Page 1
!1!
Relativity:*Time’s*not*what*it*used*to*be*
!
Jon$Ogborn$!
Introduction:*Thinking*relativistically*Be!warned:!for!this!lecture!you!are!going!to!need!some!mathematics!!Here!it!is,!
all!of!it:!
!
!
Fig$0$Pythagoras$!
Leave!now!if!that’s!too!much!!!
!
I’m!going!to!start!with!an!example!what’s!involved!in!thinking!relativistically.!
The!example!is!very!familiar,!and!has!nothing!at!all!to!do!with!Einstein.!It!may!
have!puzzled!you!as!a!child.!It!is!the!fact!that!which!direction!is!downwards!
varies!according!to!where!you!are!in!the!World.!!
!
!!
Page 2
!2!
*Fig$1$Up5down$relativity$*I!remember!that!once,!when!I!was!‘phoning!my!late!wife!from!San!Francisco!
with!an!8!hour!time!difference,!it!dawned!on!me!that!the!pair!of!us!were!
standing!at!an!angle!θ!of!more!than!90!degrees!to!one!another.!Stones!fall!downward!everywhere!in!the!world,!but!stones!dropped!in!Australia!fall!
‘upwards’!towards!stones!dropped!in!England.!All!because!the!Earth!is!round.!!
This!may!have!intrigued!you!as!a!child,!but!by!now!you!have!got!used!to!it.!
!
I’m!going!to!try!to!state!rather!abstractly!what!makes!this!an!example!of!a!kind!
of!relativity,!and!later!you!will!see!that!we!can!match!the!features!point!by!point!
with!Einstein’s!otherwise!very!different!idea.!
!
The!relativity!of!up!and!down!has!these!general!features:!
• States!of!affairs!can!differ!in!ways!that!are!arbitrary!–!just!a!matter!of!
convention!–!that!make!no!fundamental!difference!to!‘how!things!are’.!
Remember!that!at!midnight!you!are!upside!down!relative!to!where!you!
were!at!lunchtime.!
• There!simply!is!no!‘absolute!direction!of!upRandRdown’.!Any!direction!is!
as!good!as!another.!Angles!θ between!places!are!all!relations!between!pairs!of!places;!there!is!no!absolute!value!of!the!angle.!!
• There!is!a!fundamental!symmetry!behind!this.!A!symmetry!is!a!change!
that!leaves!the!essentials!unchanged.!Here!it!is!rotation!through!an!angle!
θ.!• Nevertheless,!something!is!always!the!same!for!every!particular!place.!It!
is!that!stones!(or!apples)!always!fall!towards!your!feet;!that!is,!
downwards!is!always!towards!the!centre!of!the!Earth,!wherever!and!
Page 3
!3!
whenever!it!happens.!This!is!a!physical!fact!about!how!the!world!works,!
not!just!a!convention.!
• You!can!work!out!by!calculation!what!happens!at!another!place!(rotate!
by!the!appropriate!angle).!The!calculation!uses!an!understanding!of!the!
world!that!tells!us!how!different!conventional!choices!relate!to!one!
another!(that!the!World!is!round).!
• There!was!a!time!when!people!thought!that!the!Earth!was!flat,!with!upR
andRdown!being!the!same!everywhere.!The!reason!this!was!convincing!is!
that!the!Earth!is!very!large!compared!to!us.!For!everyday!activities,!a!flat!
Earth!is!a!very!good!approximation.!
• However,!it!is!possible!to!do!experiments!to!measure!the!tiny!angle!
between!upRandRdown!at!nearby!places.!
!
When!we!get!to!Einstein!we!will!replace!‘the!direction!of!upRandRdown’!by!‘the!
flow!of!time’,!and!‘change!of!relative!angle’!by!‘change!of!relative!speed’,!but!
otherwise!the!statements!will!all!be!very!similar.!
!
A*shrinking*world*Relativity!involves!reRthinking!the!relationship!of!time!and!space.!To!repeat!the!
obvious!cliché:!rapid!worldRwide!communication!via!the!internet!has!shrunk!the!
World.!People!can!affect!one!another!almost!instantly!across!large!distances!by!
sending!messages.!A!banker!in!London!can!instantly!shift!money!in!Hong!Kong;!
you!can!‘phone!to!say!happy!birthday!to!a!relative!in!Australia;!pilots!in!the!USA!
can!fire!rockets!in!Afghanistan.!People!can!cause!good!or!bad!things!to!happen!to!
one!another!almost!as!if!they!were!in!the!same!room,!house!or!street.*!
It!is!no!accident!that!Einstein!developed!his!theory!of!relativity!while!working!as!
a!patent!inspector!over!100!years!ago.!Patents!for!the!new!telegraph!cables!and!
systems!daily!crossed!his!desk.!Amongst!the!issues!he!faced!were!ones!about!
synchronising!time!across!continents.!The!railways!had!in!the!previous!century!
put!an!end!to!the!differences!between!(say)!Oxford!time!and!London!time!(about!
10!minutes),!to!make!uniform!timetables!possible.!Now,!for!Einstein!and!the!
telegraph!engineers,!questions!about!establishing!a!universal!synchronised!time!
began!to!loom.!!As!he!himself!said:!
!
“The!work!on!satisfactory!formulation!of!technical!patents!was!a!true!
blessing!for!me.!It!compelled!me!to!be!manyRsided!in!thought,!and!also!
offered!important!stimulation!for!thought!about!physics.”!
!
Contrast!the!way!we!can!now!affect!one!another!almost!directly!wherever!we!
are,!with!the!problems!of!ruling!the!British!Empire!in!the!18th
!Century.!To!
instruct!Clive!of!India!to!start!a!battle,!levy!a!tax!or!appoint!a!magistrate,!it!
would!have!taken!weeks!or!months!for!orders!to!arrive!by!letter.!Government!at!
a!distance!was!more!or!less!impossible.!Let’s!state!just!why!this!is!the!case:!
!
One!event!can!only!affect!another!by!happening!in!the!same!place!at!the!
same!time.!Instantaneous!action!at!a!distance!is!impossible;!the!most!that!
can!be!done!is!by!sending!something!(a!message,!or!a!bullet)!across!the!
space!as!quickly!as!possible.!!It!remains!impossible!today,!but!the!time!
