Relativity

1
Relativity
Chapter
37
Relativity is an important subject that looks
at the measurement of where and when events take
place, and how these events are measured in
reference frames that are moving relative to one
another. In this Chapter we will explore with
the special theory of relativity (which we will
refer to simply as "relativity"), which only
deals with inertial reference frames (where
Newton's laws are valid). The general theory of
relativity looks at the more challenging
situation where reference frames undergo
gravitational acceleration. In 1905, Albert
Einstein stunned the scientific world by
introducing two "simple" postulates with which he
showed that the old, common-sense ideas about
relativity are wrong. Although Einstein's ideas
seem strange and counter-intuitive, e.g., rate at
which time passes depends on the speed of
reference frame, these ideas have not only been
validated by experiment, they are being used in
modern technology, e.g., global positioning
satellites.
37-
2
Some references on relativity

• The original papers of Einstein
• N. David Mermin
• American Journal of Physics 65, 476-486 (1997)
  and 66, 1077-1080 (1998)
• http//people.ccmr.cornell.edu/mermin/homepage/mi
  nkowski.pdf
• It's About TimeUnderstanding Einstein's
  RelativityN. David Mermin
• http//press.princeton.edu/titles/8112.html
• Others

3
About that speed

• conventional foot (ft) 0.3048 m.
• 1 foot (f) 0.299792458 m.
• 1 f/ns 299,792,458 m/s c, speed of light.
• (ns nanosecond 10-9 sec)

4
The Postulates
Both postulates tested exhaustively, no
exceptions found!
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5
The Ultimate Speed
Experiment by Bertozzi in 1964 accelerated
electrons and measured their speed and kinetic
energy independently. Kinetic energy ?8 as speed
? c
Ultimate Speed?Speed of Light
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6
Testing the Speed of Light Postulate
If speed of light is same for all inertial
reference frames, then speed of light emitted by
a source (pion, p0) moving relative to a given
frame (for example, a laboratory) should be the
same as the speed light that is emitted by a
source that is at rest in the laboratory). 1964
experiment at CERN (European particle physics
lab) Pions moving at 0.9975c with respect to the
laboratory decay, emitting two photons (g). The
speed of the light waves (g-rays) emitted by the
pions was measured always to be c in the lab
frame (not up to 2c!)?same as if pions were at
rest in the lab frame!
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7
Measuring an Event
Event something that happens, can be assigned
three space coordinates and one time
coordinate Where something happens is
straightforward, when something happens is
trickier (for example the sound of an explosion
will reach a closer observer sooner than a
farther observer.
Space-Time Coordinates 1. Space Coordinates
three dimensional array of measuring rods 2.
Time coordinate Synchronized clocks at each
measuring rod intersection How do we synchronize
the clocks?
All clocks read exactly the same time if you were
able to look at them all at once!
Event A x3.6 rod lenghts, y1.3 rod lengths,
z0, timereading on nearest clock
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8
The Relativity of Simultaneity
Sam observes two independent events (event Red
and event Blue) occurring at the same time,
Sally, who is running at a constant speed with
respect to Sam also observes these two events.
Does Sally also find that the events occurred at
the same time?
WARNING When we speak of observers like Sam and
Sally, we are referring to the entire space-time
coordinate system (frame of reference) in which
each is at rest. The observer's location within
their frame of reference does not affect the
relativistic physics that we discuss here.
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A Closer Look at Simultaneity

• Events Blue and Red same distance from Sam and
  Sally,
• Sam at rest?the light from two events reaches
  him at same time ?he concludes that the two
  events occurred at the same time (in his frame).
• Sally is moving to right?sees the light from Red
  event before the light from Blue event. Distance
  from Sally to B' and R' same and light travels at
  c from both events towards Sally ?Event Red must
  have occurred at an earlier time (in her frame)!
• What would a third stationary observer, Bill,
  standing to the right of Sam observe?

