Relativity

1
Relativity
Special Relativity
If we both measure the same object with the same
tools, should we get the same result?
What does it mean to measure something?
Should the laws of physics be the same for
everybody?
What does it mean to know something?
Youre in a spacecraft and a comet zips by. Are
you moving or is the comet moving?
What does it mean to be in motion?
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We have this idea that physical reality,
whatever that is, ought to be independent of
who/when/where/how a measurement is made.
Electromagnetic theory was perfected by Maxwell
and others in the late 1800's.
Water waves propagate through water, sound waves
propagate through air.
It is not critical to electromagnetic theory, but
it was believed that electromagnetic waves
propagated through the ether, relative to some
universal reference frame.
The ether, being ethereal, proved very difficult
to detect!
Imagine the ether attached to this universal
reference frame. If you are moving relative to
it, you experience an ether drift.
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T1
Newtonian Relativity Theory
You swim 200 meters downstream in a river, turn
around, and swim 200 yards upstream. It takes a
time T1.
You swim 200 meters perpendicular to the river
bank, turn around, and swim 200 yards back. It
takes a time T2.
T2
T1 and T2 are different. Newtonian relativity
theory shows you how to calculate T1 and T2.
river current
If you make the same measurement on light
moving through the ether, you ought to get the
same result, T1 and T2 are different.
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Michelson and Morley built an interferometer
capable of making such a measurement.
mirror
partial mirror
A
light source
mirror
?
B
Half the light follows path A.
hypothetical ether drift
Half the light follows path B.
detector
The dashed line portions of the paths are
oriented differently relative to the ether drift.
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If the times to travel paths A and B are the
same, the two light beams arrive in phase and
interfere constructively. If the times are
different, the beams interfere destructively. Mea
surement of changes in interference fringe
shifts allow you to deduce the time difference.
A
?
B
hypothetical ether drift
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But wait! you object. It is impossible to make
paths A and B exactly the same length. An
observed fringe shift might be due to the path
length difference, or it might be due to the
different orientations of the path relative to
the ether drift.
So you take a measurement, rotate the apparatus
90 degrees in the horizontal plane, and take
another measurement. The difference between the
two measurements allows you to very precisely
measure the time difference due only to the ether
drift.
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Michelson and Morley did the experiment in July,
1887. They found nothing.
No ether drift. (Less than 5
km/s current upper limit is 15 m/s.)
No ether.
They tried it again later, in case during the
July measurement the earth was coincidentally at
rest with respect to the ether. They got the
same results.
No universal frame of reference?
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Ive seen sources that say this result wasnt
terribly bothersome, because the ether was a
conceptual convenience, and was not required to
make EM theory work. Ive seen other sources
that say this was devastating at the time. It
certainly created a problem.
In fact, if you believe Michelson and Morley and
Maxwell, you are forced to conclude that the
speed of EM radiation is the same in all
non-accelerated reference frames, regardless of
the motion of the radiation source. A bit
difficult to accept!
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How do you reconcile the lack of a universal
reference frame with the idea that everybody's
measurement of the same thing ought to produce
the same result?
Einstein, 1905, Special Theory of Relativity
Special theory of relativity treats problems
involving inertial (non-accelerated) frames of
reference.
We believe the laws of physics work and are the
same for everybody.
Experiment demands that the speed of light be
constant.
Lets make these two things our postulates and
see where we are led. The validity of the
postulates will be demonstrated if the
predictions arising from them are verified by
experiment.
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Postulates of special theory of relativity
? the laws of physics are the same in all
inertial reference frames
? the speed of light in free space has the same
value for all inertial observers
The first makes sense. The second is required
by experiments but contradicts our intuition and
common sense.
independent of the motion of the source or
relative speeds of observers!
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Time Dilation a consequence of the two
assumptions
Lets begin with a definition, and then construct
a clock.
The time interval between two events which occur
at the same place in an observers frame of
reference is called the proper time of the
interval between the events. We use t0 to denote
proper time.
Suppose you are timing an event by clicking a
stopwatch on at the start and off at the end. In
order for the stopwatch to measure the proper
time, the start and stop events must occur at
the same place in your frame of reference.
