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1PHYSICSFROMTHEEDGEANewCosmologicalModelforInertia9122hc_9789814596251_tp.indd120/1/148:51am
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3PHYSICSFROMTHEEDGEANewCosmologicalModelforInertiaMichaelEdwardMcCullochUniversityofPlymouth,UKWorldScientificNEWJERSEY•LONDON•SINGAPORE•BEIJING•SHANGHAI•HONGKONG•TAIPEI•CHENNAI9122hc_9789814596251_tp.indd220/1/148:51am
4PublishedbyWorldScientificPublishingCo.Pte.Ltd.5TohTuckLink,Singapore596224USAoffice:27WarrenStreet,Suite401-402,Hackensack,NJ07601UKoffice:57SheltonStreet,CoventGarden,LondonWC2H9HELibraryofCongressCataloging-in-PublicationDataMcCulloch,MichaelEdward,author.Physicsfromtheedge:anewcosmologicalmodelforinertia/MichaelEdwardMcCulloch,UniversityofPlymouth,UK.pagescmIncludesbibliographicalreferencesandindex.ISBN978-9814596251(hardcover:alk.paper)1.Inertia(Mechanics)2.Momentum(Mechanics)3.Acceleration(Mechanics)I.Title.QC137.M342014531'.12--dc232013051148BritishLibraryCataloguing-in-PublicationDataAcataloguerecordforthisbookisavailablefromtheBritishLibrary.Copyright©2014byMichaelEdwardMcCullochAllrightsreserved.TypesetbyStallionPressEmail:enquiries@stallionpress.comPrintedinSingaporeKahFee-PhysicsfromtheEdge.indd112/6/20142:01:37PM
5June16,201417:119inx6inPhysicsfromtheEdgeb1771-fmTomywifev
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7June16,201417:119inx6inPhysicsfromtheEdgeb1771-fmAcknowledgementsThankstomywifeBoksunKimforsupportandencouragement,toAndrewJones,RodneyMcCullochandJeehoonKangforproofreadingearlychapters,andeditorsKellyeCurtis&Kah-FeeNgforalltheirhelp.vii
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9June16,201417:119inx6inPhysicsfromtheEdgeb1771-fmContentsAcknowledgementsviiInertia:ABlindSpotinPhysicsxi1.AHistoryofInertia1Aristotle.........................1Galileo..........................2Newton..........................6Mach...........................112.ModernPhysics15Einstein..........................15TheEquivalencePrinciple................25QuantumMechanics...................30AnInterludeAboutHoyle’sCosmology........34HawkingRadiation...................36UnruhRadiation.....................383.ProblemsatLowAcceleration41TheOrbitalBalance...................41DarkMatter.......................43MoND...........................48MoNDisNotGoodEnough..............544.ASolutionfromtheEdge:MiHsC57ThePhysicsoftheFlute................57TheCasimirEffect....................61AnasymmetricCasimireffect(aCe)..........62ix
10June16,201417:119inx6inPhysicsfromtheEdgeb1771-fmxPhysicsfromtheEdge:ANewCosmologicalModelforInertiaModifiedinertiabyaHubble-scaleCasimireffect(MiHsC).....................665.EvidenceforMiHsC73CosmicAcceleration...................73TheLarge-ScaleCMBAnomaly............77GalaxyRotationwithoutDarkMatter.........81TheFlybyAnomalies..................85TheTajmarEffect....................93ThePioneerAnomaly..................1046.FutureExperimentalTestsofMiHsC111ASpinningDiscwithaWeightAbove.........111TheCERNAcceleratorandRadioWaves.......119NEMSAccelerationsandRadioWaves........122DeviationsfromtheStefan–BoltzmannLaw......123DiscretisedVariationsinInertialMass.........123DropTowerExperiment.................124GPSSatelliteTrajectories................124SummaryofExperimentsSuggested..........1257.MiHsCandFasterThanLightTravel127FTLTest1:Man-MadeEventHorizons........129FTLTest2:ParticleAccelerators...........129FTLTest3:GalacticJets................131Conclusions133AppendixA135AppendixB139AppendixC141AppendixD143References147Index151
11June16,201417:119inx6inPhysicsfromtheEdgeb1771-fmInertia:ABlindSpotinPhysicsFeynman:Say,pop,Inoticedsomething.WhenIpullmywagon,theballrollstothebackofthewagonandwhenI’mpullingitalongandIsuddenlystop,theballrollstothefrontofthewagon.Whyisthat?Feynman’sDad:Thatnobodyknows.Thegeneralprincipleisthatthingswhicharemovingtendtokeeponmovingandthingswhicharestandingstilltendtostandstill,unlessyoupushthemhard.Thistendencyiscalled‘inertia’butnobodyknowswhyitistrue(Feynman,1988).xi
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13June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01Chapter1AHistoryofInertiaAristotleThepropertythatwenowcallinertia(explainedveryclearlybyFeynman’sfatherinthequotation)remainedhiddenfromviewforalongtime.TheGreekphilosopherAristotle,forexample,saidthatafteraforceisappliedtoobjects,theytendtocometoarestafterawhile,unlessaforceiscontinuallyapplied(Aristotle,330BC,Physics).Hethoughtthatthenaturalstatewasoneofrest.Thisisanaturalideatohavesincemostthingsinourexperienceareatrest,andofcourseforhumanstheeffortlessstateisoneatrest.Ittakessomeefforttogetupandgogetabeerfromthefridge,soitiseasytoassumenaturebehavesthesameway.Basedonthisassumption(thedefaultstateiszerovelocity),AristotlearguedthattheEarthdidnotmove.HesaidthatwhenyouthrowsomethingupintotheairitlandsonthesameplaceitwasthrownfromandiftheEarthwasmovingthethrownobjectwouldbeleftbehindbythespinningEarthandlandelsewhere.Aristotlewasarelativelygoodobserver,butatthistimeGreeksciencehadnotprogressedtowardstheideaofexperiment,aprocessbywhichphysicistsfindwaystopurgemostotherinfluencesfromasystemsothattheinfluenceyouwanttolookatstandsoutundi-luted.OneexampleofsuchanexperimentisFeynman’swagon,aballrollinginawagonfeelslessfriction,soothereffects,likeinertia,aremoreapparentinitsbehaviour.AtthemomentIamwritingthisIamwatchingtheSpaceXDragonspacecraftasitattemptstodockwiththeInternationalSpaceStation,andthisisagoodexampleofaninadvertentinertialexperi-mentsinceinspacethereisnoairtocausedragsothemotionoftheDragonveryclearlyshowsthatthingskeepgoingataconstantspeed1
14June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch012PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure1.Theacceptednaturalstateofmotionisaconstantvelocity.Peoplehaveanintuitiveunderstandingofthis,seethecartoonabove,butitwasfirstbroughtintophysicsbyGalileoGalilei.ifthereisnoforceapplied(forgettingforamomenttheellipticalorbitwhichisduetothegravitationalforcepullingtheDragontowardstheEarth).WhenDragonfiresitsengines,andtheselongburstsoflightareclearlyvisibleinthelivevideo,itdoeschangethespeedofthespacecraftandtheyhavetobecarefultheydon’tcoastforwardtoofaratconstantvelocityandbangintotheISS.Sothedefaultstateisnotrest,butconstantspeed.Aristotledidnothaveaccesstoexperimentsinthevacuumofspace,buthecouldconceivablyhaveusedrollingballs.GalileoItwasGalileowhofirstcombinedtheexperimentalmethodwithmathematicalrigour.Galileohadbeentrainedinmusicfromanearlyageandbecauseofthathebecameinterestedinmathematics,andthiswasagreatadvantage.Mathematicsallowsyoutoprovethingsinafarmoreobjectiveway:itkeepsyouhonest.TheancientGreekstendedtothinkthatonecouldarriveatthetruththroughdebate,butordinarylanguageisvagueenoughthatmistakescanbehiddeninnuanceanddefinitions,anditistooeasyforglibfellowstowinargumentsthroughcharismaorcheaptricks.Asanexample,oneearlyargumentofGalileo’stime(seeWhite,2007,p44)involvedahailstormthatoccurredinPisashortlyafterhemovedthere.HiscolleaguescitedAristotle’sideasinwhichheavier
15June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01AHistoryofInertia3objectsdroppedfromanequalheightweresupposedtoacceleratemorerapidlyandhitthegroundfirst.Galileopointedoutthatthishadnotbeenobserved,largeandsmallhailstoneshadhitthegroundatthesametime.Hisopponents(includingthethenfamousacademicGirolamoBorro)thensaidthatthisdidnotcontradictAristotlesincethesmalleronesmusthavestartedfromalesserheight.Thisisaverygoodexampleofsomethingthatalsohappenstoday,theadhocrearrangementofunknownquantitiestosaveapopulartheory.ThecluebackthenwasthearbitrarywayGalileo’sopponentshadtoarrangethedifferently-sizedhailstonestoagreewiththeobservationofanequalfalltime:therewasnoreasonforit.Thereissimilarlynoreasonknownforthedistributionofdarkmatterinahaloroundgalaxies(butmoreonthislater).Galileo,annoyedbytheslipperynatureofhisopponents’arguments,usedanicethoughtexperimenttoprovethatstonesofdifferentweightfallatthesamespeed.Hefirstassumedthattheoppositewasthecase,anddisprovedthispossibilitylogicallyasfol-lows:considerastoneAfalling(seeFigure2,left).NowbreakitintotwohalvesBandC(Figure2,right).IfAristotleisrightthenbothBandCwillfallslowerthanA.ButBandCtogetherconstituteFigure2.Galileo’sargumentthatdifferentmassesfallatthesameratesbyreductioadabsurdum.ConsiderfirstthetheoryofAristotlethatheavierobjectsfallfaster.Consideralargeblockfalling(ontheleft).Nowcutitintwodownthecentrewithalaser,withoutapplyinganysignificantexternalforce.SuddenlyyouhavetwoblocksfallingofhalftheweightsoAristotlewouldpredicttheyshouldsuddenlyfalllessfast,buttherehasbeennoforceapplied!Galileosaidtheonlyconsistentthingwecanassumeisthatfallspeedisindependentofmass(ofcourse,neglectingairresistance).
16June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch014PhysicsfromtheEdge:ANewCosmologicalModelforInertiaA.ThereforethisimpliesthatAwillfallslowerthanitself!Theonlywaytoresolvethisparadoxistoassumethatallstonesfallatthesamespeed(seealsoHoffman,1906,p25).Atthetimeitwasdifficulttoprovethisexperimentally,sinceaccurateclocksdidnotyetexistandfallingobjectsmovetoofasttobeeasilytimed.Galileosolvedthisproblembyrollingballsdowninclinedplanes,thisreducedtheapparentforceofgravitysinceonlyasmallcomponentofgravitywasdirecteddowntheslope,andhetimedthese“slowmotion”experi-mentsusingawaterclock.Heshowedthatalltheballsacceleratedequallyirrespectiveoftheirmass.ThisflatlycontradictedAristotle,whohadclaimedthata100poundballfallingfrom100cubits(58metres)heightwouldhitthegroundbeforea1poundballcouldfallonecubit(White,p59).GirolamoBorro,thenthemostfamousacademicinPisatriedthisexperimentandconfirmedAristotle’sprediction.Galileotriedtheexperimenthimselfanddisagreed(White,p59).Wenowknow(withintheaccuracyofexperiments)thatobjectsfalltogetherirrespectiveoftheirmass.Thisisbecausealthoughheav-ierobjectsaremoreattractedtotheEarthduetotheirgreatergravi-tationalmass,theyalsohavemoreinertialmassandsofindithardertoaccelerate(seeFigure3).Thesetwoeffectscancel(althoughinthisbookIwilltrytoconvinceyouthatthingsareslightlymorecom-plex).Thiswasdemonstratedelegantly,althoughnotparticularlyrigorously,bytheApolloastronautDavidScottwhenhewenttotheMoon.TheMoonhasnoatmospheresowhenhedroppedahammerandafeathertheretheyhitthelunarsurfacetogether.Galileoalsodiscoveredthelawofinertia(whichcanbesum-marisedbysayingthatthedefaultaccelerationiszero).HetriedseveralexperimentswithVshapedplanesandfoundthatiftheballswererolleddownfromaheightH,ononesideoftheV,thentheywouldrisetotheheightHontheotherside.Thisisaconsequenceoftheconservationofenergywhichcanbewritten:Potentialenergy+Kineticenergy=Constant(1.1)Thepotentialenergyanobjecthasisduetoitsbeinghighupinagravitationalfield,soithasthepotentialtomove,andthekineticenergyithasisbecauseofmovementandtheenergythatcanbe
17June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01AHistoryofInertia5releasedwhenitisstopped.Theballsstartwithlotsofpotentialenergy(PE)becauseoftheirheight,convertthispotentialenergytokineticenergy(KE)astheyrolldownonesideoftheVandthenareabletoclimbuptheothersideoftheVbyconvertingthekineticenergybackintopotentialenergy.Someenergyisalwayslostinthisprocessandbecomesinternalenergy—i.e.:heat.GalileoaskedwhatwouldhappenifthesecondhalfoftheVwashorizontal?Thentheballwouldbeabletokeepallthekineticenergyitconvertedfromitsinitialpotentialenergyandwouldrollhorizontallyforever!Inreality,theballwouldeventuallystopduetotheenergylosttointernalenergy(heat)whichwecallfriction,butGalileorealisedthatthisfrictionhadformillenniaobscuredthebasiclawwhichwasthatthingskeepgoing.Moreformally,theprincipleofinertiaisasfollows:Amovingbodywillcontinueinthesamedirectionunlessdisturbed.Figure3.Theprincipleofequivalence.Objectswithmoremass(left)areattractedmorebythegravityoftheEarth,butalsohavemoreinertialmasssofinditmoredifficulttoaccelerategiventhesameforce.Theassumptionthenisthattheupanddownarrowsarealwaysequalinlength.Thismeansaccelerationisindependentofthemass—theamountof‘stuff’intheobject.
