Download - Chapter 2 Definitions and Concepts from General Relativity

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Chapter2

DefinitionsandConceptsfromGeneralRelativity

Theprimarypurposeofthisbookistoshowhowitispossibleforthefundamentalparticlesandtheforcesofnaturetobeconceptuallyexplainedusingonly4dimensionalspacetime.Thisisnothing less than a new model of the universe. Presentation of this model would be animpossiblylargetaskforasinglepersonifeveryaspectofthemodelneedstobeanalyzedwiththedetailthatmightbeexpectedfromatechnicalpapercoveringaspecificaspectofamaturesubject. Therefore, the concepts will be introduced using simple equations that involveapproximations and ignore dimensionless constants such as 2π or½. These simplificationspermitthekeyconceptsofthisverylargesubjecttobeexplained.Later,otherscananalyzeandexpanduponthislargesubjectinmoredetail.Wewillstartby lookingatthegravitationaleffectsonspacetimeinthe limitingcaseofweakgravity.Muchoftheanalysistofollowinsubsequentchapterswilldealwiththegravityexhibitedbyasinglefundamentalparticlesuchasanelectronoraquark.Workingwithsingleparticlesorthe interaction between two fundamental particles allows the proposed structure of suchparticlestobeconnectedtotheforcesthatareexhibitedbytheseparticles.Atthesametime,theextremelyweakfieldspermitsimplificationsintheanalysis.In the following discussion, a distinction will be made between the words “length” and“distance”.Normally,thesewordsaresimilar,butwewillmakethefollowingdistinction.Lengthisaspatialmeasurementstandard.Thisisnotjustastandardizedsizesuchasameterorinch,butitalsocanincludeaqualificationsuchasproperlengthorcoordinatelength.Aunitoflengthcanbedefinedeitherbya rulerorby thespeedof lightanda time interval. Theconceptofdistanceasusedhere isbest illustratedby thephrase “thedistancebetween twopoints”. Adistancecanbequantifiedasaspecificnumberoflengthunits.This chapter starts off with a discussion of the Schwarzschild solution to the Einstein fieldequationandphysicalexamplesoftheeffectofgravityonspacetime.Thiswillseemelementaryto many scientists, but new terminology and physical interpretations are introduced.Understandingthisterminologyandperspectiveisarequirementforsubsequentchapters.Schwarzschild Solution of the Field Equation:Einstein’sfieldequationhasanexactsolutionfor the simplified case of a static, nonrotating and uncharged spherically symmetric massdistribution inanemptyuniverse.Thismassdistributionwith totalmassm is locatedat theoriginofasphericalcoordinatesystem.Thestandard nonisotropic Schwarzschildsolutioninthiscasetakestheform:


dS2 c2dτ2 1/Г2 c2dt2–Г2dR2–R2dΩ2

Г≡dt/dτ explainedbelow

dS cdτ The invariantquantitydS is the lengthof theworld linebetweentwoevents in4dimensionalspacetime.R circumferentialradius circumference/2π explainedbelow Ω asolidangleinasphericalcoordinatesystem dΩ2 dθ2 sin2θdΦ2 c thespeedoflightconstantofnaturet coordinatetime timeinfinitelyfarfromthemass‐effectivelyzerogravity τ propertime–timeintervalonalocalclockingravityCircumferential radius:Gravitywarpsthespacearoundmass,sothatthespacearoundthetestmasshasanon‐Euclidiangeometry.Thecircumferenceofacirclearoundthemassdoesnotequal2πtimestheradialdistancetothecenterofmass.Toaccommodatethiswarpedspace,theSchwarzschildequationusesaspecialdefinitionofdistancetospecifythecoordinatedistanceinthe radial direction. Names like “R‐coordinate” and “reduced circumference” are sometimesusedtodescribethisradialcoordinatethatcannotbemeasuredwithameterstickorapulseoflight.Thenamethatwillbeusedhereis“circumferentialradius”anddesignatedwiththesymbolR.Thisisadistancethatiscalculatedbymeasuringthecircumferenceofacirclethatsurroundsamass,andthendividingthiscircumferenceby2π.Ifwemeasuretheradiususingahypotheticalmeterstickortapemeasure,thenthe“proper”radialdistancewillbedesignatedbyr.WecanseefromtheSchwarzschildmetricthatifwesetdt 0,dS cdτ dranddΩ 0then:dr ГdRGravitational Gamma Γ:ThemetrichasbeenwrittenintermsofthequantityГwhichthisbookwillrefertoasthe“gravitationalgammaГ”.ThebasicdefinitionofГis:

