9|SpecialRelativity
297
measurethelengthofthetrain,measurethetimeittakestomovepastandmultiplyitbythespeed:
L
=
vT
0
. (
T
0
isthetimethatyou read onyour clock.) ) Thequestionis s nowto ndthelengthas
measuredbyapassenger.Callthelengthandthetimethatthepassengermeasures
L
0
and
T
instead
of
L
and
T
.(
L
0
isthelengththepassengerobservesforthe[tohim,stationary]train.) Thepassenger
doesn’tneedanotherclocktomeasure
T
,allthat’sneededistobeabletoreadyour clock. Ifyours
recorded
T
0
,andaccordingtothepassengeryouaremovingatspeed
v
,then(againaccordingtothe
passenger)yourclockisrunningslow.Thepassengerthensaysthatthetime
T
foryoutohavemoved
fromoneendofthetraintotheotherisgreaterthanyourmeasured
T
0
. Thecalibrationfactorbetween
theclocksis
p
v
2
=c
2
,sothepassengersaysthatthetimeinwhichyougofromoneend ofthe
traintotheotherislongerthanthe
T
0
thatyoureadonyourslowclock. Itis
T
0
=
p
v
2
=c
2. Thisis
thepassenger’s
T
andthelengthasmeasuredbythepassengeris
L
0
=
vT
. (RereadtheabovePoint
ofConfusion.)
yourview
v
T
0
passenger’sview
v
T
0
Fig.9.2
you:
L
=
vT
0
passenger:
L
0
=
vT
=
vT
0
q
v
2
=c
2=
L
q
v
2
=c
2
lengthchange:
L
=
L
0
q
v
2
=c
=
L
0
=
(9
:
2)
You seethe train as shorter r thanthe e passengerdoes. . This s length contraction is asecond peculiar
featureofspecialrelativity.
This derivation of f length contraction n (also o called the e Lorentz contraction) involves s only the
simplestalgebra,but itrequirescarefulattentiontothedetails ofwho’s looking atwhat and when.
Thereisanotherwaytogettothis result,onethatreversesthesetwodiculties. . Thisconceptsare
easy,butthealgebraismoreinvolved. Theideaissimplytotakethelightclockoftheprevioustwo
pagesandturnitonitsside,sothatthelightisgoingleft-to-rightandright-to-leftastheclockmoves
right. Itshouldreadthesametimeasitdidwhenitwasupright,andthatwillletyougureoutwhat
itslengthis. Seeproblem9.6forsomesuggestionsabouthowtodothiscalculation(orbetter,youcan
tryitonyourownrst).
It’seasytomixuptheapplicationsofthesedilationandcontractionequations,sobeforeapplying
eitherofEqs.(9.1)or(9.2)youshouldthinkaboutthequalitativebehaviorofthetimeanddistance
measurements. Ifaclocktakes s twosecondstomarkoasecond,itsownerwillthinkthatsomeone
justdidthe100meterdashinveseconds. Thismeansthatyoucan’tsimplyplugintoformulas.The
qualitativeanalysisshouldcomerst.
Amovingclockrunsslow.
Thecalibrationfactorbetweenclocksis
=1
=
p
v
2
=c
2
.
Thatfactorisgreaterthanone.
So,doesthefactorgointhenumeratororinthedenominator?
Terminology: Properlengthisthelength h ofan objectas measuredby someone whosays itis
notmoving. Proper time isthetimeintervalbetweentwoevents s asmeasuredbysomeonewhosays
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9|SpecialRelativity
298
theoccurredatthesameplace*|theclockisstandingstill. Forthetrainexample,
L
0
istheproper
lengthofthetrain,and
T
0
isthepropertimeonmyclock. Thereisnouniversalconventionabouthow
youplacesubscriptshere,sodon’tassumethatasub-0willalwaysbeproper. Forproperacceleration
seeequation(9.22).[Whatwould\propervelocity"be?]
Iflengthcontracts,shouldn’tthe
L
0
intheoriginalclockcontract? Shouldthe
L
0
inEq.(9.1)
beavariabletoo?No,butitisnotonlyalegitimatequestion,it’sanimportantone.Inthatderivation
Iassumedthatbothyouandyourfriendusedthesamevalueforthatnumber,thelengthperpendicular
tothemotion.Isthatvalid? Theanswerisyes,butnowyouhavetotestittobecertain.
The proofis by contradiction. . Assumethe e statementis false and showthatitdoesn’t work.
