Special Relativity · 2020-01-09 · Special Relativity Presentation to UCT Summer School Jan 2020...

transcript

SpecialRelativity

PresentationtoUCTSummerSchoolJan2020(Part2of3)

ByRobLouw

roblouw47@gmail.com 1

Test your understanding of simultaneity

Jan is a railway worker working for South African Railways. He has ingeniously synchronised the clocks on all South Africa’s railway stations. Motsi is on a high-speed train travelling from Cape Town to Johannesburg. As the train passes De Aar at full speed, all the clocks strike noon

According to Motsi when the Cape Town clock strikes noon, what time is it in Johannesburg? (a) noon? (b) before noon? (c) after noon?

Test your understanding of Einstein’s second postulate

Asaveryhigh-speedrocketshipfliespastyouitfiresaflashlightthatshineslightinalldirectionsAnobserveraboardthespaceshipobservesawavefrontthatspreadsawayfromthespaceshipatspeedc inalldirectionsWhatistheshapeofthewavefrontthatanearthobservermeasuresa)spherical,b)ellipsoidalwiththelongestsideoftheellipsoidalongthedirectionofthespaceship'smovementc)ellipsoidalwiththeshortestsideoftheellipsoidalongthedirectionofthespaceship’smovementd)neitherofthese?Isthewavefrontcenteredonthespaceship?

Time Dilation and Lorentz gamma (𝛾)

Inordertogainabetterunderstandingofwhatishappening,weclearlyneedtoderiveaquantitativerelationshipthatallowsustocomparetimeintervalsindifferentframesofreference

ThiswillbedoneusinganotherthoughtexperimentThiswillbedoneusinganotherthoughtexperimentAgainwewillusetrainmovingclosetothespeedoflightMavis,sittinginamovingtrainisinreferenceframeS’StanleyisstationaryonthegroundinreferenceframeSReferenceframeS’movesatconstantvelocityu,relativetoreferenceframeS,alongthecommonx– x’axisMavis,ridinginframeS’measuresthetimeintervalbetween

Inordertogainabetterunderstandingofwhatishappening,weclearlyneedtoderiveaquantitativerelationshipthatallowsustocomparetimeintervalsindifferentframesofreference

Thiswillbedoneusinganotherthoughtexperiment

Time Dilation Thought Experiment

Theobjectiveoftheexperimentistodemonstrate:

Thatobserversmeasureanyclocktorunslowifitmovesrelativetothemandastherelativespeedapproachesthespeedoflight,themovingclock’schangeintimetendstozero

ImaginewehaveatrainmovingclosetothespeedoflightalongastraightstretchofrailwaytrackMavis,sittinginamovingtrainisinreferenceframeS’StanleyisstationaryonthegroundinreferenceframeSReferenceframeS’movesatconstantvelocityu,relativetoreferenceframeS,alongthecommonx– x’axisMavis,ridinginframeS’measuresthetimeintervalbetweentwoeventsthatoccuratthesamepointinspace(a)

Imaginewehaveatrainmovingclosetothespeedoflightalongastraightstretchofrailwaytrack

Sarah,sittinginacoach,isridinginframeS’whereshemeasuresthetimeintervalbetweentwoeventsthatoccuratthesamepointinspace(a)onher‘lightclock’betweentwoeventsthatoccuratthesamepointinspace(a)

SarahSarah

Referenceframe S’

SarahMirror

Lightsource

O’(Event1occurshere)

SarahMirror

Lightsource

O’(Event2alsooccurshere)

SarahMirror

Lightsource

Sarahmeasuresaroundtriptimeof∆t0 forthelightbeam

O’(Events1and2occurhere)

Thelightbeamtravelsatotaldistanceof2dinatimeof∆t0 andsincethespeedoflight=c,d=c∆t0/2

SarahMirror

Lightsource

O’(Events1and2occurhere)

Sarahmeasuresaroundtriptimeof∆t0 forthelightbeam

Sourcemovesfromheretohere

Event1occurshere

Peterwhoisstationaryobservesthesamelightpulsefollowingadiagonalpath

Event1occurshere

Event2occurshere

Petermeasurestheround-triptimetobe∆t

Event1occurshere

Event2occurshere

(Distancetravelled)

