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http//www.nearingzero.net
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Time Dilation Length Contraction Relativity
and Electricity Magnetism
If you are out to describe the truth, leave
elegance to the tailor.A. Einstein
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Fall 2003 Digression on the constancy of c.
Recent research suggested that c may not be
constant.
Several researchers in Australia have been
studying light absorbed by distant gas clouds
about 12 billion years ago.
The fine structure in spectral lines (the spacing
of multiple lines close together) depends on the
fine structure constant
In this formula, e is the charge of an electron,
?0 is a constant you encountered in Physics 24, c
is the speed of light, and h is another constant
we will learn about soon.
Ill skip this digression. You can read the notes
yourself, if you want.
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Everything in the formula for ? is a constant.
? seems to have changed by 0.001 in 12000000000
years. Thats a change of 0.00000000000008 per
year.
The data obtained by the Australian group suggest
that ? has a larger value now than it did when
the light they observed was emitted.
If ? has been increasing over time, then either
the charge on an electron has been increasing, or
?0, h, or c have been decreasing.
According to a commentary put out by the
American Physical Society
So it is an admittedly biased opinion of the
commentary author.
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Since the effect on the laws of physics of
increasing the electronic charge are too awful to
contemplate, they figure light is going slower.
That kills relativity, but my mail indicates
nobody but physicists believe that stuff
anyway.Bob Park
The Australian researchers have reported on this
work three times in recent years, including 2001
in Physical Review Letters and August 2002 in
Nature.
It seems like I am constantly hearing new
reports of findings that c is decreasing, but it
is all essentially this one group reporting their
work as it progresses.
Arguably the most prestigious US physics journal.
The most prestigious science journal known to
man.
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To date no one else has reproduced this
result. Some possibilities ? The Australians
could have made a mistake (unlikely?). ? Their
results could be a statistical fluke
(unlikely). ? A yet-undiscovered systematic
error could have influenced their results. ?
The interpretation could be wrong. ? They are
correct. ? ???
This is important enough that others will be
investigating carefully. We should know the
results within a few years.
Another Australian group disputes the necessity
for c to have changed. See http//eprints.anu.edu
.au/archive/00000797/00/Bicknell_Scott.pdf. Oh,
and Im not picking on Australians. They are as
smart as we are.
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What if they are right
Theories you will learn in this class have
superseded theories you learned earlier (e.g.,
relativity will supersede Newtonian
mechanics). You should not think of the earlier
theories as being wrong. Rather, the new
theories are better, and incorporate the old ones
within them.
You will learn that relativistic mechanics
reduces to Newtonian mechanics in the limit of
small relative velocities. So use the simpler
Newtonian mechanics when the error introduced is
small. Use relativity only if you must!
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So if they are right
? There will be profound implications for
cosmological theories.
? Someone will have to re-think special
relativity. Someone will have to come up with a
new theory which incorporates all of special
relativity but goes beyond it to include the
slowly-changing value of c.
? This may have profound implications for mankind
(as did special relativity). It may not. Well
see.
? Newtonian mechanics will still work just fine
as long as velocities are not too big.
? Lots of physicists will have nice jobs for a
long time to come.
"If we knew what it was we were doing, it would
not be called research, would it?A. Einstein
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Fall 2004 Update on the constancy of c.
New studies, conducted using the UVES
spectrograph on Kueyen, one of the 8.2-m
telescopes of ESO's Very Large Telescope array at
Paranal (Chile), secured new data with
unprecedented quality. These data, combined with
a very careful analysis, have provided the
strongest astronomical constraints to date on the
possible variation of the fine structure
constant. They show that, contrary to previous
claims, no evidence exist for assuming a time
variation of this fundamental constant. (http//w
ww.eso.org/outreach/press-rel/pr-2004/pr-05-04.htm
l)
This web page cites journal articles on which the
above claim is based http//www.sciencenews.org/a
rticles/20040508/note10ref.asp
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Getting back on topiclets consider another
problem that time dilation helps us solve.
