einstein’s special relativity this presentation was adapted from one i prepared for my students...

Einstein’s Special Einstein’s Special RelativityRelativity

This presentation was adapted from one I prepared for my This presentation was adapted from one I prepared for my students doing the DS 3.1 in 2004 in the pilot physics program. students doing the DS 3.1 in 2004 in the pilot physics program. It was given at STAVCON Nov 2004 and PHYSCON 2005. This It was given at STAVCON Nov 2004 and PHYSCON 2005. This version has been reduced somewhat to keep its download size version has been reduced somewhat to keep its download size moderate.moderate.

It is copyright but made available for teachers free use in the It is copyright but made available for teachers free use in the classroom. Feel free to adapt it to your purposes. Please note classroom. Feel free to adapt it to your purposes. Please note that while it does aim to cover the whole DS, it can not include that while it does aim to cover the whole DS, it can not include all of the material required.all of the material required.

The first part was directed to teachers and could be omitted or The first part was directed to teachers and could be omitted or modified for students. I have also omitted the resources as they modified for students. I have also omitted the resources as they are covered in the Word doc also available from Vicphysics.org. are covered in the Word doc also available from Vicphysics.org. I hope you enjoy teaching the Relativity DS!I hope you enjoy teaching the Relativity DS!Keith Burrows, Feb 2005Keith Burrows, Feb 2005

PS – see the notes pages for some added comments.PS – see the notes pages for some added comments.The drawings from Heinemann Physics 12 are by Linus LaneThe drawings from Heinemann Physics 12 are by Linus Lane

Einstein’s Special Einstein’s Special RelativityRelativity

in VCE Physicsin VCE Physics

PHYSCON February PHYSCON February 20052005

Keith BurrowsKeith Burrows

AIP Education AIP Education Committee, Committee, Warranwood Steiner Warranwood Steiner School School

Why Relativity?Why Relativity? What is physics What is physics reallyreally all about? all about?

What physics really is all about:What physics really is all about:

Newton realised that we assume that time Newton realised that we assume that time and space are ‘straight’ – no loops allowed!and space are ‘straight’ – no loops allowed!

and that time and space are unrelated.and that time and space are unrelated.

Einstein said Einstein said that we should that we should not assume not assume this.this.

He said that He said that travel through travel through time and space time and space were were intimately intimately related.related.

Special Special Relativity is Relativity is about that about that relationship.relationship.

These illustrations from Hawking: Universe in a NutshellThese illustrations from Hawking: Universe in a Nutshell

Why study relativity?Why study relativity?

Relativity represents a ‘giant step’ in the Relativity represents a ‘giant step’ in the story of physics. Why leave out the story of physics. Why leave out the climaxclimax of the story?of the story?

It is an excellent illustration of the It is an excellent illustration of the process process and natureand nature of physics. Through it we, and of physics. Through it we, and our students, can get a our students, can get a feelfeel for real physics. for real physics.

““Imagination is more important Imagination is more important

than knowledgethan knowledge””Albert EinsteinAlbert Einstein

Einstein’s LegacyEinstein’s Legacy In many ways Einstein represents a In many ways Einstein represents a new new

way of thinkingway of thinking, not just about space and , not just about space and time, but about time, but about everythingeverything..

Will this new way Will this new way of thinking have an of thinking have an impact on the way impact on the way we see our world we see our world as profound as that as profound as that of the physicists of of the physicists of the Enlightenment? the Enlightenment?

Relativity could change the Relativity could change the way we think!way we think!

““When the ideas When the ideas involved in relativity involved in relativity have become familiar, have become familiar, as they will do when as they will do when they are taught in they are taught in schools, certain schools, certain changes in our habits changes in our habits of thought are likely of thought are likely to result, and to have to result, and to have great importance in great importance in the long runthe long run.”.”

Bertrand Russell Bertrand Russell

‘‘ABC of ABC of RelativityRelativity’’

Albert Michelson Albert Michelson (1898):(1898):

““While it is never safe to While it is never safe to affirm that the future of affirm that the future of Physical Science has no Physical Science has no marvels in store even more marvels in store even more astonishing than those of the astonishing than those of the past, it seems probable that past, it seems probable that most of the grand underlying most of the grand underlying principles have been firmly principles have been firmly established …”established …”

The view in the early The view in the early 1900’s1900’s

In 1900 the In 1900 the mechanical world view mechanical world view seemed capable of seemed capable of explaining just about explaining just about everything.everything.

