special relativity · 2020-01-09 · special relativity presentation to uct summer school jan 2020...

SpecialRelativity

PresentationtoUCTSummerSchoolJan2020(Part2of3)

ByRobLouw

[email protected] 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?

2

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)

4

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Inordertogainabetterunderstandingofwhatishappening,weclearlyneedtoderiveaquantitativerelationshipthatallowsustocomparetimeintervalsindifferentframesofreference

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

6

Inordertogainabetterunderstandingofwhatishappening,weclearlyneedtoderiveaquantitativerelationshipthatallowsustocomparetimeintervalsindifferentframesofreference

Thiswillbedoneusinganotherthoughtexperiment

Time Dilation Thought Experiment

Theobjectiveoftheexperimentistodemonstrate:

Thatobserversmeasureanyclocktorunslowifitmovesrelativetothemandastherelativespeedapproachesthespeedoflight,themovingclock’schangeintimetendstozero

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

9

Imaginewehaveatrainmovingclosetothespeedoflightalongastraightstretchofrailwaytrack

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

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Peter

SarahSarah

Referenceframe S’

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SarahMirror

Lightsource

d

S’

O’(Event1occurshere)

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SarahMirror

Lightsource

d

S’

O’(Event2alsooccurshere)

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SarahMirror

Lightsource

d

S’

Sarahmeasuresaroundtriptimeof∆t0 forthelightbeam

O’(Events1and2occurhere)

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

SarahMirror

Lightsource

d

O’(Events1and2occurhere)

S’

Sarahmeasuresaroundtriptimeof∆t0 forthelightbeam

15

Sarah

Sourcemovesfromheretohere

Event1occurshere

Peterwhoisstationaryobservesthesamelightpulsefollowingadiagonalpath

16

Sarah


Event1occurshere

Event2occurshere

17

Petermeasurestheround-triptimetobe∆t

Sarah


Event1occurshere

Event2occurshere

18


Sarah


(Distancetravelled)

Event1occurshere

Event2occurshere

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Sarah


(Distancetravelled)

Theround-tripdistanceforthelightbeaminreferenceframeS is2ℓ

Event1occurshere

Event2occurshere

Pythagorean theorem

20

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

a

b

c

Thusc2 =a2 +b2

∴ c= 𝑎$ + 𝑏$

d

21

Peter

Sarah

u∆t/2

d

22

Peter

Sarah

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)$

23


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

24


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)$

25

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

26


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)

27


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

28

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




Thequantity1/ 1 − 𝑢$/𝑐2appearssoofteninrelativitythatithasitsownsymbol 𝛾 andisreferredtoasLorentzgamma

𝛾 =1/ 1 − 𝑢$/𝑐2Lorentzgammafactor32

Notethat𝛾 isalways≥1and1/𝛾 isalways≤1!

If𝛾 appearsinthenumeratorofanyrelativisticequation,itwilltendtowardsinfinityasvelocityapproachesc

Converselyif𝛾 appearsinthedenominatorofanyrelativisticequation,itwilltendtowardszeroasvelocityapproachesc

33


If𝛾 appearsinthenumeratorofanyrelativisticequation,itwilltendtowardsinfinityasvelocity,u approachesc

Converselyif𝛾 appearsinthedenominatorofanyrelativisticequation,itwilltendtowardszeroasvelocityapproachesc

34


If𝛾 appearsinthenumeratorofanyrelativisticequation,itwilltendtowardsinfinityasvelocity,approachesc

Converselyif𝛾 appearsinthedenominatorofanyrelativisticequation,itwilltendtowardszeroasvelocity,u approachesc

35

∆t0 iscalledthepropertimeandisequaltothetimeintervalbetweentwoeventsthatoccuratthesameposition

Onlyoneinertialframe(S’)measuresthepropertimeanditdoessowithasingleclockthatispresentatbothevents

Aninertialreferenceframemovingwithvelocityurelativetothepropertimeframemustusetwoclockstomeasurethetimeinterval:Oneatthepositionofthefirsteventandoneatthepositionofthesecondevent

Byrearrangingourearlierequations,thetimeintervalintheframewheretwoclocksarerequiredisasfollows

36





37



Aninertialreferenceframemovingwithvelocityu relativetothepropertimeframemustusetwoclockstomeasurethetimeinterval:Oneatthepositionofthefirsteventandoneatthepositionofthesecondevent


