Einstein's concept of a clock and clock paradox

0.37 MB·English

by  Wang Guowen

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0 Einstein’s concept of a clock and clock paradox
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Wang Guowen
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J CollegeofPhysics,Peking University,Beijing,China

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2 January 25,2005

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p Abstract
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n A geometric illustration of the Lorentz transformations is given. According
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to similaritybetween space and timeand correspondence between a ruler and a
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clock, like the division number in a moving ruler, the tick number of a moving
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s clock is independent of its relative speed and hence invariant under the Lorentz
c
transformations. Sothe hand of the moving clock never runs slow but the time
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s intervalbetweenitstwoconsecutivetickscontracts. ThusitisEinstein’sconcept
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ofslowingofthehandsofmovingclockstocreatetheclockparadoxortwinpara-
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dox. Regrettably, theconcept oftheclockthatEinsteinretainedisequivalent to
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Newton’sconceptofabsolutetimethatherejected.ThisisablemishinEinstein’s
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otherwiseperfectspecialrelativity.

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v Key words: special relativity, time contraction, time dilation, clock paradox, twin
1 paradox,lightclock
3 PACS:03.30.+p
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0 1 Introduction
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/ Aswellknown,theconceptofslowingofamovingclockasphysicalinterpretationof
s
relative time was first stated in Einstein’s 1905 paper [1] and the lifetime dilation of
c
i movingorganismsmentionedinhis1911paper[2]. Thiskindofconcepthascreated
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y theso-calledclockparadoxortwinparadox.Thetwinparadoxconcernsapairoftwins,
h oneofwhichaboardarocketfliesintospaceforyearsataspeedfractionallylesslight
p speedandthe otherremainshome. AccordingEinstein’sconcept,the travellingtwin
:
v will be much younger than the other on returning. This seems to be a paradoxical
i consequencesincemotionisrelative.Aswellknown,Einstein“solvedtheparadoxby
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takingintoaccounttheinfluenceonclocksofthegravitationalfield,whichisrelative
ar to (the accelerated system) K′.′′ [3] He meant that the acceleration and deceleration
effectsandthegravityeffectarethecausescreatinganasymmetryofagingofthetwins.
Theparadoxhasevenledaminuteminorityofphysicists,forexample,HerbertDingle
[4],toarguethatEinstein’srelativitymustbefalse. Oneofthelongdebatedquestions
is whether general relativity is required to solve the paradox. If special relativity is
tenable as the overwhelming majority of physicists believes, what mistake might be
evermadebyEinstein?
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Insearchofananswertothequestion,thepresentauthorthinksofEinstein’sdis-
cussionoftimemeasurement.Hestated:“onedeterminestherateofthemovingclock
bycomparingthepositionofthehandsofthisclockwiththepositionsofthehandsof
otherclocksatrestandifitmoveswithavelocityapproximatingthevelocityoflight,
the hands of the clock would move forward infinitely slowly.′′ [2] He asserted also:
“Bydefinition,tomeasurethetimeintervalduringwhichaneventtakesplacemeans
tocountthenumberoftimeperiodsindicatedbytheclockfromthebeginningtillthe
end of the event in question.′′ [5] Clearly, his way to measure time is identical with
Newton’s. ItseemsthattheconceptoftheclockthatheretainedisequivalenttoNew-
ton’sconceptofabsolutetimethatherejected. Theauthordiscoversthatageometric
illustrationoftheLorentztransformationsandagraphicdisplayofsimilaritybetween
divisionsofarulerandticksofaclockcanmakethematterclear.
2 GeometricillustrationoftheLorentztransformations