Has anyone here ever felt a muon?
Does anybody even know what a muon is?
A muon is an elementary particle with a mass 207
times that of an electron, and a charge of either
e or e. Muons are created in abundance at
altitudes of 6 km or more when cosmic rays
collide with nuclei in the atmosphere.
Fortunately, muons interact only very weakly with
matter, which is why it is OK that many of them
are passing through your body right now.
This is in the upper reaches of the troposphere,
the part of the atmosphere in which we live.
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Muons travel with speeds of about 0.998 c (fast!)
and have an average lifetime of 2.2 ?s (2.2x10-6
s).
How far can an average muon travel during its
lifetime? d v t d 0.998 3108 2.210-6
0.66 km.
How can muons get through the 6 or more
kilometers of atmosphere between their birthplace
and us if they only live long enough to travel
0.66 km?
OK, some will go more than 0.66 km, and some
less, but mostly not by very much. So the
question stands.
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Time dilation!
I say the muons clock ticks slow. I say that
while the muon thinks its clock ticks 2.2 ?s, I
observe that it actually ticks
During this time the muon travels a distance d
0.998 3108 34.810-6 10.4 km, so the
average muon will reach me before decaying.
Of course, a muon doesnt think anything, but
we use words like that to help us form a mental
image of the process. If you prefer, imagine a
nano-human riding on the muon and reporting what
he/she sees.
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Double-check what is the event, who is the
observer, and who measures the proper time.
The event is the muon living.
The event does not take place at a single
location in my reference frame, so I measure the
dilated time, and the calculation was correct.
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One important aspect of relativity is that there
is only one reality. If I see the muon arrive at
the surface of the earth, the muon must agree
that it actually did arrive at the surface of the
earth.
Our average muon says there is no doubt
whatsoever that its lifetime is 2.2 ?s, and
during that time it travels 0.66 km. I say the
muon reaches the surface of the earth. The muon
says it doesnt??
I thought you said time dilation would help us
solve the muon problem. We seem to have created
a new problem. Either we have encountered two
different realities, or else there is
Relativity teaches us the connection between the
different descriptions of one and the same
reality.A. Einstein
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Length Contraction
If two observers in relative motion measure
different times for an identical event, what
makes us think they should measure the same
lengths for an identical object?
The formula for length contraction is not
terribly difficult to derive. Ill lend you a
book if you are curious. Here is the formula.
The Proper Length, L0, of an object is its length
as measured in its own rest frame.
The faster you go, the shorter you are.A.
Einstein
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An observer measuring the length of an object
moving relative to him will measure a length L
less than the length L0 he would measure if he
were not moving relative to the object.
Let me demonstrate length contraction using a
meter stick
The length contraction occurs only along the
direction of relative motion. A spacecraft
moving past an observer at nearly the speed of
light will seem to be very short in length and
normal diameter.
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A muon created at an altitude of 10.4 km would
say that during its lifetime it saw an atmosphere
of length
I say the muon gets to earth because its lifetime
is longer. The muon says it gets to earth
because the atmosphere is shorter. Different
descriptions of the same reality.
Be careful when you talk about the lifetime of a
particle moving with v close to c. You need to
specify the reference frame in which the lifetime
is measured!
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The Twin Paradox
A and B are 20 year old twins. A travels on a
spaceship at v 0.8c to a star 20 light years
away and returns.
A light year, y, is the distance light travels
in one year. Thus, y (1 year)(c). If D is a
distance expressed in light years, then the
number of years it takes to travel that distance
at a speed of v is found from time (distance) /
velocity. Thus time in years (distance in
light years) / (velocity expressed as a fraction
of c).
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B, left behind on earth, says the trip takes
220/0.8 50 years. B is 70 years old when A
returns.
B also observes that As clock (which is
identical to Bs) ticks slowly, and records less
time. If the event in question is the ticking of
As clock, then the 50 years calculated above is
the dilated time t (why?).