Event1occurshere

Event2occurshere

(Distancetravelled)

Theround-tripdistanceforthelightbeaminreferenceframeS is2ℓ

Event1occurshere

Event2occurshere

Pythagorean theorem

ThePythagoreantheoremstatesthatforaright-angletriangle,thesquareofthehypotenuse(c)isequaltothesumofthesquaresoftheremainingtwoshorterperpendicularsides(a &b)

Thusc2 =a2 +b2

∴ c= 𝑎$ + 𝑏$

u∆t/2

Using the Pythagorean theorem we can calculate ℓ

ℓ = 𝑑$ + (𝑢∆t/2)$

The speed of light is the same for both observers, so theround-trip time measured in S is

∆t = 2ℓ/c = 2/c 𝑑$ + (𝑢∆t/2)$

ℓ = 𝑑$ + (𝑢∆t/2)$

The speed of light is the same for both observers, so theround-trip time measured in S is ∆twhere

∆t = 2ℓ/c

ℓ = 𝑑$ + (𝑢∆t/2)$

The speed of light is the same for both observers so theround-trip time measured in S is ∆twhere

∆t = 2ℓ/c = 2/c 𝑑$ + (𝑢∆t/2)$

We would like to have a relationship between ∆t and ∆t0 thatis independent of d (but is dependent on u and c)

By substitution we get

∆t = 2/c (𝑐∆t0/2)$+(𝑢∆t/2)$

Squaring this equation and solving for ∆t we get

∆t = ∆t0 / 1 − 𝑢$/𝑐2

Remembering that d = 𝑐∆t0/2, then by substitution we get

∆t = 2/c (𝑐∆t0/2)$+(𝑢∆t/2)$

Squaring this equation and solving for ∆t we get

∆t = ∆t0 / 1 − 𝑢$/𝑐2)

Remembering that d = 𝑐∆t0/2, then by substitution we get

∆t = 2/c (𝑐∆t0/2)$+(𝑢∆t/2)$

Squaring this equation and then solving for ∆t we finally get

∆t = ∆t0 / 1 − 𝑢$/𝑐2

Sincethequantity 1 − 𝑢$/𝑐2 islessthan1,∆tisalwaysgreaterthan∆t0

ThusStanleymeasuresalongerround-triptimeforthelightpulsethandoesMavis

Thequantity𝟏/ 𝟏 − 𝒖𝟐/𝒄2appearssoofteninrelativitythatithasitsownsymbol andisreferredtoasLorentzgamma

𝛾 =𝟏/ 𝟏 − 𝒖𝟐/𝒄2Lorentzgammadefinition29

ThusPetermeasuresalongerround-triptimeforthelightpulsethandoesSarah

Thequantity𝟏/ 𝟏 − 𝒖𝟐/𝒄2appearssoofteninrelativitythatithasitsownsymbol andisreferredtoasLorentzgamma

Thequantity1/ 1 − 𝑢$/𝑐2appearssoofteninrelativitythatithasitsownsymbol 𝛾 andisreferredtoasLorentzgamma

𝛾 =1/ 1 − 𝑢$/𝑐2Lorentzgammafactor32

Notethat𝛾 isalways≥1and1/𝛾 isalways≤1!

If𝛾 appearsinthenumeratorofanyrelativisticequation,itwilltendtowardsinfinityasvelocityapproachesc

Converselyif𝛾 appearsinthedenominatorofanyrelativisticequation,itwilltendtowardszeroasvelocityapproachesc

If𝛾 appearsinthenumeratorofanyrelativisticequation,itwilltendtowardsinfinityasvelocity,u approachesc

Converselyif𝛾 appearsinthedenominatorofanyrelativisticequation,itwilltendtowardszeroasvelocityapproachesc

If𝛾 appearsinthenumeratorofanyrelativisticequation,itwilltendtowardsinfinityasvelocity,approachesc

Converselyif𝛾 appearsinthedenominatorofanyrelativisticequation,itwilltendtowardszeroasvelocity,u approachesc

∆t0 iscalledthepropertimeandisequaltothetimeintervalbetweentwoeventsthatoccuratthesameposition