18June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch016PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThisisalsotrueof“Aballrollingonalevelsurface...”butanyway,thisisthemodelthatisdemandedbyFeynman’swagonexperimentanditcouldhavebeenfoundoutbytheGreeksiftheyhadhadamodernkindofexperimentalattitudeorhadbeenkeenbowlsplayers.Itshowsthattheappropriatequestiontoaskisnot“whatmakesitgo”sincegoingisthenaturalstate,astheGreeksdid,butrather“whatmakesitstop”(Feynman,1985)sinceitisstoppingthatrequirestheinputenergy.ThisprincipleofinertiaisimportantbecauseitdefeatedAris-totle’sobjectiontothemovingEarth.AlthoughCopernicushadalreadyproposedthattheEarthmovedbeforeGalileofoundhislawofinertia,ithadbeenthoughtthatiftheEarthmovedtheneverythingonitwouldhavefallenbehindtomaintaintheirdefaultposition.Nowitwasunderstoodthattheimportantthingwasnotthedefaultposition,butratherthedefaultvelocityandsolongasthisdidnotchangethentheEarthcouldmovealongataconstantspeedandhappilydragtrees,waterandphilosophersalongwithit.Thisdoesnotyetconsideracceleration,butGalileo’sinertiahelpedtomakeCopernicus’hypothesismoreplausibleandhelpedthemoremathematicalmodelsofitderivedlaterbyKeplerandNewton.Galileomadethisslightlymoregeneralbyusingtheexampleofasailingship.Onboardashiptravellingataconstantspeedthelawsofdynamicsworkjustthewaytheydoinanunmovingreferenceframe,sayonland,becausethedefaultstateofmatterisaconstantvelocity(zeroacceleration)andnotafixedplace,soonecannottellwhetheroneismovingbyanydynamicsexperimentonboardtheship,butonlybylookingoutside.Einsteinlaterextendedthistoincludeelectromagneticexperimentsontheship(thisisspecialrelativity)andacceleratedmotions(withgeneralrelativity).NewtonIsaacNewtonwasbornonChristmasday,1642inWoolsthorp,nearGranthaminEngland,andhecouldbesaidtohaveinventedmodernphysicssincehewasanexpertintheuseofexperimenttoisolatesystemssotheycouldbestudiedandthesubsequentapplicationofrigidmathematics,aftertheobservationshavebeencompleted,to
19June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01AHistoryofInertia7Figure4.Inaninertial(unaccelerated)referenceframe,forexampleonashipmovingataconstantspeedonaflatsea,thelawsofphysicsarethesameastheywouldbeinastaticship,soonecannottellyouareonamovingshipifthereisnowaytolookoutside.ThatisthetheoryofGalileanrelativity,butitonlyappliestoashiponaperfectlycalmsea.Asanysailorknowstherealityisthatsurfacewavesmaketheshipaccelerate,soinertialframesrarelyoccur.describethesystem.Hewasgoodatbothoccupations.Inhisfirstlawofmotion,NewtonstatedthelawofinertiathathegotfromGalileolikethis:Unlessacteduponbyanetunbalancedforce,anobjectwillmaintainaconstantvelocity.Morecrucially,hebegantoapplyitmathematicallytoobjectsinspace,liketheMoon,where,becauseofthelackoffriction,itismoreobviouslyvalidthanontheEarth.Galileodidnotmakeadistinctionbetweengravitationalmassandinertialmass.ThiswasdonebyNewton.RicherprovidedacluewhenhefoundthattheperiodofoscillationofapendulumcouldbeincreasedbytakingitfromParistoCayenneinFrenchGuyana(Bradley,1971,p107).TheperiodofapendulumisgivenbyLT=2π(1.2)g
20June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch018PhysicsfromtheEdge:ANewCosmologicalModelforInertiawhereListhelengthofthependulum,andgistheaccelerationduetogravity.Attheequator,thecentrifugaloutwardsforceduetotheEarth’srotationcounterstheaccelerationduetogravity(g)sothatitappearstobeslightlyless,sotheperiod(T)ofapendulumisgreater.Theamountofmass,definedas:density×volume,inthependulumwasthesame,butitsdynamics(theforceonit)hadchanged.Sincebodieswithdifferentweights(balls,feathers)haveequalacceleration,thenitisclearthataccelerationisnotjustproportionaltotheforceonsomething(weight)andwehavetosayAcceleration=weight(force)/somethingelse.(1.3)This‘somethingelse’isapropertythatcouldbecalled‘resistancetoacceleration’andisnowcalledinertialmass(mi)andtheformulabecomes:F=mi×a(1.4)ThisisNewton’ssecondlawofmotion,andstatesthatwhenyouapplyagivenforceinnewtons(N)theaccelerationyougetisdilutedbytheamountofinertialmassthebodyhas.Ifyoupushacarwithyourlittlefinger,theaccelerationissosmallthatitwon’tevenovercomefriction,butifyoupushafeatherwithyourlittlefingeritwillaccelerate.Newton’sfirstlawcanthenbederivedfromthissecondone.Ifwehavenoexternalforceapplied,F=0andso:0=mi×a(1.5)Thereforetheaccelerationiszeroandwegetthefirstlawasdescribedabove.ThismakesitseemthatallyouneedactuallyisNewton’ssecondlaw,andnotthefirst.Newtonalsousedsomethingcalledthegravitationalmass.Hedefinedtheforce(F)ofgravityasGMmgF=(1.6)r2whereGisaconstantapproximatelyequalto6.6×10−11m3kg−1s−2,Misthegravitationalmassofonebody,mgisthegravitationalmassofanother,risthedistancebetweenthem.ToworkouttheiraccelerationyouthenneedNewton’ssecondlaw.Sowehavenowtwodefinitionsofmass,thegravitationalmass(mg)andtheinertialmass(mi)anditseemsalthoughtheyare
21June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01AHistoryofInertia9derivedorfoundinverydifferentways(staticallyanddynamically)theyhavethesamevalue,whichiswhyafallingbodyhasaconstantaccelerationirrespectiveofitsmass.CombiningNewton’sgravityandsecondlawGMmgF==mia(1.7)r2Ifweassumethatmi=mg(theequivalenceprinciple)thenthissimplifiestoGMa=(1.8)r2andsotheaccelerationdependsonlyontheEarth’smassandnotthemassofthefallingobject.HowGalileowouldhavelovedtoseethisformula!Itisimportantinprincipletoseehowthingscanbemea-sured.Gravitationalmassisarealthingbecauseitcanbemeasured.Youcanusetheequationabove:dropanymassaknowndistancerfromabodymassMandtheacceleration(andaknowledgeofG)willtellyouthegravitationalmassM.NotethatthisisdependentuponourknowingG,soallthatcanreallybedoneismeasurerelativegravitationalmasses.Similarlyforinertialmass.Thewaytomeasurethatistoknowtheforceonanobjectandmeasureitsacceleration.ThenuseF=mi×atogettotheinertialmass(mi).Newton’stheoryhadtwomainproblems.Thefirstwasthatamechanismforgravitywasnotapparent,whyshoulditfollowaninversesquarelawwithdistance?Whatphysicalprocesswasinvolved?Newtonrefusedtospeculateaboutthissaying“Hypothe-sisnonfingo”or“Idonotfeignhypotheses”.Hebelievedthatifthecausewasnotclearlydeterminablefromthedatathenoneshouldbesilentaboutit.Thesecondproblemwasthepailexperimentwhichisthefirstrealstatementoftheconceptualpartoftheproblemsthathaveledtotheideaofphysicsfromtheedge.InthePrincipia,Newtondescribedthefollowingexperiment.Fillabucketwithwaterandsuspenditfromaceilingwitharope,twisttheropeandthenreleaseitandletthepailspin.Atfirstthepailspinsandthewaterdoesnotbecauseitsinertiaishardtoovercome,anditssurfaceisflat.Eventuallythough,thepail’sspinistransferred
22June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0110PhysicsfromtheEdge:ANewCosmologicalModelforInertiatothewaterbyfrictionandthewaterspins,anditssurfacebecomesconcave:itbendsuptowardstheedges.Thisoccursbecauseofiner-tia,which,asNewtonwasthefirsttorealise,makesitdifficultforobjectstochangetheirspeedanddirection(theyfindithardtoaccelerate,whichincludesspinning).Thisiswhythevectorquantityvelocityisneededinsteadofthescalarquantityofspeed.Objectsliketotravelinstraightlines,thelawofinertia,butthespinningwatercannotbecauseitisheldwithinthewallsofthebucket.Thebestthewatercandoismoveoutwardstowardsthewallofthebucket.So,thewaterpilesupatthisedgeuntilapressuregradientforceduetothegreaterheightofwaterformsthatislargeenoughtopushitbacktowardsthecentreandastableconcavewatersurfaceistheresult.Newtonasked:whatcausesthecurvedsurfaceofthewater?Itcannotbetherotationofthewaterrelativetothewallsofthepail,becauseatthestartthewaterwasrotatingrelativetothepail(theFigure5.Newton’spail.Thebucketisspun,andthewaterstaysstillatfirst,thenfrictionwiththemovingwallofthebucketmakesitspintoo.Thewatertriestomoveinastraightlinesinceithasinertia,andsomovesoutwardsbutthewallsofthebucketstopitanditpilesupnearthewall.Howdoesthewatermeasureitsownmovement?Relativetothewallsofthebucket?No,itsconcavitywaszerowhenthewallwasmovinganditwasstatic.Itsconcavityismaximumwhenitismovingrelativetoitssurroundings,andNewtonsaiditwasmovingrelativetoabsolutespace—abitlikeanimaginarygrid.
23June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01AHistoryofInertia11pailwasrotatingrelativetothewaterwhichisthesamething)andtheconcavitywaszero.Theconcavitywaslargestwhentherelativerotationsofpailandwaterwaszero,theywererotatingtogether(Hoffman,p40),buttheywererotatingrelativetotheirsurroundings.SoNewtonsaidthattheconcavitymustbecausedbytherotationofthewaterwithrespecttosomethinghecalledabsolutespace.Heimaginedthatspacewaslikeastageuponwhicheventsoccurredandcouldberigidlydefinedusingx,y,zco-ordinates.MachB.G.BerkeleyandG.LeibnitzhatedNewton’sconceptofabsolutespace,butitwasErnestMachwhowasthefirsttodiscreditit.Hehatedanyconceptthatcouldnotbedirectlyobserved,andheusedNewton’sownthirdlawtoattacktheideaofabsolutespace.ThislawstatesthatForeveryactionthereisanequalandoppositereaction.IcouldneverunderstandwhythislawwasneededwhenIwasatschool,itseemsbothobviousanduseless,butithasitsuses,forexample(Gleick,2003,p133)iftheEarthpullsontheMooncausingittoorbit,thentheMoonalsotugsontheEarthcausingtheEarthalsotomove,moreslightly,andtoshowtidesinitsfluidcomponents.Macharguedthatifabsolutespacewassomehowproducingaforcethatcausestheinertiaofthewater,thensurelytheremustbeareactionofthewaterbackontothenearbybitofabsolutespace,butbydefinitionNewtonhadassumedthatabsolutespacewasunaffectedbywhateverwentonwithinit,itwasabackgroundonly,sothiswasacontradiction.InsteadMachgotaroundthisproblembysayingthatthewaterwasinteractingwithalltheothermassintheuniverse,andespeciallythemoredistantmattersincetherewasmoreofit.Thisgetsaroundtheproblembecausealthoughtheeffectofalltheothermatterintheuniverseonthewaterissignificant,thereactionaccelerationcausedbythetinyamountofwaterinabucketonalltheothermatterintheuniverseisincrediblytinysincetheinertialmassoftheuniverseissohugecomparedtothemassofthewaterinthebucket.Thissolution
24June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0112PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure6.Mach’spail.UnlikeNewton,MachabhorredtheideaoftheinertialforcecomingfromarotationoraccelerationwithrespecttoaneternalabsolutespacebecausetherewasnowaythatspacecouldcopewithareactionbackfromthebucketinclearviolationofNewton’s3rdlaw.Hesuggestedinsteadthatinertiawasaforcethatappearedwhensomethingrotatedoracceleratedwithrespecttoalltheothermatterinthecosmos.Thiswaytherecouldbeareactionofthebucketonthecosmos,butitwouldbeinsignificantbecauseofthehugemassofthecosmosrelativetothepail.doesofcourseproduceanotherproblemandthatis:howdoesthecommunicationbetweenthewaterandtherestofthedistantuniversehappen,ifspecialrelativityforbidsfasterthanlighttravel?Wewillcomebacktothisproblemlateron,andfailtoresolveit!Machalsosuggestedawaytodefineinertialmassorexperimen-tallydetermineit.Hesuggestedthatthebestwaytodothisistohavetwobodiessidebysidethatattractgravitationallyormagnetically(theoriginoftheforceisimmaterial,forgivethepun).Theywillthenmovetowardseachother,andtheratiooftheiraccelerationswillbethesameastheratiobetweentheirinertialmasses(Bradley,1971,p110).Machalsosaidthattheprincipleofinertiaisincluded
25June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch01AHistoryofInertia13Figure7.MachdidnotapproveofNewton’sconceptofabsolutespace,whichisathingthatcanneverbeseen.Hepreferredtodefinetheorieslike“we’removing”relativetoobservablethings,likesolidrocks.inthissinceifthereisnosecondmass,thenthereisnoaccelerationofthefirst,soitsvelocityisconstant.Aconstantvelocityisthedefaultmode.BeforeweleaveMachentirely,itisworthmentioningafertileattitudetosciencethathehadwhichinfluencedspecialrelativity,andisalsousedlaterinthisbook.MachstatedthatNewton’sconceptsofabsolutespaceandtimecouldnotbedefinedintermsofobservationsyoucouldevermake.Thereforetheyweremeaningless.Theywere“purelyathought-thingwhichcannotbepointedtoinexperience”(Isaacson,2007,p84).This‘don’ttrustthingsthatcan’tbeseen’attitudewasveryimportanttotheearlyEinstein,buttodiscussthatwefirstneedtotalkaboutlight.