Youve been chosen to be a timer at a track meet,
so you go stand by the finish line. You start
your stopwatch when you see the puff of smoke
from the starters gun at the starting line, and
stop it when the first runner crosses the finish
line. Did you measure the proper time for the
sprint?
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Now lets make a clock.
mirror
L0
tick
tock
?
laser with built-in light detector
It's not that I'm so smart , it's just that I
stay with problems longer.A. Einstein
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How long does a tick-tock take?
mirror
time distance / velocity
t0 2L0 / c
L0
I measure proper time because the light pulse
starts and stops at the same place.
?
laser
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Now put this clock in a transparent spacecraft
and observe as it speeds past.
I dont measure proper time because tick and
tock occur in different places.
L0
tick
tock
?
?
entire clock moves with speed v
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How long does a tick-tock take? Let the total
time be t.
distance velocity time
( L02 (vt/2)2 )1/2
( L02 (vt/2)2 )1/2
L0
?
?
vt/2
vt/2
v
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According to the second postulate of special
relativity, light travels at a speed c, so D
ct. We also know the proper time from our
stationary clock experiment t0 2L0 / c
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Solving (1) and (2) for t and replacing L0 using
(3) gives
Note that (1-v2/c2)1/2 lt 1 so t gt t0. It takes
longer for an event to happen when it takes place
(is timed) in a reference frame moving relative
to the observer than when in takes place in the
observer's reference frame. Time is dilated.
This applies to all clocks.
Everybody chant a moving clock ticks slower, a
moving clock ticks slower, a moving clock ticks
slower If a moving clock ticks slower, it
counts fewer seconds.
How to remember what dilated means. Pupils in
your eye can dilate or contract. Dilate must be
the opposite of contract, so dilate must mean
take on a larger value.
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A moving clock ticks slower.
If I time an event which starts and stops in in
my frame of reference, I measure t0. If I use my
clock to time the same event as it takes place in
a reference frame moving relative to me, I
measure tgtt0.
In the latter case, I claim my clock, which
measured t, is correct, so that an identical
moving clock, which would measure t0 in the
moving reference frame, is slow.
An event must be specified by stating both its
space and time coordinates.
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Example the Apollo 11 spacecraft that went to
the moon traveled a maximum speed of 10840 m/s.
An event observed by an astronaut in the
spacecraft takes an hour. How long does an earth
observer say the event took?
Problem Solving Step 0. Think first!
Always ask what is the reference frame of the
event? Is the observer in this reference frame
or moving relative to it? The event took place
in the spacecraft. The proper time t0 is the
time measured in the spacecraft. Thus, t0 3600
s. The observer in this problem is the person
on earth, not the astronaut! The earth observer
measures t.
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Problem Solving Step 1. Draw (if appropriate) a
fully-labeled diagram. (Include values of known
quantities.)
If it helps you to draw a sketch of the earth, a
spacecraft, and a couple of stick figures, do so!
Include values of known quantities. c 3x108, v
10840, t0 3600. If you use SI units
throughout, your answer will be in SI units, and
I only need to see units with your final
answer. If you mix systems of units, show the
units at each step. You can always show units
at each step if it helps you.
Sooner or later, if you mix units, you will
suffer pain.
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Problem Solving Step 2. OSE.
So far, we only have one relativity OSE
Put your hand on a hot stove for a minute, and
it seems like an hour. Sit with a pretty girl for
an hour, and it seems like a minute. THAT'S
relativity. A. Einstein.
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Problem Solving Steps 3 and 4. Solve
algebraically first, then substitute values.
The algebra is already done in this case.
t 3600.00000235 s.
Not a big difference, but it is measurable. The
actual experiment has been done with jets flying
around the earth, and the predicted time dilation
has been observed.1
As expected, the earth observer measures a bigger
number for the time. The moving clock on the
spacecraft measured a smaller number. The moving
clock ticks slower.
1J. C. Hafele and R. C. Keating, Science 177, 186
(1972).
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What if vgtc? It can't happen. Well see later
that it would take an infinite amount of energy
to accelerate an object up to the speed of light.