Onlyoneinertialframe(S’)measuresthepropertimeanditdoessowithasingleclockthatispresentatbothevents

Aninertialreferenceframemovingwithvelocityurelativetothepropertimeframemustusetwoclockstomeasurethetimeinterval:Oneatthepositionofthefirsteventandoneatthepositionofthesecondevent

Byrearrangingourearlierequations,thetimeintervalintheframewheretwoclocksarerequiredisasfollows

Aninertialreferenceframemovingwithvelocityu relativetothepropertimeframemustusetwoclockstomeasurethetimeinterval:Oneatthepositionofthefirsteventandoneatthepositionofthesecondevent

∆t=∆t0 / 1 − 𝑢$/𝑐2 =𝛾 ∆t0 andthus∆t≥ ∆t0

ThestretchingoutoftimeofthetimeintervaliscalledtimedilationTheequationAbovetellstwothings:Firstly,ifitwerepossibletotravelfasterthanthespeedoflightthen1– u2/c2 wouldbenegativeand 1 − 𝑢$/𝑐2wouldbeanimaginarynumber.Wedon’thaveimaginarytime!Secondly,atimedilationplotof∆t/∆t0asafunctionofrelativevelocity,uwilltendtoinfinityasu approachesc (orinotherwordsasu/capproachesone)Thisisillustratedgraphicallyinthefollowingslide 40

ThestretchingoutoftimeofthetimeintervaliscalledtimedilationTheequationAbovetellsustwothings:Firstly,ifitwerepossibletotravelfasterthanthespeedoflightthen1– u2/c2 wouldbenegativeand 1 − 𝑢$/𝑐2wouldbeanimaginarynumber.Wedon’thaveimaginarytime!Secondly,atimedilationplotof∆t/∆t0asafunctionofrelativevelocity,uwilltendtoinfinityasu approachesc (orinotherwordsasu/capproachesone)Thisisillustratedgraphicallyinthefollowingslide 41

ThestretchingoutoftimeofthetimeintervaliscalledtimedilationTheequationAbovetellstwothings:Firstly,ifitwerepossibletotravelfasterthanthespeedoflightthen1– u2/c2 wouldbenegativeand 1 − 𝑢$/𝑐2wouldbeanimaginarynumber.Wedon’thaveimaginarytime!Secondly,atimedilationplotof∆t/∆t0asafunctionofrelativevelocity, willtendtoinfinityasu approachesc (orinotherwordsasu/capproachesone)Thisisillustratedgraphicallyinthefollowingslide 42

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∆t/∆t 0=𝜸=1/√(1−u2/c

Speedu relativetothespeedoflight(u/c)

Time dilation

Asu approachesc,𝜸 approachesinfinity

∆t/∆t0=𝛾

Timedilationissometimesdescribedbysayingthatmovingclocksrunslow.Thismustbeinterpretedcarefully

Thewholepointofrelativityisthatallinertialframesareequallyvalidsothereisnoabsolutesenseinwhichaclockismovingoratrest

Timedilationissometimesdescribedbysayingthatmovingclocksrunslow.Thismustbeinterpretedcarefully

Thewholepointofrelativityisthatallinertialframesareequallyvalidsothereisnoabsolutesenseinwhichaclockismovingoratrest

Toillustratethispoint,thisimageshowstwofirecrackerexplosionsi.e.twoeventsthatoccuratdifferentpositionsinthegroundframeAssistantsonthegroundneedtwoclockstomeasurethetimeinterval∆tInthetrainreferenceframehoweverasingleclockispresentatbothevents,hencethetimeintervalmeasuredinthetrainreferenceisthepropertime∆t0

Inthissensethemovingclock(theonethatispresentatbothevents)‘runsslower’thanthetheclocksthatarestationarywithrespecttobothevents

Moregenerally,thetimeintervalbetweentwoeventsissmallestinthereferenceframeinwhichthetwoeventsoccuratthesameposition

Inthissensethemovingclock(theonethatispresentatbothevents)‘runsslower’thanthetheclocksthatarestationarywithrespecttobothevents

Moregenerally,thetimeintervalbetweentwoeventsissmallestinthereferenceframeinwhichthetwoeventsoccuratthesameposition

In deriving the time dilation equation we made use of a lightclock which made our analysis clear and easy

The conclusion is about time itself

Any clock, regardless of how it operates (e.g. a grandfatherclock, a wind-up wristwatch, alarm clock or supper accuratequartz clock (as used in GPS satellites)) behave the same!