Did this lead to the Did this lead to the materialism and materialism and economic rationalism economic rationalism of the twentieth of the twentieth century?century?

Relativity could change the Relativity could change the way we think!way we think!

Could relativity really change the way Could relativity really change the way we think?we think?

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

Imagine what could happen if we applied Imagine what could happen if we applied these new ways of thinking to, say, these new ways of thinking to, say, economics! economics!

It’s great It’s great physics!physics! Relativity is about questioning Relativity is about questioning

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

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

So why study relativity?So why study relativity?

Relativity for allRelativity for all

This is This is notnot (just) for the specialists, it is (just) for the specialists, it is for future...for future... JournalistsJournalists TeachersTeachers PoliticiansPoliticians LawyersLawyers HairdressersHairdressers Mothers and FathersMothers and Fathers CitizensCitizens

ok – so how do we do it?ok – so how do we do it?

Developing the storyDeveloping 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: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 but 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 Galilean/Newtonian “principle of relativity”:relativity”: Nothing special about a Nothing special about a

velocity of zerovelocity of zero Velocity can only be Velocity can only be

measured measured relativerelative to some to some other frame of referenceother frame of reference

No absolute velocityNo absolute velocity Force Force changeschanges velocity velocity The laws of physics are theThe laws of physics are the

same in any same in any inertialinertial frame. 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 Maxwell and the speed

of light of light In the 1830’s Michael Faraday In the 1830’s Michael Faraday

suggested that light may be suggested that light may be some sort of electromagnetic some sort of electromagnetic wave phenomenon.wave phenomenon.

In the 1860’s James Clerk In the 1860’s James Clerk Maxwell developed his famous Maxwell developed his famous electromagnetic equations.electromagnetic equations.


of light of light The equations suggested The equations suggested

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


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

c

m s

o o

1

3 10 8

/


of light of light But this expression suggested that But this expression suggested that electromagnetic waves would travel at this electromagnetic waves would travel at this

fixed speed fixed speed in in anyany frame of reference. frame of reference. Most physicists, including Maxwell, thought Most physicists, including Maxwell, thought

this must be wrong and that perhaps this this must be wrong and that perhaps this speed was relative to the aether... speed was relative to the aether...

……the aether being a the aether being a hypothetical hypothetical medium which filled medium which filled all space.all space.

1. Two principles Einstein did NOT want to give up Michelson and Morley Michelson and Morley

look for the aether look for the aether Was the speed of electromagnetic waves Was the speed of electromagnetic waves

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

Michelson and Michelson and Morley decided to Morley decided to look for evidence of look for evidence of the Earth’s motion the Earth’s motion through the aether.through the aether.


look for the aetherlook for the aether The principle of their experiment – an The principle of their experiment – an

analogy:analogy:

The principle of their experiment:The principle of their experiment:


look for the aetherlook for the aether

They couldn’t measure the actual speed They couldn’t measure the actual speed accurately enough, but they could accurately enough, but they could comparecompare speeds in two perpendicular directions very speeds in two perpendicular directions very accurately.accurately.

They knew that They knew that these speeds should these speeds should be a little different if be a little different if the Earth was the Earth was speeding through speeding through the aether at 30 the aether at 30 km/sec.km/sec.

1. Two principles Einstein did NOT want to give up Michelson and Morley look for Michelson and Morley look for

the aetherthe aether


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

5 m/s in a river flowing at 3 m/s5 m/s in a river flowing at 3 m/s


look for the aetherlook for the aether It travels at 4 m/s across the river It travels at 4 m/s across the river

and so takes 2000/4 = 500 sec to and so takes 2000/4 = 500 sec to complete a two way tripcomplete a two way trip


look for the aetherlook for the aether But it travels at 2 m/s upstream and But it travels at 2 m/s upstream and

8 m/s downstream and so takes:8 m/s downstream and so takes:


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

sec for the two way trip parallel to sec for the two way trip parallel to the water.the water.


look for the aetherlook for the aether When they rotated their apparatus When they rotated their apparatus

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


Einstein apparently knew about the Einstein apparently knew about the Michelson-Morley experiment, but Michelson-Morley experiment, but he does not seem to have been he does not seem to have been particularly interested in it…particularly interested in it…

his main interest was in Maxwell’s his main interest was in Maxwell’s equations and their predictions equations and their predictions about the speed of light and it’s about the speed of light and it’s relativity.relativity.


