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Introduction to Relativity & Time Dilation
• The Principle of Newtonian Relativity
• Galilean Transformations
• The Michelson-Morley Experiment
• Einstein’s Postulates of Relativity
• Relativity of Simultaneity
• Time Dilation
• Homework
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The Principle of Newtonian Relativity
• The laws of mechanics must be the same in all inertialframes of reference.
• An inertial frame is one in which Newton’s 1st law isvalid.
• Any frame moving with constant velocity with re-spect to an inertial frame must also be an inertial frame.
• This does not say that the measured values of physicalquantities are the same for all inertial observers.
• It says that the laws of mechanics, that relate thesemeasurements to each other, are the same.
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Two Inertial Reference Frames
• The observer in the truck sees the ball move in a ver-tical path when thrown upward.
• The stationary observer sees the path of the ball to bea parabola.
• Their measurements differ, but the measurements sat-isfy the same laws.
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Galilean Transformations
• Consider an event that occurs at point P and is ob-served by two observers in different inertial referenceframes S and S ′, where S ′ is moving with a velocityv relative to S as shown below
• The coordinates for the event as observed from thetwo reference frames are related by the equations knownas the Galilean transformation of coordinates
x′ = x − vt y′ = y z′ = z t′ = t
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Galilean Addition of Velocities
• Suppose a particle moves a distance dx in a time in-terval dt as measured by an observer in S
• The corresponding distance dx′ measured by an ob-server in S ′ is
dx′ = dx − vdt
• Since dt = dt′, we have
dx′
dt′=
dx
dt− v
oru′
x = ux − v
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Michelson-Morley Experiment
• In the 19th century, physicists believed light, like me-chanical waves, required a medium to propagate throughand they proposed the existence of such a mediumcalled the ether
• The ether would define an absolute reference framein which the speed of light is c
• The Michelson-Morley experiment was designed toshow the presence of the ether
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Michelson-Morley Experiment (cont’d)
• The ether theory claims that there should be a timedifference for light traveling to mirrors M1 and M2
• No time difference was observed!
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Einstein’s Postulates
• The Relativity Postulate: The laws of physics are thesame for observers in all inertial reference frames.
– Galileo and Newton assumed this for mechanics.– Einstein extended the idea to include all the laws
of physics.
• The Speed of Light Postulate: The speed of light in avacuum has the same value c in all directions and inall inertial reference frames.
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Tests of the Speed of Light Postulate
• Accelerated electron experiment: Bill Bertozzi (MIT)showed this in 1964 by independently measuring thespeed and kinetic energy of accelerated electrons
Speed (10 m/s)8
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1 2 3
2
4
6
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Ulti
mat
e sp
eed
• π0 → γγ decay experiment (CERN 1964)
π0v = 0.99975c
v = cγ
v = cγ
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Relativity of Simultaneity
• Two lightning bolts strike the ends of a moving box-car.
• The events appear to be simultaneous to the observerat O, who is standing on the ground midway betweenA and B.
• The events do not appear to be simultaneous to theobserver O′ riding on the boxcar, who claims the frontend of the car is struck before the rear.
• A time measurement depends on the reference framein which the measurement is made.
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Time Dilation 1
• The observer at O′ measures the time interval be-tween the two events to be
∆tp =2d
c
• The two events occur at the same location in O′s ref-erence frame, and she needs only one clock at thatlocation to measure the time interval, so we call thistime interval the proper time.
• The observer at O uses two synchronized clocks, oneat each event, and measures the time interval to be
∆t =2L
c=
2√
(
1
2v∆t
)2+ d2
c
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Time Dilation (cont’d)
∆t =2
√
(
1
2v∆t
)2+
(
1
2c∆tp
)2
c1
4c2∆t2 =
1
4v2∆t2 +
1
4c2∆t2p
(
c2 − v2)
∆t2 = c2∆t2p
∆t =c∆tp√c2 − v2
∆t =∆tp
√
1 −(
vc
)2
• It is convenient to define the speed parameter as β = vc
and the Lorentz factor as γ = 1√1−β2
• Then the time dilation expression can be written as
∆t = γ∆tp
• Since we must have v < c, γ > 1, and ∆t > ∆tp
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Time Dilation (cont’d)
• All clocks will run more slowly according to an ob-server in relative motion (this includes biological clocks).
• Time dilation has been tested and confirmed on boththe microscopic (lifetimes of subatomic particles) andmacroscopic (flying high precision clocks in airplanes)levels.
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Example
The elementary particle known as the positive kaon (K+)has, on average, a lifetime of 0.1237 µs when stationary-that is, when the lifetime is measured in the rest frameof the kaon. If a positive kaon has a speed of 0.990c inthe laboratory, how far can it travel in the lab during itslifetime?
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Example Solution
The elementary particle known as the positive kaon (K+)has, on average, a lifetime of 0.1237 µs when stationary-that is, when the lifetime is measured in the rest frameof the kaon. If a positive kaon has a speed of 0.990c inthe laboratory, how far can it travel in the lab during itslifetime?
∆t =∆tp
√
1 −(
vc
)2
∆t =0.1237 × 10−6s
√
1 −(
0.990cc
)2= 8.769 × 10−7s
d = v∆t = (0.990)(
3.00 × 108m/s) (
8.769 × 10−7s)
= 260 m
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Homework Set 16 - Due Wed. Oct. 20
• Read Sections 9.1-9.4
• Answer Questions 9.2 & 9.4
• Do Problems 9.1, 9.2, 9.6, 9.9 & 9.13
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