Transcript Power Points (Chapter 29)

Lecture Outline
Chapter 29
Physics, 4th Edition
James S. Walker
Copyright © 2010 Pearson Education, Inc.
Chapter 29
Relativity
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Units of Chapter 29
• The Postulates of Special Relativity
• The Relativity of Time and Time Dilation
• The Relativity of Length and Length
Contraction
• The Relativistic Addition of Velocities
• Relativistic Momentum
• Relativistic Energy and E = mc2
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Units of Chapter 29
• The Relativistic Universe
• General Relativity
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29-1 The Postulates of Special Relativity
The postulates of relativity as stated by Einstein:
1. Equivalence of Physical Laws
The laws of physics are the same in all inertial frames of
reference.
2. Constancy of the Speed of Light
The speed of light in a vacuum, c = 3.00 x 108 m/s, is the
same in all inertial frames of reference, independent
of the motion of the source or the receiver.
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29-1 The Postulates of Special Relativity
The first postulate is certainly reasonable; it
would be hard to discover the laws of physics if
it were not true!
But why would the speed of light be constant? It
was thought that, like all other waves, light
propagated as a disturbance in some medium,
which was called the ether. The Earth’s motion
through the ether should be detectable by
experiment. Experiments showed, however, no
sign of the ether.
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29-1 The Postulates of Special Relativity
Other experiments and measurements have
been done, verifying that the speed of light is
indeed constant in all inertial frames of
reference.
With water waves,
our measurement
of the wave speed
depends on our
speed relative to
the water:
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29-1 The Postulates of Special Relativity
But with light, our measurements of its speed
always give the same result:
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29-1 The Postulates of Special Relativity
The fact that the speed of light is constant also
means that nothing can go faster than the
speed of light – it is the ultimate speed limit of
the universe.
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29-2 The Relativity of Time and Time
Dilation
To begin to understand the implications of
relativity, consider a light clock:
The time it takes
for light to make a
round trip is:
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29-2 The Relativity of Time and Time
Dilation
Now, look at the clock moving at a speed v:
The light has to travel farther. Now the round trip
time is:
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29-2 The Relativity of Time and Time
Dilation
Therefore, a moving clock will appear to run
slowly.
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29-2 The Relativity of Time and Time
Dilation
As the speed gets closer to the speed of light, the
clocks run slower and slower:
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29-2 The Relativity of Time and Time
Dilation
This result applies to any kind of clock or
process that is time-dependent – if it did not, the
first postulate would be violated.
Definitions
Event: a physical occurrence that happens at a
specified location at a specified time.
Proper time: the amount of time separating two
events that occur at the same location.
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29-2 The Relativity of Time and Time
Dilation
Time dilation has been measured with extremely
accurate atomic clocks in airplanes, and also is
frequently observed in subatomic particles.
Another consequence of time dilation is that
different observers will disagree about the
simultaneity of events occurring at different
places.
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29-3 The Relativity of Length and Length
Contraction
The observer on Earth sees the astronaut’s
clock running slow; it takes him 25.6 years to
go from Earth to Vega, but only 3.61 years have
passed on the astronaut’s clock.
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29-3 The Relativity of Length and Length
Contraction
But how does it appear to the astronaut, who
thinks his clock is fine? He sees the distance as
contracted instead – for him, Vega is only 3.57
light-years away.
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29-3 The Relativity of Length and Length
Contraction
Proper length, L0: The proper length is the distance
between two points as measured by an observer who is
at rest with respect to them.
So in the above example, 25.3 light-years is the
proper length.
With some arithmetic, we find:
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29-3 The Relativity of Length and Length
Contraction
Length contraction as a function of v:
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29-3 The Relativity of Length and Length
Contraction
Important note:
Length contraction
occurs only in the
direction of motion.
Other directions are
unaffected.
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29-4 The Relativistic Addition of Velocities
Suppose two space ships are heading towards
each other, each with a speed of 0.6 c with
respect to Earth. How fast do the astronauts in
one ship see the other ship approach? It can’t be
1.2 c, but what is it? Here we give the answer:
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29-4 The Relativistic Addition of Velocities
So in the above example, the relative speed
would be 0.88 c.