What about time running backwards? Sorry, time
always runs forwards.
What about seeing an event before it happens?
Can't, because c is finite.
However, because of time dilation, events which
appear to be simultaneous in one reference frame
may not appear to be simultaneous in another
reference frame.
The only reason for time is so that everything
doesn't happen at once. A. Einstein
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On the constancy of c
Recent research suggests 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 (i.e., the spacing of multiple
lines close together) depends mathematically on
the fine structure constant
In this formula, e is the charge of an electron,
?0 is a constant you encountered in EM, c is the
speed of light, and h is another constant
important in quantum mechanics.
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Everything in the formula for ? is a constant.
However, the data obtained by the Australian
group suggest that ? had a larger value when the
light they observed was emitted than it does now.
? seems to have changed by 0.001 in 12000000000
years. Thats a change of 0.00000000000008 per
year.
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 in 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.
Im not picking on Australians. They are as
smart as we are. I am using the country of
origin as a convenient way to identify the
research group.
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What if they are right?
Theories you will learn in this class will
supersede theories you learned earlier (e.g.,
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. Relativistic mechanics
reduces to Newtonian mechanics in the limit of
small relative velocities. Use Newtonian
mechanics when the error introduced is small
(because it is easier to use). Use relativity
only if you must!
"If we knew what it was we were doing, it would
not be called research, would it?A. Einstein
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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.
And now, back to our show
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Lets 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. Fortunatel
y, 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, a some will go more than 0.66 km, and some
less, but not many, and not by 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
The faster you go, the shorter you are.A.
Einstein
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.
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.
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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
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. 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?).
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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The proper time, which in this case is amount of
time recorded by a clock in the spacecraft, is
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.
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We wont have completely eliminated the paradox
unless we can find a description for As reality
that agrees with Bs reality.
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 paradox, where a very fast runner
tries to put a 10 meter pole in a 5 meter barn, a
paradox arises because (Ill let you ponder that
and come back to it later).
44
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.
45
Consider a conducting wire and a positive test
charge.

What force does the test charge feel due to the
charges in the wire?
46
Repulsion, 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!
48
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 incomplete or 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.
50
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?
My 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 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.
51
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?
52
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.
53
The answer to b) 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.
54
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.
55
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 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.
56
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.
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.
57
The mathematical transformations which lead to
our relativistic equations for length and time
were derived by Lorentz 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!
58
1.7 Relativistic Momentum
We believe very strongly that momentum is
conserved. Lets see what effect a relativistic
calculation has on momentum.
Heres the essence of the calculation my text
uses Take two observers with identical
(including mass), elastic balls (elastic, so that
kinetic energy is conserved). Have the observers
stand along the y-axis, equal distances away from
the origin, and throw the balls with equal speeds
(call them VA and VB') towards the origin.
B
A
59
There is no new or exciting physics here. Using
conservation of momentum, you could easily show
that the two balls have equal and opposite
velocities after the collision.
Now, just for kicks, lets put one of the
observers in motion, with a speed v in the x
direction. Call that observer S'. To the other
observer, S, this is what the collision looks
like.
S' throws B
v
S throws A
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Let the speeds of the balls as measured by S be
VA and VB and let the y-component of the
(identical) distance each one travels be Y/2.
The travel time (to collision and back) for ball
A as measured by observer S is T0. The travel
time which observer S measures for ball B is the
dilated time T.
Y
The y-components of the distances are identical
because v has no y-component.
61
According to observer S
According to observer S'
Time dilation
According to observer S, VB is
The two boxed equations give the speeds S
observes for balls A and B.
VBltVA. Same distance, S says B takes longer so B
moves slower.
62
If we use the classical (Newtonian) definition
for momentum, S says that
If mAmB then pBltpA, as expected -- remember,
VBltVA.
If mA and mB are identical, then momentum is not
conserved.
This analysis is correct, but I find it confusing
because details are left out.
If we could somehow make ball B have more
momentum, then momentum would be conserved.
redmomentum of B
purplemomentum of A
greentotal momentum (not conserved)
Before
After
63
Non-conservation of momentum is an alarming idea.
