special relativity - summerschool.uct.ac.za · special relativity presentation to uct summer school...

SpecialRelativity

PresentationtoUCTSummerSchoolJan2020(Part3of3)

ByRobLouw

roblouw47@gmail.com 1

Suggested You Tube viewing of dr Don Lincoln of FermilabHowtotravelfasterthanlight(20)Relativisticvelocitywhen1+1=1(21)Lengthcontraction:therealexplanation(22)Twinparadox:therealexplanation(22)Relativity:howpeoplegettimedilationwrong(25)Whatisrelativityallabout?(26)Whatyouneverlearnedaboutmass(27)WhyE=mc2 iswrong(28)Relativity’skeyconcept:Lorentzgamma(29)Whycan’tyougofasterthanlight?(31)Isrelativisticmassreal?(32)Einstein’sclocks(63)HowdoesCerenkovradiationwork?(15)Cosmicinflation(73)

Gravitationallensing(66)Howfaristheedgeoftheuniverse?(2)

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 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?FirstSarah’sandthenPeter’s

Thestatementthatmovingclocksrunslowreferstoanyclockthatismovingrelativetoanobserver.SarahandherstopwatcharemovingrelativetoPeter,soPetermeasure’sstopwatchtoberunningslowandtohavetickedofffewersecondsthanhisownstopwatch.PeterandhisstopwatcharemovingrelativetoSarah,soshelikewisemeasuresPeter’sstopwatchrunningslow.Thisisconsistentwiththeprincipleofrelativitywhichstatesthatthelawsofphysicsarethesameinallinertialreferenceframes.

Test your understanding of length contraction

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

Youmeasureboththerestlengthofthestationerymeterstickandthecontractedlengthofthemovingspaceshiptobeonemeter.Therestlengthofthespaceshipisgreaterthanthecontractedlengththatyoumeasureandsomustbegreaterthanonemeter.Aminiatureobserveronboardthespaceshipwouldmeasureacontractedlengthforthemetersticktobelessthanonemeter.Notethatinyourframeofreferencethenoseandtailofthespaceshipcansimultaneouslyalignwiththetwoendsofthemeterstick,sinceinyourframeofreferencetheyhavethesamelengthof1meter.Inthespaceship’sframethesetwoalignmentscannothappensimultaneouslybecausethemeterstickisshorterthanthespaceship.Thisshouldn’tbeasurprise,twoeventsthataresimultaneoustooneobservermaynotbesimultaneoustoasecondobservermovingrelativetothefirstone.

What we have learnt so farAll speeds are relative; There is no such thing as absolute speedWhat a reference frame is; (3 spatial coordinates + a time coordinate)What an inertial reference frame is; (a frame of reference which is either stationary or moving at a fixed velocity relative to another inertial reference frame). All the laws of physics are invariant between all IRFsWhat an event is; An event has and x, y and z location and a timeMeasurements are done with clocks and meter sticks which are present at the event. All clocks are synchronized in their respective reference framesEvents which are simultaneous in one IRF may not be simultaneous when observed from a different IRFTime dilation: Observers observe clocks that are moving relative to them are running slowLength contraction: Observers observe lengths that are moving relative are to be contractedProper time: The time on a watch which is present at both of two eventsProper length: A fixed length which is present at both of two events

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https://www.forbes.com/sites/startswithabang/2019/03/01/relativity-wasnt-einsteins-miracle-it-was-waiting-in-plain-sight-for-71-years/#16235c7f644c

‘Relativitywasn’tEinstein’smiracle:Itwaswaitinginplainsightfor71years’

Seealso:UniversityPhysicsbyHDYoung&RAFreedman.14thglobaledition.Section37.1–Invarianceofphysicallawspages1242/1243

Lorentz coordinate transformations

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

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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

x’=𝛾 (x-ut)Lorentzcoordinatetransformations

t’=𝛾 (t-ux/c2)

y’=yand z’=zsincetheyareperpendiculartox

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

x’=𝛾 (x-ut)Lorentzcoordinatetransformations

t’=𝛾 (t-ux/c2)

y’=yand z’=zsincetheyareperpendiculartox

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Spaceandtimehaveclearlybecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreference

Timeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallxandttogetherthespacetimecoordinatesofanevent

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Aswesawyesterday,spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreferenceTimeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent

UsingtheLorentzcoordinatetransformationswecanderiveasetofLorentzvelocitytransformations

Theresult(withoutderivation)isshowninthenextslide19

Aswesawyesterday,spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreferenceTimeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent

UsingtheLorentzcoordinatetransformationswecanderiveasetofLorentzvelocitytransformations

Theresult(withoutderivation)isshowninthenextslide20

Aswesawyesterday,spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreferenceTimeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent

UsingtheLorentzcoordinatetransformationswecanderiveasetofLorentzvelocitytransformations

Theresult(withoutderivation)isshowninthenextslide21

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

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

Let'sconsideranexampleofthevelocitylimitwhichanyobservercanreachrelativetosomeotherobserver

IfwehadasetoffivespaceshipsstackedlikeRussiandollswhereeachshipcouldlaunchtheremainingshipsatavelocityequaltotherelativevelocityofthelaunchingshipasobservedfromearthwhatrelativevelocitiescouldthevariousshipsachieverelativetotheearthobserver?