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27June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02Chapter2ModernPhysicsEinsteinThegreatestadvancesinphysicsinthe20thcenturyare,forme,quantummechanicsandspecialrelativityandnotgeneralrelativity,forreasonsthatwillbeexplainedlater.TounderstandspecialrelativityinallitsstrangenessweneedtogobacktothelawsofmotionwearefamiliarwithandthatweresummarisedbyGalileo.ItwasproposedbyGalileothatifAisastaticobject,(inourtime)sayamanwaitingforabus,andBisabusmovingatavelocityu,thenthepositionsandthetimeofB(x,y,z,t)relativetothatofA(x,y,z,t)aregivenby(seealsoFeynman,1977,Chap.15):x=x−uty=y(2.1)z=zt=tTheseformulaeagreewithwhatweseehappeningtosolidobjectsintheworldaroundus.Ifwearedrivingonthemotorwayat70mph,andsomeonecrawlspastusatarelativespeedof30mph,thenusingthefirstequation,inonehourtheywillbeataposition(x−ut)of0−30mph×1hr=30milesaheadofus,andwillprobablyhaveapolicecarafterthem.WecandrawthisquiteclearlyasshowninFigure1.Thexaxisrepresentsthedistancetravelledandtheyaxisisthetime.Thecartravellingat70mphisthelefthanddiagonalline,andthefastercaristhelinewiththelesserslope.Ifwewanttoknowhowmuchfurtherthefastercarhastravelledafter1hourwejustneedtostart15
28June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0216PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure1.Aspacetimeplot.Spaceisplottedonthex(horizontal)axisandtimeonthey(vertical)axis.Alineontheplotrepresentsthetrajectoryofanobject.thelinesattheorigin,watchthemdivergeforanhourandmeasuretheirdistanceapartonthehorizontalaxis:inthiscase30miles.Howdoesthisapplytolight?AlthoughtheancientGreekEmpe-doclesfirstclaimedthatlighttravels(Sarton,1993)theinfluentialGreekphilosopherAristotledisagreedandeventuallytheGreeksthoughtthatlightwasemittedfromtheeyes,andsincethedistantstarscouldbeseenlightmusthaveinfinitespeed.ExperimentalproofthatlighttravelsatfinitespeedwasfirstprovidedbyOleRomerin1676.HerealisedthatwhentheEarthismovingawayfromJupiterinitsorbitthetimeelapsedbetweentheoccultationofJupiter’smoonIobyJupiteranditslateremer-gencewouldbelengthened,sincethelightfromtheemergenceofthemoonwouldhavefurthertotravelthanthelightfromtheini-tialoccultation(thetimeintervalisshortenedwhentheEarthismovingtowardsJupiter).TheamountbywhichthetimeintervalislengtheneddividedbythedistanceEarthhasmovedalongitsorbitisthespeedofthelight.Romercalculatedthespeedoflighttobeapproximately220,000km/s.Wenowknowittobe300,000km/s.Unfortunately,hewasignoredatthetime.In1873JamesClerkMaxwellwrotedownhisfourequationsthatdescribedhowtheelectricandmagneticfieldsplayoffeachotherand
29June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics17hepredictedthatthisinterplaywouldleadtoanelectromagneticwavewithaspeedof300,000km/s.Tohisamazementthiswasthespeedoflight.Thusitwasfoundthatlightwasanelectromagneticwave.ButtherewasaparadoximplicitinMaxwell’sequationsandGalileanrelativity.In1899orsoEinsteinimaginedwhatwouldhap-penifhefollowedalightbeamatitsownvelocity.Ifyoudidthatyouwouldexpecttoseetheelectricandmagneticfieldsstandingstillinspace,butherealisedthatMaxwell’stheorydoesnotseemtoallowanythinglikethat,theelectromagneticwavealwayshastotravelatthespeedoflight!OtherpeoplewonderedwhetherlightwouldfollowGalileo’srulesornotandMichelsonandMorleyintheUSAattemptedtomeasuretheeffectofthemotionoftheEarthonthelightwaves.Bydifferen-tiatingtheaboveGalileanequationswithrespecttotime,onewouldexpectthatthevelocityoftheEarthinitsorbit(about30,000m/s)whichis0.01%ofthespeedoflight(c)wouldmeanthatalightbeamgoingalongwiththeorbitwouldbe0.01%slowerthanonegoingacrosstheorbitaltrajectory.In1881MichelsonandMorleydecidedtotestthisusinganinterferometerasshowninFigure2.Here,alightbeamissplitbyahalf-silveredmirrorintotwocompo-nents,oneintotheaetherwind(againsttheflowofNewton’sabsolutespace)andoneperpendiculartoit.Thelightthattravelsupstreamanddownstream(alongtheEarth’sorbit)cangoinastraightlineFigure2.SchematicshowingtheMichelson–Morleyexperiment.
30June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0218PhysicsfromtheEdge:ANewCosmologicalModelforInertiaandalthoughitwouldbeslowedupbytheaetherwindonitsforwardjourney,itwouldbespedupbyitonitswayback,andtheseeffectscancel.However,thelightthatgoesperpendiculartotheEarth’sorbit(upanddownontheschematic)wasexpectedtotravelfurthersinceithastotravelalongadiagonalpathtogetbacktothemirror,whichhasnowmoved(youcanjustaboutseethesediagonalsontheschematic).Soitwasexpectedthatwhenthetwobeamsoflightgotbacktothehalfsilveredmirrorandwererecombined,sincetheyhavetravelledadifferentdistancethereandback,thephaseoftheindividualwavesinthelightbeamshouldnotcoincide.Instead,theyalwaysdidcoincide.Sincetheindividuallightwavesareveryshort,about10−6metres,thismethodshouldhavedetectedeventhespeedoftheEarthinitsorbitof0.01%ofc.Thiswasashockinganomaly.WerethenewMaxwell’sequationswrong?Thiswasunlikelysincetheyseemedtoagreeverywellwithotherobservations.MaybetheolderequationsofGalileowerewrong,andtherewasapowerfultabooagainstthissincepeoplehadlongtestedtheseequationssothattheyseemedtobealmostcommonsense.ThisiswhythesolutionfoundbyEinstein(1905)wassoshock-ing.Heproposedthatspacetimeitselfdistortssothatthespeedoflightisalwaysmeasuredtobethesame.Moregenerallyhesaidthat:Allthelawsofphysicsarethesameinallinertialreferenceframes.Aninertialreferenceframeisonewhichisnotaccelerating,andthephrase“thelawsofphysics”includethelawthatthespeedoflightis300,000km/s.ThismeansthattheGalileanrelativitylawsdescribedabove,whicharesoingrainedinourexperienceofsolideverydayobjects,donotworkforlightandallelectromagneticradi-ation.Thereisnorealsurprise,sincewhyshouldthemechanicsofsolidobjectsworkforlight,butthisdifferencehadescapedhumannoticeuptillthensincelightmovessofastthatittakescenturiesforsocietiestoevennoticethatitmovesatall(Adams,1990).TheexactmathematicaltransformationrequiredtomodifyspaceandtimetokeepMaxwell’sequationsthesameforallobserverswasfirstsuggestedasa‘real’distortionofobjectsbytheIrishmanGeorgeFitzGeraldandgivenmathematicalprecisionbytheDutchmanHen-drikLorentz,priortoEinstein’smorephysicalinsight.Togetherthey
31June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics19Figure3.Lorentzcontraction.InorderthatnochangeinthespeedoflightcanbedetectedbytheMichelson–Morleyexperiment,themetricofspaceitselfmustcontractinthedirectionofmovement.Forlighttravellingatthespeedoflighttheleft-rightcontraction(seetheschematic)iscompleteandthisgetsridoftheextradiagonaldistancethelighthadtotravelingoingtoandbackfromtheuppermirror.replacedtheGalileantransformationsmentionedabove,withthesetransformationsforanobjectmovingatspeeduinthex-direction:x−utx=1−u2/c2y=y(2.2)z=zt−ux/c2t=1−u2/c2wherecisthespeedoflight.ThesearenowcalledtheLorentztransformations.FitzGeraldandLorentzseemedtobelievethatthelengthchangeswererealchangesintheobjectsrelativetoabsolutespace(arealshrinking),butEinstein,followingMach,dislikedtheaether,whichwas“purelyathoughtthingwhichcanneverbemea-sured”,andalsowasscepticaloftheconceptsofspaceandtime,whichalsocannotbemeasuredwithouttheuseoflight.HeproposedthatinsteadofchangingMaxwell’selectromagneticlaws(whichonecantestinalaboratory)oneshouldbeelasticwiththefundamental
32June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0220PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure4.Lightningstrikesastationarytrain(pictureabove).Thelightgetstothepersonsittinginthecentreofthecarriageatthesametime.Theysayitissimultaneous.Iflightningstrikestheendsofthecarriageofatrainmovingforwardthelightfromthelightningatthefrontofthecarriagewillgettotheobserverbeforethelightfromthestrikeattheback,sotheywillsaythestrikesarenotsimultaneous.EinsteinwasafollowerofMachwhosaidthatifsomethingisnotobservable,itdoesnotexist.Thisalsomeansthatwhateveryoucan‘see’istherealityandthereisnosuchthingassimultaneity,orthat‘timeisintheeyeofthebeholder’.conceptsofspaceandtimewhichonecannotmeasuredirectlyany-way.Themeaningoftheaboveequationsisthatspaceandtime(xandt)changeinorderthatc(whichcanbemeasuredmoredirectly)iskeptconstantforallobservers.Forexample,iflightningstrikestheendsofatrainatrest,seeFigure4,apassengerinthemiddlecarriagewouldseeeachstrikesimultaneously,butifthetrainismovingforwardthepassengerwouldseetheforelightningstrikefirstsincehewouldbemovingtowardsthelight.Soawordlike“simultaneous”dependsonyourreferenceframe(howfastyouaremoving)andthereisnoabsoluterightanswertothequestionof“wheredidthelightningstrikefirst?”,itbecomesrelative(hencethename)toyourownmotion.ThereforejustasMachhadargued,thereisnoabsoluteNewtoniantimethat
33June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics21Figure5.Amirrorclock.Therearetwomirrors,oneaboveandonebelow(seetheshadedareas)andalightbeambouncesatthespeedoflightrepeatedlybetweenthem.Allobserversmustseethelighttravelatthespeedoflightcwhethertheyaretravellingwiththemirrorsornot(Einstein,1905)soifthemirrorsmove,thefrequencywithwhichthelightbeamhitsamirror,ameasureoftime,hastoslowdown.existsforeveryone,eachreferenceframehasitsowntimeandthesamegoesforspace.Onesimplerwaytothinkaboutspecialrelativityistoimagineaclockmadeoutofacoupleofmirrorsandabeamoflightthatbouncesbetweenthem.Onebouncemeansonetickoftheclock.ThesituationisshowninFigure5.Twomirrorshavetheirreflec-tivesurfacespointingateachotherandabeamoflaserlightbouncesbetweenthem.Now,Einstein(1905)madetheassumption,whichhasbeensuccessfullyconfirmedbyexperimentthat,providingyouarenotaccelerating,allthelawsofphysicsmustbeunchangedforyou.ThisisverysimilartoGalileo’sideathatapersononashipwouldnotnoticethemotionoftheship,unlessanaccelerationoccurs,which,onaship,Ihavefound,italwaysdoes(IwentonascientificcruisetoIce-landinastormyautumnonceandwassickformostofthetime).ThelawsthatmustremainconstantincludeMaxwell’selectromagneticlawsthatforcethespeedoflighttobec=300,000km/s.Einstein’sassumptionwasprovokedbyhisthoughtexperimentaboutchasingalightbeam,andtheapparentobservationbyMichelsonandMorley(1887)thatthespeedoflightdidnotvarybecauseoftheEarth’smotionthroughspace,whichhedefinitelyknewabout(Pais,1982).
