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∆ℓ󰀁ℓ1/3ℓ2/3Pontheuncertaintyofanydistance,whereℓisthedistancetobemeasuredandℓPisthePlancklength.Theirideaisroughlythattominimizethepositionuncertaintyofafreelyfallingmeasuringdeviceonemustincreaseitsmass,butifitsmassbecomestoolargeitwillcollapsetoformablackhole.HereweshowthatonecangobelowtheNg–vanDamboundbyattachingthemeasuringdevicetoamassiveelasticrod.Relativisticlimitationsontherod’srigidity,togetherwiththeconstraintthatitslengthexceedsitsSchwarzschildradius,implythatzero-pointfluctuationsoftherodgiveanuncertainty∆ℓ󰀁ℓP.

1Introduction

Ithaslongbeenbelievedthatquantumgravityeffectsbecomeimportantatdistancescomparable

tothePlancklength,ℓP,andanumberofargumentshavebeenpresentedtosupportthisidea[3,4].

Toshowthissortofthing,onemainlyneedstoshowthatgravityeffectsbecomeimportantatsome

fixedlengthscaledependingonlyontheconstantsc,Gand󰀁.Dimensionalanalysisdoestherest,

sinceℓPistheonlyquantitywithdimensionsoflengththatonecanconstructfromtheseconstants.

However,insituationswhereasecondlengthscalebecomesrelevant,onecannotusedimensional

analysistosettleallcontroversies.Forexample,NgandvanDam[5]haverecentlyarguedthat

quantumgravityeffectscausesurprisinglylargeuncertaintiesinthemeasurementofalargedistance

ℓ,namely

∆ℓ󰀁ℓ1/3ℓ2/3P

wherethesymbol󰀁meansthatweareignoringaconstantfactoroforderunity.Amelino-Camelia

[1]hasgoneevenfurther,arguingthat

∆ℓ󰀁ℓ1/2ℓ1/2P.

Uncertaintiesonthisscaleareonthebrinkofbeingexperimentallydetectable,lendingextrainterest

totheissue.However,inwhatfollows,wereanalyzetheNg–vanDamthoughtexperimentandshow

thatbymodifyingitsdesignwecandramaticallyreducetheuncertaintyofdistancemeasurements.

Ourmodifiedthoughtexperimentgives

∆ℓ󰀁ℓP.

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:

∆ℓ󰀁ℓ1/2ℓ1/2C.(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

∆ℓ󰀁ℓS.(5)

Finally,squaringtheuncertaintyfromequation(4)andmultiplyingtheresultbyequation(5),

weobtain(∆ℓ)3󰀁ℓℓCℓS.NotethatℓCℓSisequaltoℓ2P,thePlancklengthsquared.Thusthe

primaryresultobtainedfromtheNg–vanDamthoughtexperimentisthattheminimumuncertainty

inthiskindofmeasurementsatisfies

∆ℓ󰀁ℓ1/3ℓ2/3P,(6)

abounddependingonlyonthedistanceℓtobemeasuredandthePlancklength.

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