’)
2) By Einstein’s first postulate:
3) The wavefront along the x,x’- axis must satisfy:x = ct and x’ = ct’
4) Thus ct’ = (ct – vt) and ct = (ct’ + vt’)
5) Solving the first one above for t’ and substituting into the second...
Derivation – see pages 30-31
Youtube clips (part 2)
• Galilean/Classical Relativity Part 2 – The Cassiopeia Project http://www.youtube.com/watch?v=WgsKlSnUO0k
The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.
http://www.cassiopeiaproject.com/
To read more about the Theory of Special Relativity, you can start here:
http://www.phys.unsw.edu.au/einsteinlight/
http://www.einstein-online.info/en/elementary/index.html
http://en.wikipedia.org/wiki/Special_relativity
LORENTZ VELOCITY
TRANSFORMATIONS
Lorentz Velocity Transformationsee page 40
c
xtt
zz
yy
vtxx
''
'
'
''
c
dxdtdt
dzdz
dydy
vdtdxdx
''
'
'
''
Lorentz Velocity Transformationsee page 40
c
dxdtdt
dzdz
dydy
vdtdxdx
''
'
'
''
cdx
dt
dy
dt
dyu
cdx
dt
vdtdx
dt
dxu
y
x
''
'
''
''
Lorentz Velocity Transformationsee page 40
x
yy
x
xx
uc
u
dtdx
cdt
dtdy
dt
cdx
dt
dy
dt
dyu
uc
vu
dtdx
cdt
vdtdx
dt
cdx
dt
vdtdx
dt
dxu
'1
'
''
1'
''
'
''
'
'1
'
''
1'
''
'
''
''
Note that because of the time transformation, the y- and z-components get messed up.
A spaceship traveling at 60%c shoots a proton with a muzzle speed of 99%c at an asteroid.
What is the velocity of the proton as viewed from a ‘stationary’ space station?
MISC. LORENTZ TRANSFORMATION
EXAMPLES
Cosmic Ray Muon Lifetime electron mo=9.1e-31 kg halflife = inf
muon mo=207 * (mass e) halflife = 1.5e-6 sec
http://landshape.org/enm/cosmic-ray-basics/http://www.windows2universe.org/physical_science/physics/ atom_particle/cosmic_rays.html
Cosmic Rays
Susan BaileyNuclear NewsJan 2000, pg 32
Cosmic Ray references
http://www.ans.org/pubs/magazines/nn/docs/2000-1-3.pdf
http://pdg.lbl.gov/2011/reviews/rpp2011-rev-cosmic-rays.pdf
http://hyperphysics.phy-astr.gsu.edu cosmic rays
ashsd.afacwa.org/ radation
Cosmic Ray Muon Measurementshttp://www.youtube.com/watch?v=yjE5LHfqEQI
Cosmic Ray Muon Lifetime muon mo=207 (9.1e-31 kg) halflife = 1.5e-6 sec
Q1. Classically, how far could the muon travel during a time 1.5e-6 sec ?
Suppose muon traveling at 0.98c
Q2. What do we observe the lifetime to be ?
Q3. How far do we observe the muon to travel during that time ?
2000
met
ers
Q4. How high does the muon think the mountain is?
Simultaneity
• http://www.youtube.com/watch?v=wteiuxyqtoM
• http://www.youtube.com/watch?v=KYWM2oZgi4E
Atomic Clock Measurements
• http://www.youtube.com/watch?v=cDvmN_Pw96A
• d
Twin Paradox
http://www.youtube.com/watch?v=A0jiY-CZ6YA
http://www.phys.unsw.edu.au/einsteinlight/jw/module4_twin_paradox.htm
Video Clip
What’s the correct explanation of the paradox?
Reliable Discussion at
Spacetime DiagramsMinkowski Diagrams
x
t x
t
In SP211 course:
ux
ux
Forbidden region
Allowed region
Two events plotted on a space time diagramP=(x,y,z,t)
Simultaneity in a Stationary System
#1 Measuring location #2 Measuring location
Watching a moving system
Animated Minkowski Diagrams
• http://www.youtube.com/watch?v=C2VMO7pcWhg – (uses Minkowski space diagrams, but with time axis pointing
down, opposite from figures in textbook.)
Analysis of theTwin Paradox
usingMinkowski Diagrams
Frank sends a signal once a year.
Mary sends a signal once a year.
Invariant Quantities
Define s2 = x2 – (ct)2
Then can show s’2 = x’2 – (ct’)2 x2 – (ct)2 = s2
s2 is independent of choice of reference frame.
We can use s2 to discuss the ability for ‘events’ to impact one another.
s2 = x2 – (ct)2
If s2 < 0 then x2 < (ct)2 = c2 t2
The distance between ‘events’ is less than the time required for light to propagate between the spatial locations.
