special relativity uct summer school presentation part 3 of 3 … · 2019-12-18 · 12/17/19 1...
TRANSCRIPT
12/17/19
1
SpecialRelativity
PresentationtoUCT Summer School January 2020 (Part3of3)
ByRobLouw
1
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?
2
Test your understanding of length contraction
A10mlongspaceshipfliespastyouhorizontallyat0.99cAtacertaininstantyouobservethatthatthenoseandtailofthespaceshipalignexactlywiththetwoendsofameterstickthatyouholdinyourhandRankthefollowingdistancesinorderfromlongesttoshortest:a)therestlengthofthespaceship,b)theproperlengthofthemeterstick,c)theproperlengthofthespaceshipd)thelengthofthespaceshipmeasuredinyourreferenceframee)thelengthofthemeterstickmeasuredinthespaceship’sframeofreference?
3 4
12/17/19
2
5
Aswesawyesterday,spaceandtimehavebecomeintertwinedandwecannolongersaythatlengthandtimehaveabsolutemeaningsindependentoftheframeofreferenceTimeandthethreedimensionsofspacecollectivelyforafour-dimensionalentitycalledspacetime andwecallx,y,zandt togetherthespacetimecoordinatesofanevent
UsingtheLorentzcoordinatetransformationswecanderiveasetofLorentzvelocitytransformations
Theresult(withoutderivation)isshowninthenextslide6
6
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
7
vx’=(vx – u)/(1- uvx/c2)Lorentzvelocitytransformation
7
TheLorentzvelocitytransformationshowsthatabodywithaspeedlessthanc inoneframeofreferencealwayshasaspeedlessthanc ineveryotherframeofreference
Thisisonereasonforconcludingthatnomaterialbodymaytravelwithaspeedgreaterthanorequaltothespeedoflightinavacuum,relativetoanyinertialreferenceframe
10
10
12/17/19
3
Let'sconsideranexampleofthevelocitylimitwhichanyobservercanreachrelativetosomeotherobserver
IfwehadasetoffivespaceshipsstackedlikeRussiandollswhereeachshipcouldlaunchtheremainingshipsatavelocityequaltotherelativevelocityofthelaunchingshipasobservedfromearthwhatrelativevelocitiescouldthevariousshipsachieverelativetotheearthobserver?
Thefollowingslideshowsthevelocityprofilesofthefivespaceshipsrelativetoanearthobserver
11
11
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
12
12
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
13
13
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
14
14
12/17/19
4
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
15
15
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
16
16
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
17
17
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
Rocketspeedsrelativetospeedof
lightcasasobservedonearth
Rocketspeedsrelativetospeedoflightcobservedbysuccessiveshipobserverswhenu=v
Relative rocket ship speeds
M ot her sh ip Ro cket 1 Ro cket 2 Ro cket 3 Ro cket 4 Ro cket 5
Nomatterhowmanysuccessiverocketsarelaunchedtheirvelocitywillneverexceedc!