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The proper time, which in this case is amount of
time recorded by a clock in the spacecraft, is
found by solving our time OSE for t0
According to B (who was left back on earth), As
clock only ticked 30 years, so that A is 20 30
50 years old on return to earth.
At the end of the trip, B, left behind, is 70
years old. A, who made the trip, is 50 years
old. Can this be possible?
Yes! Absolutely! and it was verified
experimentally in the jets-around-the-world
experiment mentioned earlier.
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Now heres the paradox. A moving clock ticks
slower. This applies to all observers. A, on the
spacecraft, sees B move away and then come back.
A says Bs clock ticks slower. A does the
calculation presented on the last slide and
concludes that at the end of the trip, B is 50
and A is 70.
Thats the famous twin paradox. It would appear
that each twin rightfully claims the other aged
less. Have we discovered an example of the
existence of two different, mutually exclusive
realities?
Remember, there is no absolute reference frame
for specifying motion. Motion is relative! An
observer is free to say I am at rest you are
the one moving!
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When you encounter a paradox like this you can be
sure that someone has pulled a fast one on you.
In this case, an unwarranted calculation was made.
Special relativity applies only to observers in
inertial (non-accelerated) reference frames. A
had to accelerate (very rapidly) to leave earth
and get up to speed, and again when turning
around to head home, and a third time when
landing on earth.
A is not allowed to use the equations of special
relativity! B is, and Bs calculation is
correct A comes back 20 years younger.
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If you examine the problem carefully, its only
the turning around part that causes A trouble.
Whats poor A to do? Doesnt a moving clock tick
slower? Yes, so evidently during As period of
extreme acceleration, Bs clock (as observed by
A) would tick incredibly fast. Isnt A allowed
to use the laws of physics? Yes, but it would
have to be general relativity.
We wont have completely eliminated the paradox
unless we can find a description for As reality
that agrees with Bs reality.
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A, in the spacecraft, needs to reconsider the
distance traveled. During the out portion of
the trip, A will say that the actual distance
traveled was
and that the back portion was also 12 light
years. 24 light years at a speed of 0.8 c takes
30 years so A ages 30 years during the trip, and
comes back at age 50.
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B tells A you are younger because your clock
ticked slower.
A says I am younger because the trip covered
less distance than you thought.
Same reality, two different descriptions.
There are a number of famous paradoxes based on
relativistic calculations. Typically, someone
makes an invalid calculation (usually on purpose,
to see if they can trick you).
In another famous problem, where a very fast
runner tries to put a 10 meter pole in a 5 meter
barn, a paradox arises because
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Electricity and Magnetism
The material weve been studying is fascinating
and thought-provoking, but it is not how
Einsteins theory of relativity came into being.
What led me more or less directly to the special
theory of relativity was the conviction that the
electromagnetic force acting on a body in motion
in a magnetic field was nothing else but an
electric field.A. Einstein.
In other words, Einstein believed that what you
and I might call a magnetic force is really just
an electric force in another inertial reference
frame.
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Consider a conducting wire and a positive test
charge.

What force does the test charge feel due to the
charges in the wire?
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Attraction, because there is a closest to the
test ?
No net charge inside the conductor.
No electric field outside the conductor.

No force!
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What does the test charge see when an electric
field is applied and current flows?
E
The test charge observes that the space between
the moving electrons is contracted. There are
more electrons in the part of the conductor
nearest the test charge!
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The test charge observes that the moving
electrons are closer together than the stationary
protons, and therefore "feels" a Coulomb
attraction.
A human observer is unable to see the electrons,
and attributes the attraction to a magnetic
force generated by the moving charges.
Same reality, two different descriptions! And
both descriptions are mildly troublesome, as we
will see shortly
Beisers presentation of this material is
different, but equivalent.
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If you think about it, this presentation is
bothersome. To illustrate, I need to talk about
conservation and invariance.
A quantity is relativistically invariant if it
has the same value in all inertial frames of
reference.