34June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0222PhysicsfromtheEdge:ANewCosmologicalModelforInertiaYoucandetermineintervalsoftimebecauseeverytimethelightbeamhits,say,theuppermirrortheclockticksandyoucanarrangethingssothatitticksoncepersecond.Toproduceaclockmeasuringseconds,sincethespeedoflightissofast,themirrorswouldhavetobe150,000kmapart.ThisisroughlythedistancetotheMoon,butwecandothisinathoughtexperiment.Now,imaginethatsuddenlythismirrorclockisputontoamov-ingvehicleanddrivenpastyou(seetheschematicontheright).Nowweareforcedtokeeptheassumptionthatthespeedoflightcan’tchange,youhavetoseethebeamoflightmovingatthesamespeed(c)becausethiscanbemeasuredanditseemsitisalwaysconstant,butnow(seeFigure5)thelighthasmoredistancetotravelinthesamesecond,sincetheclockismovingsidewaysthelightisnowtravellingalongadiagonal,soforsomeonelookingatit,thetimemeasuredontheclockappearstoslowdown.Theticksoccurlessfrequentlysoitappearsthattimeontheclockhassloweddown.Also,forabuglivingontheclock,sincehisbrainworksusingelec-tromagneticwaves,histhoughtprocessesarealsosloweddownsothathedoesnotfeelthattheclockisrunningslowatall.Itallseemsthesametohim.Themainpointhereisthatyoumightthinkthattheslownessofthemirrorclockisjustanapparentthingbecauseyouareseeingitfromafar,butthisisnotthecase!Thishasbeentested.Somescientists(HafeleandKeating,1971)leftoneatomicclockathomeandtookoneforarideonafastandhighplanetoslowitdownandspeeditupbyspecialandgeneralrelativityrespectively.Whentheybroughttheclocksbacktogethertheeffectsofrelativitywerestillthere:theclocksdidn’tagree.ThisisamazingandconfirmsthephysicalintuitionofEinsteinoverthemeremathematicalapproachofLorentz,becauseitmeansthattheslowingdownoftimeis“real”(whateverthatmeans)andnotjustapparent.ThishasahugeimplicationthatgoesbacktomyearliermentionofBerkeleyandMach:thatrealityiswhatyoucanobserve.Itseemsthatbecauseitisimpossibleinourreferenceframetoeverperceive(usinglight)themovingmirrorclockgoingatthe‘normal’speed,thenitdoesn’tgoatthenormalspeed,itgoesattheonlyspeedwecanperceiveittogo,i.e.:slower.Itisnotparticularlythatwe
35June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics23ashumanscannotseeit(itisnotanthropomorphicorsubjectiveinthisway),butratherthatthiscannotbeseen“inprinciple”,amoreobjectiveview.Howdoesallthisapplytomass?EinsteinfurthershowedthattomakethechangetoNewton’slawsallyouhavetodoischangetheinertialmassinthefollowingwaym0m=(2.3)1−v2/c2wheretherestmassm0istheinertialmassofabodyinitsownrestframe(butinordertomeasureinertialmassyouhavetogetittomoveinresponsetoaknownforce,soitisuncleartomewhatm0means).Thisformulalooksinnocuousenough,butithasahugeimplication.Notethatifyouputv=cintotheformulathedenomi-natorbecomes1−1=0andtheinertialmass(m)becomesinfinite.Thisistheoriginfortherelativisticpredictionthataspaceship,oranyobject,canneverpassthroughthespeedoflightsinceitsinertialmassbecomesinfiniteandyouneedinfiniteenergytopushitacrossthespeedoflightbarrier.Thisdoesnotmeanthatthingsthatweretravellingfasterthanlightinthefirstplacearedisallowed,itjustmeansthatobjectslikeustravellingslowerthanlight,arenotabletoexceedit.Thisseemstoprecludeinterstellartravelinahumanlifetime,thoughasweshallseelatertheremaybeawayouthere.Aspeedoflightlimitisalsopuzzlingcosmologically.Whenonethinksaboutit,itseemsthatthenightskyshouldbebright.Ifweassumethatthestarsareuniformlybrightandhomogeneous(uniformlyspreadout)thenthestars’brightnessfromwhereweareshoulddecreasefollowinga1/r2law,whereristhedistance,butthenumberofstarsateachdistanceawayisproportionaltothesurfaceareaofashellarounduswhichis4πr2,sotherearefarmoreofthemfarawayandthiscompensatesforthefactthattheyaredimmer.Whatthismeansisthatthenightskyshouldbewhite,butitisdark.Thedarknessofthenightskyoccurs,itisthought,becauseoftheopaquegasanddustbutalsotheHubbleexpansionofthecosmos.Thefartherthestarisawayfromus,thefasteritisrecedingfromusandthegreateristheDopplershifttoitslight.Atsomegreat
36June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0224PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure6.AcommentontheinsightofMachandEinsteinthattheworldseemstodependtosomeextentonwhatwecansee,usinglight,aboutit.distance,theHubblescale(whichisthoughttobeabout2.7×1026m)away,thestarsarerecedingfromusatthespeedoflight,theirlightisinfinitelyredshiftedandwecannotseethem.Thisisonewaytoaccountforthedarknessofthenightsky,butraisessomequestions.Onequestionis:whathappenstoastarattheHubbleedge?JustshyoftheHubbleedgeitismovingrelativetousslowerthanlightintheHubbleexpansion,butafewminuteslaterbeyondtheHubbleedgeithasacceleratedandisnowmovingfasterthanlight.Accordingtospecialrelativity,itsinertialmassshouldhaveapproachedinfinityatthespeedoflightsomewhereinbetweenthesetwotimes,sohowdiditmanagetoacceleratepastthisspeed?Ingeneralrelativitythereasongivenforthisisthatitisspacetimeitselfthatisexpanding,butthisanswerseemsunsatisfactorytomesincehowcanyoutestexperimentallywhethertwostarsaremovingapartorspacetimeismovingthemapart?ThisseemstometobeafixandIwillcomebacktothisproblemlater.
37June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics25Figure7.TheDopplershift.Anambulancetravellingfasterthansoundwouldnotbeimmediatelydetectableusingsoundwaves,justlikestarsattheedgeofthecosmosarenotdetectablebylightbecausetheyaremovingawayfromusfasterthanlight.TheEquivalencePrincipleSpecialrelativityonlyappliestounacceleratedmotionswhichisnotthemostusefuloflimitationsbecausethissortofenvironmentonlyoccursindeepspace.EinsteinwasunabletoapplyittosituationswithgravitywhichiswhatweareusedtoontheEarth,butin1909hehadwhathedescribedashis“happiestthought”,anditwasthatifyouarebeingacceleratedatacceleration‘a’withnogravitatingmassnearby,thenyouwouldfeelexactlythesameasifyouwerenotbeingacceleratedbutclosetoagravitationalmasssothattheaccelerationitcauses,g=a.So,ifyouareinanelevatorthatbreaksfreeandfallsdownundergravitythetwoeffects,ofgravityandaccelerationwillcanceloutexactlysothatyoufloatfree,aspeopledoontheInternationalSpaceStationsincetheyarefreefallingaroundtheworld.ThisisthesameinsightthatGalileohadwhenheconsideredballsofdifferentweightsbeingdroppedfromthetowerofPisa.Aswillbedescribedinaminute,thisledtotheGeneralTheoryofRelativitywhichwasahugemathematicalandperhapsalsophysicalaccomplishment.However,aswillbecomeclearlaterinthisbook,I
38June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0226PhysicsfromtheEdge:ANewCosmologicalModelforInertiaregardthis“happiestthought”ofEinsteinasbeingrathermorebasedonaestheticalconsiderationsthananyrealdata.Itisabeautifulidea,butthereisnothingthatforcestheinertialmassandthegravitationalmasstobeequal,andIwillspendtherestofthebooktryingtoconvinceyouthatitisn’tquitetrue,butthatthedifferencebetweenthemisofsuchakindthatithasnotbeendetected.But,backtoEinstein:imagineyouareinastaticelevator.Youseeabeamoflighttravelacrossinslowmotionandittravelsinastraightline.ThisisnormalphysicsandEinsteinsaysspacetimechangestokeepthelawsthesame.Nowimaginetheelevatorisaccel-eratingupwards.Nowalightbeamtravellingacrossyourelevatorwouldappeartohaveatrajectorycurveddownwards.Ohoh!Thelawsofphysicsseemtohavechanged,lightisnotmovinginstraightlinesbutisnowmovingoncurves.SoEinsteinextendedtheprincipleofrelativitysothatthelawsofphysicsmustbeunchangedinacceler-atedreferenceframes,buthowdoyouunbendthetrajectoryofalightbeam?Well,EinsteinandMachhadnoconcernoverbendingspacesinceitcan’tbeseenanyway,soEinsteinarguedthatspaceitselfiscurved(actuallyspacetime)andthelightismovingonastraightlineinbentspace.Ifwedothistherightwaythenthepersonintheeleva-torcannevertellbymeasuringthebendinalightbeamthattheyareacceleratingsincetheirrulerswilldistorttoosoastohidethebend.ThisiswhatEinsteindidwiththeGeneralTheoryofRelativity.Atthetime(1915)therewereothertheoriesofgravityaround.In1914,forexample,aFinnishscientistGunnarNordstrom,wrotedownMaxwell’sequationsinfivedimensionsandthenfoundthatwhatcameoutwaselectromagnetismandgravity.Hehadthusuni-fiedthesetwoprocesseseasily.Hegotaroundtheproblemoftheinvisibilityoftheextradimensionbysayingitwasacirclesoifyoulookalongityoucomebacktowhereyoustartedandthecirclecouldbemadetobetoosmalltosee.However,NordstromandEinstein’stheorieswerebothtestedbyobservations,andonlyEinstein’sgeneralrelativitypredictedthebendingoflightaroundtheSunandthisbendingwassubsequentlyseenbyEddingtonin1919,althoughtheresultsatthetimewereprobablynotgoodenoughtoreallyprovethis(Smolin,2006,p39–43)subsequentmeasurementsinhighaccelerationregimeshavesincealso
39June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics27Figure8.Einstein’sinsightwasthatthelawsofphysicsshouldbethesameeveninacceleratedreferenceframes.Thisincludesthelawaboutthespeedoflightbeingthesameandlighttravellinginastraightline.So,alaserbeaminaliftshouldneverallowustomeasureitscurvature.Thecurveofspacewouldbendourrulerssowecouldnevermeasureit.supportedgeneralrelativity(thoughnot,Iwillargue,thoseinlowaccelerationregimes).ItisinterestingthatNordstrom’sapproachofaddingdimensionstotryandunifytheoriesmanagedtogiveasensiblelookinganswer,untilitwasfalsifiedbythedata,andthisshouldbeawarningtoall,sinceitispossibletogettherightanswerforthewrongreason(Ptolemy’sepicyclesareanotherexample)andonlydatacansettherecordstraight.UnfortunatelyinmyviewNordstrom’sapproachhasbeenpursuedbystringtheoristseversinceandtheyarefamouslyshyofanyobservationbasedtests.GeneralrelativitywasalsotestedontheprecessionoftheperihelionofMercuryandthebendingofstarlightbytheSun.Ithasrecentlyalsobeentestedbyexperimentsinorbit.Forexample,thesatellitesusedforGPSnavigation,aretravellingveryfast,sothattheonboardclocksshouldappearfromtheEarth
40June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0228PhysicsfromtheEdge:ANewCosmologicalModelforInertiatobegoingslowerduetospecialrelativity.Theactualformulais1v2=1−(2.4)γc2whichisthefractionalamountofchangeduetospecialrelativityforaninitialvalueof1inducedbymovingatspeedv.ForGPSsatelliteswhichmoveat3.9km/sthischangeisaverysmallnumber:a1partin1010slowing,andthishasbeenconfirmed,andisroutinelycorrectedfor.Also,thesesatellitesareinamediumEarthorbit20,000kmhighandveryhighupinthegravitationalfieldsoaccordingtogeneralrelativitythetimeasmeasuredbythemshouldspeedupbyafactorgivenby12GM=1−(2.5)γrc2ForGPSsatellitesataheightorradius(r)of20,000kmgeneralrelativitycausesaspeedupof5partsin1010andthisisalsoobservedandiscorrectedfor.GravityProbeB(GP-B)waslaunchedin2004tomeasuretwopredictionsofgeneralrelativity,frame-draggingandthegeodeticeffect(NASA,2005).TheGP-Bsatellitecontainedfourgyroscopescooledbyliquidheliumto2Ktomaketheirniobiumcoatingsuper-conducting.Thesegyroscopeswereexpectedtomaintaintheirori-entationinspaceandeachhadatelescopeonitlookingataguidestar,soanychangeintheirorientationcouldbeseen.ThegeodeticeffectisaconsequenceofthedistortionofspacetimebyalargemassliketheEarthandshouldcauseamuchlargereffectonthegyroscopes.YoucanpicturethisbydrawingacirclewiththesameradiusastheEarth.NowputtheEarthinsideandaccordingtogeneralrelativityspaceitselfbendssothatthecircumferenceofthecircleisreducedby1.1inches.Nowbecauseofthisagyroscopewillchangeitsorientationasitgoesaround(StanfordUniversity,2013).Theexpectedchangeintheirorientationduetogeneralrela-tivitywas0.0018degreesperyear.IfNewtonwasrightinsteadthenthegyroscopes(notsubjecttoanyexternalforce)wouldnotchange
41June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics29theirorientationatall.ThegeodeticeffectwasconfirmedbyGravityProbeB.Thesecondeffectistheframe-draggingeffectandiscausedbythelargemassoftheEarthdraggingspacetimearoundwithitasitspins.ItwasexpectedtocausetheorientationofthespinaxisoftheorbitinggyroscopestofollowtheEarth’srotationby0.000011degreesperyear.ThiseffecthasnotbeenconfirmedbyGP-B.Generalrelativityhasalsobeentestedusingbinarypulsars,thefirstofwhichwasdiscoveredbyHulseandTaylor(1975).Theseveryclosepulsarsarepredictedtoloseorbitalenergybygeneralrelativityintheformofgravitationalwaves.Thesearewavesinspacetimeitselfthatpropagateandtakeenergyawayfromorbitingbinaries.TheobserveddecayoftheorbitofHulseandTaylor’sbinarywasfoundtobeconsistentwithgeneralrelativity(TaylorandWeisberg,1982),butthisisonlyindirectevidenceforgravitationalwavesandgeneralrelativity.Thewavesthemselveshavenotyetbeenseendespitethemanydetectorsthathavebeenbuilt.OneoftheproblemsIhavewithgeneralrelativityisthatitstillsuffersfromtheproblemdescribedbyMachinthatitdependsonsomethingthatcannotinprinciplebeobserved:spacetime,andfur-therwhenappliedtothemotionofgalaxiesandgalaxyclustersitrequirestheintroductionof10timesasmuchmassasisseentoenableittopredicttherotationcurvescorrectly.Aswillbediscussedbelowthiscanbefixedbyaddingdarkmatterinahaloaroundthegalaxy,buttheparticlerequiredtogivethisdiffusehalohasnotbeenseen,andsinceitmustbediffuseitcannotexplainthesimilarbehaviouroftinyglobularclusters(seeChapter3).Howcanwedobetter?Acluemaylieintheinconsistencybetweengeneralrelativityandquantummechanics,especiallysincethelatterisafarbettertestedtheoryandhasagreedfarmoreaccu-ratelywithexperiment.ThisisreminiscentoftheclashbetweenGalileanmechanics(takingthepartofgeneralrelativity)andMaxwell’sequations(takingthepartofquantummechanics)inwhichMaxwell’sequationsprovedtobethemorefundamental.Isquantummechanicsmorefundamental?Well,itiscertainlymorebizarre...