Thefollowingslideshowsthevelocityprofilesofthefivespaceshipsrelativetoanearthobserver

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Let'sconsideranexampleofthevelocitylimitwhichanyobservercanreachrelativetosomeotherobserver

IfwehadasetoffivespaceshipsstackedlikeRussiandollswhereeachshipcouldlaunchtheremainingshipsatavelocityequaltotherelativevelocityofthelaunchingshipasobservedfromearthwhatrelativevelocitiescouldthevariousshipsachieverelativetotheearthobserver?

Thefollowingslideshowsthevelocityprofilesofthefivespaceshipsrelativetoanearthobserver

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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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Rocketspeedsrelativetospeedof

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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

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Rocketspeedsrelativetospeedof

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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

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Rocketspeedsrelativetospeedof

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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

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Rocketspeedsrelativetospeedof

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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

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Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v

Relative rocket ship speeds

Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5

Nomatterhowmanysuccessiverocketsarelaunchedtheirvelocitywillneverexceedc!

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Relativistic kinematics and the Doppler effect for electromagnetic waves

Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain

Asourceoflight ismovingtowardsStanleywithconstantspeeduwhoisinastationeryinertialreferenceframeS

Thesourceemitslightemitslightwavesoffrequencyf0 asMeasuredinitsrestframe

Stanleyreceiveslightwavesoffrequencyfasshownbelow

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Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain

Asourceoflight ismovingtowardsStanley,withconstantspeedu,whoisinastationeryinertialreferenceframeS

Thesourceemitslightemitslightwavesoffrequencyf0 asMeasuredinitsrestframe

Stanleyreceiveslightwavesoffrequencyfasshownbelow

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Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain

Asourceoflight ismovingtowardsStanley,withconstantspeedu,whoisinastationeryinertialreferenceframeS

Thesourceemitslightwavesoffrequencyf0 asmeasuredinitsrestframe

Stanleyreceiveslightwavesoffrequencyfasshownbelow

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Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain

Asourceoflight ismovingtowardsStanleywithconstantspeeduwhoisinastationeryinertialreferenceframeS

Thesourceemitslightwavesoffrequencyf0 asmeasuredinitsrestframe

Stanleyreceiveslightwavesoffrequencyfasshowninthenextslide

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Withanelectromagneticsourceapproaching anobserver,therelativisticblueshiftDopplerformulacanbederivedusingtheappropriateLorentztransformsandis

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Thedopplerblueshiftequationindicatesthatfincreasesi.e.thewavelengthgetsshorter(bluer)asu approachesthespeedoflight c

f= (𝐜 + 𝐮)/(𝐜 − 𝐮) f0 Dopplerformula(blueshift)

Withanelectromagneticsourceapproaching anobservertherelativisticblueshiftDopplerformulacanbederivedusingtheappropriateLorentztransformsis

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Thedopplerblueshiftequationindicatesthatfincreasesi.e.thewavelengthgetsshorter(bluer)asu approachesthespeedoflight c

f= (𝐜 + 𝐮)/(𝐜 − 𝐮) f0 Dopplerformula(blueshift)

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Withlight,unlikesound,thereisnodistinctionbetweenmotionofsourceandmotionofobserver,onlytherelativevelocityofthetwoissignificant

ThefollowingslideillustratestheDopplerblueshifteffect

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f/f 0=(𝒄+𝒖)/(𝒄−𝒖)

Doppler effect - source approaching observer

Asthesourcevelocity- uapproachesthespeedoflight,f/f0approachesinfinity(BLUESHIFT)

f/f0

Withelectromagneticwavesmovingaway fromanobserver,therelativisticredshiftDopplerformulacanbederivedusingtheappropriateLorentztransforms

Thedopplerredshiftequationindicatesthatfdecreasei.e.thewavelengthgetslonger(redder)asu approachesthespeedoflightc

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f= (𝐜 − 𝐮)/(𝐜 + 𝐮) f0 Dopplerformula(redshift)

Withelectromagneticwavesmovingaway fromanobserver,therelativisticredshiftDopplerformulacanbederivedusingtheappropriateLorentztransforms

Thedopplerredshiftequationindicatesthatfdecreasei.e.thewavelengthgetslonger(redder)asu approachesthespeedoflightc

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f= (𝐜 − 𝐮)/(𝐜 + 𝐮) f0 Dopplerformula(redshift)

NotethatinderivingtheDopplerequations,𝛾 hascancelledout

TheDopplerredshifteffectisshowninthenextfewslides

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f/f 0=(𝒄+𝒖)/(𝒄−𝒖)