Take the light clock,supposedlylength
L
0
. Yourfriend d wants tomeasureittosee if itremains
L
0
evenwhenitmovespast,soheusesaruleroflength
L
0
andholdsitupsotherulerandtheclockjust
misseachother. Tobecertainofhismeasurementheattachesapairofpaintbrushestothetwotips
oftheruler,setsothattheywillmarktheendsoftheclockasitpasses.Iftheclockshrinksbecauseof
itsmotionthenthepaintbrushononeendoftherulerwillmarktheclockwhiletheotherendmisses.
Thetroubleisthatpaintleavesapermanentmarkontheclock. Youandyourfriendcangettogether
overabeeraftertheexperimentinordertocomparedata. Ifyourclockhasonlyonepaintmark,then
thereisalaterallengthcontractionandyouweremoving. Ifithas twopaintmarks thatarenot at
theends,thenhewasmoving. Eitherwayyouhave e violatedtherstaxiom ofthetheory,thatonly
relativevelocitiescanbemeasured. Whatiflaterallengthsexpandedinsteadofcontracting? Youget
intoexactlythesamecontradiction,butinterchangingwhoismoving.
Anotherwaytopicturethis: Ifarailroadtrainisenteringatunnelthroughamountain,andif
thetunnelisjustbarelylargeenoughtoaccommodatethewidthofthetrainwhenthetrainismoving
slowly,whatwill happenwhen thetrainis movingrapidly? ? If f thereis either alateralcontraction or
expansion becauseofthespeedthen someone,eitherthetrainengineerorthepersonwhobuiltthe
tunnel,willthinkthatthetrainwillstilltwhiletheotherwillexpectacrash. Thatisaqualitative
dierence,sotheycan’tbothberight.
Paradoxes
Undernormalcircumstances,i.e.beforeyoustartedtostudy relativity,youmay havean occasionto
thinkthatyourwatchkeepsgoodtimeandthatyourfriend’sisrunningslow.Yourfriend,whothinks
justashighly ofhis timepiece,wouldthinkthatyourwatchis runningfast. . Thecalculationoftime
dilation leading toEq.(9.1)says thatyourfriend’s clockruns slowbecauseit’smoving. . Ifthe e rst
assumptionofrelativityiscorrect,thatyoucan’t tellwhichofyouismoving,thenbothof you will
believethattheother’sclockisrunningslow|andbothberight. Howcanthathappen?
Thesameparadoxappliestolengths. Ifthemovingtraincontracts,apassengeronthattrain
canlookoutandsay thatanothertrainyou thinktobestandingstillhasshrunk. . Howcanbothbe
true? ResolvingbothoftheseproblemswillhavetowaitafewpagesuntiltheLorentztransformation
appears.
Ifyoudrivealongcarandhaveashortgarage,canyoutyourcarwithinthegaragebydriving
extrafast? Someone e standingoutside and watchingyou try to do this wouldsay yes; yourcarhas
shrunk,soitnowts inside. . You,asthedriver,seethegarageapproachingyou,onlyitisnoweven
shorterthanitwasbefore.You’renevergoingtotinsidenow.Who’sright?
Iftwopeopleareapproachingeachother,oneat
3
4
c
fromtheleftandtheotherat
3
4
c
fromthe
right,andyouaskoneofthemhowfasttheotherismoving,theanswerwillnotbe
3
2
c
,but
24
25
c
. A
derivationofthiswillappearinEq.(9.13).
* Thereisanobscurewordforthis: collocal isthespatialanalogofsimultaneous.
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9|SpecialRelativity
299
Example
Beforegoingintosomemoreanalyticalproblemswithrelativity,afewelementaryexamplesareuseful.
Cosmic rays are(mostly) very highenergy protons that come from outer r space e andhit the Earth’s
atmosphere.Whensuchanenergeticprotoncollideswithanoxygenornitrogennucleusitsimplyblasts
thenucleus apartand in the process creates many thousands or r millions s of othernew w particles. . A
majorcomponentoftheresultingdebris isthemuon. . Theseareparticles s thatarebasically massive,
butunstable,electrons;
m
=207
m
e
,andthemeanlifetimeoftheseparticlesis2.2microseconds. In
thisshorttime,evenmovingatnearlythespeedoflight,themeandistancesuchaparticlegoesbefore
decayingisabout
c
=(310
8
m
=
s)(2
:
2
s)=(300m
=
s)(2
:
2
s)=660m=0
:
66km
Whatdoes \mean life"mean? ? Radioactivedecay y followsstatisticallaws, , and starting witha
populationofparticlesattimezero,thefractionofthemleftattime
t
is
e
t=
,orexpressedinterms
ofthetotalpopulation,
N
(
t
)=
N
0
e
t=
. Tondtheaveragevalueofthedecaytime,let
dN
bethe
changeinthisnumberintime
dt
,sothat 
dN
isthenumberthatdecayedbetweentime
t
and
t
+
dt
.
