Uncertainty in Measurements of Distance
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Xiv:gr-qc/0201030v1 9 Jan 2002UncertaintyinMeasurementsofDistance
JohnC.Baez
DepartmentofMathematics,UniversityofCaliforniaRiverside,California92521USA
S.JayOlson
DepartmentofPhysics,UniversityofCaliforniaRiverside,California92521USA
email:baez@math.ucr.edu,olson@math.ucr.edu
January9,2002
Abstract
NgandvanDamhavearguedthatquantumtheoryandgeneralrelativitygivealowerbound∆ℓℓ1/3ℓ2/3Pontheuncertaintyofanydistance,whereℓisthedistancetobemeasuredandℓPisthePlancklength.Theirideaisroughlythattominimizethepositionuncertaintyofafreelyfallingmeasuringdeviceonemustincreaseitsmass,butifitsmassbecomestoolargeitwillcollapsetoformablackhole.HereweshowthatonecangobelowtheNg–vanDamboundbyattachingthemeasuringdevicetoamassiveelasticrod.Relativisticlimitationsontherod’srigidity,togetherwiththeconstraintthatitslengthexceedsitsSchwarzschildradius,implythatzero-pointfluctuationsoftherodgiveanuncertainty∆ℓℓP.
1Introduction
Ithaslongbeenbelievedthatquantumgravityeffectsbecomeimportantatdistancescomparable
tothePlancklength,ℓP,andanumberofargumentshavebeenpresentedtosupportthisidea[3,4].
Toshowthissortofthing,onemainlyneedstoshowthatgravityeffectsbecomeimportantatsome
fixedlengthscaledependingonlyontheconstantsc,Gand.Dimensionalanalysisdoestherest,
sinceℓPistheonlyquantitywithdimensionsoflengththatonecanconstructfromtheseconstants.
However,insituationswhereasecondlengthscalebecomesrelevant,onecannotusedimensional
analysistosettleallcontroversies.Forexample,NgandvanDam[5]haverecentlyarguedthat
quantumgravityeffectscausesurprisinglylargeuncertaintiesinthemeasurementofalargedistance
ℓ,namely
∆ℓℓ1/3ℓ2/3P
wherethesymbolmeansthatweareignoringaconstantfactoroforderunity.Amelino-Camelia
[1]hasgoneevenfurther,arguingthat
∆ℓℓ1/2ℓ1/2P.
Uncertaintiesonthisscaleareonthebrinkofbeingexperimentallydetectable,lendingextrainterest
totheissue.However,inwhatfollows,wereanalyzetheNg–vanDamthoughtexperimentandshow
thatbymodifyingitsdesignwecandramaticallyreducetheuncertaintyofdistancemeasurements.
Ourmodifiedthoughtexperimentgives
∆ℓℓP.
12Ng–vanDamThoughtExperiment
TheelementsoftheNg–vanDamthoughtexperimentarestraightforward,andtheaimistoshow
thatthroughasimpleapplicationoftheuncertaintyprinciple,togetherwithlimitsimposedby
generalrelativity,weareledtotheconclusionthatafundamentaldistanceuncertaintyarisesthat
maybefarlargerthanthePlanckscale.
Theargumentproceedsasfollows.Firstconsidertwonearbyobjectsinfreefallapproximately
atrestrelativetooneanother:anobserverconsistingofaclockandlightemitter,andamirror.If
theobserverwantstoknowthedistancetothemirror,hemaysimplyemitaburstoflight,waita
timetforthelighttoreturn,andconcludethatthemirrorisadistanceℓ=ct/2away.
Nowweareinterestedintheuncertaintyofthismeasurement.Followinganargumentdueto
Wigner[7,9]wetreattheclockasafreequantummechanicalparticleandimposetheuncertainty
condition∆q∆p.Writing∆p=m∆vwheremisthemassoftheclock,wethusobtainthe
followingboundontheuncertaintyoftheclock’spositionattimet:
∆q(t)=∆(q+tv)
=
m∆q.(1)
Tominimizethepositionuncertaintyattimet,wefindthattheoptimalpositionuncertaintyat
timezeroshouldbe∆q=
t/m.(2)
ItisalsoconvenienttowritethisintermsofthedistancetobemeasuredandtheComptonwave-
lengthoftheclock,ℓC=/mc:
∆q(t)ℓ1/2ℓ1/2C.(3)
Thisuncertaintyinthepositionoftheclockcontributestotheuncertaintyinℓ,thedistancebetween
theclockandmirror.Wecanignoretheuncertaintyinthepositionofthemirror,whichbehaves
similarly,andobtainthislowerboundon∆ℓ:
∆ℓℓ1/2ℓ1/2C.(4)
Sofarwehaveonlyconsideredtheeffectsofquantummechanicsandthespeedoflight,withno
mentionoftheeffectsofgeneralrelativity.Next,NgandvanDamconsiderthedetailsoftheclock
itself.Theytaketheclocktoconsistoftwoparallelmirrorsadistancedapart,andconsideratick
oftheclocktobethetime2d/cthatittakeslighttotravelbackandforthbetweenthem.Sincewe
nowhavethelengthscaledandthemassscalemoftheclock,wecannowbegintoconsidergeneral
relativityeffects.Inparticular,NgandvanDamassertthatthesizeoftheclock,d,mustbelarger
thanitsSchwarzschildradiusℓS=Gm/c2.Ifthetickoftheclockisalowerboundontheaccuracy
ofitstimemeasurements,thisrequirementimpliesthat
∆ℓℓS.(5)
Finally,squaringtheuncertaintyfromequation(4)andmultiplyingtheresultbyequation(5),
weobtain(∆ℓ)3ℓℓCℓS.NotethatℓCℓSisequaltoℓ2P,thePlancklengthsquared.Thusthe
primaryresultobtainedfromtheNg–vanDamthoughtexperimentisthattheminimumuncertainty
inthiskindofmeasurementsatisfies
∆ℓℓ1/3ℓ2/3P,(6)
abounddependingonlyonthedistanceℓtobemeasuredandthePlancklength.
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