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Transcript
Einstein’s Special Relativity
This presentation was adapted from one I prepared for my
students doing the DS 3.1 in 2004 in the pilot physics program.
It was given at STAVCON Nov 2004. This version has been
reduced a little to keep its download size moderate.
It is copyright but made available for teachers free use in the
classroom. Feel free to adapt it to your purposes. Please note that
it does not cover the whole curriculum.
The first part was directed to teachers at STAVCON and could
be omitted or modified for students. There are also resources for
teachers shown at the end. Enjoy teaching the Relativity DS!
Keith Burrows, Nov 2004
PS – see the notes pages for some added comments.
The drawings from Heinemann Physics 12 are by Linus Lane
Einstein’s Special Relativity
in VCE Physics
Keith Burrows
STAVCON 2004
Why Relativity?

What is physics really all about?

What physics really is all about:


Newton realised that we assume that time is ‘straight’ – no
loops allowed!
and that time and space are unrelated.
These illustrations from Hawking: Universe in a Nutshell



Einstein said
that we should
not assume this.
He said that
travel through
time and space
were intimately
related.
Special relativity
is about that
relationship.
These illustrations from Hawking: Universe in a Nutshell
So why study relativity?


Relativity represents a ‘giant step’ in the story of
physics. Why leave out the climax of the story?
It is an excellent illustration of the process and
nature of physics. Through it we, and our
students, can get a feel for real physics.
“Imagination is more important than knowledge”
Albert Einstein
So why study relativity?



Relativity is about questioning common
assumptions and finding new ways of looking at
a situation.
What the world needs now… is a reasoned,
thoughtful, and questioning approach to the
social, political and environmental challenges we
face.
Which is just what we see in the ‘big picture’
story of physics.
Albert Michelson (1898):
“While it is never safe to affirm that the future of
Physical Science has no marvels in store even more
astonishing than those of the past, it seems
probable that most of the grand underlying
principles have been firmly established and that
further advances are to be sought chiefly in the
rigorous application of the principles to all the
phenomena which come under our notice.”
The view in the early 1900’s


In 1900 the mechanical
world view seemed
capable of explaining just
about everything.
Did this lead to the
materialism and economic
rationalism of the
twentieth century?
Relativity could change the way we think!
“When the ideas involved in
relativity have become familiar,
as they will do when they are
taught in schools, certain changes
in our habits of thought are
likely to result, and to have great
importance in the long run.”
Bertrand Russell
‘ABC of Relativity’
Relativity for all

This is not (just) for the specialists, it is for future...
Journalists
 Teachers
 Politicians
 Lawyers
 Hairdressers
 Mothers and Fathers
 Citizens

Developing the story





1. Two principles Einstein did NOT want to
give up
2. Einstein's crazy idea
3. Time is not as it seems: Time Dilation
4. If time is strange, what about space?
5. Faster than light? Momentum, Energy
and E = mc²
Summary of 1:
Two principles Einstein did NOT want to give up


The principle of relativity seems universal
Maxwell’s equations suggested
light is an electromagnetic wave
 and has a fixed speed
 which was assumed to be the speed through the
aether



Michelson and Morley could not detect the
aether.
The principle of relativity seemed inconsistent
with the predictions of Maxwell’s equations!
1. Two principles Einstein did NOT want to give up (1)

Galilean/Newtonian “principle of relativity”:





Nothing special about a velocity
of zero
Velocity can only be measured
relative to some other frame of
reference
No absolute velocity
Force changes velocity
The laws of physics are the
same in any inertial frame.
1. Two principles Einstein did NOT want to give up (1)
The principle of relativity
1. Two principles Einstein did NOT want to give up (2)
Maxwell and the speed of light


In the 1830’s Michael Faraday
suggested that light may be some sort
of electromagnetic wave
phenomenon.
In the 1860’s James Clerk Maxwell
developed his famous electromagnetic
equations.
1. Two principles Einstein did NOT want to give up (2)
Maxwell and the speed of light

The equations suggested the
possibility of electromagnetic
waves travelling through space
from an accelerated charge.
1. Two principles Einstein did NOT want to give up (2)
Maxwell and the speed of light

Maxwell’s equations predicted that electromagnetic
waves would travel at a speed given by a simple
expression involving electric and magnetic constants.
c1
 o o
 3  10 m / s
8
1. Two principles Einstein did NOT want to give up (2)
Maxwell and the speed of light

But this expression suggested that electromagnetic
waves would travel at this fixed speed in any
frame of reference.


