Post on 05-Jun-2020
transcript
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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Relative rocket ship speeds
Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5
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Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5
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Mothership Rocket 1 Rocket 2 Rocket 3 Rocket 4 Rocket 5
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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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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 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
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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
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Motionterm Restenergyterm
Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms
K= mc2
1−v2/c2– mc2
Themotiontermdependsonmotionandtherestenergytermisindependentofmotion
ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest
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Motionterm Energyterm
Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms
K= mc2
1−v2/c2– mc2
Themotiontermdependsonmotionandtheenergytermisindependentofmotion
ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest
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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
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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!
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Thesimplestexampleofthepresenceofrestenergyisthereleaseofenergyofdecayofaneutralpion(𝝿 ).
Itisanunstableparticleofmassmwhichwhenitdecays(withzerokineticenergybeforeitsdecay)releasesradiationwithanenergyexactlyequaltom𝝿 c2
Toputthingsintoperspective,agolfballofmass0.05kghasenoughrestenergytopowera100Wlightbulbfor1.3millionyears!
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Thesimplestexampleofthepresenceofrestenergyisthereleaseofenergyofdecayofaneutralpion(𝝿 ).
Itisanunstableparticleofmassmwhichwhenitdecays(withzerokineticenergybeforeitsdecay)releasesradiationwithanenergyexactlyequaltom𝝿 c2
Toputthingsintoperspective,a50ggolfballhasenoughrestenergytopotentiallypowera100Wlightbulbfor1.3millionyears!
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Withabitofmanipulationthemomentumandrestenergyequationscanbereformulatedasfollows
(p/m)2 = v2/c2
1 − v2/c2 and(E/mc2)2= 71 − v2/c2
Subtractingandrearrangingtheseequationsgiveus
E2 =(mc2)2 +(pc)2
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Withabitofmanipulationthemomentumandrestenergyequationscanbereformulatedasfollows
(p/m)2 = v2/c2
1 − v2/c2 and(E/mc2)2= 7
1−v2/c2
Subtractingandrearrangingtheseequationsgivesus
E2 =(mc2)2 +(pc)2
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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
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115
Fatmanreplica
More Relativistic phenomena in nature
Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!
ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2
CosmicexplosionsarealsodrivenbyE=mc2
InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingwhichhasledtoourunderstandingoftheexpandinguniverse
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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
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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
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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
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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
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