Page 4
!4!
needed!to!cause!an!event!at!a!distance!has!shrunk!from!weeks!to!fractions!
of!a!second.!
$Video$Clip:$Thunder$and$lightning$This!clip!of!a!thunderstorm!reminds!us!of!the!delay!between!cause!and!effect!
when!they!are!some!distance!apart.!Because!sound!travels!about!a!million!times!
slower!than!light,!we!see!the!flash!some!seconds!before!we!hear!the!thunder.!In!
sound!travelRtime!terms,!the!storm!is!a!few!seconds!away.!But!there’s!still!a!
delay!for!the!flash,!typically!a!few!microseconds!(light!travels!a!million!times!
faster!than!sound).!
!
Whether!a!causal!delay!across!space!matters!depends!on!what!things!you!are!
trying!to!do.!It!made!it!quite!tricky!to!control!the!spaceRcraft!which!in!recent!
years!have!visited!a!comet!or!landed!on!a!moon!of!Jupiter,!with!a!time!delay!of!
several!minutes.!But!it!matters!at!home!too.!Your!computer!has!a!clock!rate!of!a!
few!GHz,!meaning!that!pulses!representing!ones!and!zeroes!are!generated!and!
processed!at!intervals!of!a!fraction!of!a!nanoRsecond.!But!if!the!computer!is!1!ns!
=!300!mm!across!(about!a!foot)!then!pulses!can!arrive!too!late!if!they!have!to!
travel!across!it.!!That’s!one!very!good!reason!why!computer!chips!are!small,!only!
say!10!mm!across.!Thus!it!is!that!the!physical!size!of!a!microchip!sets!a!limit!to!
how!fast!it!can!be!made!to!act,!and!present!chips!run!not!so!very!far!from!this!
limit.!For!the!same!reason,!supercomputers!making!weather!or!climate!forecasts!
have!to!be!built!as!small!as!possible.!
!
An!important!example!is!the!development!of!the!electricity!Power!Grid.!Through!
the!grid,!the!action!of!closing!a!switch!here!can!make!something!happen!in!any!
other!part!of!the!UK!within!a!few!milliseconds.!After!the!Queen’s!Speech!on!
Christmas!Day,!millions!of!electric!kettles!switched!on!at!the!same!time!require!
power!station!generators!to!be!made!to!spin!harder!to!meet!the!increasing!load,!
all!within!a!fraction!of!a!second.!In!this!way,!cause!and!effect!are!almost!in!
touching!distance!across!the!whole!country.!One!less!happy!consequence!is!that!
there!can!be!largeRscale!power!blackouts,!as!one!failure!leads!to!another!and!
outages!propagate!rapidly!though!the!system.!
!
It!is!also!no!accident!that!power!networks!were!rapidly!spreading!at!the!start!of!
the!last!century,!when!Einstein!was!thinking!and!writing.!Electric!power!
depended!crucially!on!the!work!of!Faraday!and!Maxwell,!who!developed!the!
idea!of!a!field!carrying!energy!(and!information)!across!space.!Indeed!Einstein’s!
paper!introducing!relativity!had!the!title!“The!electrodynamics!of!moving!
bodies”.!However,!what!interested!Einstein!more!than!technical!questions!were!
fundamental!issues,!amongst!then!the!nature!of!space!and!time,!which!had!
exercised!philosophers!for!centuries.!Here’s!what!he!said!about!this:!
!
“I!sometimes!ask!myself!how!did!it!come!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.!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.!Naturally,!I!could!go!
deeper!into!the!problem!than!a!child!with!normal!abilities.”!
Page 5
!5!
!
And!also:!
!“All!of!science!is!nothing!more!than!refinement!of!everyday!thinking.”!
!
Whilst!second!quote!is!wholly!untrue!of!most!of!science,!it!is!certainly!true!of!
relativity.!What!is!Relativity?!!!
Relativity!is!a!theory!of!how!space,!time!and!cause!and!effect!are!
necessarily!linked!together.!!
!
The!rest!of!this!talk!is!in!two!parts.!In!the!first!part!I’ll!give!a!fairly!conventional!
introduction!to!relativistic!ideas!and!phenomena,!using!the!very!simplest!
approach!that!I!know,!through!the!phenomena!of!radar!echoes.!Then!in!the!
second!part!I’ll!start!again!from!scratch!and!offer!a!much!more!radical!account,!
showing!that!relativity!involves!a!complete!rethink!of!our!ideas!of!space,!time,!
cause!and!effect.!It!is!this!second!view!that!led!Einstein!towards!General!
Relativity!a!hundred!years!ago!in!1915,!ten!years!after!his!first!ideas!in!the!
Special!theory!in!1905.!
!
Part*1*Special*relativity**!
Radar:*measuring*space*by*time*The!simplest!way!into!the!new!thinking!about!space!and!time,!is!to!consider!the!
everyday!business!of!radar!and!echoRlocation,!used!by!every!speed!camera!in!
Brighton.!This!uses!effects!that!allow!distances!and!speeds!to!be!measured!using!
only!time!intervals.!!Let’s!contrast!a!few!different!distances!with!their!signal!
time!equivalents:!
! !
A!hand’s!breadth! a!nanoRsecond!(thousandths!of!a!millionth)! !
A!walk!across!town! a!few!microRseconds!(millionths)!! !
Length!of!UK! a!few!milliRseconds!(thousandths)!
Around!the!world! a!tenth!of!a!second! !
To!the!Moon! a!second!
To!the!Sun!! a!few!minutes!(8)!
To!the!nearest!stars! a!few!years! !
Across!the!Milky!Way! a!hundred!thousand!years! ! !
The!Universe! ten!thousand!million!years!
!
Perhaps!the!best!modern!example!of!time!used!to!measure!space!is!the!Global!
Positioning!System!GPS.!Satellites!carrying!extremely!stable!clocks!send!signals!
to!a!receiver!in!your!car,!which!detects!and!analyses!the!differences!in!time!
between!signals!from!different!satellites.!From!these!differing!delays!it!works!
out!where!you!are!to!within!a!few!metres!(about!10!nanoseconds).!
!