42June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0230PhysicsfromtheEdge:ANewCosmologicalModelforInertiaQuantumMechanicsWhenthetemperatureofabodyisaboveabsolutezeroontheKelvinscale,asallbodiesare,theparticleswithinitemitthermalradiation(allchargedobjectsemitradiationwhenacceleratedandtemperatureisjustaccelerationonsmallscales).Sincetheradiationisabsorbedandemittedseveraltimesbyatomsandelectronswithinthebody,bythetimeitleavesthesurfacethespectrumiscontinuousandlooksratherlikethecurvesshowninFigure9.Foracolderbodytheradiationemittedhasalongerwavelength(thecurveontheright)andlessenergyisemittedoverall(theareaunderthecurveissmaller).Forahotterbodymoreenergyisemitted(theareaunderthecurveislarger)andthewavelengthisshorter(thelefthandcurve).Theemissionofheatradiationalsodependsonthecharacterofthesurface.Ifitismirrorlike(asinathermosflask)thenboththeemissionandabsorptionofheatradiationislowandthisisageneralrule,agoodabsorberisagoodemitter.Abodywhichisaperfectemitterandabsorberiscalledablackbodybecauseitwouldlookblackevenifyoushonealightonit(Ohanian,1985,p902).Onewaytomakeablackbodyistohavealargehollowcube(acavity)Figure9.Theenergycontent(verticalaxis)ofdifferentwavelengths(horizontalaxis)ofradiationemittedfrombodiesaboveabsolutezero.Thetwopeaksshowthespectrumofradiationemittedbyahotter(left)andcooler(right)blackbody.
43June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics31Figure10.Ablackbodycavity,likeamaze,easytogetin,noteasytogetout.withasmallholeinit.Theholeisthenagoodanalogytoablackbodybecauseifradiationgoesinthroughthehole,itbouncesaroundinsidethelargeboxandisunlikelytogetoutagain,andifradiationcomesout,itstaysout.SeeFigure10.LordRayleighcalculatedtheamountofthermalenergypresentinsuchacavityandfoundthataninfinitenumberofwaveswereabletofitintothecavity—theywerelimitedbecausetheyallhadtohavenodesattheboundarybuttherewerestillaninfinitenumberofthembecauseyoucouldhaveevenveryfine,shortwavelength,highenergywaves.Accordingtotheequipartitiontheorem,eachmodeorwavehasanenergyofkT(thisisBoltzmann’sconstantktimesthetemperatureinkelvin)andsotheenergyforallofthewavestogetherwasinfinite.Thiswascalledtheultravioletcatastrophe,whichreferredtothefactthatmostoftheenergyresidedattheshortendofthewavelengthspectrum,intheultraviolet.Planck(1901)solvedthisproblembyproposingthatenergyisalwaysquantised.Ifanoscillatororwavehasafrequencyofνthentheallowedvaluesofenergyare,E=0,hν,2hν,3hν,etc.,where
44June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0232PhysicsfromtheEdge:ANewCosmologicalModelforInertiathehisnowcalledPlanck’sconstant.Allothervaluesfortheenergyareforbidden(wavesthatdonotfitnicelywithintheboundary).Thissolvestheproblembecausethethermalenergyinthewallsissharedbetweenthedifferentoscillatorsinthewalls,butfortheoscillatorswithveryhighfrequencyν,theenergyofthehigherallowedenergy(e.g.:5hν)issolargethatevenrandomvariationsoftheenergyinthewallswon’tprovideenoughenergytoexcitethem,sothehigherener-giesremainunexcited,andtheultravioletcatastrophewasaverted(Ohanian,p905).Thislooksverysimpleinhindsightbutitwasarevolutionarysuggestion,nottakenseriouslyuntilEinsteinprovidedmoredirectevidencetosupportitfiveyearslaterusingthephotoelectriceffect.FromPlanck’slawitispossibletoprovethatthespectralemit-tanceofablackbodyhasamaximumatawavelengthgivenbyβhc0.0029λ(m)=∼(2.6)kTT(K)ThisiscalledWien’slaw,whichwasderivedtheoreticallybeforeitwastestedexperimentally,andinitappearsaconstantβwhichisknowntobe0.2,Planck’sconstanthalsoappearsandis6.6×10−34Js−1,cisthespeedoflight,andBoltzmann’sconstantkis1.38×10−23JK−1.ThisformulapredictsthepeaksofFigure9andsaysthatifthetemperatureofanobject(T)islow,thenthewavesofradiationcomingoffitarelong.Conversely,ifthetemperature(T)ishigh,thentheradiationcomingoffitisshort.Forexample,theSun’ssurfacetemperatureis5778Ksotheradi-ationcomingoffitisshortwaveradiationofwavelength5×10−7m.Bynotsuchacoincidencethisisthewavelengthoflightweareusedtoseeing.Wehavedevelopedthroughevolutiontoseethemostcommonwavelengthavailable.IncontrasttheEarthisalotcooler,about14◦C(287K)soitemitslongerwaveradiationofwavelength10−5metres.In1905EinsteintookPlanck’squantamoreseriouslyandshowedthattheyexplainedsomecuriousobservationsmadeofthephoto-electriceffect.Lenardin1902showedthattheenergyofelectronsejectedfromametalbylightdependednotontheintensityofthelight,butitsfrequency.ThissupportedPlanck’squantumhypothesissinceifyouincreasetheintensityoflight,itmeansyouarefiring
45June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics33Figure11.Planck’squantumidea.Allowwavesofanywavelength(energy)andyoucanbuildupinfiniteenergywithinacavity(left).Ifyourecognizethatenergycomesinpackets(thequantumwavesmustfitexactlywithinthewalls),thenit’sunlikelythatapacketwillhaveenoughenergytoexcitetheshort(highenergy)wavesandtheproblemdisappears.morelightenergyatthemetalandcanejectmoreelectrons,butthelightenergycomesinpacketsor‘quanta’,soeachelectroncanonlytakeonepacketofenergysotheelectrons’energyisunaf-fected.Ifyouincreasethefrequencyofthelightthough,thepacketsthemselvescanholdmoreenergyandsotheejectedelectronscanobtainmore.Planckhadassumedthatonlytheoscillatorsinthewallsofthecavitywerequantised,butEinsteinproposedthatradiationitselfwasquantisedevenwhenitwaszoomingaroundintheairandthatwaveswerecomposedofsayhνor2hνamountsofenergy.Planckdidn’tlikethisverymuch,buttheideaeventuallyprevailedandthepacketsofenergywerecalledphotons.Einsteinthoughtofradiationasbeingratherlikeagasofphotons(Ohanian,p907).ThisledeventuallytotheideasofLouisdeBrogliewhoproposedthatnotonlydidradiationbehavelikeaparticle,butalsoparti-clesbehavelikewaves,sothatwhenyoufire,forexample,protonsthroughtwoslits,eventhoughtheyaresupposedtobesolid,theyinterferewitheachotherandproduceaninterferencepatternonascreenbeyondtheslits.Thisisoftencalledwave-particledualityandshowsuptheapparentschizophreniaofthecosmos.
46June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0234PhysicsfromtheEdge:ANewCosmologicalModelforInertiaAnInterludeAboutHoyle’sCosmologyWehavenowlookedatinertia,gravityandquantummechanics.Gravityandquantummechanicsaremodelledverydifferentlyinphysics.Gravityismodelledusingasmoothlycurvedspacetime,whereasquantummechanicsismodelledusingtheexchangeofvir-tualbosons,particles.Oneissmooth,theotherispointlike.Whichismorefundamental?ThereisaproblemwithNewtonianmechanicsandalsogeneralrelativityinthatifitwereperfectlytruethen,surelybynowallthematterintheuniverseshouldhaveself-attractedandcollapsedintoonehugemass.Incontrastwhatweseeisauniversethatisnotonlynotcollapsingonitself,butitisactuallyacceleratingawayfromitself!InordertoescapefromagravitationalmassM,abodymustbeexpandingawayfromitatmorethantheescapevelocity(v)whichisgivenby2GMv=(2.7)RwhereGisNewton’sgravitationalconstant,MisthemassoftheattractingbodyandRisthedistancetoitscentre.PuttingnumbersintothisequationfortheEarth,forexample,youcancalculatethattheescapevelocityisabout10.4km/s,whichisprettyfast,sorocketsareneededtoescapefromtheEarth’sgravity.Itispossibletocalculatetheescapevelocityofthevisibleuni-verse,sincewecancountthenumberofstarswecanseeandmakeacrudeestimateofitsgravitationalmass(2×1052±1kg),anddeter-minethedistancetoitsedge(cosmicradius=c/H=1.3×1026m).Usingtheequationfortheescapevelocityaboveweget2G×2×1052kgv==143,000km/s(2.8)1.3×1026GiventheuncertaintyinthevaluesusedforthemassandtheHubblescale,thisvelocitymightbealowas45,000km/sorasbigas451,000km/ssoitisconsistentwiththespeedoflightof300,000km/s.Thismaybeacoincidence,butitisalsointeresting.Whymightthespeedoflightbeanescapevelocityforthecosmos?