Any clock, regardless of how it operates (e.g. a grandfatherclock, a wind-up wristwatch, digital watch, alarm clock or asuper accurate quartz clock) behaves in the same way!

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

∆t/∆t 0=𝜸=1/√(1−u2/c

Speedu relativetothespeedoflight(u/c)

Time dilation

Asu approachesc,𝜸 approachesinfinity

∆t/∆t0=𝛾

For𝛥t/𝛥t0 =7,u/c=0.990

For𝛥t/𝛥t0=8,u/c=0.992

Faster than the speed of light?

Spaceisexpandingfasterthanthespeedoflight.Thisisbecausespacetimeitselfisexpandingandisdenyingustheopportunitytoseefurtherthan14billionlightyearsInwater,muonscantravelfasterthenthespeedoflight.ThisisknownasCherenkovlightwhichhasadistinctbluehue.Itcanbeobservedinnuclearreactors.AlthoughthisistruenothingcantravelfasterthanthespeedoflightinavacuumNeutrinosfromsupernovaexplosionsarriveatearthbeforephotonsdo.Thisisbecausethephotonstakeasignificantamountoftimetoescapefromtheexplodingstarwhileneutrinos(withnearzeromass)escapeunhinderedWeareconstantlymovingthroughspacetimeatthespeedoflightinavacuum.Weeitherexperiencespaceortimeoramixtureofboth

Hubbleultradeepfieldimage

Galaxiesasoldas13billionyearsarevisible

Spaceisexpandingfasterthanthespeedoflight.Thisisbecausespacetimeitselfisexpandingandisdenyingustheopportunitytoseefurtherthan14billionlightyearsInwater,muonscantravelfasterthanthespeedoflight.ThisisknownasCherenkovlightwhichhasadistinctbluehue.Itcanbeobservedinnuclearreactors.Althoughthisistrue,nothingcantravelfasterthanthespeedoflightinavacuumNeutrinosfromsupernovaexplosionsarriveatearthbeforephotonsdo.Thisisbecausethephotonstakeasignificantamountoftimetoescapefromtheexplodingstarwhileneutrinos(withnearzeromass)escapeunhinderedWeareconstantlymovingthroughspacetimeatthespeedoflightinavacuum.Weeitherexperiencespaceortimeoramixtureofboth

AnexampleofCherenkovradiationinsideanuclearreactorwheremuons(heavyelectrons)travelfasterthanphotonsoflightinwater

Spaceisexpandingfasterthanthespeedoflight.Thisisbecausespacetimeitselfisexpandingandisdenyingustheopportunitytoseefurtherthan14billionlightyearsInwater,muonscantravelfasterthanthespeedoflight.ThisisknownasCherenkovlightwhichhasadistinctbluehue.Itcanbeobservedinnuclearreactors.Althoughthisistrue,nothingcantravelfasterthanthespeedoflightinavacuumNeutrinosfromsupernovaexplosionsarriveatearthbeforephotonsdo.Thisisbecausethephotonstakeasignificantamountoftimetoescapefromtheexplodingstarwhileneutrinos(withnearzeromass)escapeunhinderedWeareconstantlymovingthroughspacetimeatthespeedoflightinavacuum.Weeitherexperiencespaceortimeoramixtureofboth

Time Dilation in nature

Imageofanexplodingsupernovainadistantgalaxy.Itsbrightnessdecaysatacertainratebutbecauseitismovingawayfromusatasubstantialfractionofthespeedoflight,itdecaysmoreslowlyasseenfromearth.Thesupernovaisa‘movingclockthatrunsslow.’

HighenergycosmicrayprotonsenteringourupperatmosphereinteractwiththenucleiofN2andO2 generatingpionswhichthendecayintomuons(heavyelectrons)whichmoveoffataspeedof0.994c

Thehalflifeofamuonis2.2microseconds.