Echoes**To!understand!all!this!it’s!best!to!start!with!something!simpler,!namely!a!single!
radar!echo!measurement,!first!of!distance!and!then!of!relative!speed.!Think!of!a!
speed!camera!pointing!a!radar!gun!at!your!car.!It!sends!a!pulse!of!radar!waves!
Page 6
!6!
towards!the!car,!which!is!reflected!back!to!the!gun.!Suppose!the!gun!receives!the!
reflected!pulse!2!microseconds!after!it!was!sent.!Your!car!must!have!been!1!
microsecond!of!radarRpulse!time!away!(about!300!metres).!!!
!
!
Fig$2$Speed$Camera$!
The!same!method!is!used!to!measure!distances!to!planets!such!as!Venus!in!the!
Solar!System.!But!for!that!it!takes!perhaps!ten!minutes!for!the!signal!to!go!and!
return,!making!Venus!to!be!at!that!time!5!minutes!from!Earth.!Since!5!minutes!is!
300!seconds,!that!put!Venus!300!x!300,000!km,!or!90!million!km!from!Earth.!
When!was!Venus!at!the!measured!distance?!Answer:!at!the!moment!of!
reflection,!5!minutes!after!the!signal!was!sent,!calculated!not!by!observing!the!
reflection!happening!(which!can’t!be!done!from!Earth!because!it!is!isn’t!there!to!
see!the!event,!but!by!halving!the!time!taken!for!the!outRandRback!signal).!!!
!
There!are!two!features!that!make!these!simple!sums!fully!correct!relativistic!
calculations.!The!first!is!that!all!the!measurements!are!taken!from!a!single!
moving!object,!namely!the!Earth.!They!make!no!reference!to!“how!the!Earth!is!
moving”.!Indeed,!for!this!purpose!the!Earth!is!“at!rest”.!Remember!that!the!radar!
set!and!the!Earth!are!travelling!together!and!as!we!said!earlier:!
!
!being!“at!rest”!means!simply!“travelling!together”!
!
Notice!that!what!seem!to!be!rather!large!distances!translate!into!what!seem!like!
short!or!quite!modest!times.!A!distance!down!a!motorway!becomes!a!few!
millionths!of!a!second;!a!distance!across!the!Solar!System!(some!hundred!million!
km)!becomes!several!minutes.!In!other!words,!!
!
causal!connections!can!happen!rather!quickly!across!space.!!
!
That’s!why!we!mostly!think!of!them!as!instantaneous.!!
!
There’s*no*such*thing*as*an*absolutely*fixed*place*You!know!how!unsettling!it!is!when!you!find!that!bits!of!the!world!you!knew!as!a!
child!have!changed:!your!old!home!isn’t!there!any!more,!a!new!road!has!
appeared.!Things!no!longer!seem!to!be!“in!their!proper!place”.!Because!the!
physical!world!around!us!changes!only!slowly,!we!get!to!imagine!a!kind!of!fixed!
framework!in!which!everything!belongs!and!has!a!definite!place.!!
!
Page 7
!7!
In!school,!you!could!easily!get!to!feel!that!this!is!the!right!way!to!think!about!
space,!as!a!kind!of!empty!framework!with!fixed!places!in!it!ready!to!receive!
things.!“Let!x!and!y!be!two!coordinates”!says!the!Maths!teacher,!drawing!a!pair!of!axes!(always!y!upwards!and!x!sideways!).!!But!the!teacher!is!not!then!telling!you!about!anything!real!R!he!is!just!setting!out!some!arbitrary!conventions!for!saying!
where!things!are.!!All!that!we!really!know!is!where!one!thing!is!relative!to!
others.!
!
So!it’s!impossible!to!give!an!absolute!answer!to!the!question!“Where!am!I?”!so!
that!all!we!can!say!is!that!!
!
!being!“in!the!same!place”!just!means!“being!zero!distance!away”!
!
Galilean*Relativity*of*velocity*There’s!another!reason!why!the!idea!of!an!absolute!fixed!space!is!misleading,!
which!is!to!do!with!motion.!Consider!the!very!basic!idea!of!“being!in!the!same!
place!as!before”.!What!could!be!simpler?!I!pick!up!this!book,!and!after!a!second!
or!so!put!it!down!again!exactly!where!it!was.!Wrong!!Because!the!Earth!rotates,!
at!this!latitude!the!surface!moves!East!at!about!300!m!per!second.!The!Bus!
Station!is!about!300!m!East!of!here,!so!I!picked!it!up!where!this!room!used!to!be!
and!put!it!down!near!where!the!bus!station!previously!was.!Luckily!you!
travelled!with!me!and!the!book,!so!you!were!present!at!both!the!picking!up!and!
putting!down.!Worse,!reflect!that!in!those!same!few!seconds!the!Earth!went!on!
travelling!around!the!Sun,!going!perhaps!100!km!in!that!time.!Finally!remember!
that!the!Sun!circles!the!centre!of!the!Galaxy!every!few!hundred!million!years.!So!
are!we!in!this!room!sitting!in!a!spot!in!space!where!dinosaurs!once!roamed?!
It!also!means!that!it!is!impossible!to!give!a!fixed!answer!to!the!question!“How!
exactly!am!I!moving?”!Thus!the!idea!of!“not!moving”!is!also!a!problem.!The!only!
way!out!is!to!cut!the!Gordian!knot,!and!say!that:!
!
!“moving”!is!always!relative!to!some!other!object!
being!“at!rest”!just!means!“travelling!together”!
!
So!to!say!that!we!in!this!room!are!all!at!rest!relative!to!one!another,!is!just!to!say!
that!we!are!all!travelling!together.!There!is!nothing,!anywhere!in!the!Universe,!
that!can!be!said!to!be!absolutely!“not!moving”.!!
!
This!thought!takes!a!lot!of!getting!used!to.!It!was!what!was!at!issue!between!
Galileo!and!the!Catholic!Church:!did!the!Earth!move!or!was!it!fixed!at!the!centre!
of!the!Universe?!What!Galileo!had!grasped!is!that!there!is!another!symmetry,!
namely!that:!!
!
merely!moving!makes!no!difference!to!anything!
!
