Chapter 1: Relativity

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Chapter 1 Relativity
Special Relativity all motion is relative All
motion is described relative to an inertial frame
of reference Inertial frame frame of reference
in which Newtons law of inertia holds Inertial
frames are (mathematically) simpler in Special
Relativity, and are related in a (mathematically
simple) manner. Accelerations and forces are
accounted for in S.R. General Relativity
Special Relativity Gravity (curved space time
more complex relations between inertial
reference frames) Postulates of Special
Relativity The laws of physics are the same in
all inertial frames of reference. (mathmatical
form, conservation laws, etc.) The speed of light
in free space has the same value for all inertial
frames of reference. v 3.00E8 m/s
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The First Postulate and everyday
relativity The laws of physics are the same in
all inertial frames of reference.
Conservation of momentum, kinetic energy, etc
But waves like sound waves have special
inertial frame, the frame at rest w.r.t. medium
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Second Postulate The speed of light in free
space has the same value for all inertial frames
of reference need a new relationship between
coordinates of different inertial frames linear,
invertible transformation Lorentz
Transformations
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Time Dialation the time interval between events
depends upon the inertial observer proper time
t0 the time interval between two events in a
frame of reference where the events occur at the
same place. timing with a moving mirror
c t/2
c t0/2
L0
L0
v
v t/2
v
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Time Dialation and the passage of time t0 time
interval for clock at rest relative to an
observer proper time t time interval for
clock in motion relative to an observer, t gt
t0 v speed of relative motion c speed of light
Example 1.1 A spacecraft is moving relative to
the earth. An observer finds that, according to
her clock, 3601s elapse between 1 PM and 2 PM on
the spacecraftss clock. What is the
spacecrafts speed relative to the earth?
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Doppler Effect frequency shifts due to moving
source/observer doppler effect for sound n0
emitted frequency n detected frequency
v speed of observer V speed of source c
speed of sound
But, light does not require a medium!
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Doppler effect with light three cases 1-
observer moving perpendicular to a line between
him and light source time between ticks is dilated
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Doppler effect with light three cases 2-
observer moving away from source time between
ticks is dilated next tick travels farther
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Doppler effect with light three cases 2-
observer moving away towards source time between
ticks is dilated next tick travels less distance
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Example 1.2 A driver is caught going through a
red light. The driver claims that the color she
actually saw was green (n 5.60x1014 Hz) and not
red (n 0 4.80x1014 Hz) because of the doppler
effect. How fast must she have been going?
Example 1.3 A distant galaxy Hydra is receding
grom the earth at 6.12x107 m/s. By how much is a
green spectral line of wavelength l 500 nm
emitted by this galaxy shifted towards the red
end of the spectrum?
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Length Contraction determining the extent of a
moving object
v
L vt0
Example Muons created in earths atmosphere by
cosmic rays have speeds of about 2.994x108
m/s(.998c) At rest, a muon lifetime is about 2.2
ms. -How far would these muons travel in 2.2
ms? -What is the time dialated lifetime? How far
do they travel in THIS amount of time? -What is
the length contracted distance of the previous
answer, as seen in the muons frame of
reference? -How far does the earth move relative
to the muons during the 2.2 mslifetime?
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The Twin Paradox a counter-intuitive
result Two twins on earth are separated
temporarily when one twin takes a trip at .80c to
a star twenty light years away. Earth bound twin
round trip takes 2x(20 ly / .8ly/y) 50 years
Traveling twin Earth-Star distance is length
contracted to
so round trip takes 2x(12 ly / .8ly/y) 30
years Twins age differently! Travelling twin
does not stay in one inertial reference frame,
situation is not symetric.
Example 1.4 Each twin emits a radio signal once
a year for the duration of the voyage. How many
signals does each twin recieve?
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Electricity and Magnetism
Maxwells EquationsIntegral
Form Differential Form
Lorentz Force Law
Mathematically invariant formulation electric
charge is a relativistic invariant
quantity Maxwells equations -gt Wave Equation
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Relativistic Inertia (relativistic
mass) consider elastic collision between two
identical (when at rest) masses
y
VB
Ball A moves vertically only in frame S with
speed VA , Ball B moves vertically only in frame
S with speed VB VA . Ball A rebounds in S
with speed VA , Ball B rebounds in S with
speed VB .
y
Y
VA
S
x
S
x
z
v
z
Y/2
Collision in S
Collision in S
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Example 1.5 Find the mass of an electron (m0
9.1E-31 kg)whose speed is .99c.
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Example 1.6 A stationary body explodes into
two fragments each of mass 1kg that move apart at
speeds 0.6c relative to the original body. Find
the rest mass of the original body.
Example 1.7 Solar energy reaches the earth at
a rate of about 1.4kW/m2. Given that the average
radius of the earths orbit is 1.5E11 m, how much
mass does the sun lose per second?
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General momentum-energy relations
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Electronvolts typical energy scale for atomic
physics 1eV 1.6E-19 Coulomb x 1 Volt 1.6E-19
J X-Ray energies keV Nuclear Physics
MeV Particle Physics GeV
Example What are the momenta and speeds of a
5 MeV -electron? (m0 .511 MeV/c2, -proton? (m0
938 MeV/c2, -photon? (m0 0 MeV/c2).
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Chapter 1 problems 1,2,4,5,7,9,10,11,12,16,19,20,
22,32,33,34,35,36,41,47,49