18
18
12/17/19
5
Relativistic kinematics and the Doppler effect for electromagnetic waves
33
Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain
Asourceoflight ismovingtowardsStanleywithconstantspeeduwhoisinastationeryinertialreferenceframeS
Thesourceemitslightemitslightwavesoffrequencyf0 asMeasuredinitsrestframe
Stanleyreceiveslightwavesoffrequencyfasshownbelow
34
34
36
36
Herewegowithanotherthoughtexperimentinvolvingtheuseofahigh-speedtrain
Asourceoflight ismovingtowardsStanleywithconstantspeeduwhoisinastationeryinertialreferenceframeS
Thesourceemitslightwavesoffrequencyf0 asmeasuredinitsrestframe
Stanleyreceiveslightwavesoffrequencyfasshowninthenextslide
37
37
12/17/19
6
Withanelectromagneticsourceapproaching anobserver,therelativisticblueshiftDopplerformulacanbederivedusingtheappropriateLorentztransformsandis
39
Thedopplerblueshiftequationindicatesthatfincreasesi.e.thewavelengthgetsshorter(bluer)asu approachesthespeedoflight c
f= (𝐜 + 𝐮)/(𝐜 − 𝐮) f0 Dopplerformula(blueshift)
39
40
Withlight,unlikesound,thereisnodistinctionbetweenmotionofsourceandmotionofobserver,onlytherelativevelocityofthetwoissignificant
ThefollowingslideillustratestheDopplerblueshifteffect
40
0
2
4
6
8
10
12
14
16
18
20
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
42
Withelectromagneticwavesmovingaway fromanobserver,therelativisticredshiftDopplerformulacanbederivedusingtheappropriateLorentztransforms
Thedopplerredshiftequationindicatesthatfdecreasei.e.thewavelengthgetslonger(redder)asu approachesthespeedoflightc
43
f= (𝐜 − 𝐮)/(𝐜 + 𝐮) f0 Dopplerformula(redshift)
43
12/17/19
7
NotethatinderivingtheDopplerequations,𝛾 hascancelledout
TheDopplerredshifteffectisshowninthenextfewslides
44
44
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
46
f/f0=(𝒄+𝒖)/(𝒄−𝒖)
Asthesourcevelocityuapproachesthespeedoflight,f/f0approacheszero(redSHIFT)
Doppler effect- source moving away from observer
Speedvrelativetothespeedoflightc(v/c)
f/f0
46
47
Hubblephotographofafastmoving,DopplerblueshiftedjetemanatingfromablackholeatthecentreofGalaxyM87
47
48
QueenMary2’sradarantennae
48
12/17/19
8
49
Radarequipmentinstallationatanairport
49
50
50
51
51
Relativistic particle physics
52
12/17/19
9
Relativistic particle momentum p
53
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
54
54
Thisposesuswithaproblem:SupposewelookatacollisioninaninertialcoordinatesystemSandwefindthatmomentumisconserved
WhenweusetheLorentztransformationtoobtainvelocitiesinasecondinertialsystemS’wefindthatusingtheNewtoniandefinitionofmomentum(p=mv),momentumisnotconservedinthesecondsystem
Tosolvethisproblemweneedamoregeneraliseddefinitionofmomentum
55
55
Theequationwillnotbederivedfromfirstprinciples,butitwillsimplybestatedbelowSupposewehaveamaterialparticlewitharestmassofm,whensuchaparticlehasavelocityv,thenitsrelativisticmomentum pis
56
p =mv/ 1 − (𝑣/𝑐). =𝛾mvRelativisticmomentum
56
12/17/19
10
Relativisticmomentumplaysakeyroleinunderstandingthekinematicsofparticlephysics
Particlevelocitieswillbedenotedwithv fortherestofthispresentation
Wewillnolongerbemakinguseofu,therelativevelocityofreferenceframesaswewillbethestationaryobserveronearth
RelativisticandNewtonianmomentumasafunctionofrelativespeedv/careillustratedgraphicallyinthenextfewslides 57
57
59
0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6
Particle momentum
1
8 mc
1mc
2mc
P=𝜸mv=mv/𝟏−(𝐯/𝐜)𝟐
Speedvrelativetothespeedoflightc(v/c)
Asv approachesc,relativisticmomentumapproachesinfinity
3mc
4mc
5mc
6mc
7mc
0
p=𝜸mv