47June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics35Figure12.Thetwo-slitexperimentworksforsolidparticlesaswellaspho-tons,butonlyforparticlessmallenoughthatquantummechanicaleffectsbecomevisible.Insomeregarditdoesmakesensesinceanythingfasterthanlightwouldpresumablybeabletoachieveauniversalescapevelocityandwouldnotbevisible,hencethesizeoftheobservableuniverse.ItispossibletoimaginelotsofothercombinationsofmassMandradiusRforwhichtheescapevelocityisgreaterthanthespeedoflightandthereforetheobjectwouldbeblacksinceanylightfallingonitwouldneverescapetorevealitspresencetous.Thecombinationoftheself-attractivenatureofgravityandtheeventualhugeescapevelocityistheideabehindablackhole.Rearrangingtheaboveequationgives:c2RM=(2.9)2GThisistheequationforthemassoftheobservableuniverseproposedbytheSteadyStatetheoryofHoyle(1948).ThewayIthinkaboutthistheoryisthatitisbasedonthespeedoflightbeingconstant,whichisapropertythathasbeenobserved,butthisideaisextendedtothespeedattheuniverse’sedge.IfweassumethatcmustbeconstanttherethenastheradiusRofthecosmosincreases,thentokeepthespeedoflight(theescapevelocity)constant,themassofthecosmoshastoincreasetokeepthematteratafurtherdistanceboundedtothecentre.SoHoyle’stheorypredictsthatastheuniverseexpands,itgainsgravitationalmass,followingtheequationabove.
48June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0236PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure13.Averysimplecosmology,thatisbasedontheideathatthespeedoflightisanescapevelocity.Thisagreeswiththeideathatwhensomethingismovingawayatthespeedoflightyoucannotseeit,soithasescapedfromyourknowledgeofit.Tomakethiswork,astheuniverseexpandsyoumustincreasethegravitationalmassinsideittokeepallthematterboundedtothecentreandkeepthespeed(c)constantontheedge.ThetheorywaseventuallydiscreditedbecauseitdidnotpredictorexplaintheCosmicMicrowaveBackground(CMB)radiation,whichappearstobearemnantofatimewhentheuniversewasextremelyhot,butasweshallseelatertheremaybeawaytoreconcileHoyle’sintuitivecosmologywiththeCMB.HawkingRadiationGenerallythecurrenttheoryofgravity,generalrelativityandquan-tummechanicsarethoughttobeincompatible,butthereisoneinci-dencewheretheyhavebeencombinedtomakeaprediction:blackholes.ThetheoryofQED(QuantumElectro-Dynamics)predictsthatvirtualparticles,includingvirtualelectronsareconstantlybeingcreatedallovertheplace.Thesameistrueattheboundaryofablackhole.WhatHawking(1974)noticedwasthattheseelectronsmaythenbreakupintotwovirtualphotonsattheedgeoftheblackhole,goingoffindifferentdirections,oneenteringtheblackholeandonezoomingoffintospace.Thisseparationcausedbytheeventhorizonmakestheexitingphotonsuddenlyreal.Heshowedthatthetemperatureofthisradiationisc3T=(2.10)2GMk
49June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics37Figure14.Twoblackholes.Oneissmall(ontheleft)andsoishotandradiatesshortwaveradiation(rememberWien’slaw)andtheotheroneismassive,coldandradiateslongwaveradiation(ontheright).ThederivationisgiveninAppendixB.Thisformulaisoddbecauseitsaysthatalightblackholeishot,andalargeoneiscold.Therefore,smallonesradiateheattotheirsurroundingsandtherebyloseenergyandmassandgetsmaller,gethotter,andradiatemoreuntiltheypresumablyvanishinablazeofheatradiation.Incontrastlargeonesarecoldand,iftheyarecolderthantheirsurroundingstheywillnotradiateatall.ThesurroundingtemperatureisthetemperatureoftheCosmicMicrowaveBackgroundandisabout2.7K,sowecandoaquickcalculationtoseewhatmasstheblackholehastohavesothatitdoesnotradiate:c31×10−34×(3×108)323M===5.4×10kg2GkT2×6.6×10−11×1.38×10−23×2.7(2.11)ThisislessthanthemassoftheMoon.IttellsusthatonlyblackholeswithmasseslessthanthatoftheMoonwillradiateandevaporate,biggeronesjustabsorbheatandenergyandsogetmoreandmoremassive(Adler,2006).Itwillbearguedlaterinthisbookthatouruniverseisjustsuchablackhole,growingandabsorbingheatfromitssurroundingsandgivingrisetotheaccelerationofthecosmos:whathasbeencalleddarkenergy.Figure14showstwoblackholes,oneissmallandsoishotandradiatesshortwaveradiation(rememberWien’slaw)andtheotheroneismassive,coldandradiateslongwaveradiation.
50June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0238PhysicsfromtheEdge:ANewCosmologicalModelforInertiaUnruhRadiationAblackholeisnottheonlywaytocreateaneventhorizon.Wheneveranobjectaccelerates,aso-calledRindlereventhorizonappears.Thisisbecauseifyouhaveanaccelerationa,forexampletotheright,informationfromacertaindistancetoyourlefthasnohopewhateverofcatchingupwithyou,providedyoucontinuetoaccelerate,sincethespeedofinformationtransferislimitedtothespeedoflight.Thedistanceawayfromyou(r)ofthisRindlerhorizonisgivenbyc2r=(2.12)aFulling(1973),Davies(1975)andUnruh(1976)derivedasimilarformulatoHawking’sblackholehorizon,forthisdynamicaleventhorizonandfoundthatthetemperature(T)seenbyanobjectwithaccelerationawasaT≈(2.13)2πckAderivationisgiveninAppendixC.Whatthismeansisthat,astheaccelerationincreasessodoestheUnruhtemperatureandalsothewavelengthoftheUnruhradiationshortens.WecanuseWien’slawthatwelookedatearlier,whichtellsusthewavelengthemittedfromablackbodyoftemperatureT,andwritedownanexpressionforthewavelengthintermsoftheacceleration:4π2βc28c2λ=∼(2.14)aaThisUnruhradiationusuallyhasawavelengthtoolongtoseewithourtechnology.Forexample,foraterrestrialobjectwithanacceler-ationof9.8m/s2theUnruhwavelengthis7×1016m.Thisisabouttenlightyears.However,Unruhradiationmightbevisibleforverylargeaccel-erations.Smolyaninov(2008)discussedanexperimentbyBeversluisetal.(2003)whoilluminatedagoldnanotipwithlaserlightandmea-suredanomalousinfraredphotoluminescencecomingoffthenanotip.HesuggestedthatthiswasUnruhradiation.Thisisplausiblesincetheaccelerationoftheelectronsaroundthetightcurveofthetinynanotipishuge,itisc2/r=9×1022m/s2,wherecisthespeedof
51June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch02ModernPhysics39Figure15.Anobjectacceleratingtotherightatanaccelerationa,seesaRindler2horizonformtoitsleftatadistanceawayofc/a.Figure16.Terrestrialaccelerations,forexampleapplesfallingofftrees,produceUnruhwavestoolongtobedetectedbyourpresenttechnology,butelectronsacceleratingaroundgoldnanotips,mayproduceUnruhwavesshortenoughtobedetected.lightandristheradiusofcurvatureandforsuchanaccelerationtheUnruhwavelengthis8c2/a=8×(3×108)2/(9×1022)=8×10−6m,withinthedetectableinfraredrange.Thisisclosetothewavelengththatwasseen.Themeaningofallthisrelativelyrecentworkinphysicsformeisthatwhereasfirstwehadaphysicsbasedonmechanicsandthermo-dynamics,probablybecausepeoplewereusedtothinkingintermsof
52June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0240PhysicsfromtheEdge:ANewCosmologicalModelforInertiamachinesbecauseoftheindustrialrevolution,physicsisnowmovingintoanerawhereinformationisused,thanksinparttothechangesinthinkingledbytheinnovationsofthecomputerage.Thischangeinthinkingmayhelptosolvesomeoftheproblemsthatpresentmachine-basedphysicscannotsolve.Therearetwowaysofdoingphysics,onewaspioneeredbyIsaacNewtonandinvolveslookingatdata,orbetterstillfindinga‘crucialexperiment’thatfocusesonthephenomenayouwanttounderstandandderiveatheoryfromwhatyoufind.Thisistheempiricalmethod.ThenthereisthemethodusedbyEinsteinwhereyoufindaparadoxsuchasfollowingalightbeamandnotingthatanelectromagneticwavecannotthenexist.Paradoxesaregreatplacestostartthink-ingfrombecausetheyrepresentthecollisions,likeingeologywithcontinentalplates,oftwodifferentparadigmsofthinking,neitherofwhicharetrue.Atthisinterface,thedeeperconceptdrivingthemmaybeseen,i.e.:magma.Iprefertheempiricalway,butwithatwist:theimportanceofanomalies.Theorycanonlygetussofarandcanbeanobstacletoprogress.TheimaginationofnatureisfarsuperiortoourssoitisnotenoughtotrytoimaginethemindofGodortrytouselogictoderivenewtheories.Ithinkitisnecessarytogolookingatrealdatatotryandfigureoutwhatisgoingon,andtogettonewphysicsthishastobeanomalousdata.ThatiswhereIshallgonext.
53June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03Chapter3ProblemsatLowAccelerationTheOrbitalBalanceWeliveontheEarth,andinthisenvironmentthereareaccelera-tionseverywhere.Atomsarevibratingwithheat,Brownianmotionisjostlingmolecules,soundandpressuregradientsarevibratingandmovingtheair,carsarezoomingpastandtheEarthitselfisspinningandorbitingtheSun.AtypicalaccelerationontheEarth’ssurfacedependsonlatitude.Attheequatoritisv2/r∼4602/6300,000∼0.03m/s2anditdecreasestowardsthepoles.Infreefallitis9.8m/s2.Progressinscienceoftencomesfromlookingatnewregimes.ForexampleGalileousedthenewlyinventedtelescopetolookatJupiter’smoonsforthefirsttimeandwitnessedthe,thenunknown,lawsoforbitalmotionandhespenttherestofhislifegettingintotroubleforthis.ThankstoKepler,NewtonandthesmallcorrectionofEinsteinwenowunderstandorbitalmotionverywell.Whenasmallerbodyorbitsalargeone,theyareattractedtogetherbygravity,butifthesmallerbodyisorbiting,thenitsinertialmassisalwaystryingtomakeittravelinastraightline,andthisinertialtendencypushesitoutwardsfromthelargerbody.Thisiswhatwecallacentripetalforce.Theorbitsthatwetendtoseemustbestable,otherwisetheywouldhavecollapsed,sotheremustbeabalancebetweenthesetwoopposingtendenciesgivenbyGMmgmiv2=(3.1)r2rwherethelefthandsideistheforceduetogravitypullingmasseswithgravitationalmassesofMandmgtogetheroveradistancerandtherighthandsideistheinertial(centrifugal)forcepullingthemapartandinthistermIhavewrittenthemassasmi.Thisistheinertial41
54June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0342PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure1.Asmallbody(blackcircle)orbitingalarger.Thereisabalancebetweenthegravitationalforce(G)pullingthesmallerbodyinwards,andtheinertialforce(I)tryingtogetittomoveinastraightlineandpullingitoutwards.mass,whichisusuallyconsideredtobeequaltothegravitationalmassmg,butthereisnoreasonwhytheyshouldbeequal.IfwedoassumethattheinertialandgravitationalmassesareequalthenwecansimplifytheaboveequationtoGMv=(3.2)rAndthispredictstheorbitalvelocity(v)ofthesmallerobjectmroundalargerone,anditpredictsthatitshoulddecreaseinapar-ticularwayasthedistancebetweenthem(r)increases.ThisformulahasbeenextremelywelltestedonmostoftheplanetsintheSolarSystem.Itpredicts,forexamplethatforittoremaininitspresentorbit,theorbitalvelocityoftheEartharoundtheSunmustbe6.6×10−11×2×1030v=∼30km/s(3.3)1.496×1011Which,luckily,itis.Ifitwasn’ttheEarthwouldnotbeinorbit.Assumingwehavethemassanddistancesright,Jupiter’sorbitalvelocityaroundtheSunispredictedtobe6.6×10−11×2×1030v=∼13.3km/s(3.4)5×1.496×1011
55June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration43Figure2.TheplanetsoftheSolarSystemarelockedinbyabalancebetweenthegravitationalforcetowardstheSun,andtheinertialforcepullingthemout.Forthisbalancetoworktheremustbeadropoffoforbitalspeed(v)withdistancefromtheSun,otherwiseinertialmasswouldwinandtheplanetswouldzoomoffintointerstellarspace.Thiscanbeseenintheplot,sincetheinnermostplanethasalargevelocity(largearrow),andtheouterplanethasasmallervelocity.whichisratherslower.Thepointisthattheorbitalspeeddecreasesastheorbitalradiusgoesup,becauseasthegravitationalattractioninwardslowersastheplanetgetsfurtherfromtheSun,thecentrifu-galforceoutwardsrequiredtobalanceitalsocanbesmaller,sotheorbitalspeedcanbeless.Thisformula,whichreliesontheassump-tionthatinertialmassisequaltothegravitationalmassworksverywellintheSolarSystem,althoughthereisstillsomedoubtastotheouterplanetssincesomeofthemhaven’tcompletedafullorbitsincewehavebeenmonitoringthem.Doesthisbalanceworkonlargerscales?Thereisnoguaranteeitwill,andthefirsthintoftroublecamefromaSwissastronomercalledFritzZwicky.DarkMatterZwickyhadsomethingofthecharacterofSheldoninthepopularTVprogram“TheBigBangTheory”inthathewasprobablymore
56June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0344PhysicsfromtheEdge:ANewCosmologicalModelforInertiabrilliantthanmostofhiscolleagues,buthehadahabitoftellingthem,eventotheextentofcarryingaroundwithhimapetitionthatheusedtoaskpeopletosignsayingthatheshouldbeawardedtheNobelprize.Healsousedtocallhiscolleagues“sphericalbastards”since,hesaid,theyseemedtohavethispropertynomatterwhichwayhelookedatthem.ProbablyasaresultofhisabrasivecharactermanyofhisdiscoverieswerenotreadilytakenupbyothersandhadtoberediscoveredmanydecadeslaterandhenevergothisNobelprize.ZwickyhadlookedattheComaclustersofgalaxies.GalaxiestendtocongregateingroupsorclustersandthereisarulecalledtheVirialtheoremthatstatesthattheamountofkineticenergypresentinastablesystemisusuallyabouthalfthegravitationalpotentialenergy,andthiscanbewritten:2KEavg=−GPEavg(3.5)Astronomersareverykeentoworkoutthemassofdistantsystems,andtheycandothisbylookingatthelightemitted,andassumingthatthemasstolightratioisthesameasthatoftheSun.Thisisnotaverydirectway.Whatiftheratioofbrightstellarmassestodarkgasforexampleisdifferent?OrifthestarsarenotSun-like?Thismethodwouldnotwork.AnalternativemethodistousetheVirialtheorem,sincethelefthandsideoftheaboveequation,thekineticenergy,canbedeter-minedfromtheaveragespeedofthestars(thisissomethingthatcanbedirectlyseen,soMachwouldapprove)andtherighthandsideisratherlikethegravitationalpotentialenergyontheEarth,whichisgivenbymgh,soyoucanworkoutthegravitationalmass(m)fromit(whichyouaretryingtofind).ZwickylookedatthespeedofthegalaxiesintheComaclustersandhenoticedtheyhadahugeamountofkineticenergyandheworkedoutthemassthatthisimpliedforastablesystem,anditwas400timeslargerthanthemassimpliedbythestarshecouldseeinthecluster—thelight.Theconclusionhecametowasthattherewasalotofdarkmatterinthecluster:alotofmatterthatwasgravitating,butnotlitup(Zwicky,1933).JanOortin1939alsonoticedthatthedistributionofmassthathewasabletoinferfromvelocitiesinanellipticalgalaxydidn’tlooklikethedistributionoflight.Thistoowasignored.