Equivalently to Minkowski’s formulation of special relativity, we consider two four-
dimensionalsystems(x,y,x,w)and(x′,y′,x′,w′)inrelativetranslationalmotionand
assume thatthe originof the latter movesat the light speedc in accordancewith the
followingequation:
2 2 2 2 2 2
c t =x0+y0+z0 +w0, w0 =cτ0 (1)
whereτ0 isthepropertime. Thisequationexpressesmathematicallythatafreeobject
movesatthemaximumspeedcatwhichitcantravelinthefourdimensions(x,y,x,w)
wherethevariablewismuchlikethespatialones,x,yandz[6]. FromEq.(1)wecan
draw a figure shown as Fig.1 to illustrate the Lorentz transformations. In this figure
only two-dimensional systems (x,w) and (x′,w′) are drawn for the purpose of this
article and x0 = vt is given. From the projective relations of the coordinatesin the
figure,wecanwriteintuitivelywitheasetheLorentztransformationsforthepointson
thex′axisandanidentity:
x−vt
x′ = , γ =cosφ=p1−v2/c2 (2)
γ
τ +x′v/c2
t= (3)
γ
′
x +vτ
x= (4)
γ
t−xv/c2
τ = (5)
γ
2 2 ′2 2 2 2
c t +x =c τ +x (6)
Thelastequationcanbechangedinto
c2t2−x2 =c2τ2−x′2 (7)
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Figure 1: Geometric illustration of the Lorentz transformations and graphic display
of similarity between divisions of the ruler and ticks of the clock denoted by the V
symbols.
Thus,ingeneral,includingyandz,wehave
c2t2−x2−y2−z2 =c2τ2−x′2−y′2−z′2 (8)
Foraphotonoralightpulse,wehave
c2t2−x2−y2−z2 =c2τ2−x′2−y′2−z′2 =0 (9)
whichdescribestheprincipleofconstancyoflightspeed. Theτ isoftenwrittenast′
intheliteratureofrelativity.
Let a ruler be located on the x′ axis. From Fig.1 we can see directly the length
contractionofthemovingruler:
ρ=Lγ =Lcosφ (10)
Obviouslythelengthcontractionoftherulerisareciprocalprojectioneffectbetween
the two equivalentinertialsystems in relative motion and the division numberof the
rulerisindependentoftherelativespeedvandhenceinvariantundertheLorentztrans-
formations.Thelengthintervalbetweentwoneighboringdivisionsofthemovingruler
at the speed v in the x direction contracts by the factor γ. Similarly, let a clock be
locatedattheoriginofthemovingsystemandsynchronizedwithoneclockatrestin
thestationarysystem whenthe originsofthetwo systemsarecoincideatt = 0. For
themovingclock,inhis1905paper,EinsteinderivedfromEq.(3)thetimecontraction
equation[1]
τ0 =tγ =tcosφ (11)
whichissimilartoEq.(10)inform.ThiscontractioncanalsobeseendirectlyinFig.1.
Itsinverseprocessisreferredtoasdilationofthetimeτ0. Accordingtosimilaritybe-
tweenspaceandtimeandcorrespondencebetweenarulerandaclock,itisevidentthat
theticksofthe movingclockare nothingbutthe correspondencesofthe divisionsof
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theruler. Obviouslythelengthcontractionoftherulerismerelyattributedtocontrac-
tionofthelengthintervalbetweenitstwoneighboringdivisions,so,similarly,thetime
contractionofthemovingclockismerelyattributedtocontractionofthetimeinterval
betweenitstwoconsecutiveticksandthenumberoftheticksisalsoindependentofthe
speedv andhenceinvariantundertheLorentztransformations. Butweareunableto
employanexistingclocktorecordthetimeτ whichdependsonboththespeedv and
thelocationx′. Therefore,inordertorecordit,weneeddeviseanalternativeclock.
3 Time τ and τ-clock
Theclocksthatwecandevisetoindicatethetimeτ areshowninFig.2. Inthefigure,
onthelargestcirclesontheclockfaces,thearclengthsweptoutbythehandrepresents
Figure 2: The τ-clocks moving in the x direction, for example, runningat γ = 0.5:
(a)attheoriginofthemovingsystem,(b)and(c)atthepositionsalight-secondaway
fromtheoriginonthenegativeandthepositivex′ half-axises.
timetinthestationarysystem. Theγvalues(0,0.25,0.5and0.75)ontheclock’shand
representthedifferentrelativespeeds. Thereadingsinsecondsonthecircletowhich
theslidingarrowpointsrepresentthedurationofaneventinthemovingsystemjudged
by an observer in the stationary system. For example, for the case of relative speed
0.866c, namely, γ = 0.5, Fig.2(a) shows that the clock at the origin of the moving
systemticks0.5secondsandFig.2(b)andFig.2(c)showthattheclocksalight-second
awayfromtheorigintickthesamebutrun0.866secondslaterandearlierthantheone
attheorigin,respectively.Thecounterclockwiseandclockwiseangulardisplacements
ofthereadingsτ dependonthetermx′v/c2inEq.(3).Thusthiskindofclockhasdual
functions for recording both t and τ simultaneously and hence meets the relativistic