Both!friends!and!enemies!of!Galileo!found!it!hard!to!accept!that!the!Earth!
rotated!and!travelled!round!the!Sun.!Why!did!they!not!feel!the!wind!in!their!hair,!
they!asked?!Because!the!air!was!moving!with!them,!was!the!notRsoRobvious!
answer.!!
Page 8
!8!
!
Effects*of*relative*speed*There!are!absolutely!no!effects!due!to!the!‘absolute!speed’!of!travel!of!an!object,!
But!there!are!definitely!effects!due!to!the!relative!speed!of!two!objects.!The!most!
important!is!the!Doppler!effect,!which!you!hear!every!time!a!Formula!1!race!is!
on!TV,!in!the!“WheeehRooow”!sound!a!car!makes!as!it!approaches,!passes!and!
recedes.!The!same!effect!is!used!by!the!speed!camera’s!radar!as!it!checks!how!
fast!you!are!going!towards!it.!
!
We’ll!look!at!a!series!of!calculations!of!distance!and!relative!speed,!to!see!how!
very!simple!they!are.!I’ll!use!a!graphic!way!of!showing!what!is!going!on,!which!
you!need!to!get!used!to,!since!later!I!will!use!it!to!talk!about!the!idea!of!SpaceR
Time.!But!first,!here’s!a!model!of!motion!on!a!twoRdimensional!flat!surface.!!
!
!
!
Fig$3$x5y$motion$!
As!I!change!the!size!or!direction!of!the!object’s!velocity,!so!it!travels!around!in!
the!space.!If!an!object!isn’t!moving,!it!just!sits!in!one!place.!Notice!that!time!is!not!
represented!here,!except!as!moments!captured!by!the!object!leaving!a!mark!
where!it!was!at!a!given!moment.!I!can!change!which!object!we!travel!with,!that!
is,!which!is!“at!rest”.!If!the!first!object!circled!round!the!second,!then!after!the!
change!we!see!the!second!object!circling!round!the!first.!This!was!the!problem!
faced!by!Ptolemy!and!Copernicus!in!giving!an!account!of!motions!in!the!Solar!
system.!
!
SpaceDTime*diagrams*Now!I!want!to!show!time!and!position!on!the!same!graph.!So!I’ve!simplified!by!
keeping!just!one!spatial!dimension,!labelled!x,!and!used!the!vertical!axis!to!plot!time.!!Here!is!what!“not!moving!relative!to!one!another”!now!looks!like.!
Page 9
!9!
!
!
Fig$4A$Space5Time$graph$no$motion$!
Now!we!see!time!marching!forward,!even!when!objects!are!at!rest!relative!to!
one!another.!They!sit!at!the!same!places,!but!those!places!move!forward!in!time.!
How!does!the!graph!show!that!they!are!at!rest!relative!to!each!other!–!that!they!
travel!together?!Just!by!the!vertical!traces!which!stay!the!same!distance!x!apart.!Now!lets!see!what!movement!on!the!xRaxis!looks!like:!!
!
Fig$4B$Space5Time$graph$relative$motion$!
A!constant!speed!moving!apart!gives!a!trace!sloping!up!to!the!right.!A!constant!
speed!coming!together!gives!a!trace!sloping!up!and!to!the!left.!The!slower!the!
speed,!the!more!nearly!vertical!is!the!trace.!The!greater!the!speed,!the!more!the!
distance!increases!in!a!given!time,!so!the!trace!slopes!more!horizontally.!
!
Page 10
!10!
!
Fig$4C1$Space5Time$graph$Grey$“at$rest”$So!far!you’ve!seen!the!motion!as!seen!travelling!with!the!grey!object!(grey!one!
“at!rest”).!I!can!also!show!the!same!motion!as!seen!travelling!with!the!purple!
object!(purple!one!“at!rest”).!
!
!
Fig$4C2$Space5Time$graph$purple$“at$rest”$!
Distance*from*echoes*So!that’s!how!these!diagrams!of!space!and!time!work.!Let’s!see!what!a!radar!
echo!measurement!of!distance!looks!like!with!these!graphs.!It’s!convenient!to!
choose!the!scales!so!that!distance!is!measured!in!units!of!signal!travel!time.!Then!
since!the!pulses!travel!1!unit!of!travel!time!in!1!unit!of!time,!the!pulses!travel!
diagonally!up!the!diagram!at!45!degrees.!This!simple!idea!will!be!very!important!
later!on.!!
!
!
!
Page 11
!11!
!
Fig$5$Radar$echo$gives$distance$!
Suppose!the!radar!is!being!used!in!surveying,!and!the!time!units!are!
microseconds.!Then!the!pulse!echoes!back!from!the!object!whose!distance!is!
being!surveyed!in!8!microseconds.!At!the!moment!of!reflection!it!must!have!
been!half!that!–!4!microseconds!away.!You!can!see!it!if!I!rewind!the!motion.!At!
300!metres!per!microsecond!that!puts!it!1200!metres!away.!Police!radar!might!
just!pick!up!a!car!at!that!distance,!too.!It’s!not!a!plausible!picture!of!anything!
much!if!time!is!in!seconds,!because!there!are!no!planets!or!asteroids!or!
spacecraft!that!are!not!moving!relative!to!the!Earth.!Of!course,!they!still!reflect!
echoes,!and!echoes!from!moving!objects!can!show!how!fast!the!objects!are!
moving!as!well!as!how!far!away!they!are.!!
!
These!simple!sums!are!fully!relativistically!correct.!One!reason!is!we!did!the!
“obvious”!thing!and!split!the!echo!time!equally!in!two!to!work!out!how!far!away!
the!car!(or!the!planet)!was!when!the!reflection!occurred.!That’s!equivalent!to!
assuming!that!the!speed!of!the!radar!pulse!is!the!same!going!out!as!it!is!coming!
back:!that!it!is!a!constant,!independent!of!relative!motion!of!the!source!and!
echoing!object.!A!second!reason!is!that!all!the!times!were!measured!by!the!same!
clock!in!the!same!place.!
!
Measuring*speed:*Doppler*effect*!
Here’s!a!simplified!version!of!a!radar!measurement!of!speed.!I’m!sending!two!
pulses,!one!after!the!other!to!reflect!from!a!distant!object.!First,!when!the!
distance!doesn’t!change,!the!pulses!echo!back!at!the!same!time!interval!as!they!
were!sent!out.!That!sameness!tells!me!that!there!is!no!relative!motion,!no!change!
in!distance,!that!source!and!reflecting!object!are!travelling!together.!