What can we do to fix this situation?
Making ball B have more mass would conserve
momentum!
then the problem would be fixed. Kind of.
If
64
I say kind of because both balls A and B would
be moving. We would really like to compare the
mass of a moving object with the mass of an
identical, non-moving object. If we let VA?0
then we have the proper condition for comparison,
and you can show that,
where m is the mass of the ball at rest, and m(v)
is the mass it needs to have when it is moving,
if we believe in conservation of momentum.
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In the old days we then said OK, m is the mass
of the object at rest and m(v) is its mass when
it is moving. Lets call m0 the rest mass and m
the mass when it is moving (relativistic
mass). This notation is consistent with our
equations for time dilation and length
contraction, so we have
Not an OSE this semester. Dont use it!
This was Einsteins original approach, but later
he said it is not good.
When I studied relativity in college, and in the
previous edition of our text.
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The new approach is to say Look, mass is mass.
We believe it is something fundamental. If we
believe in conservation of momentum, we had
better change our definition of momentum.
If we define momentum as
where
Then mass is mass, momentum is conserved in our
thought experiment (and in real life), and
relativistic momentum reduces to classical
(Newtonian) momentum in the limit v?0.
More satisfying than saying mass changes with
velocity.
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So in this most recent version of our text, we
are always going to use the symbol m for mass.
Its what we called proper mass or rest mass
in the old days.
In the old days, rest mass was relativistically
invariant. Now mass is relativistically
invariant. Same reality, just different use of
words.
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Lets make this new notation official.
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More consequences of this new definition of
momentum
For finite F, a?0 as v?c. No finite force can
accelerate an object having nonzero mass up to
the speed of light!
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When can I use rest mass, and when do I have to
use
Object v v/c m(v)/m
jogger 10 km/h .000000009 1
space shuttle 104 m/s 0.000033 1.0000001
electron 106 m/s 0.0033 1.001
electron 108 m/s 0.333 1.061
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1.8 Mass and Energy
From your first-semester physics course
Use the definition of ? and integrate by parts to
get
Assuming potential energy is zero (we can always
choose coordinates to do this), we interpret ?mc2
as total energy.
The box indicates an OSE.
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When an object is at rest KE 0, and any energy
that remains is interpreted as the objects rest
energy E0.
When an object is moving, its total energy is
This is really just a variation of the OSE on the
previous slide.
This is the closest youll come to seeing Emc2
in this class. In the old days, E?mc2 would
have been written Emc2.
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These equations have a number of interesting
implications.
Mass and energy are two different aspects of the
same thing.
Conservation of energy is actually conservation
of mass-energy.
The c2 in E0mc2 means a little mass is worth a
lot of energy.
Your lunch an example of relativity at work in
everyday life.
Total energy is conserved but not
relativistically invariant. Rest (or proper) mass
is relativistically invariant. Mass is not
conserved! (But it is for the purposes of
chemistry.)
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Example when 1 kg (how much is that?) of
dynamite explodes, it releases 5.4x106 joules of
energy. How much mass disappears?
This is actually a conservation of mass-energy
problem. If this material were presented in
Physics 23, I would make you start with your
conservation of mass-energy OSE and derive the
appropriate equations from there.
For Physics 107, it is sufficient to realize that
the problem is just asking what is the mass
equivalent of 5.4x106 joules of energy?
Conservation of mass is a very good approximation!
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If we are to claim relativistic mechanics as a
replacement theory for Newtonian mechanics, then
relativistic mechanics had better reduce to
Newtonian mechanics in the limit of small
relative velocities.
Our text shows that for vltltc,
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When can I get away with using KE mv2/2, and
when do I have to use KE ?mc2 - mc2?
Use Newtonian KE every time you can get away with
it! Use relativistic KE only when you must!
If v 1x107 m/s (fast!) then mv2/2 is off by
only 0.08. Probably OK to use mv2/2. If v
0.5 c, then mv2/2 is off by 19. Better use
relativity.
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Energy and Momentum
Total energy and magnitude of momentum are given
by
With a bit of algebra, you can show
The quantities on the LHS and RHS of the above
equation are relativistically invariant (same for
all inertial observers).