Themeantimetodecayisthenasumoverallthesetimesdividedbythetotalnumberofparticles:
1
N
0
Z
t
dN
)=
1
N
0
Z
1
0
t
dN
dt
dt
=
Z
1
0
1
te
t=
dt
=
Startingfromanaltitudeof20kmandmovingstraightdown,thefractionoftheseparticlesthat
wouldreachtheEarth’ssurfaceis
e
20
=
0
:
66
10
13
.Despitethis,somanyreachthesurfacethatyou
willhaveathousandormoreofthempassingthroughyoueachminute. Thisisnotbecausethereare
somanycosmicraysthathittheEarth,butbecauseoftimedilation. Themuonsaremovingsofast
thatthetime dilation eectislarge,andtheirlifetime is farlongerthan thetwomicrosecondsthat
theyhavewhenatrest. Afactorof1
=
p
v
2
=c
=10istypical,andyoumightguessthatsucha
factorwouldleadtoachangeinthenumberreachingthesurfacebyaboutafactorof10.Notso. This
exponentialchangesthefractionhittingthesurfacefrom
e
ct=c
=
e
20km
=
0
:
66km
=710
14
to
e
20
=
6
:
6
=0
:
05
(9
:
3)
Thatisafactorofabout10
12
,andthisisamajorpartofthebackgroundradiationthatyoulivewith
allyourlife.
Iusedthe\meanlifetime"todescribethisandyoumaybemorefamiliarwiththe\half-life". They
areproportional. Seeproblem9.3tounderstandtherelationbetweenthesetwoideas:
t
1
=
2
=0
:
693
.
Also,notallmuons willhave thesame speed,sothey won’t all havethe sametimedilationfactor.
Howdoyouhandlethat? Seeproblem9.8.
Example
The GlobalPositioning System is socommonly used thatit is standardequipmentinsome cars,
andisavailableforpersonalusealmosteverywhere.Itdependsonpreciselymeasuringthesignalsfrom
severalsatellitesinorbitataradiusof26600km(measuredfromtheEarth’scenter). Theorbitalperiod
isabouthalfaday;itsorbitalspeedisabout3.8km/s. Atthisspeedthetimedilationeectis
=
1
p
v
2
=c
2
1+
v
2
2
c
2
;
with
v
2
2
c
2
=
1
2
3
:
8
300000
2
=810
11
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9|SpecialRelativity
300
Inoneday =86400s,theclockerrorwouldbe86400s810
11
=7
s. Atthespeedof f light,
thedistanceerrorfromthisisthen7
s
.
300000km
=
s=2km.Ifnocorrectionhadbeenmadeforthis
eecttheGPSwouldhavebeenworthless. Theeectsfromgeneralrelativityhavetheoppositesign
butareevenlarger,givingacombineddriftthat,ifuncorrectedfortheserelativisticeects,wouldbe
about10km/day.
9.2Space-TimeDiagrams
Aswithalmosteverythingelse inphysics,theabilitytodrawasketchofyourproblemisinvaluable.
Sometimesit’sadrawingandsometimesit’sagraph. Here,thegoalistoturnconceptually y dicult
questionsaboutmeasurements with clocks andrulersintoconceptually simplerproblems inanalytic
geometry. Whenyoudescribethemotionofaparticle,youcangiveitspositionasafunctionoftime
and graph that. . Fornow,onedimension n ofspace(
x
)willbeenough. The
y
and
z
coordinateswill
comealonglater.
Dealing with objects that move near r or at the e speed d of f light t makes it t necessary y to o choose
convenientunits. Thespeedoflightis s 310
8
m
=
s=300m
=
s=1ft
=n
s. Thetricktomakethis
easy is to measure time in microsecondsand measurespace in unitsof300meters. . Inotherwords,
pickthe units so that
c
hasthe numericalvalueone. . If f you preferlight-years andyears orfeetand
nanoseconds,that’so.k. too. Choosethetimeaxisup*andthe
x
-axisleftandrightandthepictures
willlooklikethese:
t
x
t
x
t
x
t
x
stationary
x
=
x
0
simultaneous
t
=
t
0
x
=
ct
x
=
vt;v<c
Thesecondonerepresentsasetofeventsthatoccuratthesametime,theequationis
t
=
t
0
.