Г≡dt/dτ

whereГ gravitationalgammaandφ –Gm/RgravitationalpotentialГ dt/dτinthestaticcasewhendR 0anddΩ 0.gooisametriccoefficientcommonlyusedingeneralrelativityThesymbolofuppercasegammaГwaschosenbecausethisequationcanalsobewrittenasfollows:


Г whereVe≡2

escapevelocity

ThesimilaritybetweenГandγofspecialrelativityisobvioussince:γ

Thisanalogybetweentheescapevelocityingeneralrelativityandtherelativevelocityinspecialrelativityextendsfurtherintheweakgravityapproximation.Thetimedilationduetogravityapproximatelyequalsthetimedilationduetorelativemotionwhentherelativemotionisequaltothegravitationalescapevelocity weakgravity .TheSchwarzschilduniversewithonlyonemasshaseffectively “zerogravity”atany locationinfinitelyfarfromthemass.AtsuchalocationГ 1.TheoppositeextremeofthemaximumpossiblevalueofГistheeventhorizonofablackholewhereГ ∞.Iftheearthwasinanemptyuniverse,thenthesurfaceoftheearthwouldhaveagravitationalgammaof:Г 1 7 10‐10.ItisimportanttorememberthatГisalwayslargerthan1whengravityispresent.AstationaryclockinfinitelyfarawayfromthemassinSchwarzschild’suniverseisdesignatedthe “coordinate clock” with a rate of time designated dt. In the same stationary frame ofreference,thelocalrateoftimeingravity nearthemass isdesignateddτ.Therelationshipis:Г dt/dτThereisausefulapproximationofΓthatisvalidforweakgravity.

Γ 1 weakgravityapproximationofГ

Gravitational Magnitude β: ThegravitationalgammaΓhasarangeofpossiblevaluesthatextends from 1 to infinity. There is another related concept where the strength of thegravitationaleffectonspacetimerangesfrom0to1where0isalocationinzerogravityand1istheeventhorizonofablackhole.Thisdimensionlessnumberwillbecalledthe“gravitationalmagnitudeβ”andisdefinedas:

β≡1– 1 1– 1–Гβ gravitationalmagnitude

Inweakgravitythefollowingapproximationisaccurate:

β andГ 1 βweakgravityapproximations


Forexample,theearth’sgravity,intheabsenceofanyothergravityisβ 7 10‐10.Thesun’ssurfacehasβ 2 10‐6andthesurfaceofahypotheticalneutronstarwithescapevelocityequaltohalfthespeedoflightwouldhaveβ 0.13.In common usage, the strength of a gravitational field is normally associated with theacceleration of gravity. However, the acceleration of gravity depends on the gradient ofgravitationalpotential.Incontrast,thegravitationalmagnitudeβandthegravitationalgammaГ are measurements of the effect of gravity on time and distance without regard for thegravitational gradient. For example, there is an elevation in Neptune’s atmosphere whereNeptunehasapproximatelythesamegravitationalaccelerationastheearth.However,Neptunehasroughly16timestheearth’smassandroughly4timestheearth’sradius.ThismeansthatNeptune’sgravitationalmagnitudeβisroughly4timeslargerthanearth’satlocationswherethegravitationalaccelerationisaboutthesame.The gravitational magnitude approximation β Gm/c2r will be used frequently with weakgravity.Forexample,thisapproximationisaccuratetobetterthanonepartin1036forexamplesthatwillbepresentedlaterinvolvingthegravityofasinglefundamentalparticleatanimportantradial distance. Therefore, this approximation will be considered exact when dealing withfundamentalparticles.NotethatthisapproximationincludesthesubstitutionofproperradialdistancerforthecircumferentialradiusR. r R .Usingtheapproximationβ Gm/c2rwealsoobtainthefollowingequalitiesforβinweakgravity:

β φ –Gm/RandRs Gm/c2 Schwarzschildradius* seenotebelow

β therateoftimeapproximationforweakgravity

All of these approximationswill be considered exactwhendealingwith the extremelyweakgravityofsinglefundamentalparticlesinsubsequentchapters.