Asthesourcevelocityuapproachesthespeedoflight,f/f0approacheszero(redSHIFT)

Doppler effect- source moving away from observer

Speedvrelativetothespeedoflightc(v/c)

f/f0

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Hubblephotographofafastmoving,DopplerblueshiftedjetemanatingfromablackholeatthecentreofGalaxyM87

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QueenMary2’sradarantennae

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Radarequipmentinstallationatanairport

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Relativistic particle physics

Relativistic particle momentum p

Newton’s laws of of motion have the same form in all inertialframes of referenceUsing Lorentz transformations to change from one inertialframe to another, the laws should be invariantTheprincipleoftheconservationofmomentumstatesthatwhentwobodiesinteract,thetotalmomentumisconstantprovidingthatthereisnonetexternalforceactingonthebodiesinaninertialreferenceframeConservationofmomentummustthereforebevalidinallinertialframedofreference

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Newton’s laws of of motion have the same form in all inertialframes of referenceUsing Lorentz transformations to change from one inertialframe to another, the laws should be invariantTheprincipleoftheconservationofmomentumstatesthatwhentwobodiesinteract,thetotalmomentumisconstantprovidingthatthereisnonetexternalforceactingonthebodiesinaninertialreferenceframeConservationofmomentummustthereforebevalidinallinertialframedofreference

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Newton’s laws of of motion have the same form in all inertialframes of referenceUsing Lorentz transformations to change from one inertialframe to another, the laws should be invariantTheprincipleoftheconservationofmomentumstatesthatwhentwobodiesinteract,thetotalmomentumisconstantprovidingthatthereisnonetexternalforceactingonthebodiesinaninertialreferenceframeConservationofmomentummustthereforebevalidinallinertialframedofreference

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Newton’s laws of of motion have the same form in all inertialframes of referenceUsing Lorentz transformations to change from one inertialframe to another, the laws should be invariantTheprincipleoftheconservationofmomentumstatesthatwhentwobodiesinteract,thetotalmomentumisconstantprovidingthatthereisnonetexternalforceactingonthebodiesinaninertialreferenceframeConservationofmomentummustthereforebevalidinallinertialframesofreference

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Thisposesuswithaproblem:SupposewelookatacollisioninaninertialcoordinatesystemS andwefindthatmomentumisconservedWhenweusetheLorentztransformationtoobtainvelocitiesinasecondinertialsystemS’wefindthatusingtheNewtoniandefinitionofmomentum(p=mv),momentumisnotconservedinthesecondsystemTosolvethisproblemweneedamoregeneraliseddefinitionofmomentum

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Thisposesuswithaproblem:SupposewelookatacollisioninaninertialcoordinatesystemSandwefindthatmomentumisconserved

WhenweusetheLorentztransformationtoobtainvelocitiesinasecondinertialsystemS’wefindthatusingtheNewtoniandefinitionofmomentum(p=mv),momentumisnotconservedinthesecondsystem

Tosolvethisproblemweneedamoregeneraliseddefinitionofmomentum

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Thisposesuswithaproblem:SupposewelookatacollisioninaninertialcoordinatesystemSandwefindthatmomentumisconserved

WhenweusetheLorentztransformationtoobtainvelocitiesinasecondinertialsystemS’wefindthatusingtheNewtoniandefinitionofmomentum(p=mv),momentumisnotconservedinthesecondsystemTosolvethisproblemweneedamoregeneraliseddefinitionofmomentum

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Theequationwillnotbederivedfromfirstprinciples,butitwillsimplybestatedbelowSupposewehaveamaterialparticlewitharestmassofm(m>0), whensuchaparticlehasavelocityv,thenitsrelativisticmomentumpisp =mv/ 1 − (𝑣/𝑐). =𝛾mvRelativisticmomentum

p=momentumm=particle(rest)massv=particlevelocityc=speedoflight𝛾 =Lorentzfactorforaparticle

Relativisticmomentumplaysakeyroleinunderstandingthekinematicsofparticlephysics 63

Theequationwillnotbederivedfromfirstprinciples,butitwillsimplybestatedbelowSupposewehaveamaterialparticlewitharestmassofm,whensuchaparticlehasavelocityv,thenitsrelativisticmomentum pis

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p =mv/ 1 − (𝑣/𝑐). =𝛾mvRelativisticmomentum

Relativisticmomentumplaysakeyroleinunderstandingthekinematicsofparticlephysics

Particlevelocitieswillbedenotedwithv fortherestofthispresentation

Wewillnolongerbemakinguseofu,therelativevelocityofreferenceframesaswewillbethestationaryobserveronearth

RelativisticandNewtonianmomentumasafunctionofrelativespeedv/careillustratedgraphicallyinthenextfewslides

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Relativisticmomentumplaysakeyroleinunderstandingthekinematicsofparticlephysics

Particlevelocitieswillbedenotedwithv fortherestofthispresentation

Wewillnolongerbemakinguseofu,therelativevelocityofreferenceframesaswewillbethestationaryobserveronearth