Therstandfourthgraphspresentconstantvelocitymotionofaparticle,oneofthemwith
v
=0and
theotherwith
v
=300m
=
4
s=
c=
4.
Thethirdisagraphofthemotionofaphoton,
v
=
c
.
The
t
and
x
coordinateaxeshaveequationsthatarerespectively
x
=0and
t
=0.
Thetwopointsinthethirdgraphrepresenteventsat(
x;t
)=(2
:
5
;
0
:
5)and(1
:
5
;
2
:
5)intheunitsof
300m,1
s.
ButFirst:
Whenyouaredealingwithordinarytwo-dimensionalrectangularcoordinate
x
and
y
,sometimes
youneedtoswitchtoarotatedcoordinatesystem. Ifyou’redoinganelementarymechanicsproblem
and
~g
isdown,youmaychoose
x
horizontaland
y
vertical,oryoumaynot.Perhapssomeotheraspect
oftheproblem,maybeahill,suggestsadierentcoordinatesystemwith
x
0
alongtheinclineofthehill
and
y
0
perpendiculartoit.Whatittherelationshipbetweenthesecoordinates?Justexpresseachpair
ofcoordinatesintermsof
r
andtheangles. Theanglethehillmakeswiththehorizontalis
,andthe
* Whyistimeupandspaceleftandright? ? ForthesamereasonthatpeopleintheU.S.driveon
therightandthoseintheU.K.driveontheleft.That’sthecustom.Besides,ifyouhave
x
,
y
,and
t
,
thenlayingoutthe
x
-
y
planesort-ofhorizontallyseemsbetter.
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9|SpecialRelativity
301
singlepointPhascoordinates
x
x
0
y
y
0
r
P
x
=
r
cos
y
=
r
sin
x
0
=
r
cos(
)
y
0
=
r
sin(
)
(9
:
4)
=)
x
0
=
r
cos
cos
+
r
sin
sin
=
x
cos
+
y
sin
y
0
=
r
sin
cos
r
cos
sin
x
sin
+
y
cos
(9
:
5)
Theseequationsshowhowtorelatethecoordinatesthattwodierentpeopleusetodescribethesame
point,andallthatyouneededwasacoupleofstandardtrigidentities. [Lookattheexercisesatthe
endofthischapter,page325.]
Atrivial-soundingquestion: Whataretheequationsforthevariousaxes?Asimpleanswer: the
x
-axisequationis
y
=0andthe
y
-axisequationis
x
=0.Similarly,the
x
0
-and
y
0
-axeshaveequations
y
0
=0and
x
0
=0.Ifthisseemstrivial,justwait.
Thereasonforlookingattheserotatedcoordinatesystemsisthatsomethingverymuchlikethis
canbedonewithspace-timecoordinatesystems. Thetransformationswillnotberotations,butthey
willhavemanypropertiesanalogoustothem. Forspaceandtime,thekeystepistorealizethatyou
havetoprovideaprecisestatementofwhatthesecoordinatesmean.You’vealreadyseenthattimesand
lengthsarenolongersuchsimpleconcepts,sothecoordinatesthatbuildonthemhavetobedened
withcare.
Onemethodtodene thesecoordinates uses toolsthatareconceptuallyverysimple: : aclock
andaradarset.Theradarsetsendsoutelectromagneticpulsesthatcanbouncefromadistanceobject
andaclocktimesthere ectedsignal,tellinghowfarawaytheobjectis.
Let
t
1
bethetimetheradarpulseissentand
t
2
thetimeitreturns. Thetotaltraveltimeofthe
pulseis
t
2
t
1
,halfineachdirection,sothetimeatwhichthepulsehitstheobjectis
t
1
+
1
2
(
t
2
t
1
)=
1
2
(
t
1
+
t
2
)
;
andthedistancetoitis
c
2
(
t
2
t
1
)
Translatethisintotheanalyticgeometryofspace-timediagrams.Thelighttravelsatconstantspeed,
sothatgoingoutitisastraightlineinthepicture.Returning,it’sanotherstraightline.
x
c
(
t
t
2
)
x
=
c
(
t
t
1
)
t
x
t
1
t
2
E
x
E
=
c
2
(
t
2
t
1
)
t
E
=
1
2
(
t
2
+
t
1
)
(9
:
6)
Theeventatwhichtheradarpulsehits theobjectislabeledE.Perhapsitisapoliceradarhittinga
speedingcar. Inanycase,thespace-timecoordinatesoftheeventaredenedbythetwomeasurements
oftimebackattheradarset.
