Special Relativity UCT Summer School presentation part 3 of 3 … · 2019-12-18 · 12/17/19 1...
Transcript of Special Relativity UCT Summer School presentation part 3 of 3 … · 2019-12-18 · 12/17/19 1...
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SpecialRelativity
PresentationtoUCT Summer School January 2020 (Part3of3)
ByRobLouw
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Test your understanding of time dilationPeter,whoisstandingontheground,startshisstopwatchthemomentthatSarahfliesoverheadinaspaceshipataspeedof0.6cAtthesameinstantSarahstartsherstopwatchAsmeasuredinPeter’sframeofreference,whatisthereadingonSarah’sstopwatchattheinstantpeter’sstopwatchreads10s?a)10s,b)lessthan10sorc)morethan10s?AsmeasuredinSarah’sframeofreference,whatisthereadingonPeter’sstopwatchattheinstantthatSarah’sstopwatchreads10s?a)10s,b)lessthan10sorc)morethan10s?Whosestopwatchisreadingpropertimeintheabovetwoexamples?
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Test your understanding of length contraction
A10mlongspaceshipfliespastyouhorizontallyat0.99cAtacertaininstantyouobservethatthatthenoseandtailofthespaceshipalignexactlywiththetwoendsofameterstickthatyouholdinyourhandRankthefollowingdistancesinorderfromlongesttoshortest:a)therestlengthofthespaceship,b)theproperlengthofthemeterstick,c)theproperlengthofthespaceshipd)thelengthofthespaceshipmeasuredinyourreferenceframee)thelengthofthemeterstickmeasuredinthespaceship’sframeofreference?
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Aswesawyesterday,spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreferenceTimeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent
UsingtheLorentzcoordinatetransformationswecanderiveasetofLorentzvelocitytransformations
Theresult(withoutderivation)isshowninthenextslide6
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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)Lorentzvelocitytransformation
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TheLorentzvelocitytransformationshowsthatabodywithaspeedlessthanc inoneframeofreferencealwayshasaspeedlessthanc ineveryotherframeofreference
Thisisonereasonforconcludingthatnomaterialbodymaytravelwithaspeedgreaterthanorequaltothespeedoflightinavacuum,relativetoanyinertialreferenceframe
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Let'sconsideranexampleofthevelocitylimitwhichanyobservercanreachrelativetosomeotherobserver
IfwehadasetoffivespaceshipsstackedlikeRussiandollswhereeachshipcouldlaunchtheremainingshipsatavelocityequaltotherelativevelocityofthelaunchingshipasobservedfromearthwhatrelativevelocitiescouldthevariousshipsachieverelativetotheearthobserver?
Thefollowingslideshowsthevelocityprofilesofthefivespaceshipsrelativetoanearthobserver
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Rocketspeedsrelativetospeedof
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
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Rocketspeedsrelativetospeedof
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
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Rocketspeedsrelativetospeedof
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
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Rocketspeedsrelativetospeedof
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
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Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
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M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
Nomatterhowmanysuccessiverocketsarelaunchedtheirvelocitywillneverexceedc!
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Relativistic kinematics and the Doppler effect for electromagnetic waves
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Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain
Asourceoflight ismovingtowardsStanleywithconstantspeeduwhoisinastationeryinertialreferenceframeS
Thesourceemitslightemitslightwavesoffrequencyf0 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)
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Withlight,unlikesound,thereisnodistinctionbetweenmotionofsourceandmotionofobserver,onlytherelativevelocityofthetwoissignificant
ThefollowingslideillustratestheDopplerblueshifteffect
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0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 142Speedvrelativetothespeedoflightc(v/c)
f/f0=(𝒄+𝒖)/(𝒄−𝒖)
Doppler effect- source approaching observer
Asthesourcevelocity- uapproachesthespeedoflight,f/f0approachesinfinity(BLUESHIFT)
f/f0
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Withelectromagneticwavesmovingaway fromanobserver,therelativisticredshiftDopplerformulacanbederivedusingtheappropriateLorentztransforms
Thedopplerredshiftequationindicatesthatfdecreasei.e.thewavelengthgetslonger(redder)asu approachesthespeedoflightc
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f= (𝐜 − 𝐮)/(𝐜 + 𝐮) f0 Dopplerformula(redshift)
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NotethatinderivingtheDopplerequations,𝛾 hascancelledout
TheDopplerredshifteffectisshowninthenextfewslides
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f/f0=(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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Relativistic particle momentum p
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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:SupposewelookatacollisioninaninertialcoordinatesystemSandwefindthatmomentumisconserved
WhenweusetheLorentztransformationtoobtainvelocitiesinasecondinertialsystemS’wefindthatusingtheNewtoniandefinitionofmomentum(p=mv),momentumisnotconservedinthesecondsystem
Tosolvethisproblemweneedamoregeneraliseddefinitionofmomentum
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Theequationwillnotbederivedfromfirstprinciples,butitwillsimplybestatedbelowSupposewehaveamaterialparticlewitharestmassofm,whensuchaparticlehasavelocityv,thenitsrelativisticmomentum pis
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p =mv/ 1 − (𝑣/𝑐). =𝛾mvRelativisticmomentum
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Relativisticmomentumplaysakeyroleinunderstandingthekinematicsofparticlephysics
Particlevelocitieswillbedenotedwithv fortherestofthispresentation
Wewillnolongerbemakinguseofu,therelativevelocityofreferenceframesaswewillbethestationaryobserveronearth
RelativisticandNewtonianmomentumasafunctionofrelativespeedv/careillustratedgraphicallyinthenextfewslides 57
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Particle momentum
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P=𝜸mv=mv/𝟏−(𝐯/𝐜)𝟐
Speedvrelativetothespeedoflightc(v/c)
Asv approachesc,relativisticmomentumapproachesinfinity
3mc
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p=𝜸mv
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Particle momentum
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P=𝜸mv=mv/𝟏−(𝐯/𝐜)𝟐
Speedvrelativetothespeedoflightc(v/c)
Newtonianmechanicsincorrectly predictsthatmomentumonlyreachesinfinityifvbecomesinfinite
3mc
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p=𝜸mv
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Force F and acceleration a
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ThegeneralformofNewton’ssecondlawisF=dp/dt=ma