Note: The Schwarzschild radius of a nonrotating black hole is rs 2Gm/c2 and theSchwarzschildradiusofmaximallyrotatingblackhole,suchasaphotonblackhole, isRs Gm/c2.Thisbookwilloftenignorenumericalconstantsnear1,butthereisanotherreasonforusingRs Gm/c2forparticles.TheparticlemodelproposedlaterinthisbookhasenergyrotatingatthespeedoflightwhichwouldhavegravitationaleffectsscalewithRsratherthanrs.

Zero Gravity: Schwarzschild assumedanemptyuniversewithonly a singlemass. Suchauniverseapproacheszerogravityasthedistancefromthemassapproachesinfinity.However,isthere anywhere in our observable universe that can truly be designated as a zero gravitylocation?Therearevastvolumeswithvirtuallynogravitationalaccelerationcomparedtothecosmicmicrowavebackground.However,thisisnotthesameassayingthatthesevolumeshaveagravitationalgammaofГ 1 ahypotheticalemptyuniverse .Everywhereintherealuniverse


thereisgravitationalinfluencefromallthemass/energyintheobservableuniverse.Later,inthechaptersoncosmology,anattemptismadetoestimatethebackground omni‐directional gravitationalgammaofourobservableuniversecompared toahypotheticalemptyuniverse.WhilethepresenceofauniformbackgroundГfortheuniversehasimplicationsforcosmology,wecanonlymeasuredifferencesinГ.ThereisampleevidencethatgeneralrelativityworkswellbysimplyignoringtheuniformbackgroundГoftheuniverse.EffectivelyweareassigningГ 1to the background gravitational gammaof the universe and proportionally scaling from thisassumption.Therefore,adistantlocationwhichwedesignateashavingβ 0orΓ 1willbereferredtoasa“zerogravitylocation”orsimply“zerogravity”.Theterm“zerogravity”incommonusageusuallyimpliestheabsenceofgravitationalaccelerationasmightbeexperiencedinfreefall.However,inthisbook“zerogravity”literallymeansthatweareusingtheSchwarzschildmodelofadistantlocationwhichhasbeenassignedcoordinatevaluesofβ 0andГ 1.Therateoftimeatthiscoordinate locationwillbedesignateddtandcalled“coordinaterateoftime”. Aclockatthislocationwillbedesignatedasthe“coordinateclock”.Gravitational Effect on the Rate of Time: The equation dt Γdτ is perhaps themostimportantandeasiesttointerpretresultoftheSchwarzschildequation.ItsaysthattherateoftimedependsonthegravitationalgammaГ.Thisequationhasbeenprovencorrectbynumerousexperiments.TodaytheatomicclocksinGPSsatellitesareroutinelycalibratedtoaccountforthedifferentrateoftimebetweenthelowergammaattheGPSsatelliteelevationandthehighergammaattheEarth’ssurface. Withoutaccountingforthisgravitationalrelativisticeffect,theGPSnetworkwouldaccumulate errors and cease to functionaccurately after aboutoneday.Thereisalsotimedilationcausedbytherelativemotionofthesatellite.Thisisamuchsmallercorrectionthanthegravitationaleffectandintheoppositedirection. Thedifference in therateof timewithrespect toradialdistance ingravitywillbecalled the“gravitationalrateoftimegradient”.Thegravitationalrateoftimegradientisnotatidaleffect.Anacceleratingframeofreferencehasnotidaleffects,yetitexhibitsarateoftimegradient.Ourobjectiveistoprovideanequationthatrelatestheaccelerationofgravity“g”totherateoftimegradientandutilizesproperlengthintheexpressionoftherateoftimegradient.Forexample,inside a closed room it is possible to measure the gravitational acceleration. If we cannotmeasure any tidal effects, there is no information about themass anddistanceof the objectproducingthegravity.Isitpossibletodeterminethelocalrateoftimegradient expressedusingproperlengthandpropertime fromjustthegravitationalacceleration?It has been shown1 that a uniform gravitational fieldwith proper acceleration g measuredlocally ,hasthefollowingrelationshipbetweenredshiftandgravitationalacceleration:

1 Edward A. Desloge, “The Gravitational Redshift in a Uniform Field” Am. J. Phys. 58 (9), 856-858 (1990)


υ/υo 1–g /c2where:υo frequencyasmeasuredatthesourcelocationwithrateoftimedτoυ frequencyasmeasuredatthedetectorlocationwithrateoftimedτ verticaldistanceusingproperlengthbetweenthesourceanddetector

g accelerationofgravityReference 1 showsthatthisequationisexactifthefollowingqualificationsareplacedontheabovedefinitions.Thesequalificationsare:1 thesourcelocation subscripto shouldbeatalowerelevationthanthedetectorlocation.2 theseparationdistance shouldbetheproperlengthasmeasuredbyatimeofflightmeasurement radarlength measuredfromthesourcelocation.Aslightlydifferentradarlengthwouldbeobtainedifthisdistancewasmeasuredfromthedetectorelevationormeasuredwitharuler.However,inthelimitofagradient infinitelysmall ,thisdiscrepancydisappears.Thereforewiththesequalifications:υ/υo 1–g /c2isexact.Itshouldbenotedthattheheightdifference isaproperdistanceradarlengthmeasuredfromthesource andnotcircumferentialradius.Aswillbeshowinthenextchapter,thegravitationalredshiftisreallycausedbyadifferenceintherateoftimeatdifferentelevations.Thereisnoaccumulationofwavelengths,soυo 1/dτoandυ 1/dτ.Aftermakingthesesubstitutions,thisequationbecomes:

g c2

This is also an exact equation if the above qualifications are observed. Here the ratiodτ‐dτo /dτd willbereferredtoasthegradientintherateoftime.Therearetwopointstobenoticed.First,thegradientintherateiftimeisabletobedeterminedfromtheaccelerationofgravitywithnoknowledgeaboutthemassordistanceofthebodyproducingthegravity.Forexample,agravitationalaccelerationofg 1m/s2 isproducedbyarateof timegradientof1.113 10‐17seconds/secondpermeter.Theearth’sgravitationalaccelerationof9.8m/s2neartheearth’ssurfaceiscausedbyarateoftimegradientofabout10‐16seconds/secondpermeterofelevationdifferenceintheearth’sgravity.Themostaccurateclockpresentlyavailable 2015 isan87Sropticallatticeclock2whichhasanaccuracyof2.1x10‐18.Aclockwiththisaccuracyhasaresolutioncomparableto2cmelevationchangeintheearth’sgravity.Thesecondimportantpointisthattherateoftimegradientisafunctionofproperlengthintheradial direction. Even though dt/dτ is a function of circumferential radius, the relationshipbetweenrateoftimegradientandgravitationalaccelerationisnotafunctionofcircumferentialradius.Thisfactwillbecomeimportantinthenextchapterwhenweexaminehownaturekeepsthe laws of physics constant when there is an elevation change. The connection between

2 http://www.nature.com/ncomms/2015/150421/ncomms7896/full/ncomms7896.html


gravitationalaccelerationandrateoftimegradientwillalsobeanimportantconsiderationwhenweexaminethecosmologicalmodeloftheuniverse chapters13&14 .Previously,wedefinedtheconceptof“gravitationalmagnitudeβ”as:β 1–dτ/dt.It isalsopossibletorelatetheaccelerationofgravitytothegradientinthegravitationalmagnitude.