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10
The Relativity of Time
37-
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The Relativity of Time, cont'd
In previous example, who measures the proper time?
Speed Parameter
Lorentz factor
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12
The Relativity of Time, cont'd
Lorentz factor g as a function of the speed
parameter b
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13
Two Tests of Time Dilation
1. Microscopic Clocks. Subatomic particles called
muons are unstable and decay (transform into
other particles). The average time from when a
muon is produced to when it decays (Dt) depends
on how fast the muon is moving. Muon stationary
in lab (production and decay in same place, at
muon itself) Dt02.200 ms If muon is moving at
speed 0.9994c with respect to the lab (production
and decay in different places in the lab frame)
the lifetime measured by laboratory clocks will
be dilated
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Two Tests of Time Dilation, cont'd
1. Macroscopic Clocks. Super precision atomic
clocks (large systems) flown in airplanes
b7x10-7 (Hafele and Keating in 1977 within 10,
and U. Maryland a few years later within 1 of
predictions) repeated the muon lifetime
experiment on a macroscopic scale If the clock on
the U. Maryland flight registered
15.00000000000000 hours as the flight duration,
how much would a clock that stayed on earth (lab
frame) have measured for the duration? More or
less? Does it matter whether airplane returns to
same place?
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15
The Relativity of Length
37-
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Does a moving object really shrink?
Must measure front and back of moving penguin
simultaneously to get its length in your frame.
Let's do this by having two lights flash
simultaneously in the rest frame when the front
and back of the penguin align with them. In
penguin's frame, your measurements did not occur
simultaneously. You first measured the front end
(light from front flash reaches moving observer
first as in slide 37-7) and then the back (after
the back has moved forward), so the length that
you measure will appear to be shorter than in the
penguin's rest frame.
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Proof of Eq. 37-13
Sam is sitting on bench at train station. Using a
tape measure, Sam determines the length of the
station in his frame, which is the proper length
L0. Sally is sitting on a train that passes
through the station. What is the length L of the
train station that Sally measures?
v
According to Sam, Sally moves through the station
(time interval between passing point A and then
point B, different places in Sam's frame) in time
DtL0/v
Sally
Train
B
A
Sam
length of train station
For Sally, the platform moves past her. She
passes points A and B at the same place in her
reference frame (proper time) in time Dt0
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18
The Lorentz Transformation
How are coordinates x, y, z, and t reporting an
event in frame S related to the coordinates x',
y', z', and t' reporting the same event in moving
frame S'?
Gallilean Transformation Equations
Origins coincide at t t' 0
Lorentz Transformation Equations
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19
The Lorentz Transformation, cont'd
What about S coordinates in terms of S'
coordinates?
Switch from one frame to the other by letting v?
-v
What about position and time intervals for pairs
of events?
37-
20
Some Consequences of the Lorentz Equations
Simultaneity
Time Dilation
Length Contraction
37-
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The Relativity of Velocities
37-
22
Doppler Effect for Light
Let f0 represent the proper frequency (frequency
in the source's rest frame)
If source and detector moving towards one another
b ? - b Note Unlike Doppler shift with sound,
only relative motion matters since there is no
ether/air to be moving with respect to.
Low Speed Doppler Effect
For bltlt1
Same as for sound waves
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Doppler Effect for Light, cont'd
Astronomical Doppler Effect
Proper wavelength l0 associated with rest frame
frequency f0.
Replacing bv/c and using l-l0 Dl Doppler
shift
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Doppler Effect for Light , cont'd
Transverse Doppler Effect
Classical theory predicts no Doppler shift
observed at point D when source S is at point P.
For low speeds (bltlt1)
Transverse Doppler effect another test of time
dilation (T1/f)
Proper period T01/f0)
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Doppler Effect for Light , cont'd
The NAVSTAR Navigation System
v1
f03
f01
v3
vairplane
v2
f02
Given v1, v2, v3, f01, f02, f03, and measured f1,
f2, f3, can determine vairplane,
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A New Look at Momentum
relativistic expression using DtDt0 g, where the
time Dt0 to move a distance Dx is measured in the
moving observer's frame
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27
A New Look at Energy
Mass energy or rest energy
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A New Look at Energy , cont'd
Total energy
37-
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A New Look at Energy, cont'd
Kinetic energy
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A New Look at Energy, cont'd
Momentum and kinetic energy
37-