59
61
0 0 .2 0 .4 0 .6 0 .8 1 1 .2 1 .4 1 .6
Particle momentum
1
8 mc
1mc
2mc
P=𝜸mv=mv/𝟏−(𝐯/𝐜)𝟐
Speedvrelativetothespeedoflightc(v/c)
Newtonianmechanicsincorrectly predictsthatmomentumonlyreachesinfinityifvbecomesinfinite
3mc
4mc
5mc
6mc
7mc
0
p=𝜸mv
61
Force F and acceleration a
62
12/17/19
11
ThegeneralformofNewton’ssecondlawisF=dp/dt=ma
Experimentsshowthisresultisstillvalidinrelativisticmechanicsprovidedweuserelativisticmomentum.ThustherelativisticallycorrectversionofNewton’ssecondlaw is
F=ma/{ 𝟏 − (𝒗/𝒄)𝟐}3=𝛾 3ma Forceformula
63
63
Rearrangingthepreviousequationwecanestablishwhathappenstotheaccelerationa ofaparticleofrestmassmwhichissubjectedtoaconstantforcea=(F/m 𝟏 − (𝒗/𝒄)𝟐 3=F/m𝛾 𝟑 Accelerationformula
64
64
InNewtonianmechanicsifaconstantforceF isappliedtoaparticleofrestmassm itwillcontinuetoaccelerateataconstantaccelerationa regardlessofitsspeedvInrelativisticmechanics,whenaparticleofrestmassmissubjectedtoaconstantforceF,itsaccelerationdecreasestozeroasitsvelocitytendstowardthespeedoflightInfactitdoesnotmatterhowbigtheforceornonzeromassis,accelerationwillalwaysdecreasetozeroastheparticlespeedincreasestowardsthespeedoflightTherelativisticeffectofincreasedspeedontheaccelerationofaparticleofrestmassmwhensubjectedtoaconstantforceF isillustratedinthenextfewslides
65
65
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/m𝜸3
Particle 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
69
69
12/17/19
12
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/m𝜸3
Particle acceleration a1F/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
70
70
Relativistic Work and Particle Energy
75
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
76
76
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
77
77
12/17/19
13
0 0 .5 1 1 .5 2 2 .5
Particlekineticenergy
79Speedvrelativetothespeedoflightc(v/c)
0
0.5mc2
1mc2
1.5mc2
2mc2
2.5mc2
3mc2
3.5mc2
4mc2
K=(𝜸
–1)mc2 (Kineticenergy)
K
Relativistickineticenergybecomesinfiniteasv approachesc
79
0 0 .5 1 1 .5 2 2 .5
Particlekineticenergy
81Speedvrelativetothespeedoflightc(v/c)
0
0.5mc2
1mc2
1.5mc2
2mc2
2.5mc2
3mc2
3.5mc2
4mc2
K=(𝜸
–1)mc2 (Kineticenergy)
K
Newtonianmechanicsincorrectly predictsthatkineticenergyonlybecomesinfiniteifv becomesinfinite(K=1/2mv2)
81
Total particle energy E, Rest energy (E = mc2) and Massless energy (E = pc)
82
Torecall,therelativistickineticenergyequationforamovingparticleincludestwoterms
K= mc2
1−v2/c2– mc2
Themotiontermdependsonmotionandtheenergytermisindependentofmotion
ItseemsthatthekineticenergyofaparticleisthedifferencebetweensometotalenergyEandanenergymc2 thatithasevenatrest
83
Motionterm Energyterm
83
12/17/19
14
A particle’s total energy E can thus be expressed as follows
E = K +mc2 = mc2
1−v2/c2= 𝜸mc2 Total particle energy
84
84
Tosummarise,thetotalenergyEofaparticleisthesumofitsKineticenergyplusitsrestenergy
Whatisapparentisthatevenwhenaparticleisatrestitstillhasenergy
Thisiscalleditsrestenergywhichisproportionaltoitsrest(andonlyrest)mass
Thishasbeenexperimentallyconfirmed.Whenunstablefundamentalparticlesdecay,thereisalwaysanenergychangeconsistentwiththeassumptionofarestenergyofmc2witharestmassofm 85
85
Thesimplestexampleofthepresenceofrestenergyisthereleaseofenergyofdecayofaneutralpion(𝝿 ).
Itisanunstableparticleofmassmwhichwhenitdecays(withzerokineticenergybeforeitsdecay)releasesradiationwithanenergyexactlyequaltom𝝿 c2
Toputthingsintoperspective,a50ggolfballhasenoughrestenergytopotentiallypowera100Wlightbulbfor1.3millionyears!