Most physicists, including Maxwell, thought this
must be wrong and that perhaps this speed was
relative to the aether...
the aether being a
hypothetical medium
which filled all space.
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether



Was the speed of electromagnetic waves relative to an
absolute frame of reference – an aether?
But this would be contrary to the principle of Galilean
relativity!
Michelson and Morley
decided to look for
evidence of the Earth’s
motion through the
aether.
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

The principle of their experiment – an analogy:
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

The principle of their experiment:
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether


They couldn’t measure the speed accurately enough, but
they could compare speeds in two perpendicular
directions very accurately.
They knew that these
speeds should be a little
different if the Earth was
speeding through the
aether at 30 km/sec.
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

A water analogy: The boat travels at 5 m/s in a
river flowing at 3 m/s
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

It travels at 4 m/s across the river and so takes
2000/4 = 500 sec to complete a two way trip
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

But it travels at 2 m/s upstream and 8 m/s
downstream and so takes:
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

1000/2 + 1000/8 = 500 + 125 = 625 sec for the
two way trip parallel to the water.
1. Two principles Einstein did NOT want to give up
Michelson and Morley look for the aether

When they rotated their apparatus they found
NO DIFFERENCE in the speed of light in
the two perpendicular directions!
1. Two principles Einstein did NOT want to give up


Einstein knew about the Michelson-Morley
experiment, but he does not seem to have been
particularly interested in it…
his main interest was in Maxwell’s equations and
their predictions about the speed of light and it’s
relativity.
1. Two principles Einstein did NOT want to give up



Einstein thought the principle of relativity was
so fundamental it should apply in all areas of
physics – including electromagnetism
He also thought Maxwell’s equations were so
elegant they, and their prediction about the
speed of electromagnetic waves, had to be true.
But how could these two great ideas be
reconciled? Surely the speed of light (like every
other speed) should depend on one’s frame of
reference!
Summary of 2: Einstein’s crazy idea



Einstein thought about light and decided that it
should be impossible to catch up to it.
He also thought the aether did not make sense
and scrapped it.
This led to his two postulates:
I
II

No law of physics can identify a state of absolute rest.
The speed of light is the same for all observers.
The implication of taking these postulates at face
value is that time seems to be relative!
2. Einstein's crazy idea (1)

Einstein had spent a
lot of time wondering
about the nature of
light…
2. Einstein's crazy idea (2)

... and concluded that you could not catch up to
light – or the electromagnetic waves would be
‘frozen’ in space. This was something that had
never been seen – and seemed impossible.
2. Einstein's crazy idea (2)



Einstein did all his experiments in
his head – Gedanken experiments.
(Much neater than messy
apparatus!)
This might sound odd – but many
of the greatest discoveries in
physics were through Gedanken
experiments…
Including Newton’s three laws!
2. Einstein's crazy idea (2)


Einstein realised
that the principle
of relativity was
an extremely
‘elegant’ principle.
The real world
would be very
messy without it!
2. Einstein's crazy idea (2)

If there was a way
to find an absolute
velocity (a ‘veelo’) –
whose system is
absolute?

It seemed more
likely that the
universe was truly
democratic!
2. Einstein's crazy idea (2)


All motion is
relative – there is
no absolute frame
of reference.
However
changes of
velocity are
absolute –
acceleration is
absolute.
2. Einstein's crazy idea (2)


If this is the case
how could light
move in some
aether which
permeated all
space?
Einstein decided
that it can not be
possible to use
light to determine
an absolute
velocity.
2. Einstein's crazy idea (2)


He therefore
scrapped the idea
of the aether…
and concluded that
any measurement
of the speed of
light must give the
same result:
3 x 108 m/s
2. Einstein's crazy idea (3)

I
He decided to keep these two
great principles and so put
forward two postulates which
embodied them:
No law of physics can identify a state
of absolute rest.
II The speed of light is the same for all
observers.
2. Einstein's crazy idea (4)