RelativisticandNewtonianmomentumasafunctionofrelativespeedv/careillustratedgraphicallyinthenextfewslides

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Relativisticmomentumplaysakeyroleinunderstandingthekinematicsofparticlephysics

Particlevelocitieswillbedenotedwithv fortherestofthispresentation

Wewillnolongerbemakinguseofu,therelativevelocityofreferenceframesaswewillbethestationaryobserveronearth

RelativisticandNewtonianmomentumasafunctionofrelativespeedv/careillustratedgraphicallyinthenextfewslides

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Particle momentum

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Speedvrelativetothespeedoflightc(v/c)

Asv approachesc,relativisticmomentumapproachesinfinity

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Speedvrelativetothespeedoflightc(v/c)

Newtonianmechanicsincorrectly predictsthatmomentumonlyreachesinfinityifvbecomesinfinite

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Force F and acceleration a

ThegeneralformofNewton’ssecondlawisF=dp/dt=ma

Experimentsshowthisresultisstillvalidinrelativisticmechanicsprovidedweuserelativisticmomentum.ThustherelativisticallycorrectversionofNewton’ssecondlaw is

F=ma/{ 𝟏 − (𝒗/𝒄)𝟐}3=𝛾 3ma‘Relativistic’ forceF istheforcem istheparticlemassa istheparticleaccelerationv istheparticlevelocityc isthespeedoflightinavacuum𝛾 is LorentzgammaF,aandv areactinginthesameline 71

ThegeneralformofNewton’ssecondlawisF=dp/dt=ma

Experimentsshowthisresultisstillvalidinrelativisticmechanicsprovidedweuserelativisticmomentum.ThustherelativisticallycorrectversionofNewton’ssecondlaw is

F=ma/{ 𝟏 − (𝒗/𝒄)𝟐}3=𝛾 3ma Forceformula

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Rearrangingthepreviousequationwecanestablishwhathappenstotheaccelerationa ofaparticleofrestmassmwhichissubjectedtoaconstantforceFa=(F/m{ 𝟏 − (𝒗/𝒄)𝟐 }3=F/m𝛾 𝟑Relativisticaccelerationa =accelerationF=forcem=particlerestmassv=particlevelocityc=speedoflightinavacuum𝛾 =Lorentzgamma

InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedv 73

Rearrangingthepreviousequationwecanestablishwhathappenstotheaccelerationa ofaparticleofrestmassmwhichissubjectedtoaconstantforcea=(F/m 𝟏 − (𝒗/𝒄)𝟐 3=F/m𝛾 𝟑 Accelerationformula

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InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedvInrelativisticmechanics,whenaparticleofrestmassmissubjectedtoaconstantforceF,itsaccelerationdecreasestozeroasitsvelocitytendstowardthespeedoflightInfactitdoesnotmatterhowbigtheforceornonzeromassis,accelerationwillalwaysdecreasetozeroastheparticlespeedincreasestowardsthespeedoflightTheeffectofincreasedspeedontheaccelerationofaparticleofrestmassmwhensubjectedtoaconstantforceFisillustratedinthenextfewslides

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InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedvInrelativisticmechanics,whenaparticleofrestmassmissubjectedtoaconstantforceF,itsaccelerationdecreasestozeroasitsvelocitytendstowardthespeedoflightInfactitdoesnotmatterhowbigtheforceornonzeromassis,accelerationwillalwaysdecreasetozeroastheparticlespeedincreasestowardsthespeedoflightTheeffectofincreasedspeedontheaccelerationofaparticleofrestmassmwhensubjectedtoaconstantforceFisillustratedinthenextfewslides

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InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedvInrelativisticmechanics,whenaparticleofrestmassmissubjectedtoaconstantforceF,itsaccelerationdecreasestozeroasitsvelocitytendstowardthespeedoflightInfactitdoesnotmatterhowbigtheforceornonzeromassis,accelerationwillalwaysdecreasetozeroastheparticlespeedincreasestowardsthespeedoflightTheeffectofincreasedspeedontheaccelerationofaparticleofrestmassmwhensubjectedtoaconstantforceFisillustratedinthenextfewslides

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InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedvInrelativisticmechanics,whenaparticleofrestmassmissubjectedtoaconstantforceF,itsaccelerationdecreasestozeroasitsvelocitytendstowardthespeedoflightInfactitdoesnotmatterhowbigtheforceornonzeromassis,accelerationwillalwaysdecreasetozeroastheparticlespeedincreasestowardsthespeedoflightTherelativisticeffectofincreasedspeedontheaccelerationofaparticleofrestmassmwhensubjectedtoaconstantforceFisillustratedinthenextfewslides

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0.9

1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Speedvrelativetothespeedoflightc(v/c)

a=F/m𝜸3Particle acceleration a

Accelerationofaparticleapproacheszeroasitsspeedapproachesthespeedoflightregardlessofthemagnitudeoftheforceapplied