Einstein thought the principle of relativity Einstein thought the principle of relativity was so fundamental it should apply in all was so fundamental it should apply in all areas of physics – including areas of physics – including electromagnetismelectromagnetism

He also thought Maxwell’s equations He also thought Maxwell’s equations were so elegant they, and their prediction were so elegant they, and their prediction about the speed of electromagnetic about the speed of electromagnetic waves, had to be true.waves, had to be true.

But how could these two great ideas be But how could these two great ideas be reconciled? Surely the speed of light (like reconciled? Surely the speed of light (like every other speed) should depend on every other speed) should depend on one’s frame of reference!one’s frame of reference!

Summary of 2: 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:II No law of physics can identify a state of absolute No law of physics can identify a state of absolute rest.rest.

IIII The speed of light is the same for all observers.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 Einstein had spent a lot of spent a lot of time wondering time wondering about the nature about the nature of light…of light…


... and concluded that you could not ... and concluded that you could not catch up to light – or the catch up to light – or the electromagnetic waves would be electromagnetic waves would be ‘frozen’ in space. This was ‘frozen’ in space. This was something that had never been seen something that had never been seen – and seemed impossible.– and seemed impossible.


Einstein did all his Einstein did all his experiments in his head – experiments in his head – Gedanken experiments. Gedanken experiments. (Much neater than messy (Much neater than messy apparatus!)apparatus!)

This might sound odd – but This might sound odd – but many of the greatest many of the greatest discoveries in physics were discoveries in physics were through Gedanken through Gedanken experiments…experiments…

Including Newton’s three Including Newton’s three laws!laws!


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

The real world The real world would be very would be very messy without messy without it!it!


If there was a If there was a way to find an way to find an absolute velocity absolute velocity (a ‘veelo’) – (a ‘veelo’) – whose system whose system is absolute? is absolute?

It seemed more It seemed more likely that the likely that the universe was universe was truly democratic!truly democratic!


All motion is All motion is relative – relative – there is no there is no absolute frame absolute frame of reference. of reference.

However However changeschanges of of velocity are velocity are absolute – absolute – acceleration acceleration is absoluteis absolute..

2. Einstein's crazy idea (2) But if this were But if this were

the case, how the case, how could light move could light move in some aether in some aether which which permeated all permeated all space? space?

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


He therefore He therefore scrapped the scrapped the idea of the idea of the aether…aether…

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

3 x 103 x 1088 m/s m/s


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

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

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

2. Einstein's crazy idea (4) At first sight these seem quite simple At first sight these seem quite simple

and straightforward – except that in and straightforward – except that in classical physics they are inconsistent!classical physics they are inconsistent!

Einstein said the aether was Einstein said the aether was unnecessary, but this still left the unnecessary, but this still left the problem of the relative velocity of light problem of the relative velocity of light in different frames.in different frames.


There was another difficulty: Einstein realised There was another difficulty: Einstein realised that there was a problem with ‘simultaneity’ if that there was a problem with ‘simultaneity’ if allall observers saw light with the same speed... observers saw light with the same speed...

Ana and Ana and Ben see the Ben see the light hit the light hit the ends at the ends at the same same time...time...


but Chloe sees the light travelling at but Chloe sees the light travelling at cc as as well, and so it takes longer to reach the well, and so it takes longer to reach the front of the train.front of the train.

So what So what Ana and Ana and Ben saw as Ben saw as simultaneousimultaneous, Chloe s, Chloe saw as saw as separate separate events!events!

2. Einstein's crazy idea (6) This can only mean that there is something This can only mean that there is something

strange about strange about timetime! Time appears to be ! Time appears to be relative.relative.

Salvador Dali – The Persistence of Memory

Summary of 3: Time is not as it Summary of 3: Time is not as it seemsseems

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 = t = γγttoo where where

11

22

vc

3. Time is not as it seems 3. Time is not as it seems (1)(1)

We can look at the train situation We can look at the train situation qualitatively:qualitatively:

For Ana & Ben the time for the light For Ana & Ben the time for the light to get from the centre and back again to get from the centre and back again is 2is 2ll/c./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 (But note that we cancelled the l’s in the two expressions to obtain this!)expressions to obtain this!)


For Chloe the time is 2For Chloe the time is 2ll/c /c 1/(1 – 1/(1 – vv22/c/c22). Remember that Chloe sees ). Remember that Chloe sees the light travelling at c, not c the light travelling at c, not c ± v.± v.

We can write this as We can write this as Tc = TA γ²

where where

11

22

vc


We need to note here that to obtain We need to note here that to obtain this result we cancelled the two this result we cancelled the two l’l’s in s in the expressions for the times in the the expressions for the times in the two frames of reference.two frames of reference.


