Einstein’s Special Relativity in VCE Physics PHYSCON February 2012 Keith Burrows AIP Education...

Einstein’s Special Relativity

in VCE Physics

PHYSCON February 2012Keith BurrowsAIP Education Committee

Some reasons for including the Relativity DS in Unit 3:

Students fascinated by the idea of Einstein and Relativity. Expands their thinking and opens their minds – and not only in physics. It’s what they want from Physics!

We need to attract ‘non-physics’ students – who will end up as journalists, teachers, business people, and even politicians. (Especially politicians!)

Relativity is an excellent example of the way science works: Apparent inconsistencies creative thinking experiments new theories Science as creative and imaginative.

Fundamental physics: Newton’s laws, frames of reference, mass and gravity, relative motion, waves and light, nature of space and time, electromagnetism.

Some reasons for including the Relativity DS in Unit 3:

‘Classic’ physics experiments: Michelson-Morley, speed of light, Eddington’s eclipse, travelling clocks, muon lifetimes, etc.

Much modern technology can not be understood without relativity: magnetism, nuclear energy, GPS, the synchrotron.

And – it is (almost certainly) to be included in the Australian National Curriculum.

Why Relativity? What is physics really all about?

What physics really is all about:

Newton realised that we assume that time and space are ‘straight’...

and that time and space are unrelated.

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 NutshellThese illustrations from Hawking: Universe in a Nutshell

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 can get a feel for real physics.

“Imagination is more important than knowledge”

Albert Einstein

Einstein’s Legacy In many ways Einstein represents a

new way of thinking, not just about space and time, but about everything. Will this new way of thinking have an impact on the way we see our world as profound as that of the physicists of the Enlightenment?

(Not yet apparently )

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’

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 …”

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!

Could relativity really change the way we think?

Science is not about collecting facts, but finding new ways of thinking about them. (Bragg)

Imagine what could happen if we applied new ways of thinking to, say, energy production and consumption!

It’s great physics! Relativity is about questioning

common assumptions and finding new ways of looking at a situation. It’s great thinking!

What the world needs now… Our students – future leaders!

So why study relativity?

Relativity for all

This is not (just) for the specialists, it is for future... Journalists Teachers Politicians Lawyers Hairdressers Mothers and Fathers Citizens

ok – so how do we do it?

Developing the story: Overview

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 This sequence includes all of the

points in the SD, but orders them in a more historically logical way.

Summary of 1: Two principles Einstein did NOT want

to give up 1. The principle of relativity (no absolute

zero of velocity) seems universal. 2. Maxwell’s (very ‘elegant’) equations

suggested: light is an electromagnetic wave and has a fixed speed whatever frame of

reference! this speed was assumed to be the speed

through the aether (to make it consistent with the principle of relativity)

But Michelson and Morley could not detect the aether – and Maxwell’s equations didn’t want it.

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 thesame 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 Faraday’s idea into his famous electromagnetic equations.

PS: Ideal examples of the excellent experimentalist and the excellent theoretician

of light The equations suggested

the possibility of electromagnetic waves travelling through space from an accelerated charge.

of light Maxwell’s equations predicted that electromagnetic waves would travel at a speed given by a simple expression involving electric and magnetic constants.

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 really relative to the ‘aether’ – an absolute frame of reference?

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:

look for the aether The principle of their experiment – an

analogy:

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 actual 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

look for the aether A water analogy: The boat travels at

5 m/s in a river flowing at 3 m/s

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

look for the aether But it travels at 2 m/s upstream (5 – 3)

and 8 m/s downstream (5 + 3) and so takes:

look for the aether 1000/2 + 1000/8 = 500 + 125 = 625

sec for the two way trip parallel to the water.

look for the aether So the average speed seems faster

going across than going up/down the stream

500 sec 625 sec

look for the aether They arranged to compare the speed

of light in perpendicular directions to the Earth’s motion through the aether.

look for the aether When they rotated their apparatus

they found NO DIFFERENCE in the speed of light in the two perpendicular directions!

Einstein apparently 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.

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 the whole idea.

This led to his two famous postulates:I No law of physics can identify a state of absolute rest.II 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…

... 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.

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 made through ‘Gedanken’ experiments…

Including Newton’s three laws!

I wonder what would

happen if…

Einstein realised that the principle of relativity was an extremely ‘elegant’ principle.

The real world would be very messy without it!

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!

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) But then, what

is the point of the aether?

Einstein decided that it should not be possible to use an aether or light to determine an absolute velocity.