1F/m

0.9F/m

0

0.1F/m

0.2F/m

0.3F/m

0.4F/m

0.5F/m

0.6F/m

0.7F/m

0.8F/m

a

79

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

Speedvrelativetothespeedoflightc(v/c)

a=F/m𝜸3Particle acceleration a

1F/m

0.9F/m

0

0.1F/m

0.2F/m

0.3F/m

0.4F/m

0.5F/m

0.6F/m

0.7F/m

0.8F/m

a

Newtonianmechanicswrongly predictsthataparticle’saccelerationwillremainconstantwhenaconstantforceisapplied

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Relativistic Work and Particle Energy

The kinetic energy of a particle equals the net work W doneon it in moving it from rest to speed vInrelativistic termsthekineticenergyKofaparticleofrestmassm becomes

K= mc2

1−v2/c2– mc2 =(𝜸 – 1)mc2 ‘Relativistic’ kineticenergy

K=particlekineticenergym=particlerestmassc=speedoflightinavacuumv=speedofparticle𝜸 =Lorentzgammafactorrelatingrestframeofparticleandtheframeoftheobserver82

The kinetic energy of a particle equals the net energy done onit in moving it from rest to speed vInrelativistic termsthekineticenergyKofaparticleofrestmassm becomes

K= mc2

1−v2/c2– mc2 =(𝜸 – 1)mc2 Relativistickineticenergy

Asthespeedoftheparticle,v approachesthespeedoflightsoitskineticenergyKapproachesinfinity

InNewtoniantermsKonlybecomesinfiniteifv isinfinite83

The kinetic energy of a particle equals the net energy done onit in moving it from rest to speed vInrelativistic termsthekineticenergyKofaparticleofrestmassm becomes

K= mc2

1−v2/c2– mc2 =(𝜸 – 1)mc2 Relativistickineticenergy

Asthespeedoftheparticle,v approachesthespeedoflightsoitskineticenergyKapproachesinfinity

InNewtoniantermsKonlybecomesinfiniteifv isinfinite84

The kinetic energy of a particle equals the net energy done onit in moving it from rest to speed vInrelativistic termsthekineticenergyKofaparticleofrestmassm becomes

K= mc2

1−v2/c2– mc2 =(𝜸 – 1)mc2 Relativistickineticenergy

Asthespeedoftheparticle,v approachesthespeedoflightsoitskineticenergyKapproachesinfinity

InNewtoniantermsKonlybecomesinfiniteifv isinfinite85

0 0.5 1 1.5 2 2.5

Particlekineticenergy

86Speedvrelativetothespeedoflightc(v/c)

0

0.5mc2

1mc2

1.5mc2

2mc2

2.5mc2

3mc2

3.5mc2

4mc2K=(𝜸

–1)mc2 (Kineticenergy)

K

Relativistickineticenergybecomesinfiniteasv approachesc

0 0.5 1 1.5 2 2.5

Particlekineticenergy

87Speedvrelativetothespeedoflightc(v/c)

0

0.5mc2

1mc2

1.5mc2

2mc2

2.5mc2

3mc2

3.5mc2

4mc2K=(𝜸

–1)mc2 (Kineticenergy)

K

Newtonianmechanicsincorrectly predictsthatkineticenergyonlybecomesinfiniteifv becomesinfinite(K=1/2mv2)

Total particle energy E, Rest energy (E = mc2) and Massless energy (E = pc)

Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms

K= mc2

1−v2/c2– mc2

Thefirsttermdependsonmotionandasecondenergytermthatisindependentofmotion

ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest

89

Motionterm Restenergyterm

Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms

K= mc2

1−v2/c2– mc2

Themotiontermdependsonmotionandtherestenergytermisindependentofmotion

ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest

90

Motionterm Energyterm

Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms

K= mc2

1−v2/c2– mc2

Themotiontermdependsonmotionandtheenergytermisindependentofmotion

ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest

91

Motionterm Energyterm

A particle’s total energy E can thus be expressed as follows

E = K + mc2 = mc2

1−v2/c2= 𝜸mc2 Total particle energy

92

Tosummarise,thetotalenergyEofaparticleisthesumofitsKineticenergyplusitsrestenergy

Whatisapparentisthatevenwhenaparticleisatrestitstillhasenergy

Thisiscalleditsrestenergywhichisproportionaltoitsrest(andonlyrest)mass

Thishasbeenexperimentallyconfirmed.Whenunstablefundamentalparticlesdecay,thereisalwaysanenergychangeconsistentwiththeassumptionofarestenergymc2withtherestmassm 93

Tosummarise,thetotalenergyEofaparticleisthesumofitsKineticenergyplusitsrestenergy