At first sight these seem quite simple and
straightforward – except that in classical physics they
are inconsistent!
Einstein said the aether was unnecessary, but this
still left the problem of the relative velocity of light
in different frames.
2. Einstein's crazy idea (5)

There was another problem. Einstein realised that there
was a problem with ‘simultaneity’ if all observers saw light
with the same speed...
Ana and Ben
see the light hit
the ends at the
same time...
2. Einstein's crazy idea (5)

but Chloe sees the light travelling at c as well and so it
takes longer to reach the front of the train.
So what Ana
and Ben saw as
simultaneous,
Chloe saw as
separate events!
2. Einstein's crazy idea (6)

This can only mean that there is something
strange about time! Time appears to be relative.
Salvador Dali – The Persistence of Memory
Summary of 3: Time is not as it seems




We can put numbers into this flash-in-the-train
situation and find out how different the times are.
The light clock enables us to generalise.
We find that time in a moving frame appears to
run slow (to the stationary observers)
The time dilation equation: t
where

 1
2
v
1  c2
= γto
3. Time is not as it seems (1)


We can look at the train situation qualitatively:
For Ana & Ben the time for the light to get from
the centre and back again is 2l/c.
The maths of Chloe’s view. Time taken for light to get
to (either) end of carriage and back again:
Tc
= t1 + t2
= l/(c + v) + l/(c – v)
= l(c – v + c + v)/(c² – v²)
= 2lc/(c² – v²)
= 2lc/[c²(1– v²/c²)]
= 2l/[c(1 – v²/c²)]
= 2l/c  1/(1 – v²/c²)
So:
Tc = TA  γ² where γ = 1/√(1 – v²/c²)
(But note that we cancelled the l’s in the two
expressions to obtain this!)
3. Time is not as it seems (2)


For Chloe it is 2l/c  1/(1 – v2/c2). Remember
that Chloe sees the light travelling at c, not c ± v.
We can write this as Tc = TA  γ²
where
 1 v
1
2
c2
3. Time is not as it seems (3)

We need to note here that to obtain this result we
cancelled the two l’s in the expressions for the
times in the two frames of reference.
3. Time is not as it seems (4)


As gamma, γ = 1/√(1 – v²/c²) must always be
greater than one, we see that time as measured
from the rest frame is greater than that within the
moving frame.
Chloe sees Ana and Ben’s clocks going slowly!
3. Time is not as it seems (5)


To be sure about this ‘time dilation’ we need to be a
little more careful about the way we measure time. A
‘light clock’ overcomes the problems of possible
changes in lengths because it uses light itself.
We will call TA the time
for a one way trip of
the light pulse.
3. Time is not as it seems (6)



As we watch a moving clock the light travels further
between bounces – but still at speed c!
TC is the time for one ‘zig’ in this frame.
The rocket travels vTC in the time for one ‘zig’.
The maths of the light clock:





For A: d = cTA
For C: Pythagoras tells us (cTc)2 = (vTc)2 + d2
Eliminate d gives: (cTc)2 = (vTc)2 + (cTA)2
A little reorganising gives: (Tc/TA)2 = c2/(c2 – v2)
Or Tc/TA = γ = 1/√(1 – v²/c²)
3. Time is not as it seems (7)


We find Tc/TA = γ = 1/√(1 – v²/c²)
Or more generally t = γto where t is the time as
measured from a stationary frame and to is the
time as measured in the moving frame.
Einstein’s time
dilation
equation:
3. Time is not as it seems (8)


Earlier we wrote Tc = TA  γ² for the time ratio.
Why the difference?
To obtain this we cancelled the length of the
train as seen in one frame with that seen in the
other. This assumed they were the same!
 But
if time behaves strangely so
maybe does space...
Summary of 4: Space is strange too!

We cancelled the l’s as seen by A&B on the one
hand and by C on the other.

But the l Chloe saw had shrunk! – by a factor of




γ
The length contraction equation: l = lo/γ
Moving objects appear shorter because of their
motion.
This is because space itself contracts, not the
object.
The twins ‘paradox’ illustrates the strange nature
of spacetime.
4. Space is strange too!