g c2

Inertial Frame of Reference: Theconcept thatgravitycanbe simulatedbyanacceleratingframeof reference sometimes leads to theerroneous interpretation thatan inertial frameofreferenceeliminatesalleffectsofgravity.Beinginfreefalleliminatestheaccelerationofgravity,butthegravitationaleffectontherateoftimeandthespatialeffectsofthegravitationalfieldremain.AnotherwayofsayingthisisthattheeffectsofthegravitationalgammaΓonspacetimeare still present, even if amass is in an inertial frame of reference. A clock in free fall stillexperiencesthelocalgravitationaltimedilation.Arigorousanalysisfromgeneralrelativityconfirmsthispoint,buttwoexampleswillbegiventoalsoillustratetheconcept.Supposethattherewasahollowcavityatthecenteroftheearth.Aclockinthiscavitywouldexperiencenogravitationalaccelerationandwouldbeinaninertialframeofreference. Thegravitationalmagnitudeβinthiscavityisabout50%largerthanthegravitationalmagnitudeonthesurfaceoftheearth ~10.5 10‐10comparedto7 10‐10 .Forexample,ignoringairfriction,theescapevelocitystartingfromthiscavityishigherthanstartingfromthesurfaceoftheearth.Theclockinthecavityhasaslowerrateoftimethanaclockonthesurface.Theinertialframeofreferencedoesnoteliminatetheothergravitationaleffectsontherateoftimeandthegravitationaleffectonvolume.Asecondexampleisinterestingandillustratesaslightlydifferentpoint.TheAndromedagalaxyis2.5millionlightyears ~2.4 1022m awayfromEarthandhasanestimatedmassofabout2.4 1042kg includingdarkmatter .Thegravitationalaccelerationexertedbythisgalaxyatthedistance of the Earth is only about 2.8 10‐13m/s2. To put thisminute acceleration inperspective,a10,000kgspacecraftwouldaccelerateataboutthisratefromthe“thrust”ofthelight leavinga1watt flashlight. In spiteof theminutegravitationalacceleration, thedistantpresenceofAndromedaslowsdowntherateoftimeonthesurfaceoftheearthabout100timesmore than the Earth’s own gravity. This is possible because the gravitational magnitudeβ Gm/c2r forweakgravity decreasesatarateof1/rwhile thegravitationalaccelerationdecreaseswith1/r2.Attheearth’ssurface,Andromeda’sgravitationalmagnitudeisabout:β Gm/c2r G/c2 2.4 1042kg/2.4 1022m 7 10‐8Andromeda’sβatearthSincetheearth’sgravityproducesβ 7 10‐10atthesurface,Andromeda’seffectontherateoftimeattheearth’ssurfaceisabout100timesgreaterthantheeffectoftheearth’sgravity.Itdoes


notmatterwhetheraclockisinfreefallrelativetoAndromedaorwhethertheclockisstationaryrelativetoAndromedaandexperiencestheminutegravitationalacceleration.Inbothcasesthegravitationaleffectontimeandvolumeexist.Thisexamplealsohintsthatmass/energyinotherpartsoftheuniversecanhaveasubstantialcumulativeeffectonourlocalrateoftimeandourlocalvolume.Thisconceptwillbedevelopedlaterinthechaptersdealingwithcosmology.Schwarzschild Coordinate System:ThestandardSchwarzschildsolutionusescoordinatesthatsimplifygravitationalcalculations.ThissphericalcoordinatesystemusescircumferentialradiusRascoordinatelengthintheradialdirectionandusescircumferentialradiustimesanangleΩfor the tangentialdirection. While there isnodistinction inproper length for theradialandtangentialdirections,wewilltemporarilymakeadistinctionbydesignatingproperlengthintheradial direction as LR and designating proper length in the tangential direction as LT. Thisdistinction does not exist in reality since: cdτ dL dLT dLR. However, using thesedesignations,therelationshipbetweenproperlengthandSchwarzschild’scoordinatelengthis:dLR ΓdRradiallengthLRconversiontoSchwarzschildradialcoordinateRdLT RdΩtangentiallengthLTtoSchwarzschildtangentialcoordinatelengthThe equation dLR ΓdR is obtained by setting dt 0. Ifwe are using the proper distancebetweentwopointsasmeasuredbyaruler,orthecalculatedcircumferentialradius,thenthiszerotimeassumptionisjustified.NextwewillcalculatethecoordinatespeedoflightCfortheradialandtangentialdirectionsbystartingwiththestandardSchwarzschildmetric:dS2 1/Г2 c2dt2–Г2dR2–R2dΩ2forlightsetdS2 0,1/Г2 c2dt2 Г2dR2 R2dΩ2