RELATIVISTIC MOMENTUM
Style 1. Sandin’s Development
Style 2. Rex & Thorton’s Development
The following 4 slides present Sandin’s treatment of momentum
in Special Relativity
Forces and Momentum (Sandin version)
- a first look
y
x
yy
x
xx
uuc
uu
vuc
vuu
'1
'1
'
'1
'
yoy
yoy
umum
ummu
''1
'
v u’x , u’y , u’z
ux , uy , uz
p’x , p’y , p’z
px , py , pz
mo
If want p to ‘look like’ “mv” ;
Then are forced to let omm BUT
Forces and Momentum (Sandin version)
amFtot
dt
pdFtot
u
dt
dm
dt
dum
dt
mudFtot
Suppose F ┴ u
maamdt
dum
dt
dumF
dt
dm
changetdoesnmm
constconstspeed
oo
o
0
'
So why are we complaining about the phrase ‘relativistic mass’?
Forces and Momentum (Sandin version)
amFtot
dt
pdFtot
u
dt
dm
dt
dum
dt
mudFtot
Suppose F || u
mau
dt
dm
dt
dumF
mdt
dm
dt
d
dt
dm
constconstspeed
3
2/12
...
...1
BECAUSE Newton’s Eqn Motion is different depending on the direction of the force.
Forces & Momentum (Sandin version)
PRO-
Relativistic Mass People
ANTI-
Relativistic Mass People
Definitions m = mo
p=mu
No such thing
p=mu
Newton’s Eqn of Motion
if F ┴ u
F=ma
with m mo
F=dp/dt
Newton’s Eqn of Motion
if F || u
F=ma
with m mo
but throw in an extra 2F=dp/dt
The following 9 slides present Rex & Thorton’s
treatment of momentum in Special Relativity
2.11: Relativistic Momentum
Because physicists believe that the conservation of momentum is fundamental, we begin by considering collisions where there do not exist external forces and
dP/dt = Fext = 0
Frank (fixed or stationary system) is at rest in system K holding a ball of mass m. Mary (moving system) holds a similar ball in system K that is moving in the x direction with velocity v with respect to system K.
Relativistic Momentum
• If we use the definition of momentum, the momentum of the ball thrown by Frank is entirely in the y direction:
pFy = mu0
The change of momentum as observed by Frank is
ΔpF = ΔpFy = −2mu0
Relativistic Momentum
According to Mary
• Mary measures the initial velocity of her own ball to be u’Mx = 0 and u’My = −u0.
In order to determine the velocity of Mary’s ball as measured by Frank we use the velocity transformation equations:
K’
K
Relativistic MomentumBefore the collision, the momentum of Mary’s ball as measured by Frank becomes
Before
Before
For a perfectly elastic collision, the momentum after the collision is
After
After
The change in momentum of Mary’s ball according to Frank is
(2.42)
(2.43)
(2.44)
K
K
The conservation of linear momentum requires the total change in momentum of the collision, ΔpF + ΔpM, to be zero. The addition of Equations (2.40) and (2.44) clearly does not give zero.
Linear momentum is not conserved if we use the conventions for momentum from classical physics even if we use the velocity transformation equations from the special theory of relativity.
There is no problem with the x direction, but there is a problem with the y direction along the direction the ball is thrown in each system.
Relativistic Momentum
• Rather than abandon the conservation of linear momentum, let us look for a modification of the definition of linear momentum that preserves both it and Newton’s second law.
• To do so requires reexamining mass to conclude that:
Relativistic Momentum
Relativistic momentum (2.48)
Some physicists like to refer to the mass in Equation (2.48) as the rest mass m0 and call the term m = γm0 the relativistic mass. In this manner the classical form of momentum, m, is retained. The mass is then imagined to increase at high speeds.
Most physicists prefer to keep the concept of mass as an invariant, intrinsic property of an object. We adopt this latter approach and will use the term mass exclusively to mean rest mass. Although we may use the terms mass and rest mass synonymously, we will not use the term relativistic mass. The use of relativistic mass to often leads the student into mistakenly inserting the term into classical expressions where it does not apply.
Relativistic Momentum
RELATIVISTIC KINETIC ENERGY
The following 5 slides present Rex & Thorton’s
treatment of kinetic energy
in Special Relativity
2.12: Relativistic Kinetic Energy
Newtonian KE=1/2 m u2 which came from
KE = Work = ∫ F•ds with F = dp/dt = m dv/dt = ma
Relativistic Kinetic EnergyStart from rest and accelerate until u
duvuvdvu
partsbynIntegratio
Relativistic Kinetic Energy
duvuvdvu
partsbynIntegratio
u
duumuumKE
0
Start from rest and accelerate until u
22
2/12
22/1
2
21
11
cd
c
cu
duu
22
22 1 mcc
umcimitsevaluate
222
2 1mcmcmuKE
Relativistic Kinetic Energy
222
2 1mcmcmuKE
22222
2222
222 111
ucucc
uucucu
2
22
1 mcKE
mcmcKE
Which reduces to the Newtonian expression for u small
ComparisonRelativistic and Classical Kinetic Energy
Formula
Equation (2.58) does not seem to resemble the classical result for kinetic energy, K = ½mu2. However, if it is correct, we expect it to reduce to the classical result for low speeds. Let’s see if it does. For speeds u << c, we expand in a binomial series as follows:
where we have neglected all terms of power (u/c)4 and greater, because u << c. This gives the following equation for the relativistic kinetic energy at low speeds:
which is the expected classical result. We show both the relativistic and classical kinetic energies in Figure 2.31. They diverge considerably above a velocity of 0.6c.