The speed of light is relativistically invariant.
Time is not relativistically invariant.
Length is not relativistically invariant.
Electric charge is relativistically invariant.
All observers agree on the total amount of
charge in a system.
A quantity is conserved if it has the same value
before and after some event. Dont confuse
conservation with invariance.
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It is a fact that electric charge is both
conserved and relativistically invariant.
Our thought experiment with the conductor and
test charge suggests that a conductor which is
electrically neutral in one reference frame might
not be electrically neutral in another. How can
we reconcile this with charge invariance?
Our modern physics textbook author claims there
is no problem, because you have to consider the
entire circuit. Current in one part of the
circuit will be balanced by opposite current in
another part.
Although the explanation is correct, I dont find
it satisfying. Maybe the pole-in-barn paradox
will help us understand.
It seems logical that if moving electrons are
closer together in one part of the circuit, they
ought to be closer in other parts of the circuit
too, so that the conductor is no longer neutral
and charge is not conserved.
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The Pole-Barn Paradox
A speedy runner carrying a 10-meter pole
approaches a barn that is 5 meters long (short
barn!), with open doors at each end. A farmer
stands nearby, where he can see both front and
back door at the same time.
a) How fast does the runner have to go for the
farmer to observe that the pole fits entirely in
the barn?
b) What will the runner observe?
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The answer to a) involves a simple length
contraction calculation.
For the pole to fit in the barn, the farmer must
measure a contracted length L 5 m for the pole
of proper length L0 10 m.
The result is v 0.866 c. If the runner is
going that fast, or faster, the farmer observes
the pole to fit inside the barn.
Length contraction is often called the Lorentz
contraction, named after the scientist who
discovered the mathematical transformations which
lead to the equation for length contraction.
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The answer to b) (what will the runner observe?)
starts with another length contraction
calculation.
The runner is moving
no, the runner isnt moving. The runner sees
the barn moving towards him at a speed of v
0.866 c.
The runner says the speeding barn has a length
equal to
The pole cant possibly fit inside the barn.
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How do we explain this paradox? Which
observation is the physical reality?
The answer both observations are correct!
A detailed calculation (I can lend you the book
it is in, if you are interested) shows that the
runner observes the rear end of the barn
arriving at the front end of the pole long
before the front end of the barn arrives at the
rear end of the pole. The pole doesnt fit!
Events which are simultaneous in the farmers
frame of reference (front pole arriving at back
barn and back pole arriving at front barn) are
not simultaneous in the runners frame of
reference.
Remember, the runner sees the barn moving past
him.
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Simultaneity is not a universal physical
reality.
Now Im no longer worried about the
test-charge-plus-conductor example. At a certain
instant in time I may observe an excess of moving
negative charge in the portion of the circuit
nearest me, but that does not mean I can claim
there is a net excess of moving negative charge
in the entire circuit at that instant in time.
Now where were we before this interruption
started
Because simultaneity is a relative concept and
not an absolute one, physical theories that
require simultaneity in events at different
locations cannot be valid.Beiser, Modern
Physics, page 45.
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An observer who doesnt know about relativity, or
even one who knows about relativity but invokes
charge invariance, will claim that the conductor
has a neutral charge density and invents a
magnetic force to explain the attraction
between test charge and current-carrying wire.
But the magnetic force is present only when
current is flowing. It is not valid to talk
about a separate magnetic force. You must talk
about the electromagnetic force.
What you call magnetic force is just a
manifestation of the Lorentz contraction and
Coulombs law, and is not a separate force of
nature.
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The mathematical transformations which lead to
our relativistic equations for length and time
were actually derived by Lorentz in order to make
Maxwells equations invariant in inertial
reference frames.
Because Maxwells equations are invariant in
inertial reference frames, special relativity
does not demand that we correct them.
On the other hand, when it comes to Newtons Laws
Part of Einsteins genius was realizing that
Lorentz was on to something big!