c2 Г4 Г2

Ifweseparatethiscoordinatespeedof light into itsradialcomponent CR and its tangentialcomponent CT ,weobtain:R dR/dt c/Г2 R coordinatespeedoflightintheradialdirection dΩ 0 T RdΩ/dt c/Г T coordinatespeedoflightinthetangentialdirection dR 0

Thisapparentdifferenceinthecoordinatespeedoflightfortheradialandtangentialdirectionsisnotexpressingaphysicallymeasurabledifferenceintheproperspeedoflight.Thedifferencefollows fromthestandard nonisotropic form of theSchwarzschildmetric. Ifwechoose theisotropic formof theSchwarzschildmetric thedifferencewilldisappearand R T c/Γ2.


However,theisotropicformhasitsownsetofcomplexities,sowewillbeusingthestandardSchwarzschildmetric.

The Shapiro Experiment:Next,wearegoingtoswitchtoadiscussionaboutthegravitationaleffectonproperlength,propervolumeandthecoordinatespeedoflight.In1964,IrwinShapiroproposedanexperimenttomeasuretherelativisticdistortionofspacetimecausedbytheSun’sgravity.Thisnon‐NewtoniantimedelayisobtainedfromtheSchwarzschildsolutiontoEinstein’sfield equation. The Sun is a good approximation of an isolated mass addressed by theSchwarzschildsolution.Theimplicationisthatgravityaffectsspacetimesothatittakesmoretimeforlighttomaketheroundtripbetweentwopointsinspacewhenthemass gravity ispresentthanwhenthemass gravity isabsent.ShapiroandhiscolleaguesusedradartotracktheplanetVenusforabouttwoyearsasVenusandtheEarthorbitedtheSun.Duringthistime,VenuspassedbehindtheSunasseenfromtheEarth nearlysuperiorconjunction .TheorbitsofVenusandtheEarthareknownaccurately,so itwaspossibletomeasuretheadditionaltime


delayintheroundtriptimefromtheearthtoVenusandback.Theeffectofthesun’sgravityonthisroundtriptimecouldbecalculatedfrommultiplemeasurementsmadeoverthetwoyeartimeperiod.Figure2‐1showsShapiro’sgraphoftheexcesstimedelayoverthetwoyearperiod.Thepeakdelayatsuperiorconjunctionwas190μsonahalfhourroundtriptransittime.Variationsofthisexperimenthavebeenrepeatednumeroustimesinthenormalcourseofthespaceprogram.SpacecraftontheirwaytotheouterplanetsoftenstartwithanorbitalpaththatatsomepointresultsinnearlysuperiorconjunctionrelativetotheEarth. ThemostaccuratemeasurementtodatewaswiththeCassinispacecraft.Itwasequippedwithtranspondersattwodifferentradarfrequencies,thereforeitwaspossibletodetermineandremovetheeffectoftheSun’scoronaonthetimedelay.Theresultwasanagreementwiththetimedelaypredictedbygeneralrelativityaccurateto1partin50,000.Withthistypeofagreement,itwouldseemasifthereareno remainingmysteriesabout thiseffect and thephysical interpretation shouldbeobvious.Howexactlydolength,timeandthespeedoflightcombinetoproducetheobservedtimedelayintheShapiroeffect?Forsimplicity,wewouldliketolookatthetimedelayassociatedwitharadarbeamtravelingonlyintheradialdirection.Inthelimit,wecanimaginereflectingaradarbeamoffthesurfaceoftheSun.Thispathwouldbepurelyradial.Tomakeameasurementweneedtohavearoundtrip,butforsimplicityofdiscussion,wewilltalkaboutthetimedelayforaonewaytrip.ForlightdS 0,sothemetricequationgivesusthatforlightmovingintheradialdirection:

Гcdt ГdR→dt

ГdR

WecancomputethetimeΔtittakestomovebetweentwodifferentradii:r1andr2 wherer2 r1 .

cΔt

cΔt 2 Г

Гweakgravity:

Г

Г 0

Δt

SubstitutingtheSun’smass,radiusanddistancegivesΔt 50μs.Therefore,inadditiontoanon‐relativistictimedelay about8minutes ,theonewayrelativistictimedelaywouldbeaboutanadditional50μs.The190μsdelayobservedbyShapiroisroughly4timestheoneway50μsdelayfromtheearthtothesunbecauseoftheadditionallegtoVenusandthentheroundtripdoublingofthetime.


Normally,onEarthwewouldinterpreta50μsdelayinaradarbeamasindicatinganadditionaldistanceofabout15km.Howmuchdoesthesun’sgravitydistortspaceandincreasetheradialdistancebetweentheearthandthesun’ssurfacecomparedtothedistancethatwouldexistifwehad Euclidian flat space? The additional non Euclidian path length will be designated ΔL .Startingfrom:dLR ΓdR

ΔL

ΔL setr2 1.5 1011m,r1 7 108mandm 2 1030kg

ΔL 7.5kmnon‐EuclidianadditionalproperdistancebetweentheEarthandSunSupposethatitwaspossibletostretchatapemeasurefromtheearthtothesurfaceofthesun.Thedistancemeasuredbythetapemeasure properdistance wouldbeabout7.5kmgreaterthanadistanceobtainedfromanassumptionofflatspaceandaEuclidiangeometrycalculation.Theuseofatapemeasuremeansthatweareusingproperlengthasastandard.Gravity Increases Volume: Ifweuseproper length as our standardof length rather thancircumferentialradius,thenwemustadopttheperspectivethatgravityincreasesthevolumeoftheuniverse.However,aninterpretationbasedonpropervolumeisoftenignoredsincetheuseofcircumferentialradiusascoordinatelengtheliminatesthisvolumechangecausedbygravity.In theShapiroexperiment,wecalculatedthat therewasanon‐Euclidian increase indistancebetweentheearthandthesunof7.5km.Supposethatweimagineasphericalshellwitharadiusequaltotheaverageradiusoftheearth’sorbit.Thisisaradialdistanceequaltooneastronomicalunit AU 1.5 1011m .Thesun’smassis 2 1030kgandthesun’sSchwarzschildradiusisrs 2950meters.Whatisthechangeinvolume ΔV insidethissphericalshellifwecomparetheEuclidianvolumeoftheshell Vo andthenon‐Euclidianvolumeoftheshell V whentheSunisatthecenteroftheshell?V 4 Г Afterintegration,thedifferenceΔV V–Voisapproximately:ΔV 5π/3 rs r22–r12 setr2 1.5 1011m,r1 7 108mandrs 2950metersΔV 3.46 1026m3nonEuclidianvolumeincreaseToputthisnonEuclidianvolume increase inperspective, thesun’sgravityhas increasedthepropervolumewithinaradiusof1AUbyabout3.5 1026m3whichismorethan300,000timeslargerthanthevolumeoftheearth earth’svolumeis 1.08 1021m3 .Statedanotherway,the volume increase is about 20% smaller than the volume obtained by multiplying thenon‐Euclidianradiallengthincrease ~7,500m timesthesurfaceareaofthesphericalshell