!
Page 12
!12!
!
Fig$6A$Radar$Doppler$travel$together$!
If!the!distant!object!moves!relative!to!the!radar!source,!there!is!a!difference.!If!
the!two!are!moving!apart,!the!pulses!come!back!more!widely!spaced!in!time.!The!
faster!they!move!apart,!the!greater!the!increase!in!pulseRspacing.!It!might!sound!
like!this:!
“beep…..beep………………………beep……………beep!
!
!
Fig$6B$Radar$Doppler$travel$apart$!
So!this!is!what!the!Doppler!effect!looks!like!in!these!SpaceRTime!graphs.!The!
speed!can!be!found!from!the!times!of!sending!and!arrival!of!the!pulses,!by!
finding!the!distance!twice!over,!at!two!different!times,!and!seeing!how!much!it!
changes!in!this!increase!in!time.!What!is!actually!observed,!the!phenomenon,!is!
that!the!interval!between!outgoing!pulses!is!stretched!when!the!pulses!return.!
Everything!is!observed!while!travelling!with!one!object!(“at!rest”!beside!it.)!
Page 13
!13!
!
The!oneRway!effect!of!the!relative!motion!on!the!signals!from!one!to!the!other!is!
the!wellRknown!Doppler!shift.!Let!the!Doppler!stretch!factor!be!k,!which!depends!on!the!ratio!v/c.!!!
Now!we!have!to!take!a!big!step,!with!important!consequences.!We!must!answer!
the!question:!“What!must!stay!the!same!if!we!shift!point!of!view!from!one!object!
to!the!other?”!Thinking!relativistically!we!have!to!insist!that!the!relative!speed!of!
two!objects!must!be!the!same,!looked!at!by!radar!from!either!object.!That!is,!the!
Doppler!stretch!factor!k!is!the!same!(the!same!experiment!done!at!either!end!gives!the!same!result).!!By!the!same!reasoning!the!pulse!spacing!must!be!
stretched!by!the!same!factor!k!when!returning!to!the!sending!object.!So,!if!the!oneRway!Doppler!stretch!factor!is!k,!the!two!way!radar!stretch!factor!is!k!twice!over,!k!x!k!or!k2.!!
!
Fig$7$Doppler$stretch$factor$k$!
Now!we!will!do!an!actual!calculation!of!speed!just!from!time!intervals.!I!will!take!
a!specially!simplified!case,!making!the!two!objects!start!at!the!same!place,!which!
helps!keep!the!arithmetic!a!bit!easier.!You!can!read!straight!off!the!diagram!that!
at!the!moment!of!reflection,!the!purple!object!was!3!seconds!of!radar!travel!time!
from!the!grey!one,!by!the!grey!one’s!clock.!You!can!also!see!that!by!the!grey!one’s!
clock!the!two!objects!must!have!moved!apart!by!this!distance!in!5!seconds.!3!
seconds!worth!of!distance!in!5!seconds!–!the!relative!speed!is!3/5!of!the!radar!
signal!speed.!!That!in!essence!is!how!the!speed!camera!works!out!the!speed!of!
your!car.!
!
Page 14
!14!
!
Fig$8$Radar$Doppler$v=0.6$*It’s!worth!noticing!that!everything!comes!from!the!two!observed!times:!2!s!for!
the!outgoing!pulse!interval!and!8!s!for!the!returning!pulse!interval.!The!distance!
3!s!is!just!!
3!s!=!½(8!s!–!2!s)!
and!the!5!s!time!to!reflection!is!
!!
5!s!=!2!s!+!3s!=!2s!+!½(8!s!–!2!s)!=!½(8!s!+!2!s)!
!!
Thus!the!speed!v/c!=!3/5!is!calculated!from:!!
!
!! =
½(8!s!– !2!s)½(8!s!+ !2!s) =
½!x!6½!x!10 =
35!
!
Time*dilation*All!very!simple!and!clear,!I!hope.!But!wait!!This!diagram!contains!a!big!surprise:!
namely!that!time!cannot!be!passing!at!the!same!rate!for!the!two!objects.!We!can!
Page 15
!15!
see!this!by!thinking!about!the!Doppler!shift!in!two!ways!which!give!different!
answers,!one!right!and!one!wrong.!
!
First*(correct)*calculation:*The!two!time!intervals!whose!values!we!really!know!are!2!seconds!for!sending!a!
pulse!and!8!seconds!for!the!delayed!echo!return.!This!ratio!8/2!is!made!up!of!the!
two!Doppler!shifts!k,!one!for!going!out!and!one!for!coming!back.!So!k!x!k!=!8/2!=!4,!and!k!is!just!equal!to!2.!The!pulse!spacing!doubles!on!each!part!of!the!radar!trip.!
k2!=!8/2!=!4!k!=!2!
The!Doppler!shift!must!be!the!same!both!ways,!if!we!assume!that!physical!
phenomena!(like!the!Doppler!shift)!cannot!distinguish!which!object!is!“really!
moving”!in!space.!
!!
Second*(wrong)*calculation:*Now!suppose!as!one!usually!does!time!is!universal,!the!same!in!all!places!at!
once.!Then!for!the!purple!travelling!object,!the!2!seconds!recorded!at!the!grey!
object!becomes!5!seconds!at!the!purple!object.!But!this!is!just!a!singleRtrip!
Doppler!shift!k.!So!now!we!get!a!different!answer:!k!=!5/2!!=!2.5.!For!the!return!pulses,!the!Doppler!shift!k!is!different:!5!seconds!is!stretched!to!8!seconds!with!k!=!8/5!=!1.6.!This!gives!two!different!values!for!the!Doppler!shift:!
k!=!2.5!k$=!1.6!
which!must!be!wrong.!The!two!Doppler!shifts!would!be!different.!Note!that!these!
two!different!calculations!would!be!correct!if!we!were!using!sound!waves!in!air!for!the!signals,!and!the!grey!object!was!not!moving!through!the!air!but!the!
purple!one!was!(as!detected!by!the!air!rushing!past!it!)!