After660metershalfthemuonswouldhavedecayedbutataspeedof0.994cthehalflifeis20microseconds.

About25%ofthemuonscreatedreachtheground.

Iftherewasnotimedilationonly1/220muonswouldreachtheearth

HighenergycosmicrayprotonsenteringourupperatmosphereinteractwiththenucleiofN2andO2 generatingpionswhichthendecayintomuons(heavyelectrons)whichmoveoffataspeedof0.994c.

Thehalflifeofamuonis2.2microseconds

After660metershalfthemuonswouldhavedecayedbutataspeedof0.994c thehalflifeis20microseconds

About25%ofthemuonscreatedreachtheground

HighenergycosmicrayprotonsenteringourupperatmosphereinteractwiththenucleiofN2andO2 generatingpionswhichthendecayintomuons(heavyelectrons)whichmoveoffataspeedof0.994c

Thehalflifeofamuonis2.2microseconds

After660metershalfthemuonswouldhavedecayedbutataspeedof0.994cthehalflifeis20microseconds

About25%ofthemuonscreatedreachtheground

Youcanbuildyourownmuondetector!

Allyouneedisamobilephonewithacamera+astripofblackinsulationtape

ForaniPhonedownloadtheappfromcosmicrayapp.com.Forotherphonesthereareequivalentapps

Tapeupthecameralensandyouarereadytogo

Justfollowtheapp’sinstructions

Why don’t we experience time dilation in our everyday lives?

Thesunwiththeearthintowistravellingaroundthecentreofthemilkywayataspeedofapproximately220000m/s

Atthisspeed𝜸 fortheearthisonly1.00000027aroundthecentreofourgalaxy

Atsuchalowvalueof𝜸, thesurfaceoftheearthistoallintentsandpurposesaninertialreferenceframe

Ahighvelocityriflebullethasa𝜸 ofonly1.000000000001

Itisnotsurprisingthatwedon’texperiencerelativityIoureverydaylives! 74

Thesunwiththeearthintowistravellingaroundthemilkywayataspeedof217261m/s

Atthisspeed𝜸 fortheearthisonly1.0000003asitmovesaroundthecentreofourgalaxy

Whenbloodhoundfinallyreachesitstargetspeedof1000mph,its𝜸 willonlybe1.0000000000006 78

Time Dilation in Practice

Cathoderaytubeinwhichelectronsreach30%ofthespeedoflight

Length contraction

Relativity of length

Wealsoneedtoderiveaquantitativerelationshipbetweenlengthsindifferentcoordinatesystems(i.e.differentreferenceframes)usinganotherthoughtexperiment

Onceagain,wehaveatraintravellingneartothespeedoflightalongastretchofstraightrailwaytrack

SarahistravellinginthecarriageinreferenceframeS’

Nexttoherontheseatisaruler,alightsourceandamirrorasillustrated

Onceagain,wehaveatraintravellingneartothespeedoflightalongastretchofstraightrailwaytrackSarahistravellinginthecarriageinreferenceframeS’

By using logic like the derivation of time dilation we get

In special relativity a length ℓ0 measured in the frame inwhich the body is at rest is called a proper length

Lengths measured perpendicular to the direction of travel arenot contracted (the velocity in the y and z direction is zero)

ℓ =ℓ0/𝛾 Lengthcontractionformula

Rearranging the previous equation we get

What this tells us is that observers measure any ruler tocontract in length if it moves relative to them

To the traveler her ruler will continue to show the properlength ℓ0 as she is at rest in her reference frame

What the equation also tells us is that as a travelerapproaches the speed of light her ruler will contract to zeroas observed by a stationary observer as shown in the nextslide 92

ℓ/ℓ0 = 1/𝛾

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

𝓵 /𝓵 0=1/𝛄=√(1−u2/c

Speedurelativetothespeedoflightc(u/c)

Length contraction

Asu approachesc,1/𝛄 approacheszero

ℓ/ℓ0 =1/𝛾

Tarringroadsreducesthedistance!AnadvertseeninJohannesburginternationalairport

Ausefulrelationshiptoremember:

∆t0/∆t=l/l0 = 1/𝛾

Tarringroadsreducesthedistance!AnadvertseeninJohannesburginternationalairport

Ausefulrelationshiptoremember:

∆t0/∆t=ℓ/ℓ0 = 1/𝛾

Length contraction of a cube as it would appear at various relative velocitiesMeasuredlengthVisualAppearance

00.5 c 0.99c

MeasuredlengthVisualAppearanceMeasuredlengthVisualAppearance

00.5 c 0.99c

Length Contraction in Practice

Electronsreachaspeedofjust1cm/slessthancinthe3kmbeamlineoftheSLACnationalacceleratorAsmeasuredbytheelectronthebeamlinewhichstretchesfromthetoptowardsthebottomofthephotoisonly15cmlong!

Experimental proof of time dilation and length contraction

RicardFeynmanoncesaidthatnomatterhowbeautifulyourtheory,nomatterhowcleveryouareorwhatyournameis,ifitdisagreeswithexperiment,it’swrong!Let'sseeifthisappliestotimedilationandlengthcontractionAmuon(heavyelectron)hasahalflifeof2.2microsecondswhenatrestScientistshaveacceleratedabeamofmuonscirculatingarounda14mdiameterringto99.94%ofthespeedoflightattheAGSSynchrotroninNewYorkWithouttimedilationtheywouldonlylastfor15lapsoftheringTheylastfor400laps!aps

RicardFeynmanoncesaidthatnomatterhowbeautifulyourtheory,nomatterhowcleveryouareorwhatyournameis,ifitdisagreeswithexperiment,it’swrong!Let'sseeifthisappliestotimedilationandlengthcontractionofamuon(heavyelectron)whichhasahalflifeof2.2microsecondswhenatrestScientistshaveacceleratedabeamofmuonscirculatingarounda14mdiameterringto99.94%ofthespeedoflightattheAGSSynchrotroninNewYorkWithouttimedilationtheywouldonlylastfor15lapsoftheringTheylastfor400laps! 107

RicardFeynmanoncesaidthatnomatterhowbeautifulyourtheory,nomatterhowcleveryouareorwhatyournameis,ifitdisagreeswithexperiment,it’swrong!Let'sseeifthisappliestotimedilationandlengthcontractionAmuon(heavyelectron)hasahalflifeof2.2microsecondswhenatrestScientistshaveacceleratedabeamofmuonscirculatingarounda14mdiameterringto99.94%ofthespeedoflightattheAGSSynchrotroninNewYorkWithouttimedilationtheywouldonlylastfor15lapsoftheringTheylastfor400laps! 108

RicardFeynmanoncesaidthatnomatterhowbeautifulyourtheory,nomatterhowcleveryouareorwhatyournameis,ifitdisagreeswithexperiment,it’swrong!Let'sseeifthisappliestotimedilationandlengthcontractionAmuon(heavyelectron)hasahalflifeof2.2microsecondswhenatrestScientistshaveacceleratedabeamofmuonscirculatingarounda14mdiameterringto99.94%ofthespeedoflightattheAGSSynchrotroninNewYorkWithouttimedilationthemuonswouldonlylastfor15lapsoftheringTheylastfor400laps! 109

RicardFeynmanoncesaidthatnomatterhowbeautifulyourtheory,nomatterhowcleveryouareorwhatyournameis,ifitdisagreeswithexperiment,it’swrong!Let'sseeifthisappliestotimedilationandlengthcontractionAmuon(heavyelectron)hasahalflifeof2.2microsecondswhenatrestScientistshaveacceleratedabeamofmuonscirculatingarounda14mdiameterringto99.94%ofthespeedoflightattheAGSSynchrotroninNewYorkWithouttimedilationthemuonswouldonlylastfor15lapsoftheringInpracticetheylastedfor400laps! 110

Thismeansthattheirlifetimehadbeenincreasedbyafactorof29tojustover60microseconds

Thisresultagreesexactlywiththeory(𝛾 =29)

Ifyoujoinedthemuonyouwouldofcoursecirculatethering400timesaswell

Theproblemhereisthatyourwatchwouldonlymeasure2.2microsecondsbecauseyouwouldbestandingstillinthemuonsreferenceframe

Thisresultagreesexactlywiththeory(𝛾 =29)Ifyoujoinedthemuonyouwouldofcoursecirculatethering400timesaswell

Theproblemhereisthatyourwatchwouldonlymeasure2.2microsecondsbecauseyouwouldbestandingstillinthemuon’sreferenceframe

Youcouldnotcirculatethering400timesin2.2microseconds!

