http://www.nearingzero.net
Time Dilation
Length Contraction
Relativity and Electricity &
Magnetism
“If you are out to describe the truth, leave elegance to the tailor.”—A.
Einstein
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.
The fine structure in spectral lines (the spacing of multiple
lines close together) depends on the fine structure constant:
e2
=
20hc
In this formula, e is the charge of an electron, 0 is a
constant you encountered in Physics 201, c is the speed of
light, and h is another constant we will learn about soon.
Everything in the formula for  is a constant.
e2
=
20hc
 seems to have changed by 0.001% in
12000000000 years. That’s 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.
“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.
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 I’m not
picking on Australians. They are as smart as we are.
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!
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 slowlychanging value of c.
 This may have profound implications for mankind (as did
special relativity). It may not. We’ll 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
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://www.eso.org/outreach/press-rel/pr-2004/pr-05-04.html)
This web page cites journal articles on which the above claim
is based: http://www.sciencenews.org/articles/20040508/note10ref.asp
Getting back on topic…let’s 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.
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=vt
d = 0.998 · 3·108 · 2.2·10-6 = 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.
Time dilation!
I say the muon’s clock ticks slow. I say that while the muon
thinks* its clock ticks 2.2 s, I observe that it actually
ticks
t=
2.2 ×10-6
1 - (0.998)2
t  34.8s
During this time the muon travels a distance
d = 0.998 · 3·108 · 34.8·10-6 = 10.4 km,
so the average muon will reach me before decaying.
*Of course, a muon doesn’t “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.
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.
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 doesn’t??
“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
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. I’ll lend you a book if you are curious. Here is the
formula.
OSE:
L  L 0 1  v 2/c2
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
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.
A muon created at an altitude of 10.4 km would say that
during its lifetime it saw an atmosphere of length
L  L 0 1  v 2/c2  (10.4)  1  0.9982  0.66 km
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!
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
B
A
20 light years
*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).
A
B
A
20 light years
B, left behind on earth, says the trip takes 2·20/0.8 = 50 years.
B is 70 years old when A returns.
B also observes that A’s clock (which is identical to B’s) ticks
slowly, and records less time. If the event in question is the
ticking of A’s clock, then the 50 years calculated above is the
dilated time t (why?).
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:
t0 = t 1 - v2 c2
t 0 = 50 1 - 0.82
t 0 = 30
According to B (who was left back on earth), A’s 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.
Now here’s 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 B’s 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.
That’s 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!”
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 (nonaccelerated) 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 B’s calculation is correct: A comes back
20 years younger.
If you examine the problem carefully, it’s only the turning
around part that causes A trouble.
What’s poor A to do? Doesn’t a moving clock tick slower?
Yes, so evidently during A’s period of extreme acceleration, B’s
clock (as observed by A) would tick incredibly fast. Isn’t A
allowed to use the laws of physics? Yes, but it would have to
be general relativity.
We won’t have completely eliminated the paradox
unless we can find a description for A’s reality that
agrees with B’s reality.
A
B
A
20 light years 12 light years
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
L = L0 1 - v 2 c2 = 20  1 - 0.82 =12 light years,
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.
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…
Electricity and Magnetism
The material we’ve been studying is fascinating and thoughtprovoking, but it is not how Einstein’s 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.
Consider a conducting wire and a positive test charge.
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What force does the test charge “feel” due to the charges in
the wire?
Attraction, because there is a – closest to the test +?
No net charge inside the conductor.
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No electric field outside the conductor.
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No force!
What does the test charge see when an electric field is
applied and current flows?
E
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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!
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…
Beiser’s presentation of this material is different, but equivalent.
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. Don’t confuse conservation with invariance.
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 don’t 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.
The Pole-Barn Paradox
A speedy runner carrying a 10meter 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?
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.
L = L 0 1 - v 2 /c 2
5 = 10 1 - v 2 /c 2
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.
The answer to b) (what will the runner observe?) starts with
another length contraction calculation.
The runner is moving …
…no, the runner isn’t 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
L = L 0 1 - v 2 /c 2
L = 5 1 - 0.8662
L = 2.5 m.
The pole can’t possibly fit inside the barn.
How do we explain this paradox? Which observation is
physical reality?
the
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 doesn’t fit!
Events which are simultaneous in the farmer’s frame
of reference (front pole arriving at back barn and back
pole arriving at front barn) are not simultaneous in the
runner’s frame of reference.
*Remember, the runner sees the barn moving past him.
Simultaneity is not a “universal physical reality.”
Now I’m no longer worried about the test-charge-plusconductor 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.
An observer who doesn’t 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 Coulomb’s law, and is not a separate
force of nature.
The mathematical transformations which lead to our
relativistic equations for length and time were actually derived
by Lorentz in order to make Maxwell’s equations invariant in
inertial reference frames.*
Because Maxwell’s equations are invariant in inertial reference
frames, special relativity does not demand that we correct
them.
On the other hand, when it comes to Newton’s Laws…
*Part of Einstein’s genius was realizing that Lorentz was on to something
big!