As gamma,As gamma, γ = 1/√(1 – v²/c²) must must always be greater than one, we see always be greater than one, we see that time as measured from the rest that time as measured from the rest frame is greater than that within the frame is greater than that within the moving frame.moving frame.

Chloe sees Ana and Ben’s clocks Chloe sees Ana and Ben’s clocks going slowly!going slowly!


To be sure about this ‘time dilation’ we To be sure about this ‘time dilation’ we need to be a little more careful about the need to be a little more careful about the way we measure time. A ‘light clock’ way we measure time. A ‘light clock’ overcomes the problems of possible overcomes the problems of possible changes in lengths because it uses light changes in lengths because it uses light itself.itself.

We will call TWe will call TAA the time the time

for a one way trip of for a one way trip of

the light pulse.the light pulse.


As we watch a moving clock the light travels As we watch a moving clock the light travels further between bounces – further between bounces – but still at speed c!but still at speed c!

TTCC is the time for one ‘zig’ in this frame. is the time for one ‘zig’ in this frame.

The rocket travels vTThe rocket travels vTC C in the time for one ‘zig’.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 We find Tc/TA = γ = 1/√(1 – v²/c²)

Or more generally t = Or more generally t = γγttoo where t is where t is the time as measured from a the time as measured from a stationary frame and tstationary frame and too is the time as is the time as measured in the moving frame.measured in the moving frame.Einstein’s Einstein’s

time dilation time dilation equation:equation:

t = t = γγttoo


Earlier we wrote Earlier we wrote Tc = TA γ² for the time ratio. Why the difference?

To obtain this we cancelled the length To obtain this we cancelled the length of the train as seen in one frame with of the train as seen in one frame with that seen in the other. that seen in the other. This assumed This assumed they were the same!they were the same!

But if time behaves strangely But if time behaves strangely so maybe does space...so maybe does space...

Summary of 4: Space is Summary of 4: Space is strange too!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! 4. Space is strange too! (1)(1)

Einstein’s time dilation equation is Einstein’s time dilation equation is correct!correct!

The extra gamma in the earlier The extra gamma in the earlier equation is due to the fact that the equation is due to the fact that the ll ’s were different by the same gamma ’s were different by the same gamma factor.factor.

In fact we see a moving space ship In fact we see a moving space ship contracted in the direction of its contracted in the direction of its motion.motion.


Einstein’s train was also contracted Einstein’s train was also contracted in the direction of its motion – this is in the direction of its motion – this is why the time appeared to have why the time appeared to have slowed by more than slowed by more than γγ. In fact the . In fact the light pulse didn’t go as far as we light pulse didn’t go as far as we thought.thought.


Lengths are contracted by the Lengths are contracted by the gamma factorgamma factor

Einstein’s Einstein’s length length contraction contraction equation:equation:

l = ll = loo//γγ


Remember that this contraction is all Remember that this contraction is all relativerelative

From Mr Tompkins in Wonderland by George Gamow


Remember that this contraction is all Remember that this contraction is all relativerelative


A A two to threetwo to three dimensional analogy dimensional analogy for a for a three to fourthree to four dimensional dimensional situation: How far is it from Sydney situation: How far is it from Sydney to Perth?to Perth?


It is hard to picture space that is not It is hard to picture space that is not ‘straight’‘straight’

But do our X-Y-Z But do our X-Y-Z axes eventually axes eventually bend around bend around and join up and join up again? ... again? ... just as a 2-d grid just as a 2-d grid does on the does on the Earth’s surface.Earth’s surface.

4. Space is strange too! 4. Space is strange too! (5a)(5a)

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

Imagine one twin Imagine one twin travels to Vega, 25 l.y. travels to Vega, 25 l.y. away at 99.5% of c (away at 99.5% of c (γγ = 10)= 10)

His trip will take 25.1 His trip will take 25.1 years as measured years as measured from Earth (although from Earth (although we won’t get his signal we won’t get his signal for 50.2 years)for 50.2 years)

4. Space is strange too! 4. Space is strange too! (5b)(5b)

But although, from But although, from Earth, the traveller Earth, the traveller takes 25.1 years to get takes 25.1 years to get to Vega ... to Vega ...

the traveller will only the traveller will only experience 2.5 years – experience 2.5 years – because his time, as we because his time, as we see it, is going slowly.see it, is going slowly.