!
Using*relativistic*thinking*Which!is!right?!!Relativistic!thinking!insists!that!it!must!be!the!first!calculation.!
Note!that!it!is!based!on!events!happening!all!at!the!same!place,!as!that!place!
travels!forward!in!time.!You!can!always!trust!the!clock!sitting!beside!you!(what!
else!could!you!do?)!
!
This!means!that!k!for!the!outward!trip!must!be!2.!The!time!recorded!at!the!purple!object!when!the!pulse!arrives!must!be!k!x!2!s!=!4!s,!not!the!5!s!shown!on!the!clock!travelling!with!the!grey!object.!For!every!5!seconds!the!grey!clock!ticks,!
the!purple!one!ticks!4!seconds.!!The!purple!clock!runs!slow!relative!to!the!grey!
one.!
!
Page 16
!16!
!
Fig.$9$time$dilation$!
Einstein!came!to!this!radical!conclusion:!that!time!must!run!slower!for!an!object!
moving!relative!to!another,!as!judged!by!the!other.!The!ratio!of!the!clock!rates!is!
called!gamma!γ!and!is!here!an!increase!of!25%!!
!Time!stretching!factor!γ!=!5/4!=!1.25!!!
The!effect!is!called!“time!dilation”.!
!
In!summary,!here!is!the!argument:!
• Relativistic!thinking!says:!
– time!of!reflection!must!be!half!way!between!sending!and!
receiving,!because!the!signal!must!travel!at!the!same!speed!both!
ways:!so!time!=!5!s!
– time!of!reflection!must!be!k!=!2!times!the!sending!interval!2!s,!because!the!two!oneRway!Doppler!shifts!must!be!the!same!(k!x!k!=!4):!so!time!=!k!x!2s!=!4!s!
• Both!are!true!!!
– t!=!5!s!on!the!radar!measuring!clock!
– t!=!4!s!on!a!clock!travelling!with!the!reflecting!object!
– the!“moving”!clock!runs!slow!!
!
To!think!this!thought!required!a!huge!leap!of!the!imagination,!born!out!of!years!
of!thought!and!puzzlement.!Einstein!himself!wrote!that:!
!
“Imagination$is$more$important$than$knowledge.$For$knowledge$is$limited,$whereas$imagination$embraces$the$entire$world,$stimulating$progress,$giving$birth$to$evolution.”$$
!
[Just!to!be!complete,!without!showing!you!how!to!derive!the!equations,!I’ll!show!you!how!
the!calculations!work!out!in!general,!for!relative!speed!v!and!radar!speed!c.!Close!your!
Page 17
!17!
eyes!and!take!a!doze!if!a!few!equations!turn!you!off.!They!aren’t!essential!for!
understanding.!
!
The!radar!echo!twoRway!Doppler!stretch!k2!is!given!by:!!
!! = 1 + !/!1 − !/!!
With!v/c!=!3/5!we!get:!
!! = 1 + 3/51 − 3/5 =
5 + 35 − 3 =
82 = 4!
so:!
k!=!2!!
The!time!dilation!factor!γ!is!given!by:!!
! = 1
(1 − !!!!)
!
or!
! = !!! − !!
!
!
With!v/c!=!3/5!we!get:!
! = 55! − 3!
!
!
Since!52!–!3
2!=!4
2!(remember!the!3R4R5!right!angled!triangle!)!
!
!
! = 54 = 1.25!
!
This!triangle!will!play!an!essential!role!later!on,!so!remember!it!]!
!
Does*time*really*run*at*different*rates?*The!extremely!radical!result!that!times!differ!when!objects!move!relatively!on!
different!paths!seems!to!have!been!conjured!out!of!thin!air,!with!extremely!
simple!–!almost!naïve!R!arguments.!Does!it!really!happen?!The!answer!is!“Yes!it!
really!does”.!A!very!clear!case!is!that!when!one!of!a!pair!of!highly!accurate!
atomic!clocks!was!flown!round!in!circles!in!an!aeroplane,!and!then!returned!to!
join!its!twin,!it!had!recorded!less!time!elapsed!than!the!stayRatRhome!one!had.!
This!delicate!experiment!has!only!recently!become!possible,!thanks!to!Global!
Positioning!System!technology.!It!is!also!the!case!that!the!GPS!clocks!have!to!be!
corrected!for!the!time!dilation!due!to!their!satellites!moving!relative!to!the!
Earth.!Typically!the!result!is!that!the!satellite!clocks!run!slow!by!about!7!microR
seconds!per!day!because!of!this!effect!(there!are!others,!too).!That’s!7000!nanoR
seconds!per!day,!or!5!ns!per!minute.!The!GPS!system!has!a!precision!of!about!20!
Page 18
!18!
ns!(5!–!10!m),!so!the!effect!would!be!to!make!its!measurements!of!position!drift!
noticeably!wrongly!in!less!than!10!minutes.!
!
The!effect!is!not,!however,!just!a!matter!of!clocks.!The!same!thing!happens!with!
the!fundamental!particles!generated!at!accelerators!like!CERN.!FastRmoving!
particles!live!longer!before!they!decay,!so!travelling!further!through!the!lab,!
which!has!to!be!built!longer!than!it!would!have!been!without!these!effects.!It!is!
in!these!high!energy!physics!experiments!that!time!dilation!becomes!a!really!big!
effect:!for!the!protons!at!CERN!the!factor!γ!is!about!8000,!for!example.!The!largest!known!example!is!the!very!highRenergy!protons!found!in!Cosmic!Rays.!
The!most!energetic!ever!seen!have!energy!1020
!electronRvolts,!resulting!in!a!γ!of!10
11
.!Suppose!one!came!from!the!other!side!of!the!Milky!Way,!105
!light!years!
away,!for!which!the!trip!takes!105
!years!at!very!nearly!the!speed!of!light.!Time!
dilation!reduces!that!time!for!the!proton!by!the!factor!1011
,!which!means!that!it!
crosses!the!galaxy!in!a!millionth!of!a!year,!which!is!about!half!a!minute.!
!
Ageing*twins*The!most!shocking!consequence!of!time!dilation!is!that!travel!keeps!you!
younger.!Imagine!twins,!one!of!whom!takes!a!space!trip!out!and!back!again!at!