(1)
Einstein’s time dilation equation is correct!
The extra gamma in the earlier equation is due
to the fact that the l’s were different by the same
gamma factor.
In fact we see a moving space ship contracted in
the direction of its motion.
4. Space is strange too!

(2)
Einstein’s train was also contracted in the
direction of its motion – this is why the time
appeared to have slowed by more than γ. In fact
the light pulse didn’t go as far as we thought.
4. Space is strange too!

(3)
Lengths are contracted by the gamma factor
Einstein’s length
contraction
equation:
l = lo/γ
4. Space is strange too!

(4)
Remember that this contraction is all relative
From Mr Tompkins
in Wonderland by
George Gamow
4. Space is strange too!

(4)
Remember that this contraction is all relative
4. Space is strange too!

(4)
A two to three dimensional analogy for a three to
four dimensional situation: How far is it from
Sydney to Perth?
4. Space is strange too!

(4)
It is hard to picture space that is not ‘straight’

But do our X-Y-Z
axes eventually
bend around and
join up again?
... just as a 2-d grid
does on the Earth’s
surface.
4. Space is strange too!



Einstein’s ‘Twins Paradox’
illustrates the strange
relationship between space
and time.
Imagine one twin travels
to Vega, 25 l.y. away at
99.5% of c (γ = 10)
His trip will take 25.1
years as measured from
Earth (although we won’t
get his signal for 50.2
years)
(5a)
4. Space is strange too!



Although, from Earth,
the traveller takes 25.1
years to get to Vega ...
the traveller will only
experience 2.5 years –
because his time, as we
see it, is going slowly.
But we both agree on
what his clock says, that
is, 2.5 years.
(5b)
4. Space is strange too!


(5c)
Note that he does not feel that time has slowed
down. From our point of view his time has
slowed.
But what he sees is that the space he is
travelling through is contracted by 10 times
and so it only takes him 1/10 of the time we
calculate.
4. Space is strange too!

When he gets to Vega he
doesn’t like the Vegans
and so turns around and
comes straight back,
taking another 2.5 years
for the return trip.
(He is a Gedanken
traveller and doesn’t
get squashed by the
acceleration
(5d)
4. Space is strange too!


(5e)
But what took him 5
years, we saw over 50
years! He returns 5 years
older but his brother is
50 years older!
This is not a paradox – it
is true! Clocks flown
around the Earth, and in
satellites have confirmed
it.
From Hawking: Universe in Nutshell
Summary of 5: Momentum & Energy





As the speed of an object gets greater γ
approaches infinity. Time slows to a stop and
length contracts to nothing.
Why can’t we accelerate past c?
Momentum also increases with γ, which makes it
appear that mass does also. Hence more impulse
increases momentum – but the m, not the v.
Total energy also increases in a similar way, but
there is a ‘rest mass’ component: E = mc²
Energy has mass.
5. Momentum & Energy

(1)
The Lorentz factor (gamma) approaches infinity
as the velocity approaches the speed of light

1

2
v
1  c2
v/c
γ
1%
1.00005
10%
1.005
90%
2.29
99%
7.09
99.9%
22.4
99.999%
224
5. Momentum & Energy


So what happens to time and length?
Time slows down to a standstill!
(2)
5. Momentum & Energy



So what happens to time and length?
Time slows down to a standstill!
Length contracts to nothing!
(2)
5. Momentum & Energy

(3)
But why can’t we just keep accelerating beyond
the speed of light?
5. Momentum & Energy

Einstein showed that momentum was also
affected by the Lorentz factor
po
p


(4)
2
v
1  c2
or p = γpo
As the speed increases it is as though the mass
increases toward infinity: mv = γmov
5. Momentum & Energy



(5)
Only massless photons can travel at the speed
of light.
For photons time has slowed to nothing and
length to zero.
A photon crosses the universe in no time – in its
frame of reference. (Which is why it lasts
forever!)
5. Momentum & Energy



(6)
Einstein also showed that the kinetic energy of a
mass is given by: Ek = (γ – 1)moc²
This looks odd, but (with the help of the
binomial theorem) does reduce to Ek = ½mv² at
normal speeds. (Remember the v is in the γ.)
Reorganising the expression gives:
γmoc² = Ek + moc²
which he said was the ‘total energy’.
5. Momentum & Energy