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

4. Space is strange too! 4. Space is strange too! (5c)(5c)

Note that he does not feel that time Note that he does not feel that time has slowed down. It is from our point has slowed down. It is from our point of view his time has slowed.of view his time has slowed.

But what he sees is that the But what he sees is that the space he is travelling through is space he is travelling through is contracted by 10 times and so it contracted by 10 times and so it only takes him 1/10 of the time only takes him 1/10 of the time we calculate.we calculate.

4. Space is strange too! 4. Space is strange too! (5d)(5d)

When he gets to When he gets to Vega he doesn’t Vega he doesn’t like the Vegans like the Vegans and so turns and so turns around and comes around and comes straight back, straight back, taking another 2.5 taking another 2.5 years for the years for the return trip.return trip.

(He is a Gedanken (He is a Gedanken traveller and traveller and doesn’t get doesn’t get squashed by the squashed by the acceleration acceleration involved!)involved!)

4. Space is strange too! 4. Space is strange too! (5e)(5e)

But what took him 5 But what took him 5 years, we saw over years, we saw over 50 years! He returns 50 years! He returns 5 years older but his 5 years older but his brother is 50 years brother is 50 years older!older!

This is not a paradox This is not a paradox – it is true! Clocks – it is true! Clocks flown around the flown around the Earth, and in Earth, and in satellites have satellites have confirmed it.confirmed it. From Hawking: Universe in Nutshell

Summary of 5: Momentum Summary of 5: Momentum & Energy & 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 5. Momentum & Energy (1)(1)

The Lorentz factor (gamma) The Lorentz factor (gamma) approaches infinity as the velocity approaches infinity as the velocity approaches the speed of lightapproaches the speed of light

11

22

vc

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


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


Einstein showed that momentum was Einstein showed that momentum was also affected by the Lorentz factoralso affected by the Lorentz factor

or or p = p = γγppoo

As the speed increases it is as though As the speed increases it is as though the mass increases toward infinity: the mass increases toward infinity: mv mv = = γγmmoovv

ppo

vc

1 2

2


Only massless photons can travel at Only massless photons can travel at the speed of light.the speed of light.

For photons time has slowed to For photons time has slowed to nothing and length to zero.nothing and length to zero.

A photon crosses the universe in no A photon crosses the universe in no time – in its frame of reference. time – in its frame of reference. (Which is why it lasts forever!)(Which is why it lasts forever!)


Einstein also showed that the kinetic energy Einstein also showed that the kinetic energy of a mass is given by: of a mass is given by: EEkk = (γ – 1)m = (γ – 1)mooc²c²

This looks odd, but (with the help of the This looks odd, but (with the help of the binomial theorem) does reduce to binomial theorem) does reduce to EEkk = = ½mv² at normal speeds. (Remember the v ½mv² at normal speeds. (Remember the v is in the is in the γγ.).)

Reorganising the expression gives: Reorganising the expression gives:

γmγmooc²c² = = EEkk ++ m mooc²c²

which he said was the ‘total energy’.which he said was the ‘total energy’.


The ‘total energy’: The ‘total energy’: E ETotTot = = γmγmooc²c² = = EEkk ++ m mooc²c²

So what is the So what is the mmooc²c² ? ? Einstein said it is the energy associated with Einstein said it is the energy associated with

the mass of an object.the mass of an object. In fact energy and mass are different In fact energy and mass are different

manifestations of the same thing:manifestations of the same thing: ‘mass-‘mass-energy’energy’

So E = mc² (or ESo E = mc² (or ETotTot = =γmγmooc²c²)) actually actually represents the total energy of an object represents the total energy of an object (including kinetic).(including kinetic).

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


But, in some nuclear reactions the energy But, in some nuclear reactions the energy released is so great that there is a released is so great that there is a significant decrease of mass:significant decrease of mass:

When uranium splits into ‘fission When uranium splits into ‘fission fragments’ the mass of the fragments is fragments’ the mass of the fragments is about 1% less.about 1% less.

When hydrogen fuses to produce helium, When hydrogen fuses to produce helium, the helium has less mass than the the helium has less mass than the hydrogen.hydrogen.

We find that the energy released is just We find that the energy released is just equal to equal to

E = E = ΔΔmc² where mc² where ΔΔm is the lost mass.m is the lost mass.


However it is important to realise that However it is important to realise that this mass has this mass has notnot been ‘converted’ into been ‘converted’ into energy!energy!