Whatisapparentisthatevenwhenaparticleisatrestitstillhasenergy

Thisiscalleditsrestenergywhichisproportionaltoitsrest(andonlyrest)mass

Thishasbeenexperimentallyconfirmed.Whenunstablefundamentalparticlesdecay,thereisalwaysanenergychangeconsistentwiththeassumptionofarestenergymc2withtherestmassm 94

Tosummarise,thetotalenergyEofaparticleisthesumofitsKineticenergyplusitsrestenergy

Whatisapparentisthatevenwhenaparticleisatrestitstillhasenergy

Thisiscalleditsrestenergymc2whichisassociatedwithitsrestmass,m

Thishasbeenexperimentallyconfirmed.Whenunstablefundamentalparticlesdecay,thereisalwaysanenergychangeconsistentwiththeassumptionofarestenergymc2withtherestmassm 95

Tosummarise,thetotalenergyEofaparticleisthesumofitsKineticenergyplusitsrestenergy

Whatisapparentisthatevenwhenaparticleisatrestitstillhasenergy

Thisiscalleditsrestenergywhichisproportionaltoitsrest(andonlyrest)mass

Thishasbeenexperimentallyconfirmed.Whenunstablefundamentalparticlesdecay,thereisalwaysanenergychangeconsistentwiththeassumptionofarestenergyofmc2witharestmassofm 96

Thesimplestexampleofthepresenceofrestenergyisthereleaseofenergyofdecayofaneutralpion(𝝿 ).

Itisanunstableparticleofmassmwhichwhenitdecays(withzerokineticenergybeforeitsdecay)releasesradiationwithanenergyexactlyequaltom𝝿 c2

Toputthingsintoperspective,agolfballofmass0.046kghasenoughrestenergytopowera100Wlightbulbfor1.3millionyears!

97

Thesimplestexampleofthepresenceofrestenergyisthereleaseofenergyofdecayofaneutralpion(𝝿 ).

Itisanunstableparticleofmassmwhichwhenitdecays(withzerokineticenergybeforeitsdecay)releasesradiationwithanenergyexactlyequaltom𝝿 c2

Toputthingsintoperspective,agolfballofmass0.05kghasenoughrestenergytopowera100Wlightbulbfor1.3millionyears!

98

Thesimplestexampleofthepresenceofrestenergyisthereleaseofenergyofdecayofaneutralpion(𝝿 ).

Itisanunstableparticleofmassmwhichwhenitdecays(withzerokineticenergybeforeitsdecay)releasesradiationwithanenergyexactlyequaltom𝝿 c2

Toputthingsintoperspective,a50ggolfballhasenoughrestenergytopotentiallypowera100Wlightbulbfor1.3millionyears!

99

Withabitofmanipulationthemomentumandrestenergyequationscanbereformulatedasfollows

(p/m)2 = v2/c2

1 − v2/c2 and(E/mc2)2= 71 − v2/c2

Subtractingandrearrangingtheseequationsgiveus

E2 =(mc2)2 +(pc)2

100

Withabitofmanipulationthemomentumandrestenergyequationscanbereformulatedasfollows

(p/m)2 = v2/c2

1 − v2/c2 and(E/mc2)2= 7

1−v2/c2

Subtractingandrearrangingtheseequationsgivesus

E2 =(mc2)2 +(pc)2

101

Formasslessparticles(m=0),thepreviousexpressionbecomes

E = pc

All massless particles thus travel at the speed of light andhave both energy and momentum such

Photons are massless

Theonlyotherknownmasslessparticleisthegluon102

Formasslessparticles(m=0)thepreviousexpressionbecomes

E=pc

Allmasslessparticlesthustravelatthespeedoflightinavacuumandhavebothenergyandmomentum

Photons are massless

Theonlyotherknownmasslessparticleisthegluon103

Formasslessparticles(m=0)thepreviousexpressionbecomes

E=pc

AllmasslessparticlesthustravelatthespeedoflightinavacuumandhavebothenergyandmomentumsPhotons, thequantumofelectromagneticradiationaremasslessPhotonsareemittedandabsorbedduringchangesofstateofanatomicornuclearsystemwhentheenergyandmomentumofthesystemchange 104

Formasslessparticles(m=0)thepreviousexpressionbecomes

E=pc

Allmasslessparticlesthustravelatthespeedoflightandhavebothenergyandmomentum

Photons, thequantumofelectromagneticradiationaremasslessPhotonsareemittedandabsorbedduringchangesofstateofanatomicornuclearsystemwhentheenergyandmomentumofthesystemchange 105

Formasslessparticles(m=0)thepreviousexpressionbecomes

E=pc

Allmasslessparticlesthustravelatthespeedoflightandhavebothenergyandmomentum

Photons, thequantumofelectromagneticradiationaremasslessPhotonsareemittedandabsorbedduringchangesofstateofanatomicornuclearsystemwhentheenergyandmomentumofthesystemchange 106