57June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration45Finally,inthe1960sand1970s,VeraRubin,managedtoshowmoreconclusivelythatthevelocitydistributionoftheAndromedagalaxyshowedaconstantorbitalvelocitywithradius.ThisisverydifferenttowhatispredictedbytheformulaeusedfortheSolarSys-temabovewhichpredictsadecreaseoforbitalvelocitywithincreas-ingradius.Whatcouldcauseaconstantrotationcurve?CombiningNewton’ssecondlawandhislawofgravitybutkeep-ingthegravitationalmassandinertialmassdistinct,wecanwriteGMmgv=(3.6)rmiZwicky,OortandRubinsawthatthevelocitiesofgalaxiesandgalaxyclusterswerefartoolargeattheiredges.Thereareseveralwaystoexplainthis.ThemostpopularsolutionandtheonesuggestedbyZwickyistoincreasethemassMintheformulaabove,bysayingthatthereissimplymoremattertherethanwecansee.Thisseemsreasonableenough(butreasonablenessisnotaproofthatitistrue).Ifmoredarkmatterisinthegalaxythentheouterstarscanorbitfarmorequicklyanddespitetheirgreatercentrifugaloutwardsacceleration,theycanstillbeboundedgravitationallytothesystem.Thiskindofthinkingalsohasaprecedent.Inthe19thcenturytheorbitofUranusshowedananomalywhichleadAlexisBouvardtopropose,andUrbainLeVerriertomoreaccuratelypredict,thatanunknownmass(aplanet)wasperturbingUranus’orbitgravita-tionally.Thisplanet(Neptune)wasdiscoveredclosetothelocationwhereLeVerrierhadpredicteditwouldbe.Thehypothesisofdarkmatterisverysimilartothis,butwhereasinthecaseofNeptune,itrequiredtheadditionofasmallamountofnormalmatterintheplausibleshapeofaplanet,galacticdarkmatter(becauseofthesquarerootintheequationabove)needstheadditionofatleasttentimesasmuchmatterasisobservedintheMilkyWay,inanewformofmatterandwithnewphysicstogowithittoexplainwhyittakesupabizarrehalo-likeconfigurationaroundtheedgesofgalaxies.Newphysicswouldhavetobeinventedtoexplainwhyitdoesn’tconvergeonthegalaxyandspininthesamepancake-likeformation.
58June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0346PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThefollowing(repeated)storyisnotintendedtodisprovedarkmatter,thatisverydifficult:asatheoryitisnotfalsifiable.Ratherthisstoryisintendedtoshowthedangersinherentinthiskindofthinking.TherewasasimilarargumentinGalileo’stime.Aristotlehadsaidheavierobjectsfallfaster,butGalileonoticedinahailstormthatbigandsmallhailstonesfelltogetherandlateronheshowed,usingballsandinclinedplanes,whythismustbe:itrequiresinertialandgravitationalmassestobeequal.Hiscontemporaries,toplaysafeandsupportAristotle,said:“Aha!Thebighailstonesmusthavefallenfromagreaterheight”.Thiscouldhavebeenright,butthecluewasintheadhocwaytheyhadtosetuptheinitialheightofthevarioussizesofhail:therewasnoreasonforit.Thereisalsonoreasonwhydarkmattershouldbeinadiffusehaloaroundagalaxy.Thereisanotherinconclusive,butphilosophicalobjectiontotheideaofdarkmatter,inthatitisnotfalsifiableandnotpredictive.Darkmattercanbeplacedanywheretomakewhatevermodelyouwantfitthegalaxyrotationdata.Itcouldbeusedtosupportanytheoryatall.Ineffectithasaninfinitenumberofadjustableparam-eters,soitcouldnoteitherbesaidtobepredictive.Thismeansthatifyouweregivenapictureshowingthelightdistributionofagalaxy,thenusingdarkmatteryouwouldnotbeabletopredicttherotationcurve.Whatyouhavetodoisassumethatgeneralrel-ativityiscorrect,andworkoutthedarkmatterdistributionthatwouldenablegeneralrelativitytopredicttherotationcurve.Youhavenotpredictedtherotationthough,youhaveonlypredictedthedarkmatterthatsupportsgeneralrelativity,andyoucan’ttestthissolutionbecauseyoucan’tseethedarkmatter.Thisiscertainlynotgoodphysicsaswehaveeverknownit.Itisunpredictiveandnotfalsifiable.Theonlychancethoseproposingdarkmatter,i.e.:justabouteveryone,havetoreturnwithintherealmsofscience,istodetectsome.Thisisdifficultbecauseifitexists,itdoesn’tappeartofeelthestrongorweakforcesortheelectromagneticforce.Ifitdidwewouldhaveseenordetecteditbynow.Butifdarkmatterparticleshaveinertialmasstheyshouldpushthingsthattheybumpinto.Thisiswhyhugetanksofliquidxenon,forexample,havebeensetupasdarkmatterdetectorsunderground.Thisliquidisverycold,soany
59June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration47recoilcausedbyaparticleofdarkmatter,canbemoreeasilyseenabovethebackgroundofthermalnoiseandtheexperimentsarealsoplaceddeepdowninminestoavoidcollisionsfromenergeticcosmicrays.ThereisonesuchexperimentintheUK,1100metresdeepdowntheBoulbymineinWestYorkshireandoneintheUS,innorthernMinnesotaat700metresdeep(Brooks,2009).Despiteacoupleofdecadesoflookinginthisway,theyhavefoundnothing.Thisdoesnotofcoursedisprovetheexistenceofdarkmatter,anditdoesnotmeanthatithasbeenawasteoftimeeitherbecauseveryoftenexperimentsdesignedtolookforonethingoftenturnupanotherthingwhichisjustasuseful,andalsoanegativeresultisjustasusefulasapositiveone.However,thelackofdetecteddarkmatterparticlessofardoesmeanthatitishightimealternativesweresuggestedandlookedat.LookingbackatEq.(3.6),therearesomeotherwaystoincreasetheorbitalvelocityv.WecouldincreasethegravitationalconstantGforlargerdistanceslikethoseonthescaleofgalaxiesandthishasbeentriedintheconformalgravitytheoryofMannheim(1990).IhaveactuallymetMannheimandIalwaysrememberhimfondlybecausein2006Iwasproudlyattendingmyfirstastronomicalcon-ferenceattheobservatoryinEdinburgh,aworkshopon“AlternativeGravities”.TheRoyalObservatory,forobviousreasons,isontopofaverylargehill,andIwasdraggingmysuitcaseslowlyupthehilltothemeeting.Mannheimkindlystoppedhiscarandgavemealiftupthehill,sayingthathewasverysympathetictotheplightofpoorcivilservantslikemyself(asIwasthen).SothatnighttoreturnthefavourIreadallabouthistheory.Ididn’tunderstandmuchofit,butIgatheredthatgravitybecomesrepulsiveatlargescales.Anothersolutiontothegalaxyrotationproblem,istodecreasetheinertialmassatlowaccelerationswhileleavingthegravitationalmassalone.ThisisratherproblematicbecauseaccordingtoEinsteintheinertialandgravitationalmassesaresupposedtobethesame,butinertiahasalwaysbeenaratherneglectedconceptinphysics,andnotwellunderstood.In1983MordehaiMilgromfromIsraelpresentedanempiricalmodelthatcandoeitherofthesethings,i.e.:itmodifieseitherGorinertialmass.ItiscalledModifiedNewtonianDynamics,orMoNDforshort.
60June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0348PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure3.Theplightofalternativesolutionstothegalaxyrotationproblem.MoNDNewtoniandynamics,thatisNewton’ssecondlawandhisgravitylawswereinspiredwhenNewtonlookedatfallingapplesontheEarthandconceivedthelinkbetweenthatandtheorbitoftheMoon.Hedevisedamodelthatfitthemboth.Themodelthatfitturnedouttobeaforceofgravitythatvariedaccordingtotheinversesquareofdistance.Theaccelerationsinthesesystemsarerelativelyhigh,being9.8m/s2ontheEarthand1.6m/s2atthedistanceoftheMoon.AttheedgesofspiralgalaxiesliketheMilkyWay,theaccelerationsaretenordersofmagnitudesmaller,ontheorderof10−10m/s2,sotomodelthesesystemsweareextrapolatingNewton’slawsdowntoaccelerationsthatareequivalenttotheaccelerationofPlutoduetothegravityofMercury(Scarpaetal.,2006).Thisisabigassumptionandthelasttimesuchanextrapolationwasmadeoversucharangeofscaleswaswhenclassicalphysicswhichholdsatscalesof1metreorsowasappliedtotheatomwhichhasascaleof10−10metres,andthatdidn’tworkatallandtheverydifferentquantummechanicshadtobedeveloped(Unzicker,2008).
61June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration49Figure4.Therotationcurveofatypicalgalaxy.TheexpectedNewtoniandropoffoforbitalvelocityisshownbythedashedlineinthegraph.Thesolidlineistheobservedvelocitywhichdoesnotdropoff.Theaccelerationatwhich−102thetwolinesdivergeisalwaysa0=2×10m/s.Thisisdifficulttoexplainusingdarkmatter.Milgrom(1983)noticedapeculiarpatternintheaberrantbehaviourofgalaxyrotationcurveswhichisshowninFigure4.ThegraphshowstheNewtonianvelocity(dashedline)whichpredictsadropoffofvelocitywithradius.Theobservedvelocity(asshownfirstbyRubin)doesn’tdropoffandthedivergenceofthetwocurves,thedeviationfromNewtoniangravity,wasalwaysobservedwhenandonlywhenthegravitationalaccelerationcomputedconsideringonlybaryons(visiblematter)fellbelowavaluecloseto2×10−10m/s2whichiscalleda0(Scarpaetal.,2006).MakinguseofthisobservationMilgrom(1983)developedanempiricalmodelthathastwoversions,thenormalNewtonianpartforhighaccelerationsandanewoneforaccelerationsbelowa0.Foraccelerationsthatarebelowthecrucialaccelerationof2×10−10m/s2eitherNewton’ssecondlawismodifiedtoreducetheinertia,orNewton’sgravitylawismodifiedtoincreasegravity.