witharadiusof1AU.ObviouslythisnonEuclidianvolumeincreasewouldbemuchlargerifwehad chosen a larger shell radius for example, the size of the observable universe . Theimplicationsofthiswillbeexploredinthechaptersoncosmology.Concentric Shells Thought Experiment:Theconceptthatgravityincreasesthevolumeoftheuniverseisimportantenoughthatanotherexamplewillbegiven.Supposethattherearetwoconcentric spherical shells around an origin point in space. The inside spherical shell is4.4 109mincircumference aboutthecircumferenceoftheSun .Theoutsideshellis2πmeterslargercircumference.Withnomassattheoriginandinfinitelythinshells,thismeansthatthereisexactlya1metergapbetweentheshells.Wecouldconfirmthe1meterspacingwithameterstickorapulseoflightandaclock.NextweintroducetheSun'smassattheorigin.ThisintroducesgravityintothevolumebetweenthetwoshellswithanaveragevalueofaboutГ 1 2 10‐6.Thecircumferenceofeachshellproperlength doesnotchangeafterweintroducegravity.However,thedistancebetweenthetwoshellswouldnowbeabout1 2 10‐6meters.Thisis2micronslargerthanthezerogravitydistance.This2micron increase inseparation increases thepropervolumebetweenthe twoshellsbyroughly1013m3.Inthisexampleoftwoconcentricshellsweacceptedtheproperlengthofthecircumferenceoftheshells tangentialproperlength .Thequestioniswhethertherewasalsoadecreaseofthistangentiallength relativetoa“flat”coordinatesystem whenthesun’smasswasintroducedattheorigin.Itispossibletoconsiderbothradialandtangentialdirectionsaffectedequally.Thisresultsingravityproducinganevenlargerincreaseinpropervolumethanpreviouslycalculated.Thiswillbediscussedfurtherinthechaptersoncosmology.Connection Between the Rate of Time and Volume: WearegoingtocomputetheeffectofthegravitationalgammaГonpropervolumeusingthestandardSchwarzschildmetric.TheuseofthismetricmeansthatthestandardSchwarzschildconditionsapply:astaticnonrotatinganduncharged spherically symmetric mass distribution in an empty universe. This calculationinvolvesterminologyfromgeneralrelativitythatisnotexplainedhere.Readersunfamiliarwithgeneral relativity should skip the shaded calculation section below and move on to theconclusion.


Ifweknowmetricequations,wecancomputethe3‐dimensionalvolumeandthe4‐dimensionalvolume includestime .Theeasiestwaytodoitistousethefollowingdiagonalmetric:dS2 –g00 dx0 2 g11 dx1 2 g22 dx2 2 g33 dx3 2Thenthe3‐dimensionalvolumedV 3 is:dV 3 g11 g22 g33 1/2dx1dx2dx3Andfor4‐dimensionalvolumedV 4 dV 4 –g00 g11 g22 g33 1/2dx0dx1dx2dx3IntheparticularcaseofthestandardSchwarzschildmetric:g00 –1/Г2;g11 Г2;g22 R2,g33 R2sin2θThedifferentialsof4dimensionalcoordinatesinthiscaseare:dx0 cdt; dx1 dR; dx2 dθ; dx3 dΦSo:dV 3 Г2 R2 R2sin2θ 1/2dR dθ dΦdV 3 ГR2sinθdRdθdΦnotethatvolume 3 scaleswithГV 4 – –1/Г2 Г2 R2 R2sin2θ 1/2 cdt dR dθ dΦdV 4 R2csinθdtdRdθdΦnotethatthisisindependentofГThe above calculation shows that proper volume 3 spatial dimensions scales with thegravitational gamma Г. This supports the previous examples involving the volume increaseinterpretationoftheShapiroexperimentandalsothevolumeincreasethatoccursinthethoughtexperimentwithtwoconcentricshells.Whenweincludethetimedimensionandcalculatetheeffectof thegravitygeneratedbyasinglemassonthesurroundingspacetime,weobtaintheanswerthatthe4dimensionalspacetimevolumeisindependentofgravitationalgammaГ.Theradialdimensionincreases Г dLR/dR andthetemporaldimensiondecreases Г dt/dτ .Theseoffseteachotherresultinginthe4dimensionalvolumeremainingconstant.


Thereisasimplerwayofexpressingthisconcept.SinceГ therefore:

dτdLR dtdRThis concept will be developed further when we develop particles out of 4 dimensionalspacetimeanditalsohasapplicationtocosmology.

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