Theexpressionalsosaysthatforparticlesatrest(p=0),thetotalenergyequationreducesto

107

E=mc2 Einstein’sfamousrestenergyequation

Conservation of mass energy

From the preceding points it is clear that energy and mass areinterchangeable

It is also clear that the principles of conservation of mass andenergy should be restated in terms of a broader principlewhich is The law of the conservation of mass and energy

This law is the fundamental principle involved in thegeneration of nuclear power. When a uranium or plutoniumnucleus undergoes fission in a nuclear reactor, the sum of therest masses of the resulting fragments is less than the mass ofthe parent nucleus. An amount of energy is released whichequals E = mc2where m equals the lost mass 110

From the preceding points it is clear that energy and mass areinterchangeable

It is also clear that the principles of conservation of mass andenergy should be restated in terms of a broader principlewhich is the law of the conservation of mass and energy

This law is the fundamental principle involved in thegeneration of nuclear power. When a uranium or plutoniumnucleus undergoes fission in a nuclear reactor, the sum of therest masses of the resulting fragments is less than the mass ofthe parent nucleus. An amount of energy is released whichequals E = mc2where m equals the lost mass 111

From the preceding points it is clear that energy and mass areinterchangeable

It is also clear that the principles of conservation of mass andenergy should be restated in terms of a broader principlewhich is The law of the conservation of mass and energy

This law is the fundamental principle involved in thegeneration of nuclear power. When a uranium or plutoniumnucleus undergoes fission in a nuclear reactor, the sum of therest masses of the resulting fragments is less than the mass ofthe parent nucleus. An amount of energy is released whichequals E = mc2where m equals the lost mass 112

113

Block 111 Virginia – class nuclear attack submarine

114

115

Fatmanreplica

More Relativistic phenomena in nature

Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!

ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2

CosmicexplosionsarealsodrivenbyE=mc2

InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingwhichhasledtoourunderstandingoftheexpandinguniverse

117

Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!

ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2

CosmicexplosionsarealsodrivenbyE=mc2

InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingwhichhasledtoourunderstandingoftheexpandinguniverse

118

Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!

ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2

CosmicexplosionsarealsodrivenbyE=mc2

InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingwhichhasledtoourunderstandingoftheexpandinguniverse

119

Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!

ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2

CosmicexplosionsarealsodrivenbyE=mc2

InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingfromuswhichhasledtoourunderstandingoftheexpandinguniverse

120

Theheatgeneratedbythedecayofradioactiveelementsintheinnerlayersoftheearthprovidesmorethan50%oftheheattokeeptheselayersmolten

Themovementoftectonicplatesdependsonhavingamoltenmassonwhichtheycan‘float’

Thisishowourcontinentsandmountainsareformed

Theearth’srotatingmoltencorealsocreatestheearth’smagneticfieldwhichisvitalinprotectingusfromharmfulradiation 121

Theheatgeneratedbythedecayofradioactiveelementsintheinnerlayersoftheearthprovidesmorethan50%oftheheattokeeptheselayersmolten

Themovementoftectonicplatesdependsonhavingamoltenmassonwhichtheycan‘float’

Thisishowourcontinentsandmountainsareformed

Theearth’srotatingmoltencorealsocreatestheearth’smagneticfieldwhichisvitalinprotectingusfromharmfulradiation 122

Theheatgeneratedbythedecayofradioactiveelementsintheinnerlayersoftheearthprovidesmorethan50%oftheheattokeeptheselayersmolten

Themovementoftectonicplatesdependsonhavingamoltenmassonwhichtheycan‘float’

Thisishowourcontinentsandmountainsareformed

Theearth’srotatingmoltencorealsocreatestheearth’smagneticfieldwhichisvitalinprotectingusfromharmfulradiation 123

Theheatgeneratedbythedecayofradioactiveelementsintheinnerlayersoftheearthprovidesmorethan50%oftheheattokeeptheselayersmolten

Themovementoftectonicplatesdependsonhavingamoltenmassonwhichtheycan‘float’

Thisishowourcontinentsandmountainsareformed

Theearth’srotatingmoltencorealsocreatestheearth’smagneticfieldwhichisvitalinprotectingusfromharmfulradiation 124

125

Untilrecentlymarinershavereliedheavilyonthemagneticcompassfornavigation

126

Auroraborealis

You’veprobablynotgivenitmuchthought,butthereasonwhygoldisyellow(orrather,golden) isdeeplyingrainedinitsatomicstructureandit’sbecauseofsomethingcalledrelativisticquantumchemistry

Simplyput,gold’selectronsmovesofast(± c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden

Thesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasily

127

128

Simplyput,gold’selectronsmovesofast(±c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden

You’veprobablynotgivenitmuchthought,butthereasonwhygoldisyellow(orrather,golden) isdeeplyingrainedinitsatomicstructure— andit’sbecauseofsomethingcalledrelativisticquantumchemistry

Simplyput,gold’selectronsmovesofast(± c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden

Thesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasily

129

Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincrease.Withmercury,thebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit.

130

Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincrease.Withmercury,thebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit.

131

Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincrease.Withmercury,thebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit.