(2.59)
Relativistic Kinetic Energy
Total Energy
21 mcKE
22 mcmcKE
22 mcKEmc
2mcEEnergyTot tot
Relationship between Total Energy & Momentummup
4242222 cmcmcp
Square, mult c2, convert u, use =(1-2)1/2 to subst 4
42222 cmEcp tot
42222 cmcpEtot
an invariant
Youtube clips (part 3)
• Galilean/Classical Relativity Part 3 – The Cassiopeia Project http://www.youtube.com/watch?v=W6o_-yTa168
The Cassiopeia Project is an effort to make high quality science videos available to everyone. If you can visualize it, then understanding is not far behind.
http://www.cassiopeiaproject.com/
To read more about the Theory of Special Relativity, you can start here:
http://www.phys.unsw.edu.au/einsteinlight/
http://www.einstein-online.info/en/elementary/index.html
http://en.wikipedia.org/wiki/Special_relativity
Examples
Example 2.11Electrons in a television set are accelerated by a potential difference of 25000 Volts before striking the screen.
a). Calculate the speed of the electrons and b). Determine the error in using the classical kinetic energy result.
http://express.howstuffworks.com/exp-tv1.htm
http://www.o-digital.com/wholesale-products/2227/2285-4/LCD-TV-LDT32-225837.html
mc2 = 0.511 MeV m = 9.1e-31 kg |q| = 1.6e-19 Coul
Example 2.13A 2-GeV proton hits another 2-GeV proton in a head-on collision in order to create top quarks.
• For each of the initial protons, calculate– Speed v– – Momentum p– Rest-mass Energy– Kinetic Energy KE
– Total Energy Etot
http://www.fnal.gov
mc2=938 MeV
Example 2.16The helium nucleus is built from 2 protons and 2 neutrons.The binding energy is the difference in rest mass-energy of the nucleus from the total rest mass-energy of it’s component parts.
Calculate the nuclear binding energy of helium.
mHe = 4.002603 amu mp = 1.007825 amu mn = 1.008665 amu
http://www.dbxsoftware.com/helium/
Hints: 1 amu = 1.67e-27 kg or c2 = 931.5 MeV/amu
Example 2.17The molecular binding energy is called dissociation energy.It is the energy required to separate the atoms in a molecule. The dissociation energy of the NaCl molecule is 4.24 eV.
Determine the fractional mass increase of the Na and Cl atoms when they are not bound together in NaCl.
http://www.ionizers.org/water.html
Hints: 1 amu = 1.67e-27 kg or c2 = 931.5 MeV/amu
mNa = 22.98976928 amu
Average mCl = 35.453 amu
Sandin 5.30
A spaceship has a length of 100 m and a mass of 4e+9 kg as measured by the crew. When it passes us, we measure the spaceship to be 75 m long.
What do we measure its momentum to be?
RHICThe diameter of an gold nucleus is 14 fm.
If a Au nucleus has a kinetic energy of 4000 GeV, what is the apparent ‘thickness’ of the nucleus in the laboratory?
http://www.bnl.gov/rhic/
Length contraction
mc2=197*931.5 MeV
Sandin 5.22At the Stanford Linear Accelerator, 50 GeV electrons are produced
• For one of these electrons, calculate– Speed v– – Momentum p– Rest-mass Energy– Kinetic Energy KE
– Total Energy Etot
http://www.flickr.com/photos/kqedquest/3268446670/
http://www.daviddarling.info/encyclopedia/L/linear_accelerator.html mc2 = 0.511 MeV
Sandin 5.25A cosmic ray pion (rest mass 140 MeV/c2) has a momentum of 100 MeV/c.
• Calculate– Speed v– – Momentum p– Rest-mass Energy– Kinetic Energy KE
– Total Energy Etot
http://www.mpi-hd.mpg.de/hfm/CosmicRay/Showers.html
http://www2.slac.stanford.edu/vvc/cosmicrays/cratmos.html
Sandin 4.26
Spaceship A moves past us at 0.6c followed by Spaceship B in the same direction at 0.8c
What do they measure as their relative speed of approach?
What do we measure as their relative speed of approach?
B A
Sandin 4.28
Spaceship A approaches us from the right at at 0.8c
Spaceship B approaches us from the left at 0.6c
What do they measure as their relative speed of approach?
What do we measure as their relative speed of approach?
B A
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