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9|SpecialRelativity
302
Nowcomestheinterestingpart: Bringyourfriendbackintothepicture,movingatvelocity
v
.
Usingthesamedenitionstodothecomputation,whatcoordinatesdoesyourfriendndforthissame
event? Fornow,makethevelocity
v
aconstantbecauseitmakesthemathematicsfareasier,andthe
motionisastraightline inthepicture,
x
=
vt
. Insteadofhavingaseparateradarandclock,which
wouldhavetobecalibrated,simplyusethesamesetforboth.Thenitisaquestionofcomputingwhat
measurementsthemovingobserverwillget.
Theoutgoingandthereturningradarpulsespassyourfriendattwopointslabeledaandb.The
timesatwhichtheypassare
t
a
and
t
b
. Computingeachofthosetimesisnowndingtheintersection
oftwolines,solvingthetwopairsofsimultaneousequationsontheleft.
b:
x
c
(
t
t
2
)
x
=
vt
a:
x
=
c
(
t
t
1
)
x
=
vt
t
x
a
b
t
a
t
b
x
=
vt
E
t
b
=
ct
2
=
(
c
+
v
)=
t
2
=
(1+
v=c
)
t
a
=
ct
1
=
(
c
v
)=
t
1
=
(1 
v=c
)
(9
:
7)
These times are when you think theradarpulse passed the movingobserverwhen it goes outand
returns.Theyarenotthetimesthemovingclockwillread. Thatclockisrunningslow,remember,by
afactor
p
v
2
=c
2,sotheactualtimesthatheseestheradarpulsepassbyarethesmallertimes
t
0
1
and
t
0
2
:
t
0
1
=
t
a
q
v
2
=c
2
=
t
1
(1 
v=c
)
q
v
2
=c
2
t
0
2
=
t
b
q
v
2
=c
=
t
2
(1+
v=c
)
q
v
2
=c
2
(9
:
8)
Nowallthat’sleftisalgebra,eliminatingalltheunwantedvariables,(
t
1
,
t
2
,
t
a
,
t
b
,
t
0
1
,
t
0
2
). The
newcoordinatesoftheeventare
x
0
E
=
c
2
(
t
0
2
t
0
1
)
t
0
E
=
1
2
(
t
0
2
+
t
0
1
)
andsolvingEqs.(9.6)for
t
1
and
t
2
!
t
1
=
t
E
x
E
=c
t
2
=
t
E
+
x
E
=c
(9
:
9)
x
0
E
=
c
2
t
2
(1+
v=c
)
t
1
(1 
v=c
)
q
v
2
=c
2
=
c
2
t
E
+
x
E
=c
1+
v=c
t
E
x
E
=c
v=c
p
=
1
2
(
ct
E
+
x
E
)(1 
v=c
)
v
2
=c
2
(
ct
E
x
E
)(1+
v=c
)
v
2
=c
2
p
=
x
E
vt
E
p
v
2
=c
2
t
0
E
=
1
2
t
2
(1+
v=c
)
+
t
1
(1 
v=c
)
q
v
2
=c
2
=
1
2
t
E
+
x
E
=c
1+
v=c
+
t
E
x
E
=c
v=c
p
=
1
2
(
t
E
+
x
E
=c
)(1 
v=c
)
v
2
=c
2
(
t
E
x
E
=c
)(1+
v=c
)
v
2
=c
2
p
=
t
E
vx
E
=c
2
p
v
2
=c
2
9|SpecialRelativity
303
JustasEq.(9.5)referredtothepointPwithoutwritingtheletter,theselookneateras
x
0
=
x
vt
p
v
2
=c
2
=
(
x
vt
)
t
0
=
t
vx=c
2
p
v
2
=c
2
=
(
t
vx=c
2
)
y
0
=
y
z
0
=
z
(9
:
10)
TheseLorentztransformationequationsareanalgebraiccodicationofthetwoaxiomswrittenonthe
rstpageofthischapter.