requirement of the equivalence of reciprocal “time measurements′′ by two different
inertialobservers.Thiskindofclockmaybecalledas“τ-clock′′todistinguishitfrom
traditionalclockswhichrecordonlythetimeinthestationarysystemorsystemsatvery
lowrelativespeed.Withτ-clocksbothofthetwinscanthinktheother’soneticksaless
timeperiod. Noparadox! Theτ-clockisuselessinpracticebutessentiallyimportant
forproperlyunderstandingspecialrelativity.
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4 Behavior of a light clock
The discussion of the behavior of a moving light clock is also significant for under-
standingspecialrelativity. Imaginealightclockwhichiscomposedofapairofparal-
lelmirrorsreflectinglightpulsesseparatedbythedistanced=cT /2thatlighttravels
p
in half a second, here T = 1 second. There are a source of light pulses and a light
p
detectorononemirrorforproducingeverytick signal. When theverticallightclock
movesatuniformspeedvinthehorizonalxdirection,accordingtoEq.(9),wehave
c2(T /2)2−(vT /2)2−y2(T /2)=0 (12)
p p p
c2(τ /2)2−y′2(τ /2)=0 (13)
p p
where y′ = y. Thus, differentfromthe reallength d, the “relativepath′′ of the pule
betweenthetopandbottommirrorsintheydirectionis
′
d =y(T /2)=cγT /2=cτ /2=γd=dcosφ (14)
p p p
thatsatisfiestheequationoftherighttriangle
′2 2 2
d +v =c (15)
here d′ is the contracted light path in the y direction judged by an observer in the
stationarysystem. Itis necessaryto emphasizeagainthatthe time contractionof the
Figure3: Illustration ofthe behaviorof the lightclock byrelative travelpathsof the
lightpulsefordifferentγvalues.
moving light clock is merely attributed to the contraction of the period between its
two consecutive ticks. This means that the light pulse hits the top mirror when the
relative light path equals d′ and the detector produces a tick signal when it gets to
bottommirror. Fig.3illustratesthebehaviorofthelightclockbyrelativetravelpaths
ofthe lightpulse for differentγ values. Thusthe explanationoftime dilationby the
lightclockintextbooks,suchasFeynmanLecturesonPhysicsvolume1[7],iswrong
becauseofignoringtherelativisticcontractionofthetravelpathofthelightpulse.
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5 Conclusion
We have seen that, like the division number in a moving ruler, the tick number of a
movingclockisindependentofitsrelativespeedandhenceinvariantundertheLorentz
transformations.Sothehandofthemovingclockneverrunsslowbutthetimeinterval
betweenitstwoconsecutivetickscontracts.ThusitisEinstein’sconceptofslowingof
thehandsofmovingclockstocreatetheclockparadoxortwinparadox. Regrettably,
hiswaytomeasuretimeτ isidenticalwithNewton’s. Inotherwords, theconceptof
the clock that he retained is equivalentto Newton’s conceptof absolute time that he
rejected.
Similartotheclock’sticking,theheartbeatingofthetravellingtwinneverrunsre-
allyslowbutaccordingtospecialrelativitythebeatingperiodbetweentwoconsecutive
beatscontracts. Sincethehumanlifetimemaybeconsideredasbeingbasicallydeter-
minedbythetotalnumberoftheheartbeats,nooneofthetwinswillbecomeyounger
thantheotherwhentheyreunite. Thus,thegeneralrelativityisnotrequiredtoresolve
thetwinparadox.ItisEinstein’sassertionofslowingofthehandsofmovingclocksto
createtheparadoxandhencemakemanyphysicistsbelieveslowingofallphysicalpro-
cesseswiththeincreasedspeed,includingchemistryreactions,nuclearreactions,life
processandothers. ThisisablemishinEinstein’sotherwiseperfectspecialrelativity.
Finally, the authorwouldsay that if the presentconclusionis correct, it will pro-
hibits us from believing the assertion that some related experiments [8,9] have con-
firmed the Einstein’s prediction of slowing of moving clocks that creates the clock
paradox.
References
[1] A.Einstein,Ann.derPhys.17,891(1905)
[2] A.Einstein,Naturforsch.Ges.56,1(1911)
[3] A.Einstein,Naturwissenschaften6,697(1918)
[4] H.Dingle,Nature216,119(1967)
[5] A.Einstein,Arch.Sci.Phys.Natur.29,5(1910)
[6] G.Wang,inProceedingsoftheInternationalConferenceonHighSpeedPhotog-
raphyandPhotonics,WangDaheng,Ed.(SPIE,Bellingham,Washington,1988),
vol.1032,pp.428-431
[7] R.P.Feynman,R.B.LeightonandM.Sands,TheFeynmanLectureonPhysics
(Addison-Wesley,Reading,MA,1963),vol.1,sec.15.4
[8] D.H.FrischandJ.H.Smith,Am.J.Phys.31,342(1963)
[9] J.C.HafeleandR.E.Keating,Science177,168(1972)
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