38





39

∆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

1

2

3

4

5

6

7

8

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

2 )

Speedu relativetothespeedoflight(u/c)

Time dilation

Asu approachesc,𝜸 approachesinfinity

∆t/∆t0=𝛾

43

Timedilationissometimesdescribedbysayingthatmovingclocksrunslow.Thismustbeinterpretedcarefully

Thewholepointofrelativityisthatallinertialframesareequallyvalidsothereisnoabsolutesenseinwhichaclockismovingoratrest

44

Timedilationissometimesdescribedbysayingthatmovingclocksrunslow.Thismustbeinterpretedcarefully

Thewholepointofrelativityisthatallinertialframesareequallyvalidsothereisnoabsolutesenseinwhichaclockismovingoratrest

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46

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

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48


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Inthissensethemovingclock(theonethatispresentatbothevents)‘runsslower’thanthetheclocksthatarestationarywithrespecttobothevents

Moregenerally,thetimeintervalbetweentwoeventsissmallestinthereferenceframeinwhichthetwoeventsoccuratthesameposition

50

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!

51



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!

52



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!

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0

1

2

3

4

5

6

7

8

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

2 )

Speedu relativetothespeedoflight(u/c)

Time dilation

Asu approachesc,𝜸 approachesinfinity

∆t/∆t0=𝛾

56

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

59

Hubbleultradeepfieldimage

Galaxiesasoldas13billionyearsarevisible

60

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

61

62

AnexampleofCherenkovradiationinsideanuclearreactorwheremuons(heavyelectrons)travelfasterthanphotonsoflightinwater


63


64

Time Dilation in nature

65

66

Imageofanexplodingsupernovainadistantgalaxy.Itsbrightnessdecaysatacertainratebutbecauseitismovingawayfromusatasubstantialfractionofthespeedoflight,itdecaysmoreslowlyasseenfromearth.Thesupernovaisa‘movingclockthatrunsslow.’

HighenergycosmicrayprotonsenteringourupperatmosphereinteractwiththenucleiofN2andO2 generatingpionswhichthendecayintomuons(heavyelectrons)whichmoveoffataspeedof0.994c

Thehalflifeofamuonis2.2microseconds.

After660metershalfthemuonswouldhavedecayedbutataspeedof0.994cthehalflifeis20microseconds.

About25%ofthemuonscreatedreachtheground.

Iftherewasnotimedilationonly1/220muonswouldreachtheearth

67

HighenergycosmicrayprotonsenteringourupperatmosphereinteractwiththenucleiofN2andO2 generatingpionswhichthendecayintomuons(heavyelectrons)whichmoveoffataspeedof0.994c.

Thehalflifeofamuonis2.2microseconds




68



After660metershalfthemuonswouldhavedecayedbutataspeedof0.994c thehalflifeis20microseconds



69




About25%ofthemuonscreatedreachtheground


70

HighenergycosmicrayprotonsenteringourupperatmosphereinteractwiththenucleiofN2andO2 generatingpionswhichthendecayintomuons(heavyelectrons)whichmoveoffataspeedof0.994c

Thehalflifeofamuonis2.2microseconds

After660metershalfthemuonswouldhavedecayedbutataspeedof0.994cthehalflifeis20microseconds

About25%ofthemuonscreatedreachtheground


71

Youcanbuildyourownmuondetector!

Allyouneedisamobilephonewithacamera+astripofblackinsulationtape

ForaniPhonedownloadtheappfromcosmicrayapp.com.Forotherphonesthereareequivalentapps

Tapeupthecameralensandyouarereadytogo

Justfollowtheapp’sinstructions

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

73

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

79

80

Cathoderaytubeinwhichelectronsreach30%ofthespeedoflight

Length contraction

Relativity of length

83

Wealsoneedtoderiveaquantitativerelationshipbetweenlengthsindifferentcoordinatesystems(i.e.differentreferenceframes)usinganotherthoughtexperiment

Onceagain,wehaveatraintravellingneartothespeedoflightalongastretchofstraightrailwaytrack

SarahistravellinginthecarriageinreferenceframeS’

Nexttoherontheseatisaruler,alightsourceandamirrorasillustrated


84


Onceagain,wehaveatraintravellingneartothespeedoflightalongastretchofstraightrailwaytrackSarahistravellinginthecarriageinreferenceframeS’