=1
=
p
v
2
=c
asbefore,and
=
v=c
isoftenusedtoo.
Whatabouttheothercoordinates,
y
and
z
?Theydon’tchange,andthereasonisnothingbut
theargumentonpage298,showingthatthereisnocontractioninthelateraldirection.
Thiswasafairamountofalgebratoarriveattheserathersimple-lookingequations,andifyou
look atotherintroductory y texts on relativity youwill ndother, , farless s complicated derivationsof
thesametwoequations. WhythendidIchoosesuchaninvolvedwaytogettoaresultthatanother
authormay choose to do in afewlines? ? Therstreasonis s that this methodmay be algebraically
morecomplex,butitusesonlythesimpleconceptofdetermininghowyoumeasure
x
and
t
.Asecond
reasonisthatthismethodisnotrestrictedtomotionatconstantvelocity;afterall,ifanobservercan’t
makemeasurementswhileundergoinganaccelerationevenassmallas
g
,thenit’sanawfullylimited
theory. There’snoconceptualchangeinhavingan n acceleratedreferenceframe,justabigchangein
thequantityofmathematics.* Seeproblems9.44and9.45.
As with the coordinate change Eq. . (9.5) and the accompanyingdiagram, , the picture of this
transformationisimportant. Asyoucaneasily y see,itisnot arotation,andthenewaxesdon’tlook
likethoseonpage301.
x
-axis:
t
=0
t
-axis:
x
=0
x
0
-axis:
t
0
=
(
t
vx=c
2
)=0
t
0
-axis:
x
0
=
(
x
vt
)=0
x
t
x
0
t
0
simultaneous
0
simultaneous
Thedashedlinesllinthecoordinatesasdenedby theequations
x
0
=constantand
t
0
=constant.
Thepicturehas
v
=
c=
2.
Simultaneity
The phrase \at the same time"refers to events thatoccurredatthe same value of
t
. The e moving
observerhoweverusesdierentcoordinates,so\atthesametime"referstothesamevaluesof
t
0
,the
dashed linesparallelto the
x
0
-axis in the abovedrawing. . That t simultaneity depends on the motion
oftheobserveristhecentralradicaldepartureinspecialrelativity. Itisthisdeparturefromcommon
intuitionthatallowstheparadoxesinthetheorytoberesolved.
Howbigistheeect?Foranastronomicalexample,theEarthorbitsthesunataspeedofabout
30km/s. Thisis10
4
c
. Atthedistanceofthestarnearestthesun,aboutfourlight-years,howbiga
discrepancydoesthismake?
t
0
=0=
t
vx=c
2
;
!
t
=
vx=c
2
=10
4
c
.
4lt-yr
=c
2
=4
.
10
4
yr
.
10
7
s
1yr
=12000s3
:
5hr
Thisappearsinsignicant,butundersomecircumstances amuchsmallerspeedcanproducea
verylargeeect. Thecurrentinanordinaryelectricwireconsistsofelectronsinmotion. Theaverage
* anddespitewhatyoumayseeinsomebooks,thishasnothingtodowithgeneralrelativity.
9|SpecialRelativity
304
drift velocityofthoseelectrons is surprisingly small,andinhomewiringitsmagnitudeiscommonly
lessthan0.1millimeterspersecond. Howcansuchasmallspeedaccountforalargeelectriccurrent?
Therearealotofelectrons.
Comebacktothequestionofcurrentinamoment. First,lookfurtheratsimultaneityanduse
theLorentztransformationtorederivelengthcontraction. It’saconsistencycheck,anditbettergive
thesameanswerasbefore. Someoneismovingpastataspeed
v
andheclaimstohaveanobjectof
length
L
0
. Notice: Hesaysthatitisn’tmoving,sohegetstosaythatitsproperlengthis
L
0
. What
arethealgebraicequationsdescribingthefrontandbackendofthisobject?
x
0
=0
;
x
0
=
L
0
forall
t
0
|whatcouldbesimpler?
(9
:
11)
Itooktheleftendatzerotosavealgebra.Thetwolinesofconstant
x
0
producetwoparallellinesinthe
x
-
t
coordinatesystem. SimplycombinetheLorentzequations,Eq.(9.10)withtheequation
x
0
=
L
0
andndtheintersectionofthatlinewiththeline
t
=0.Thatisthedotinthegure.
x
0
=
L
0
t
=0
x
0
=
(
x
vt
)
9
>
=
>
;
!