Experimentsshowthisresultisstillvalidinrelativisticmechanicsprovidedweuserelativisticmomentum.ThustherelativisticallycorrectversionofNewton’ssecondlaw is
F=ma/{ 𝟏 − (𝒗/𝒄)𝟐}3=𝛾 3ma Forceformula
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Rearrangingthepreviousequationwecanestablishwhathappenstotheaccelerationa ofaparticleofrestmassmwhichissubjectedtoaconstantforcea=(F/m 𝟏 − (𝒗/𝒄)𝟐 3=F/m𝛾 𝟑 Accelerationformula
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InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedvInrelativisticmechanics,whenaparticleofrestmassmissubjectedtoaconstantforceF,itsaccelerationdecreasestozeroasitsvelocitytendstowardthespeedoflightInfactitdoesnotmatterhowbigtheforceornonzeromassis,accelerationwillalwaysdecreasetozeroastheparticlespeedincreasestowardsthespeedoflightTherelativisticeffectofincreasedspeedontheaccelerationofaparticleofrestmassmwhensubjectedtoaconstantforceF isillustratedinthenextfewslides
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Speedvrelativetothespeedoflightc(v/c)
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Accelerationofaparticleapproacheszeroasitsspeedapproachesthespeedoflightregardlessofthemagnitudeoftheforceapplied
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Newtonianmechanicswrongly predictsthataparticle’saccelerationwillremainconstantwhenaconstantforceisapplied
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Relativistic Work and Particle Energy
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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
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As the speed of the particle v approaches the speed of light soits kinetic energy K approaches infinity
In Newtonian terms K only becomes infinite if v is infinite
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Relativistickineticenergybecomesinfiniteasv approachesc
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Newtonianmechanicsincorrectly predictsthatkineticenergyonlybecomesinfiniteifv becomesinfinite(K=1/2mv2)
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Total particle energy E, Rest energy (E = mc2) and Massless energy (E = pc)
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Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms
K= mc2
1−v2/c2– mc2
Themotiontermdependsonmotionandtheenergytermisindependentofmotion
ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest
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Motionterm Energyterm
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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,thereisalwaysanenergychangeconsistentwiththeassumptionofarestenergyofmc2witharestmassofm 85
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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= 7
1−v2/c2
Subtractingandrearrangingtheseequationsgivesus
E2 =(mc2)2 +(pc)2
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Formasslessparticles(m=0)thepreviousexpressionbecomes
E=pc
Allmasslessparticlesthustravelatthespeedoflightandhavebothenergyandmomentumsuch
Photons, thequantumofelectromagneticradiationaremassless
Theonlyotherknownmasslessparticleisthegluon88
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Theexpressionalsosaysthatforparticlesatrest(p=0),thetotalenergyequationreducesto
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E=mc2 Einstein’sfamousrestenergyequation
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Conservation of mass energy
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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
It may appear that the foundations of Newtonian mechanicshave been destroyedNewtonian mechanics are not wrong, they are simplyincomplete
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Block 111 Virginia – class nuclear attack submarine
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Fatmanreplica
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More Relativistic phenomena in nature
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Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!
ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2
CosmicexplosionsarealsodrivenbyE=mc2
InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingfromuswhichhasledtoourunderstandingoftheexpandinguniverse
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Theheatgeneratedbythedecayofradioactiveelementsintheinnerlayersoftheearthprovidesmorethan50%oftheheattokeeptheselayersmolten
Themovementoftectonicplatesdependsonhavingamoltenmassonwhichtheycan‘float’
Thisishowourcontinentsandmountainsareformed
Theearth’srotatingmoltencorealsocreatestheearth’smagneticfieldwhichisvitalinprotectingusfromharmfulradiation 97
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Untilrecentlymarinershavereliedheavilyonthemagneticcompassfornavigation
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Auroraborealis
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You’veprobablynotgivenitmuchthought,butthereasonwhygoldisyellow(orrather,golden) isdeeplyingrainedinitsatomicstructureandit’sbecauseofsomethingcalledrelativisticquantumchemistry
Simplyput,gold’selectronsmovesofast(± c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden
Thesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasily
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Simplyput,gold’selectronsmovesofast(±c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden
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You’veprobablynotgivenitmuchthought,butthereasonwhygoldisyellow(orrather,golden) isdeeplyingrainedinitsatomicstructure—andit’sbecauseofsomethingcalledrelativisticquantumchemistry
Simplyput,gold’selectronsmovesofast(± c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden
Thesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasily
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Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincrease.Withmercury,thebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit.
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Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincreaseWithmercurythebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit 104
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More Practical applications of special relativity
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Inparticleacceleratorsmanyparticleshaveveryshorthalflives.Atspeedsclosetothespeedoflighthalflivesaresignificantlyincreasedgivingresearcherstheopportunitytostudythem
Moderncomputerchips.Thisalittlemoreesoteric,butdesigningsolid-stateelectronicsdependsonbeingabletomodelelectronbandstructures.Thatoftenrequiresrelativisticcorrectionstodosoaccurately
Inmedicine,manybodyscannersrelyonrelativisticsciencefortheiroperation
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PPet Scanner
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Positron emission tomog-raphy(PET) scanner
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Special relativity conclusions
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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! 110
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The end
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