Thecircumferenceoftheringmusthaveshrunkfromtheviewpointofthemuon

Thelengthoftheoftheringasdeterminedbythemuonmustshrinkbythesameamountthatthemuon’slifeincreases(29times)

Bothspaceandtimehavebecomemalleable!

Theeffectsarereal! 115

Thelengthoftheoftheringasdeterminedbythemuonmustshrinkbythesameamountthatthemuon’slifeincreases(29times)

Infact,thelengthoftheoftheringasdeterminedbythemuonshrinksbythesameamountthatthemuon’slifeincreases(29times)

Thelengthoftheoftheringasdeterminedbythemuonshrinksbythesameamountthatthemuon’slifeincreases(29times)

Bothspaceandtimehavebecomemalleable

Thelengthoftheoftheringasdeterminedbythemuonshrinksbythesameamountthatthemuon’slifeincreases(29times)

Relativistic paradoxes

Givenapairoftwinswhereonetravelsintospaceatnearthespeedoflightforsaytenyears,whenthetravellingtwinreturnscantheystillbethesameage?

Givenapairoftwinswhereonetravelsintospaceatnearthespeedoflightforsaytenyears,whenthetravellingtwinreturnscantheystillbethesameage?Atraintravellingnearthespeedoflightapproachesatunnelwhichmeasures80%ofitslengthwhentheyarestationeryrelativetoeachother.Canthetrainfitintothetunnel?

Givenapairoftwinswhereonetravelsintospaceatnearthespeedoflightforsaytenyears,whenthetravellingtwinreturnscantheystillbethesameage?Atraintravellingnearthespeedoflightapproachesatunnelwhichmeasures80%ofitslengthwhentheyarestationeryrelativetoeachother.Canthetrainfitintothetunnel?ToanswerthesequestionsweneedtousetwoimportantrelativisticequationscalledtheLorentztransformsnamedaftertheDutchphysicistHendrikLorentzwhodevelopedthemandfromwhichEinsteinbenefitted!

Givenapairoftwinswhereonetravelsintospaceatnearthespeedoflightforsaytenyears,whenthetravellingtwinreturnscantheystillbethesameage?Atraintravellingnearthespeedoflightapproachesatunnelwhichmeasures80%ofitslengthwhentheyarestationeryrelativetoeachother.Canthetrainfitintothetunnel?ToanswerthesequestionsweneedtousetwoimportantrelativisticequationscalledtheLorentztransformsnamedaftertheDutchphysicistHendrikLorentzwhodevelopedthemTheLorentztransformsarealsorequiredtoresolvesimultaneityissuesandarethemostusefulsetofequationsusedinrelativisticproblemsolving

Lorentz coordinate transformations

Whenaneventoccursatpoint(x,y,z)attime tasobservedinaframeofreferenceS,whatarethecoordinates(x’,y’,z’)andtimet’oftheeventasobservedinasecondframeS’movingrelativetoSwithavelocityofu inthe+xdirection?

Withoutperformingadetailedderivation,thetransformationofaneventwithspacetimecoordinatesx,y,zand tinframeSandx’,y’,z’andt’inframeS’isdonebyviathefollowingLorentzcoordinatetransformations

x’=𝛾 (x-ut)Lorentzcoordinatetransformations

t’=𝛾 (t-ux/c2)

Whereu isvelocityofS’relativetoS inthepositivex– x’axisc isthespeedoflight and𝛾 istheLorentzfactorrelatingframesS andS’y’=yand z’=zsincetheyareperpendiculartox

Withoutperformingadetailedderivation,thetransformationofaneventwithspacetimecoordinatesx,y,zand tinframeSandx’,y’,z’andt’inframeS’isdonebyviathefollowingLorentzcoordinatetransformations

t’=𝛾 (t-ux/c2)

y’=yand z’=zsincetheyareperpendiculartox

Withoutperformingadetailedderivation,thetransformationofaneventwithspacetimecoordinatesx,y,zand tinframeSandx’,y’,z’and t’inframeS’isdonebyviathefollowingLorentzcoordinatetransformations

t’=𝛾 (t-ux/c2)

y’=yand z’=zsincetheyareperpendiculartox

Spaceandtimehaveclearlybecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreference

Timeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallxandttogetherthespacetimecoordinatesofanevent

Spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreference

Timeandthethreedimensionsofspacecollectivelyformafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent

UsingtheLorentzcoordinatetransformationswecanderiveasetofLorentzvelocitytransformations

Theresult(withoutderivation)isshowninthenextslide131

Aswesawyesterday,spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreferenceTimeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent

In the extreme case where vx = cwe get

vx’ = (c-u)/(1-uc/c2) = c(1-u/c)/(1-u/c) = c

This means that anything moving at c measured in S isalso travelling at c when measured in S’ despite therelative motion of the two frames

vx’=(vx – u)/(1- uvx/c2)Lorentzonedimensionalvelocitytransformation

vx’ = (c-u)/(1-uc/c2) = c(1-u/c)/(1-u/c) = c

vx’=(vx – u)/(1- uvx/c2)Lorentzonedimensionalvelocitytransformation

vx’ = (c-u)/(1-uc/c2) = c(1-u/c)/(1-u/c) = c

vx’=(vx – u)/(1- uvx/c2)Lorentzvelocitytransformation

TheLorentzvelocitytransformationshowsthatabodywithaspeedlessthanc inoneframeofreferencealwayshasaspeedlessthanc ineveryotherframeofreference

Thisisonereasonforconcludingthatnomaterialbodymaytravelwithaspeedgreaterthanorequaltothespeedoflightinavacuum,relativetoanyinertialreferenceframe

TheLorentzvelocitytransformationshowsthatabodywithaspeedlessthanc inoneframeofreferencealwayshasaspeedlessthanc ineveryotherframeofreference

Thisisonereasonforconcludingthatnomaterialbodymaytravelwithaspeedgreaterthanorequaltothespeedoflightinavacuum,relativetoanyinertialreferenceframe

Let'sconsideranexampleofthevelocitylimitwhichanyobservercanreachrelativetosomeotherobserver

IfwehadasetoffivespaceshipsstackedlikeRussiandollswhereeachshipcouldlaunchtheremainingshipsatavelocityequaltotherelativevelocityofthelaunchingshipasobservedfromearthwhatrelativevelocitiescouldthevariousshipsachieverelativetotheearthobserver?

Thefollowingslideshowsthevelocityprofilesofthefivespaceshipsrelativetoanearthobserver

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Rocketspeedsrelativetospeedof

lightcasasobservedonearth

Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Nomatterhowmanysuccessiverocketsarelaunchedtheirvelocitywillneverexceedc!

Test your understanding of time dilationPeter,whoisstandingontheground,startshisstopwatchthemomentthatSarahfliesoverheadinaspaceshipataspeedof0.6cAtthesameinstantSarahstartsherstopwatchAsmeasuredinPeter’sframeofreference,whatisthereadingonSarah’sstopwatchattheinstantpeter’sstopwatchreads10s?a)10s,b)lessthan10sorc)morethan10s?AsmeasuredinSarah’sframeofreference,whatisthereadingonPeter’sstopwatchattheinstantthatSarah’sstopwatchreads10s?a)10s,b)lessthan10sorc)morethan10s?Whosestopwatchisreadingpropertimeintheabovetwoexamples?

Test your understanding of length contraction

Aminiaturespaceshipfliespastyouhorizontallyat0.99cAtacertaininstantyouobservethatthatthenoseandtailofthespaceshipalignexactlywiththetwoendsofameterstickthatyouholdinyourhandRankthefollowingdistancesinorderfromlongesttoshortest:a)theproperlengthofthemeterstick;b)theproperlengthofthespaceship;c)thelengthofthespaceshipmeasuredinyourreferenceframe;d)thelengthofthemeterstickmeasuredinthespaceship’sframeofreference?

Special Relativity · 2020-01-09 · Special Relativity Presentation to UCT Summer School Jan 2020...

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