85






86





87

Sarah

88

Sarah

Peter

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)

89

ℓ =ℓ0/𝛾 Lengthcontractionformula




90





91


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





ℓ/ℓ0 = 1/𝛾

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

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

2 )

Speedurelativetothespeedoflightc(u/c)

Length contraction

Asu approachesc,1/𝛄 approacheszero

ℓ/ℓ0 =1/𝛾

96

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

0.0c

00.5 c 0.99c

MeasuredlengthVisualAppearanceMeasuredlengthVisualAppearance

Length Contraction in Practice

102

103

Electronsreachaspeedofjust1cm/slessthancinthe3kmbeamlineoftheSLACnationalacceleratorAsmeasuredbytheelectronthebeamlinewhichstretchesfromthetoptowardsthebottomofthephotoisonly15cmlong!

104

Electronsreachaspeedofjust1cm/slessthancinthe3kmbeamlineoftheSLACnationalacceleratorAsmeasuredbytheelectronthebeamlinewhichstretchesfromthetoptowardsthebottomofthephotoisonly15cmlong!

Experimental proof of time dilation and length contraction

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

106

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

111


Thisresultagreesexactlywiththeory(𝛾 =29)Ifyoujoinedthemuonyouwouldofcoursecirculatethering400timesaswell


112





113




Theproblemhereisthatyourwatchwouldonlymeasure2.2microsecondsbecauseyouwouldbestandingstillinthemuon’sreferenceframe

114

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?

121

Givenapairoftwinswhereonetravelsintospaceatnearthespeedoflightforsaytenyears,whenthetravellingtwinreturnscantheystillbethesameage?Atraintravellingnearthespeedoflightapproachesatunnelwhichmeasures80%ofitslengthwhentheyarestationeryrelativetoeachother.Canthetrainfitintothetunnel?

122

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

123

Givenapairoftwinswhereonetravelsintospaceatnearthespeedoflightforsaytenyears,whenthetravellingtwinreturnscantheystillbethesameage?Atraintravellingnearthespeedoflightapproachesatunnelwhichmeasures80%ofitslengthwhentheyarestationeryrelativetoeachother.Canthetrainfitintothetunnel?ToanswerthesequestionsweneedtousetwoimportantrelativisticequationscalledtheLorentztransformsnamedaftertheDutchphysicistHendrikLorentzwhodevelopedthemTheLorentztransformsarealsorequiredtoresolvesimultaneityissuesandarethemostusefulsetofequationsusedinrelativisticproblemsolving

124

Lorentz coordinate transformations

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

126

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

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


t’=𝛾 (t-ux/c2)

y’=yand z’=zsincetheyareperpendiculartox

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


t’=𝛾 (t-ux/c2)

y’=yand z’=zsincetheyareperpendiculartox

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Spaceandtimehaveclearlybecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreference

Timeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallxandttogetherthespacetimecoordinatesofanevent

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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

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


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


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


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


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

TheLorentzvelocitytransformationshowsthatabodywithaspeedlessthanc inoneframeofreferencealwayshasaspeedlessthanc ineveryotherframeofreference

Thisisonereasonforconcludingthatnomaterialbodymaytravelwithaspeedgreaterthanorequaltothespeedoflightinavacuum,relativetoanyinertialreferenceframe

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TheLorentzvelocitytransformationshowsthatabodywithaspeedlessthanc inoneframeofreferencealwayshasaspeedlessthanc ineveryotherframeofreference

Thisisonereasonforconcludingthatnomaterialbodymaytravelwithaspeedgreaterthanorequaltothespeedoflightinavacuum,relativetoanyinertialreferenceframe

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Let'sconsideranexampleofthevelocitylimitwhichanyobservercanreachrelativetosomeotherobserver

IfwehadasetoffivespaceshipsstackedlikeRussiandollswhereeachshipcouldlaunchtheremainingshipsatavelocityequaltotherelativevelocityofthelaunchingshipasobservedfromearthwhatrelativevelocitiescouldthevariousshipsachieverelativetotheearthobserver?

Thefollowingslideshowsthevelocityprofilesofthefivespaceshipsrelativetoanearthobserver

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Rocketspeedsrelativetospeedof

lightcasasobservedonearth

Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

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Nomatterhowmanysuccessiverocketsarelaunchedtheirvelocitywillneverexceedc!

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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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