(7)
The ‘total energy’: γmoc² = Ek + moc²
So what is the moc² ?
Einstein said it is the energy associated with the
mass of an object.
In fact energy and mass are different
manifestations of the same thing: ‘massenergy’
So E = mc² (or γmoc²) actually represents the
total energy of an object (including kinetic).
Usually, however, the kinetic energy is a
5. Momentum & Energy





(8)
But it is the kinetic with which we are normally
concerned as we can’t change the mass into energy.
But, in some nuclear reactions the energy released is so
great that there is a significant decrease of mass:
When uranium splits into ‘fission fragments’ the mass
of the fragments is about 1% less.
When hydrogen fuses to produce helium, the helium
has less mass than the hydrogen.
We find that the energy released is just equal to
E = Δmc² where Δm is the lost mass.
5. Momentum & Energy




(9)
However it is important to realise that this mass
has not been ‘converted’ into energy!
This is a common misconception created by
popular accounts of the meaning of E = mc²
Energy and mass are different ‘manifestations’
of the same thing.
The energy associated with bonding (whether
nuclear or chemical) has mass and it is this mass
that is ‘lost’ when energy is released by a
reaction.
5. Momentum & Energy
(10)

If we could contain an atom bomb in a (very strong!)
box, would it get lighter after the explosion?

The answer is…
No – the energy is still in
the box, and so is its
mass.
But as the hot box
radiated energy away it
would lose mass.


The relevance of relativity (1)




Is it all just something of concern for space
travellers? Of course not!
It tells us something very fundamental about the
nature of our universe. It takes us beyond the
‘clockwork universe’ picture.
It has many practical consequences.
Mostly though, it shows us the power of human
reason and gives us cause to wonder in awe at
the mysteries of the universe in which we live.
The relevance of relativity (2)




Some of the practical consequences include:
The GPS system.
The Synchrotron – which could be a few
centimetres in diameter if it weren’t for the huge
increase in mass of the electrons at 99.9999% of
the speed of light.
Nuclear energy
The relevance of relativity (3)


Curiously enough, magnetism can only be
understood properly with relativity.
In the late 1800’s it was realised that there was a
problem with the theory of magnetism.
The relevance of relativity (3)

A moving charged particle is deflected by a
magnetic field. This is the origin of the force
that drives all electric motors.
Faraday’s original ‘motor’
A modern AC Induction motor
The relevance of relativity (3)


A moving charged particle is deflected by a
magnetic field...
whether in a wire or in free space
The relevance of relativity (4)



A moving charged particle is deflected by a
magnetic field...
but what if we observe it from a frame of
reference moving at the same speed?
In this frame it is not moving (although still in a
magnetic field) and so...

should not experience a force!

This can’t be true – if it experiences a force in
one frame it must also in another.
The relevance of relativity (5)



In fact Einstein’s paper was called
“On the Electrodynamics of Moving Bodies”
It starts with the question of how a force could
be velocity dependent and not contravene the
‘principle of relativity’.
The answer is that it can’t! So what about
F = qvB?
The relevance of relativity (5)

Relativity says that magnetism and electricity are
aspects of the same force.
The relevance of relativity (6)

We often think of relativity in terms of high
speeds. But actually it is needed to explain the
magnetic force between currents moving at
millimeters per second!

But the electrons in one wire are at rest with
respect to those in the other wire!
The relevance of relativity (6)

If we look at the situation from the point of view of
the electrons, the positives are moving but the
electrons are at rest and should not experience a
magnetic force (F = qvB)

The moving positives create a magnetic field, but as v
= 0 there should be no force on the electrons!
The relevance of relativity (6)

Actually there is a balance between the forces
between the electrons and the positive atoms in
the wires

Separately, these two forces are HUGE!
… but slightly different because:

The relevance of relativity (6)

The positive atoms are in motion relative to the
electrons and are therefore Lorentz contracted –
that is, their density increases and the electrons
see more positives than negatives

And are therefore attracted to the other wire!
The relevance of relativity (6)

The magnetic attraction in one frame of
reference is simply the electrostatic attraction in
another frame of reference. Magnetism and
electrostatics are the same after all!
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“The most beautiful thing we can
experience is the mysterious. It is the
source of all true art and science.”
Albert Einstein