86
86
Withabitofmanipulationthemomentumandrestenergyequationscanbereformulatedasfollows
(p/m)2 = v2/c2
1 − v2/c2 and(E/mc2)2= 7
1−v2/c2
Subtractingandrearrangingtheseequationsgivesus
E2 =(mc2)2 +(pc)2
87
87
12/17/19
15
Formasslessparticles(m=0)thepreviousexpressionbecomes
E=pc
Allmasslessparticlesthustravelatthespeedoflightandhavebothenergyandmomentumsuch
Photons, thequantumofelectromagneticradiationaremassless
Theonlyotherknownmasslessparticleisthegluon88
88
Theexpressionalsosaysthatforparticlesatrest(p=0),thetotalenergyequationreducesto
89
E=mc2 Einstein’sfamousrestenergyequation
89
Conservation of mass energy
90
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
91
91
12/17/19
16
92
92
Block 111 Virginia – class nuclear attack submarine
93
93
94
Fatmanreplica
94
More Relativistic phenomena in nature
95
12/17/19
17
Thestructureofspacetimeisresponsiblefortheforceofgravityandthestrangeideathattheearthisfallinginastraightlinearoundthesun!
ThesunandallthestarsgettheirenergyprincipallyfromhydrogenfusionbecauseE=mc2
CosmicexplosionsarealsodrivenbyE=mc2
InastrophysicstheredorblueDopplershiftofcelestialbodiestellushowfaststarsareapproachingorrecedingfromuswhichhasledtoourunderstandingoftheexpandinguniverse
96
96
Theheatgeneratedbythedecayofradioactiveelementsintheinnerlayersoftheearthprovidesmorethan50%oftheheattokeeptheselayersmolten
Themovementoftectonicplatesdependsonhavingamoltenmassonwhichtheycan‘float’
Thisishowourcontinentsandmountainsareformed
Theearth’srotatingmoltencorealsocreatestheearth’smagneticfieldwhichisvitalinprotectingusfromharmfulradiation 97
97
98
Untilrecentlymarinershavereliedheavilyonthemagneticcompassfornavigation
98
99
Auroraborealis
99
12/17/19
18
You’veprobablynotgivenitmuchthought,butthereasonwhygoldisyellow(orrather,golden) isdeeplyingrainedinitsatomicstructureandit’sbecauseofsomethingcalledrelativisticquantumchemistry
Simplyput,gold’selectronsmovesofast(± c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden
Thesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasily
100
100
101
Simplyput,gold’selectronsmovesofast(±c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden
101
You’veprobablynotgivenitmuchthought,butthereasonwhygoldisyellow(orrather,golden) isdeeplyingrainedinitsatomicstructure—andit’sbecauseofsomethingcalledrelativisticquantumchemistry
Simplyput,gold’selectronsmovesofast(± c/2)inordertoavoidbeingsuckedintothenucleusthattheyexhibitrelativisticcontraction,shiftingthewavelengthoflightabsorbedtoblueandreflectingtheoppositecolour:golden
Thesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasily
102
102
Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincrease.Withmercury,thebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit.
103
103
12/17/19
19
Theouterelectrongets’trapped’intheinnerorbitalsnearerthenucleusandisthereforenotfreelyavailabletoreactwithotherelementsIncontrastLithium,whichisinthesamecolumnintheperiodictable,isveryreactiveThesesamequantumrelativisticeffectsarealsothereasonwhygolddoesnotcorrodeeasilyLikegold, mercuryisalsoaheavyatom,withelectronsheldclosetothenucleusbecauseoftheirspeedandconsequentmassincreaseWithmercurythebondsbetweenitsatomsareweak,somercurymeltsatlowertemperaturesandistypicallyaliquidwhenweseeit 104
104
105
105
More Practical applications of special relativity
106
Inparticleacceleratorsmanyparticleshaveveryshorthalflives.Atspeedsclosetothespeedoflighthalflivesaresignificantlyincreasedgivingresearcherstheopportunitytostudythem
Moderncomputerchips.Thisalittlemoreesoteric,butdesigningsolid-stateelectronicsdependsonbeingabletomodelelectronbandstructures.Thatoftenrequiresrelativisticcorrectionstodosoaccurately
Inmedicine,manybodyscannersrelyonrelativisticsciencefortheiroperation
107
107
12/17/19
20
PPet Scanner
108
Positron emission tomog-raphy(PET) scanner
108
Special relativity conclusions
109
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
110
The end
Email address:
113
113