L
0
=
x
x
=
L
0
q
v
2
=c
2
x
t t
0
x
0
x
0
=
L
0
Thedirectapplicationofthetransformationequationsthenreproducesthelengthcontractionequation
correctly. Canyoudothesamesortofanalysisfortimeinsteadoflength? Yes,youcan,seeproblem
9.5.
If you happen n to o have a ruler(stationary according to you) ) that t is exactly this length
L
=
L
0
p
v
2
=c
thenwhatdoesyourmovingfriendsee?(Youwouldcallthis
L
aproperlength,butthe
symbol
L
0
isalreadytaken.) Theequationfortheright-handendofyourruleris
x
=
L
,thedashed
lineinthenextpicture. Whatisthevalueof
x
0
atthetime
t
0
=0?Whatdoyouexpectittobe? That
isthedotinthispicture. Again,theLorentztransformationappears.
x
t t
0
x
0
x
0
=
L
0
x
=
L
t
0
=0
x
=
L
and
t
0
=
(
t
vx=c
2
)
x
0
=
(
x
vt
)
Eliminate
x
and
t
toget
x
0
=
(
L
v
.
vL=c
2
)=
L
q
v
2
=c
2
andtheLorentzcontractionappearsagain! Eachpersonsaysthattheother’srulershaveshrunk.
Current
Anordinary copperwirecarryingacurrent
I
has twoparts: : thestationary y positivechargesandthe
movingnegativeones.Therearecasessuchas uorescentlightswherebothtypesofchargemove,but
that’san unneededcomplication. . Infactitdoesn’tmatterwhetherthepositivechargesaremoving
rightorthenegativechargesaremovingleft.Bothrepresentacurrenttotheright.* Ifit’sconvenient
tosimplifytheexplanationsbyassumingthatthepositivechargesaremovingrightandthenegative
chargesarestationary,that’so.k. Itmattersnotwhich. Thewirehasnonetcharge,sothedistance
betweenthepositivechargesisonaveragethesame asthedistance between thenegativeones. . At
* Itwasn’tuntil1879,whenEdwinHalldiscoveredtheHalleect,thatitwaspossibletotellthe
dierence,andtheelectronitselfwasn’tdiscovereduntilthelate1890s.
9|SpecialRelativity
305
t
=0,the
x
-axis,foreverypositivechargethereisanegativeone. Inthispicturethepositivecharges
aremovingright(
x
=const.+
vt
)andthenegativeonesarestationary(
x
=const.)
x
t
x
0
t
0
{
+
{
+
Fig.9.3
Positiveandnegativechargesinawire
Dashedlinesarethecharges.
Withoutsolvingasingleequation,youcanlookatthispictureandseethatthe
movingobserver, who says s thatthe
x
0
-axisis represents simultaneous points,
concludesthatthepositivechargesarefartherapartthanthenegativeones. That
impliesthatthemovingobserverseesanetnegativechargedensityonthewire.
Ifthe\movingobserver"isacharge,itwillfeelaforcebecauseofthenetchargeonthewire.
Ifthatmovingchargemovesfasteritwillseemorechargeonthewireandwillfeelalargerforce. You
willnoticethatthisforceisperpendiculartothevelocityvector.Thisvelocitydependentforceisthere
eventhoughyou,asastationary observer, , saythat thereisnothingto push thecharge;thewireis
neutral. Insteadyoucalltheeecta\magneticeld",andconcludethatitseectis s proportionalto
velocity.Thisishowyouderivemagnetismfromthecombinationofelectriceldsandrelativity.
Thereareafewdetailstollintogetthefamiliar\
q~v
~
B
"expression,buttheessentialideas
arealreadyhere.Theconsequenceoftheanalysisisthatjustasspaceandtimebecomeintertwinedin
relativity,sodoelectricandmagneticelds. Whatisan
~
E
tomewillbeacombinationof
~
E
and
~
B
toyou. Similarly,my
~
B
willbeyourmixtureof
~
B
and
~
E
. TherelevantequationsarenottheLorentz
transformationthistime. They’requitedierent.
FutureandPast
Ifsimultaneityhasbecomemalleable,whathappenstothemeaningofthewordsfutureandpast?Are
theyaected?Yes. Accordingtoastationaryobserver,us,thefutureisdenedbytheinequality
t>
0.