3/5!the!speed!of!radar!signals,!while!the!other!twin!just!stays!at!home.!Whilst!
the!stayRatRhome!twin!ages!by!10!years,!when!they!meet!again!the!travelling!
twin!has!only!aged!by!8!years.!
!
Here’s!how!it!works!out!moment!by!moment.!The!diagram!shows!radar!signals!
sent!once!each!year!by!both!twins.!One!twin!sends!out!10!pulses,!in!the!time!in!
which!the!other!sends!out!only!8.!I’m!using!figures!that!keep!the!arithmetic!
simple,!though!the!example!is!unrealistic.!Don’t!forget!that!exactly!the!same!kind!
of!events!have!actually!been!observed!in!experiments!with!clocks!on!aeroplanes!
and!satellites.!!
!!
Page 19
!19!
!
Figure$10$Ageing$twins$v/c$=$3/5$!
There!will!be!more!to!say!about!this!effect!from!the!SpaceRTime!point!of!view.!In!
particular,!the!obvious!point!that!the!distance!travelled!by!either!twin!depends!
on!the!path!taken!(just!like!any!circuitous!car!journey)!has!to!be!extended!to!say!
that!the!time!taken!is!also!pathRdependent.!As!it!turns!out,!it!is!the!straightest!
path!that!takes!the!longest!time.!You!age!fastest!when!sitting!still!!
!
The*factor*gamma*Let’s!see!how!the!timeRstretch!factor!γ!depends!on!relative!speed.!!
Page 20
!20!
*Fig$11.1$graph$of$gamma$!
For!all!speeds!up!to!about!4/10!the!speed!of!light,!gamma!remains!very!close!to!
1.!At!v/c!=!0.6!the!factor!is!1.25,!as!we!previously!found.!Above!0.8!c!it!really!takes!off,!approaching!10!by!the!time!the!speed!is!0.99!c.!After!that,!the!sky!is!the!limit!–!the!graph!goes!off!to!infinity!as!speed!approaches!c.!*
The*ultimate*speed*c*“Nothing!can!travel!faster!than!light”!is!the!wellRknown!relativistic!slogan.!One!
thing!that!ensures!this!is!the!fact!that!the!factor!gamma!goes!off!to!infinity!as!the!
speed!approaches!c.!I!can!at!least!show!you!why!you!won’t!ever!detect!such!a!thing,!again!using!Radar!as!the!example.!Just!consider!what!happens!if!I!set!the!
relative!speed!equal!to!the!radar!signal!speed!in!the!Radar!model.!The!signals!
never!catch!up!the!moving!object,!and!there!is!no!reflection!to!record.!The!
Doppler!stretch!factor!k!becomes!infinite.!!
0!
1!
2!
3!
4!
5!
6!
7!
8!
0! 0.2! 0.4! 0.6! 0.8! 1!
factor*gamma*
speed*v/c*
Page 21
!21!
!
Fig$12$ultimate$speed$!
Yes,*but….*If!you!are!anything!like!me,!you!are!feeling!a!bit!cheated,!as!if!you!have!been!
watching!a!conjuring!act.!You!can!perhaps!see!that!time!dilation!might!happen,!
but!you!haven’t!the!faintest!idea!how!the!trick!is!done.!The!lady!vanishes,!but!
how!and!why?!It!is!to!sketch!in!some!responses!to!this!feeling,!and!to!point!the!
way!forward,!that!in!the!second!part!of!the!talk!I’ll!start!to!tackle!the!nature!of!
SpaceRTime!itself.!!
!
* *
Page 22
!22!
Part*2*Constructing*a*new*idea*of*SpaceDTime*I’ll!now!try!to!show!you!how!a!completely!new!concept!of!SpaceRTime!became!
necessary!to!complete!Einstein’s!thinking.!This!new!conception,!in!which!space!
and!time!are!woven!together,!became!the!basis!of!Einstein’s!General!theory!of!
Relativity.!!
!
How*space,*time*and*cause*became*a*problem*Before!Einstein,!time!seemed!not!to!be!a!problem.!It!was!generally!assumed!with!
Newton!that:!
!
“Absolute,!true!and!mathematical!time,!of!itself!and!from!its!own!
nature,!flows!equably!without!relation!to!anything!external.”!
!
Einstein’s!genius!was!to!recognise!that!time!is!indeed!a!problem.!He!realised!
that!there!is!no!such!thing!as!a!universal!time,!the!same!for!everybody!and!
everything.!To!put!it!briefly,!as!we!shall!see,!the!rate!of!flow!of!time!depends!on!
one’s!path!through!SpaceRTime.!Oddly!enough,!it!passes!fastest!for!those!who!
don’t!change!their!motion;!they!get!older!quicker!than!everyone!else.!But!this!is!
to!anticipate!some!quite!subtle!arguments;!arguments!which!go!to!the!heart!of!
the!nature!of!space,!time!and!causality.!
!
However,!any!new!conception!of!time!must!preserve!one!essential!feature:!
!
Causes!always!occur!before!their!effects.!
!
As!Woody!Allen!had!it:!
“Time$is$Nature’s$way$of$stopping$everything$happening$all$at$once”$!
We!shall!see!how!this!gets!builtRin!to!the!new!conception!of!SpaceRTime.!It!all!
started!when!Faraday,!followed!by!Maxwell,!had!the!idea!of!a!field!of!force!(i.e.!
cause!of!motion)!filling!space,!and!took!it!seriously!as!a!real!entity.!
!
Fields*solve*one*problem*and*create*another*I!said!before!that!action!at!a!distance!is!impossible,!that!causes!have!to!‘touch’!
their!effects.!Newton!had!exactly!this!difficulty!with!his!theory!of!gravitation.!He!
just!decided!to!ignore!it,!and!propose!that!the!Sun!acted!on!the!Earth!or!the!
Earth!on!the!Moon,!through!a!force!of!gravity!exerted!across!empty!space,!acting!
instantaneously.!He!explicitly!said!that!he!made!no!pretence!to!understanding!
how!it!worked.!But!work!it!did.!Calculations!based!on!it!predicted!the!motions!of!
planets!round!the!Sun!with!extraordinary!precision,!so!that!for!example!the!
arrival!of!comets!or!the!occurrence!of!eclipses!could!be!predicted!even!centuries!
ahead!of!time!or!centuries!back!in!the!past.!