This is a common misconception created This is a common misconception created by popular accounts of the meaning of E by popular accounts of the meaning of E = mc= mc²²

Energy and mass are different Energy and mass are different ‘manifestations’ of the same thing.‘manifestations’ of the same thing.

The energy associated with bonding The energy associated with bonding (whether nuclear or chemical) has mass (whether nuclear or chemical) has mass and it is this mass that is ‘lost’ when and it is this mass that is ‘lost’ when energy is released by a reaction.energy is released by a reaction.


If we could contain an atom bomb in a (very strong!) If we could contain an atom bomb in a (very strong!) box, would it get lighter after the explosion?box, would it get lighter after the explosion?

The answer is…The answer is… No – the energy is No – the energy is

still in the box, and still in the box, and so is its mass.so is its mass.

But as the hot box But as the hot box radiated energy radiated energy away it would lose away it would lose mass.mass.

The relevance of The relevance of relativity relativity (1)(1)

Is it all just something of concern for Is it all just something of concern for space travellers? Of course not!space travellers? Of course not!

It tells us something very fundamental It tells us something very fundamental about the nature of our universe. It about the nature of our universe. It takes us beyond the ‘clockwork takes us beyond the ‘clockwork universe’ picture.universe’ picture.

It has many practical consequences.It has many practical consequences. Mostly though, it shows us the power of Mostly though, it shows us the power of

human reason and gives us cause to human reason and gives us cause to wonder in awe at the mysteries of the wonder in awe at the mysteries of the universe in which we live.universe in which we live.


Some of the practical consequences Some of the practical consequences include:include:

The GPS system.The GPS system. The Synchrotron – which could be a The Synchrotron – which could be a

few centimetres in diameter if it few centimetres in diameter if it weren’t for the huge increase in weren’t for the huge increase in mass of the electrons at 99.9999% of mass of the electrons at 99.9999% of the speed of light.the speed of light.

Nuclear energy Nuclear energy


Curiously enough, Curiously enough, magnetismmagnetism can can only be understood properly with only be understood properly with relativity.relativity.

In the late 1800’s it was realised In the late 1800’s it was realised that there was a problem with the that there was a problem with the theory of magnetism.theory of magnetism.


A moving charged particle is A moving charged particle is deflected by a magnetic field. This is deflected by a magnetic field. This is the origin of the force that drives all the origin of the force that drives all electric motors.electric motors.

Faraday’s original ‘motor’ A modern AC Induction motor


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

whether in a wire or in free spacewhether in a wire or in free space


but what if we observe it from a but what if we observe it from a frame of reference moving at the frame of reference moving at the same speed?same speed?

In this frame it is not moving In this frame it is not moving (although still in a magnetic field) (although still in a magnetic field) and so...and so...

should not experience a force!should not experience a force! This can’t be true – if it experiences This can’t be true – if it experiences

a force in one frame it must also in a force in one frame it must also in another.another.


In fact Einstein’s paper was calledIn fact Einstein’s paper was called

““On the Electrodynamics of Moving On the Electrodynamics of Moving Bodies”Bodies”

It starts with the question of how a It starts with the question of how a force could be velocity dependent and force could be velocity dependent and not contravene the ‘principle of not contravene the ‘principle of relativity’.relativity’.

The answer is that it can’t! So what The answer is that it can’t! So what about about

F = qvB?F = qvB?


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


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

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


If we look at the situation from the point of If we look at the situation from the point of view of the electrons, the positives are view of the electrons, the positives are moving but the electrons are at rest and moving but the electrons are at rest and should not experience a magnetic force (F = should not experience a magnetic force (F = qvB)qvB)

The moving positives create a magnetic The moving positives create a magnetic field, but as v = 0 there should be no force field, but as v = 0 there should be no force on the electrons!on the electrons!


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

Separately, these two forces are Separately, these two forces are HUGE!HUGE!

… … but slightly different because … but slightly different because …


… … the positive atoms are in motion the positive atoms are in motion relative to the electrons and are relative to the electrons and are therefore Lorentz contracted – that is, therefore Lorentz contracted – that is, their density increases and the electrons their density increases and the electrons see more positives than negativessee more positives than negatives

And are therefore attracted to the And are therefore attracted to the other wire!other wire!


The magnetic attraction in one The magnetic attraction in one frame of reference is simply the frame of reference is simply the electrostatic attraction in another electrostatic attraction in another frame of reference. Magnetism and frame of reference. Magnetism and electrostatics are the same after all!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

einstein’s special relativity this presentation was adapted from one i prepared for my students...

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