Below is a plot of the speed a rocket would have
if it increased its speed by 0.1 c every time it
fired its rockets.
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29-5 Relativistic Momentum
If adding more and more energy to a rocket only
brings its speed closer and closer to c, how can
energy and momentum be conserved?
The answer is that momentum is no longer
given by p = mv.
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29-5 Relativistic Momentum
As the speed gets closer and closer to c, the
momentum increases without limit; note that the
speed must be close to the speed of light before
the difference
between
classical and
relativistic
momentum is
noticeable:
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29-6 Relativistic Energy and E = mc2
If the momentum increases without limit, the
energy must increase without limit as well:
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29-6 Relativistic Energy and E = mc2
The rest energy of ordinary objects is immense!
In nuclear reactors, only a fraction of a percent
of the mass of fuel becomes kinetic energy, but
even that is enough to create enormous amounts
of power.
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29-6 Relativistic Energy and E = mc2
Every elementary particle, such as the electron,
has an antiparticle with the same mass but
opposite charge. The antiparticle of the electron
is called the positron.
Mass:
Charge:
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29-6 Relativistic Energy and E = mc2
When an electron
and a positron
collide, they
completely
annihilate each
other, emitting only
energy in the form
of electromagnetic
radiation.
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29-6 Relativistic Energy and E = mc2
We can find the relativistic kinetic energy by
subtracting the rest energy from the total
energy:
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29-6 Relativistic Energy and E = mc2
At ordinary speeds, the relativistic kinetic energy
and the classical kinetic energy are
indistinguishable.
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29-7 The Relativistic Universe
It may seem as though relativity has nothing to
do with our daily lives. However, medicine
makes use of radioactive materials for imaging
and treatment; satellites must take relativistic
effects into account in order to function
properly; and space exploration would be a
disaster if relativistic effects were not handled
properly.
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29-8 General Relativity
Einstein thought about the distinction between
gravitational force and acceleration, and
concluded that within a closed system one
could not tell the difference.
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29-8 General Relativity
This leads to the principle of equivalence:
All physical experiments conducted in a uniform
gravitational field and in an accelerated frame of
reference give identical results.
Therefore, the people in the elevators on the
previous page cannot, unless they are able to
see outside the elevators, tell if they are in a
gravitational field or accelerating uniformly.
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29-8 General Relativity
When the elevator is moving at a constant
speed, the light from the flashlight travels in a
straight line. When the elevator accelerates,
the light bends.
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29-8 General Relativity
The principle of equivalence then tells us that
light should bend in a gravitational field as well.
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29-8 General Relativity
This gravitational
bending of light can be
observed during a
solar eclipse, when
stars appearing very
close to the Sun can
be seen.
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29-8 General Relativity
If the gravitational field is strong enough, light
may be bent so much that it cannot escape. An
object that is this dense is called a black hole.
Calculations show that the radius of a black
hole of a given mass will be:
Plugging in the numbers shows us that the
Earth would have to have a radius of about
0.9 cm in order to be a black hole.
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29-8 General Relativity
One way to visualize the bending of light around
massive objects is to imagine that space itself is
bent (there is a deeper truth to this as well). The
region around a black hole then might look like
this:
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Summary of Chapter 29
• The laws of physics are the same in all inertial
frames of reference.
• The speed of light in a vacuum is the same in all
inertial frames of reference, independent of the
motion of the source or the receiver.
• Clocks moving with respect to one another keep
time at different rates. An observer sees a
moving clock running slowly:
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Summary of Chapter 29
• Length in the direction of motion appears
contracted:
• Relativistic velocity addition:
• It is impossible to increase the speed of an
object from less than c to greater than c.
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Summary of Chapter 29
• Relativistic momentum:
• Total relativistic energy:
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Summary of Chapter 29
• Rest energy:
• Relativistic kinetic energy:
• Principle of equivalence: All physical
experiments conducted in a gravitational field
and in an accelerated frame of reference give
identical results.
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Summary of Chapter 29
• For an object of mass M and radius R to be a
black hole, its radius must be less than the
Schwarzschild radius:
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