!
So!the!matter!rested!until!the!19th
!century,!when!Maxwell,!inspired!by!the!ideas!
of!Michael!Faraday,!created!a!field!theory!of!electricity!and!magnetism.!You’ll!all!
have!felt!the!force!between!a!pair!of!magnets,!perhaps!those!in!the!clasp!of!your!
bag.!It’s!just!as!easy!to!feel!an!electric!force.!Take!a!pair!of!plastic!drinking!
Page 23
!23!
straws!and!wipe!them!with!a!clean!dry!tissue.!You’ll!find!that!they!now!repel!one!
another.!
!
Space!had!become!filled!at!each!and!every!point!(all!the!infinite!number!of!them)!
with!numbers!representing!the!strength!of!potential!causes.!Hardly!an!easily!
believable!picture,!but!that’s!how!it!was!(and!still!remains!today).!
!
In!Maxwell’s!theory,!movement!of!a!charge!at!one!place!and!time!produced!a!
magnetic!force!at!other!places!at!later!times.!Similarly!changes!in!the!magnetic!
field!at!one!place!and!time!produced!electrical!forces!on!objects!at!other!places,!
again!at!later!times.!These!changes!were!understood!as!carried!by!electric!and!
magnetic!fields!filling!space.!Changes!in!the!fields!propagated!at!a!speed!Maxwell!
could!predict.!This!speed!turned!out!to!be!the!same!as!the!measured!speed!of!
light!waves,!and,!later,!as!the!measured!speed!of!radio!waves!generated!by!
oscillating!electric!charges!or!currents.!
!
This!speed,!c!=!300,000!km!sR1,!(actually!exactly!299,792,458!m!sR1)!gave!thinkers!a!huge!problem.!What!was!it!relative!to?!Einstein!worried!about!this!question,!
and!it!gave!him!his!starting!point!for!the!theory!of!relativity.!He!began!by!
noticing!that!in!Maxwell’s!theory,!if!you!travelled!with!the!waves,!they!ceased!
even!to!exist.!So!it!seemed!to!be!impossible!to!travel!that!fast.!
!
The*fastest*link*between*cause*and*effect*Ultimately,!people!realised!that!this!speed!was!only!accidentally!the!speed!of!
light.!It!represented!instead!the!limit!on!how!causes!could!produce!effects!across!
space.!It!is!the!ultimate!causal!connection!speed.!Changes!in!gravity!propagate!at!
this!speed!too,!though!it!is!only!very!recently!that!physicists!claim!to!have!
detected!the!actual!existence!of!gravitational!waves!(February!2016).!
!
Thus!we!now!start!from!a!new!principle!about!what!remains!the!same!from!
every!point!of!view:!
!
The!maximum!causal!connection!speed!c$=!300,!000!km!sR1,!is!the!same!for!every!possible!causal!connection!
!
This!idea!leads!to!a!reRthink!of!the!relationship!of!space!and!time,!which!is!the!
theory!of!relativity.!
!
Space*and*mapDmaking*Before!trying!to!construct!a!new!idea!of!SpaceRTime,!I!want!to!show!you!the!real!
essentials!of!what!is!involved!in!the!idea!of!Space!alone.!
!
Where!is!the!Lewes!bus!station?!I!could!tell!you!its!exact!latitude!and!longitude!
as!used!by!the!GPS!system!(if!I!knew!them),!or!I!could!say,!“It’s!about!300!metres!
East!of!here”.!Perhaps!most!simply:!“It’s!just!in!front!of!Waitrose”.!The!first!two!
both!depend!on!several!arbitrary!conventions:!amongst!them!that!longitudes!
start!at!zero!in!Greenwich,!or!that!we!have!assigned!the!label!‘East’!to!some!
Page 24
!24!
direction,!chosen!from!where!the!Sun!tends!to!rise,!or!90!degrees!clockwise!
from!the!compass!North.!!
!
The!second!depends!just!on!you!knowing!where!some!other!object!is.!It!may!
surprise!you,!after!years!of!mathematics!and!Geography!lessons!about!x!and!y!coordinates,!North!and!South,!latitude!and!longitude,!that!the!structure!of!a!
space!needs!none!of!them.!“Just!in!front!of!Waitrose”!is!enough.!All*that*is*needed*is*a*table*of*the*straightDline*distances*between*pairs*of*objects.!One!advantage!of!having!a!fairly!elderly!audience!is!that!they!will!remember!
those!triangular!tables!of!distances!between!towns!at!the!back!of!the!AA!
Gazetteer.!That’s!all!you!need!to!construct!a!map!of!the!country,!of!the!space!in!
which!the!town!exist.!I’ll!show!you!how!it!can!be!done.!Here’s!a!small!set!of!
distances!between!some!towns!in!France.!
*Distances*in*km*between*six*French*towns.*!
! Bordeaux! Calais! Lyon! Marseille! Paris! Brest!
Bordeaux! –! 696! 426! 498! 492! 489!
Calais! ! –! 612! 879! 231! 534!
Lyon! ! ! –! 270! 390! 759!
Marseille! ! ! ! –! 651! 939!
Paris! ! ! ! ! –! 486!
Brest! ! ! ! ! ! –!
!
To!be!sure!the!map!fits!on!the!paper,!I’ll!make!a!start!with!the!largest!distance,!
the!939!km!between!Marseille!and!Brest.!I’ll!choose!an!arbitrary!direction!across!
the!paper!and!draw!a!line!to!scale!for!that.!Here!it!is:!
Page 25
!25!
!
!
France$map$1$!
CalaisRMarseille!is!the!next!biggest!distance!879!km.!I!see!that!Calais!is!534!km!
from!Brest,!so!Calais!must!be!where!circles!of!these!radii!centred!on!Marseille!
and!Best!happen!to!cross.!!But!this!can!be!above!or!below!the!line!from!Marseille!
to!Brest.!!I’ll!choose!the!side!you!don’t!expect,!with!Calais!coming!below!the!line,!
just!to!show!later!that!it!doesn’t!matter.!