132

Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincreaseWithmercury,thebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit 133

Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincreaseWithmercurythebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit 134

135

More Practical applications of special relativity

Inparticleacceleratorsmanyparticleshaveveryshorthalflives.AtspeedsclosetothespeedoflighthalflivesaresignificantlyincreasedgivingresearcherstheopportunitytostudythemModerncomputerchips.Thisalittlemoreesoteric,butdesigningsolid-stateelectronicsdependsonbeingabletomodelelectronbandstructures.ThatoftenrequiresrelativisticcorrectionstodosoaccuratelyCathoderaytubes– electronstravellingat± 30%ofthespeedoflight.Relativistic

137

Inparticleacceleratorsmanyparticleshaveveryshorthalflives.Atspeedsclosetothespeedoflighthalflivesaresignificantlyincreasedgivingresearcherstheopportunitytostudythem

Moderncomputerchips.Thisalittlemoreesoteric,butdesigningsolid-stateelectronicsdependsonbeingabletomodelelectronbandstructures.Thatoftenrequiresrelativisticcorrectionstodosoaccurately

Inmedicine,manybodyscannersrelyonrelativisticsciencefortheiroperation

138

Inparticleacceleratorsmanyparticleshaveveryshorthalflives.Atspeedsclosetothespeedoflighthalflivesaresignificantlyincreasedgivingresearcherstheopportunitytostudythem

Moderncomputerchips.Thisalittlemoreesoteric,butdesigningsolid-stateelectronicsdependsonbeingabletomodelelectronbandstructures.Thatoftenrequiresrelativisticcorrectionstodosoaccurately

Inmedicine,manybodyscannersrelyonrelativisticsciencefortheiroperation

139

Leadacidbatteries

Withoutrelativityleadwouldbeexpectedtobehaveliketin,sotin-acidbatteriesshouldworkjustaswellasleadacidbatteriesusedincars

However,calculationsshowthat10Vofthe12Vproducesbya6cellbatteryarisespurelyfromrelativisticeffects!

140

PPet Scanner

141

Positron emission tomog-raphy(PET) scanner

Special relativity conclusions

It may appear that the foundations of Newtonian mechanicshave been destroyed. Newtonian mechanics are not wrong,they are simply incomplete. Newton’s laws are approximatelycorrect when speeds are small in comparison to cRather than destroying them, relativity generalises themEven special relativity is not complete!The general theory of relativity goes further and deals withhow the geometric properties of space are affected by thepresence of matterDon’t forget that all speeds are relative! (Except the speed oflight)You cannot travel faster then the speed of light! 143

It may appear that the foundations of Newtonian mechanicshave been destroyed. Newtonian mechanics are not wrong,they are simply incomplete. Newton’s laws are approximatelycorrect when speeds are small in comparison to cRather than destroying them, relativity generalises themEven special relativity is not complete!The general theory of relativity goes further and deals withhow the geometric properties of space are affected by thepresence of matterDon’t forget that all speeds are relative! (Except the speed oflight)You cannot travel faster then the speed of light! 144

It may appear that the foundations of Newtonian mechanicshave been destroyed. Newtonian mechanics are not wrong,they are simply incomplete. Newton’s laws are approximatelycorrect when speeds are small in comparison to cRather than destroying them, relativity generalises themEven special relativity is not complete!The general theory of relativity goes further and deals withhow the geometric properties of space are affected by thepresence of matterDon’t forget that all speeds are relative! (Except the speed oflight)You cannot travel faster then the speed of light! 145

It may appear that the foundations of Newtonian mechanicshave been destroyed. Newtonian mechanics are not wrong,they are simply incomplete. Newton’s laws are approximatelycorrect when speeds are small in comparison to cRather than destroying them, relativity generalises themEven special relativity is not complete!The general theory of relativity goes further and deals withhow the geometric properties of space are affected by thepresence of matterDon’t forget that all speeds are relative! (Except the speed oflight)You cannot travel faster then the speed of light! 146

It may appear that the foundations of Newtonian mechanicshave been destroyed. Newtonian mechanics are not wrong,they are simply incomplete. Newton’s laws are approximatelycorrect when speeds are small in comparison to cRather than destroying them, relativity generalises themEven special relativity is not complete!The general theory of relativity goes further and deals withhow the geometric properties of space are affected by thepresence of matterDon’t forget that all speeds are relative! (Except the speed oflight)You cannot travel faster then the speed of light! 147

It may appear that the foundations of Newtonian mechanicshave been destroyed. Newtonian mechanics are not wrong,they are simply incomplete. Newton’s laws are approximatelycorrect when speeds are small in comparison to cRather than destroying them, relativity generalises themEven special relativity is not complete!The general theory of relativity goes further and deals withhow the geometric properties of space are affected by thepresence of matterDon’t forget that all speeds are relative! (Except the speed oflight)You cannot travel faster then the speed of light! 148

The end

Email address:

roblouw47@gmail.com

149