He therefore scrapped the idea of the aether…

and concluded that any measurement of the speed of light must give the same result:

I No law of physics can identify a state of absolute rest.

II The speed of light is the same for all observers.

He decided to keep these two great principles and so put forward two postulates which embodied them:

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 difficulty: 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...

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!

Also see collection of train slides etc at end of this ppt.

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 = γto where

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 = γto where

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: (Remembering that TA = 2l/c)

Tc = TA γ² where γ = 1/√(1 – v²/c²)

(But note that we cancelled the l’s in the two expressions to obtain this!)

For Chloe the time 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

For Chloe the time 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

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.

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!

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.

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)

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:

t = γto

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!

So perhaps the extra gamma in the earlier equation is due to the fact that the l’s were different by the same gamma factor.

The time was too long – so maybe the length was actually shorter by a factor γ?

This would account for the extra γ.

So Einstein’s train was 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.

Einstein showed that in fact, as observed from one frame of reference, something moving will appear shorter by a factor γ.

This is not because the object itself shrinks – it is the space it is in that contracts.

In fact we see everything in a moving frame (relative to us) contracted in the direction of its motion.

A space ship moving at close to c will appear contracted in the direction of its motion.

Lengths are contracted by the gamma factor

Remember that this is in the direction of motion only – other dimensions are unchanged.

Einstein’s length contraction equation:

l = lo/γ

Remember that this contraction is all relative

From Mr Tompkins in Wonderland by George Gamow

Remember that this contraction is all relative

How can space be different when seen from a moving (relatively) frame?

We need to be careful to remember that we only assume that large scale space is an extrapolation of small scale space! We have no experience of it.

Einstein pointed out that space and time are inextricably connected in a 4 dimensional world.

We can not imagine 4 dimensional space, but we can imagine a 2D to 3D analogy –

A two to three dimensional analogy for a three to four dimensional situation:

How far is it from Sydney to Perth?

It is hard to picture space that is not ‘straight’ – but not so hard to picture curved 2D ‘space’.

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! (5a)

Einstein’s ‘Twins Paradox’ illustrates the strange relationship between space and time.

Imagine one twin travels to Vega, 25 ly. 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)

4. Space is strange too! (5b)

But 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.

Note that we both agree on what his clock says, that is, 2.5 years.

4. Space is strange too! (5c)

Note that he does not feel that time has slowed down. It is 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! (5d)

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 involved!)

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

This is true for train travel too – it’s just hard to measure!

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

v/c γ

1% 1.00005

10% 1.005

90% 2.29

99% 7.09

99.9% 22.4

99.999% 224

So what happens to time and length? Time slows down to a standstill! Length contracts to nothing!

But why can’t we just keep accelerating beyond the speed of light?

Einstein showed that momentum was also affected by the Lorentz factor

or simply p = γpo

Einstein just used momentum, but we can go a little further and suggest that it seems as though the mass of a speeding object increases.

p = γpo

or mv = γmov So m = γmo

The increase of gamma, and hence mass, with speed.

This is why we can’t reach the speed of light

a = F/m

So a = F/γmo

So the acceleration decreases to zero as the ship approaches c!

Remember that this slowing down of acceleration is what we see watching the rocket.

The pilot will see the world (and us) going by at a speed which approaches, but again cannot reach c. He will also deduce that his relative acceleration has decreased.

But he will not feel heavier! This is where we must leave ‘special

relativity’ and wait for ‘general relativity’ which deals with accelerated frames of reference

– and gravity.

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!)

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’.

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: ‘mass-energy’

So E = mc² (or γmoc²) actually represents the total energy of an object (including kinetic).

Usually, however, the kinetic energy is a miniscule part of the total.

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.

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.

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.

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

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.

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

A moving charged particle is deflected by a magnetic field...

whether in a wire, a TV tube, or in free space

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.

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?That is, force depends on charge, velocity and magnetic field.

Relativity says that magnetism and electricity are aspects of the same force.

We often think of relativity in terms of high speeds. But actually it is needed to explain the magnetic force between parallel currents moving at millimeters per second!

But the electrons in one wire are at rest with respect to those in the other wire!

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!

Actually there is a balance between the forces between the electrons and the positive atoms in the wires

But separately, these two forces are HUGE!

… but slightly different because …

…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 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!

“The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”

Albert Einstein

Resources

Library ‘must have’ Very good biography

Resources

Activities

Download these presentations from:

And watch for further developments

.orgwww.

“The most beautiful thing we can experience is the mysterious. It is the source of all true art and science.”

Albert Einstein

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