62June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0350PhysicsfromtheEdge:ANewCosmologicalModelforInertiaForreasonsthatarebasedonobservation,Iwillfocusonthepossibilitythattheinertiallawistheonethatneedstobechanged.Milgromdidnotspecifywhichpropertyshouldbechanged,butdidworkoutthechangesnecessaryfortheinertialpossibility.Atfamiliaraccelerationsthereistheequationthateveryoneknows:F=ma.Foraccelerationsbelowa0MilgromsuggestedthefollowingF=mgµ(a/a0)a(3.7)wheremgisthegravitationalmass,µisafunctionthatwhenaislargeisequalto1,sowegetF=maback,andwhentheaccelerationisbelowa0(aa0)thefunctionisa/a0sothatwegetthenewforceformulaof:F=ma2/a0.Herea0istheverysmallaccelerationmentionedabove.Whathappensinbetweenthesetwofunctions,howtofitthemtogetherforintermediatevaluesofa,isamatterofsomedebate,butthisdoesn’taffecttheresultsmuch.IsaacAsimovoncesaidthatthemostsignificantwordsinsciencearenot“Eureka!”whichmeans“Ihaveit!”,butrather:“That’sfunny...”.No,IdonotmeantosuggestforamomentthatMoNDorModifiedNewtonianDynamicsisfunny,butratherthatthereisaninterestingobservedpatternherethatthedeviationfromexpectedbehaviouralwaysstartsattheaccelerationa0.ThevalueofMoNDisnotthatitexplainthis,itdoesn’t,butthatitpointedoutthispattern,thatiscryingoutforanexplanation.Thevaluefora0hasanumberofintriguingqualitiesaboutit.Forexample,itisclosetothespeedoflighttimestheHubbleconstantwhichisthefractionbywhichtheuniverseisexpanding(i.e.:a0=2×10−10m/s2,andcH0∼5×10−10m/s2).Thecriticalaccelerationa0hastheproperty,mentionedbyMilgrom,thatifwestartwithaparticleatrestatthesupposedbeginningoftheuniverse,13billionyearsago,andgaveitthisacceleration,thenbynowitwouldhaveaspeedclosetothatofthespeedoflight.Thisisatremendouslysug-gestiveproposal,suggestingalinkbetweendynamicsandcosmology.Theaccelerationa0isalsothesimplestaccelerationonecanwritedownusingthecosmologicalparameters,candtheHubblescaleΘ:c2/Θ(Smolin,2006).Allthesecoincidences,andtheycouldofcoursebecoincidences,giveonetheimpressionthatMoNDissayingsome-thingveryfundamentalthatisasyettoobigtocomprehend.
63June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration51Theproblemwiththinkingaboutallthisisthatalltheparam-etersarelinkedtogether.Forexample:arewesosureabouttheageoftheuniverse?Ofcoursenot.Noone,exceptperhapsMarvintheParanoidAndroid(Adams,1979),hasbeenstandingaroundfor13billionyearswithastopwatch.Ifnot,howcanwebesurethata0givesyouthespeedoflightinthelifetimeoftheuniverse?Ifyouadjustoneofthecosmologicalparameters,theotherswillhavetobeadjustedtooandalsotheyareallratherindirectmeasurements(Machwouldputtheminthecategoryofbeing‘thoughtthings’andnotdirectlymeasurable).MyfeelingwhenIfirstreadaboutitwasthatMoNDistellingussomething,butallthecosmologicalparame-tersareuncertainandmaybeallofthemarewrong,butthelinkagesbetweenthemthatareimpliedbyMoNDfitabettermodelthatwaswaitingtobediscovered.Perhapsagoodideawouldbe,inthespiritofMach,tocomebacktowardsconceptsthatcanbedirectlyobserved(i.e.:directobserva-tions)andlinkthemtojustoneofthecosmologicalparameters.Thenwearenotassumingwhatthemodelthatbindsthemalltogethercosmologicallyis(becausewearenotconfidentthatweknowwhatitisyet).SohowabouttheobservedgalaxyrotationcurveswhichcanbeseenandtheHubblescaleΘ,whichcanbefairlydirectlydeduced?ItwasalreadyclearfromthedatathatMoNDonlyseemstoapplytotheouteredgesofgalaxies.Whentheorbitalaccelerationofstarsgoesbelowa0,thenthedynamicsbecomenon-Newtonian.Thisisdifficulttoexplainusingdarkmatterbecauseitmeansyouhavetoputallthedarkmatterintothegalaxymodelasadiffusehaloaroundthegalaxy.Howcanthisbejustified?Darkmatterdoesnotfeelanyforceapartfromgravity,sohowdoesitstaydiffuselyspreadoutoversuchahugehalo?Youhavetoinventawholenewphysicsforit,thatenablesittoremainspreadout.But,ofcourse,thisisnotproofthatdarkmatterdoesn’texist,justthatitrequiresnewphysicsalsoandthisviolatestheprincipleofOccam’srazorthatthesimplestsolutionisthebest.Ithenreadonepaperthatconvincedme,thatthesolutiontogalaxyrotationwasnotdarkmatter.ThiswasbyScarpaetal.,(2006)andtherehavebeenothermorecomprehensiveandconclusive
64June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0352PhysicsfromtheEdge:ANewCosmologicalModelforInertiaonespublishedsince,andignoredbyastrophysicists,butthiswastheoneIhappenedtoreadfirst.TheylookedatglobularclusterswhicharesmallareaswithintheMilkyWay,wherethestarsarearrangedslightlymoredenselythaninsurroundingareas.Sincedarkmatterisseentobesodiffuse,itshouldnotaffectthedynamicsofsmallsub-galacticcollectionsofstarslikeglobularclusters.Youcan’thaveitbothwaysandarguethatdarkmatterhastostayspreadoutonhugescalestofitgalaxyrotationsandthensayitexistsinsmalllumpswithinglobularclusters.Thedarkmattershouldbeuniformatthesescalessoshouldnotaffectdynamics.But,Scarpaetal.(2006)foundexactlythesamethinghappenedwiththeseglobularclustersaswithlargergalaxies:whenevertheirinternalaccelerationsdroppedbelowa0,theirdynamicsbecamenon-Newtonian.IstillhavethisgloriouspaperandIwroteonmycopy:“Brilliantstuff:tellsmewhichwaytogo!”.Also,thefactthatitisjusttheinternalaccelerationsthatmatterfordynamics,andnottheexternalgravityfield,showsthatitisnotMoNDthatisatworkeither,butthatisforlater.Ifthechangeinbehaviourisnotduetodarkmatter,itisalsodifficulttoexplainusingMoNDianvariationsinG,sincewhyshouldGincreaseforlowaccelerations?Ofcourse,itwouldbepossibletoinventamodelforthis,butthereisapossiblephysicalreasonthatcanexplain,insomesense,achangeofinertiaata0.HereisaquotefromMilgrom(2005):CanwespotaninklingofMoNDinertiaintheUnruheffect?Whentheaccelerationofaconstanta[acceleration]observerbecomessmallerthana0...theUnruhwavelengthbecomeslargerthantheHubbledistance.Weexpectthensomebreakintheresponseofthevacuumwhenwecrossthea0barrier...Iwasverytakenwiththisideaatthetime,andIcalledit“Milgrom’sbreak”sincetheHubblescaleis‘breaking’theprocessofinertia.Wecandothecalculationverysimply.WediscussedthetemperatureofUnruhradiation(Eq.(2.13)).WealsodiscussedWien’slawinthesectiononquantummechanics(Eq.(2.6)).Puttingthesetwotogethergives4βπ2c2a=(3.8)λ
65June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration53Figure5.Astheaccelerationofanobject(say,arocketship)decreases,thewavelengthoftheUnruhwavesitseesincreases.MilgromproposedthatwhentheyexceedtheHubblescale(thesizeoftheobservableuniverse)theymightvanishandthiscanbeunderstoodusingtheideasofMachwhosaidthatifsomethingcannotbeseeninprinciple,thenitdoesnotexist.Nowwecandothecalculation.Puttinginthesizeoftheobserv-ableuniversewhichis2.7×1026m,wegetanaccelerationof20×10−10m/s2.NowadmittedlythisistentimesthevalueneededforMoND,butitisinterestinglyclose.ThiswaswhereMilgromlefttheproblem.HedidnotspecifywhythereshouldbeabreakintheUnruhradiationspecifically,anddidnotmanagetoderiveMoNDfromthisideaortestitagainstobservations.Theideaofacutoffisalsonotabletoexplaingalacticdynamicssinceitimpliesasuddenchangeininertiaataparticularradius,nottheslowerchangethatisseeninthedata.IfwegobacktoMach,hesaidthatifsomethingcouldnotbeobserved,thenitcannotexist,anditispossibletojustifyMilgrom’sbreakbythiskindofargument.IfanUnruhwaveislargerthantheHubbledistance,thenitcannotbeobservedinprincipleandthereforecannotexist,andsinceinertiaisrelatedtotheaccelerationwhichinturndependsontheUnruhwave,theninertiaalsodisappears.Putlikethatitsoundsratherbatty,butIlikethelogicofthis:theideasoundsright,andsomedaythinkingmoreintermsofinformationwemayunderstandthecontextbetter.
66June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0354PhysicsfromtheEdge:ANewCosmologicalModelforInertiaAnyway,fornow,nomatterhowsuggestiveMilgrom’sbreakis,itdoesnotexactlyfittheobservations.Ifitwastruethenassoonasastarmovedfarenoughoutinagalaxy,itwoulddropbelowtheaccel-erationofa0,itsinertiawoulddissipatecompletely,andinertialess,itwouldzoomoffintothecosmosatinfinitespeed.Thisisnotwhatisseen:thechangeingalacticdynamicsismoregradualthanthatandtheedgesofgalaxiesarefilledwithstarsthat,althoughtheyarebehavingstrangely,arestillboundedtothegalaxyandhavesomeinertialmass.Soatheoryisneededthatproducesamoregradualdecreaseininertia.InsteadofMilgrom’sbreak,wewillconsideraslightbend.MoNDisNotGoodEnoughWhetherornotitisUnruh-related,MoNDitself,asanempiricalmodelhassomeproblems.ThemainproblemwithMoNDisthatitisnotatheory.Itisanempiricalmodelthathasbeendetermined,ratherlikePlanck’sblackbodyformula,byfittingaformulatogalaxyrotationdatawithanadjustableconstanta0.Thisisnotsatisfyingandanexplanationneedstobegivenforthevalueofa0andwhy,ifnotacoincidence,itlooksverymuchlikecH0orc2/Θ.MoNDalsohasproblemswithglobularclusters.AsIhintedabove,theseglobularclusters,whichareembeddedwithintheMilkyWaygalaxy,shownon-Newtonianbehaviourwhenandonlywhentheinternalaccelerationfallsbelowabout1.8±0.4×10−10m/s2(Scarpaetal.,2006).AccordingtoMoNDitisthetotalacceleration(includingtheexternalacceleration)thatisimportant,andtheseclustersareembeddedinpartsoftheMilkyWaywherethetotalaccelerationduetothegalaxyisgreaterthana0,andyettheseclus-tersstillshowanomalousbehaviourwhentheirinternalaccelerationonlygoesbelowa0.ThisdoesnotagreewithMoNDatall.Ifinternalaccelerationsonlyarethekeythenitisgoodnewsinasensebecauseitmeanswecansetupanexperimentforwhichinternalaccelerationsarelessthana0,eveniftheexternalaccelera-tionsduetothegravityoftheSunortheaccelerationsduetopassingcarsaremuchlarger,andstilldetecttheanomalousbehaviour.Suchexperimentshavealreadybeendone,andinsomeofthemanomalous
67June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch03ProblemsatLowAcceleration55Figure6.Itispossibleforasystemsuchasaglobularcluster,oracoupleofpeopleinarollercoastertohaveahugetotalacceleration(relativetosome-thingexternal),butstillhaveatinyinternalacceleration.Thesetwofriends’accelerationrelativetotheEarthiscausingoneofthemconcern,buttheyarehardlyacceleratingrelativetoeachother.Forglobularclusters,itistheinternalaccelerationthatappearstomatter.behaviourhasbeenseen,buttheyhaveprovendifficulttoreproduceinotherlabssotheseresultareinterestingbut,rightly,controversial.IwilldiscusstheseresultslateronandIdopaymoreattentiontothemthanmostscientistsbecausethereisatendencyinmodernphysicstodiscountexperimentsthatdisagreewithacceptedtheory,andinscience,itshouldbetheotherwayaround.Anyway,allthisimpliesthatweneedatheoryforwhichthelocalaccelerationsareusedtodetermineinertia.Thefirstsystemsinwhichapparentdarkmatterwasdetectedweregalaxyclusters,whicharegravitationallyboundedvillagesofgalaxiesonahugescale.Zwickyfoundthathugeamountsofdarkmatter(400timesthevisiblematter)wereneededtokeepthesegalaxyclustersbounded(nowweknowitismorelike100times).
68June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0356PhysicsfromtheEdge:