Rearranging
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Is it possible for a particle to have no mass?
If m 0, what are E and p?
For a particle with m 0 and v lt c, then E0 and
p0. A non-particle. No such particle.
But if m 0 and v c, then the two equations
above are indeterminate. We cant say one way or
the other.
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If m 0 and v c, we must use
The energy of such a particle is E pc. We
could detect this particle! It could exist.
Do you know of any massless particles?
? photon
? neutrino
? graviton
graviton is to gravity as photon is to EM field
Maybe. Nobel prize for you if you show
mneutrino 0.
Maybe. Nobel prize for its discoverer.
Problem gravitational fields much, much weaker
than EM fields.
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• Looking ahead
• Particles having KE gtgt E0 (or pc gtgt mc2) become
  more photon-like and behave more like waves.
• The momentum carried by massless particles is
  nonzero (E pc).

Could you stop a freight train with a flashlight?
Could you stop a beam of atoms with a laser beam?
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A note on units. We will use the electron volt
(eV) as an energy unit throughout this course.
Variations on the eV 1 meV 10-3 eV (milli) 1
keV 103 eV (kilo) 1 MeV 106 eV (mega) 1 GeV
109 eV (giga)
Because mass and energy are convenient, we
sometimes write masses in energy units.
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An electron has a rest mass of 9.11x10-31 kg. If
you plug that mass into E0 mc2, you get an
energy of 511,000 eV, or 511 keV, or 0.511 MeV.
We sometimes write the electron mass as 0.511
MeV/c2.
It is also possible to express momentum in
energy units. An electron might have a
momentum of 0.3 MeV/c.
If you are making a calculation with an equation
like
and you want to use 0.511 MeV/c2 for the electron
mass, please do. It often simplifies the
calculation. But watch out
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What is the total energy of an electron that has
a momentum of 1.0 MeV/c?
Notice the convenient cancellation of the cs in
the 2nd step.
Avoid the common mistake dont divide by an
extra c2 or multiply by an extra c2 in the 2nd
step.
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General Relativity
Something to think about. Is the mass that goes
in F ma (or the relativistic version) the same
thing as the mass that goes in F Gm1m2/r2?
Not necessarily!
Experimentally, the two kinds of mass are the
same to within better than one part in 1012, and
must of us believe they are the same anyway, so
An observer in a closed laboratory cannot
distinguish between the effects of a
gravitational field or an acceleration of the
lab.
The principle of equivalence.
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The principle of equivalence leads one to
conclude that light must be deflected by a
gravitational field.
Experimental observation of this effect in 1919
was one of Einsteins great triumphs.
We investigate more about light and gravity in
modern physics classes.
"If A equals success, then the formula is
AXYZ. X is work. Y is play. Z is keep your
mouth shut.A. Einstein
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More Einstein quotes As far as the laws of
mathematics refer to reality, they are not
certain, and as far as they are certain, they do
not refer to reality." "Relativity teaches us the
connection between the different descriptions of
one and the same reality". "I sometimes ask
myself how it came about that I was the one to
develop the theory of relativity. The reason, I
think, is that a normal adult never stops to
think about problems of space and time. These are
things which he has thought about as a child. But
my intellectual development was retarded, as a
result of which I began to wonder about space and
time only when I had already grown up." "The
secret to creativity is knowing how to hide your
sources." "The important thing is not to stop
questioning. "Only two things are infinite, the
universe and human stupidity, and I'm not sure
about the former." "Things should be made as
simple as possible, but not any
simpler." "Sometimes one pays most for the things
one gets for nothing." "Common sense is the
collection of prejudices acquired by age
18. "Strange is our Situation Here Upon
Earth" "If you are out to describe the truth,
leave elegance to the tailor." "I never think of
the future. It comes soon enough. "Not
everything that counts can be counted, and not
everything that can be counted counts." "The
faster you go, the shorter you are." "The
wireless telegraph is not difficult to
understand. The ordinary telegraph is like a very
long cat. You pull the tail in New York, and it
meows in Los Angeles. The wireless is the same,
only without the cat. "