The past is
t<
0. Someone e in motion uses the sameconcept, but itbecomes
t
0
>
0and
t
0
<
0
instead,andtheseareindicatedbythevariousshadedregionsinthedrawings.
t<
0
t>
0
t
0
<
0
t
0
>
0
x
t
x
0
v>
0
Fig.9.4
Thefamiliarpartis dened bythehorizontalline,the
x
-axis thatseparatesourpastfromour
future. Theunfamiliarpart t isthe pastand futureforthemovingobserver.
t
0
>
0is tiltedbecause
t
0
=
(
t
xv=c
2
),sothedemarcationbetweenpastandfutureistheline
t
=
vx=c
2
. Rememberthe
unitsI’musingtodrawthepictures:
c
=300m
=
s,theline
x
=
ct
isat45
.
c
=1inunitsof300m
and1
s,sotheboundarybetweenpastandfuturewillnotgetanysteeperthan45
. Ifthemotion
isintheoppositedirection,
v<
0,thetiltisreversedandgoesfromupperlefttolowerright.
9|SpecialRelativity
306
Thispictureimpliesthatwhatisinthefutureformemaybeinthepastforyouandviceversa.
ThetwodotsintheprecedingFigure9.4representeventsdoingthat.Noticethatthedotontheleftis
aneventthatoccurredbeforetheoneontherightaccordingtous,butitoccursafter theoneonthe
rightforthemovingobserver. Theirtime-orderisreversed. Doesthismeanthatcauseandeectcan
bereversed?No,becausesignalscan’ttravelfasterthanlight,andthatimpliesthatthesetwodotsare
farenoughapartthatlightcan’tgetfromonetotheotherinthetimebetweenthem.
Thereare two types of future and two types of past: : relative e and absolute. . Absolute e future
issomethingthateverybodyagreeson. Relativefuturewillmeandierentthingstodierentpeople,
dependingontheirrelativevelocities. Thetwodotsaboverepresentacasewherefutureandpastare
relativetotheobserver,butrememberthatthe
x
0
-axiscantiltonlyupto45
andnofarther.There
isalimitonjusthowfartherelativefutureandpastcanbepushed.
absolutepast
absolutefuture
relativepast
relativefuture
t
x
x
y
t
Fig.9.5
Theeventsintheabsolutefutureareinthefutureaccordingtoeveryone,andtheyarealsoevents
thatcanbereachedbyasignalthatstartsattheorigin. Suchasignalhastheequation
x
=
vt
with
j
v
j
c
,anditcanpassthroughanypointwithintheabsolutefuture. Inthenextpicture,jumpingup
adimensionsothattherearetwospacialcoordinates
x
and
y
togoalongwith
t
,theabsolutefuture
iswithintheupperpartofthecone. Thisconeisthe\lightcone",
r
=
p
x
2+
y
=
c
j
t
j.
Theconceptsofcauseandeectinvolvetheorderoftime. Acauseprecedesaneect. Ifthe
pastandthefuturecanbecomemuddledbecauseofrelativity,thenitputssomeconstraintsonthe
waythatthingscaninteract. Somethingcan n causeaneectonly iftheeectisinthefutureofthe
causeaccordingtoeveryone. Thismeansthatsomethingthathappensattheorigin(
x
=0
;t
=0)can
onlycausesomethingonorwithinthefuturelightcone,atadistance
r
ct
. Outsidethatcone,the
pastandfuturebecomerelativeconcepts,andtherelativepastandfuturethereareperhapsbestcalled
\elsewhere".
9.3RelativeVelocity
Youareatrest;thereisanotherperson,amovingobserverwithvelocity
v
;someotheranddierent
objecthasvelocity
u
.Whatistheirrelativevelocity?Thisquestionwillhavedierentanswersdepending
onjusthowyoudenethephrase\relativevelocity". Foronepossibledenitionyoucanaskhowyou
wouldmeasuretheirdistanceapartandhowfastthatdistanceischanging.Inyourtimeinterval
t
the
movingobservergoesadistance
v
t
. Theobjectgoesadistance
u
t
. Thedistancebetweenthose
twohaschangedbytheamount
u
t
v
t
.Youthensaythattherelativevelocityis
u
t
v
t
t
=
u
v
Whatcouldbesimpler?Thisisexactlywhatyougetifyouknownothingaboutrelativity,anditisnot
wrong. Itisjustnotthedenitionofrelativevelocityneededhere;thereisasubtledierenceinhow
itisstated.
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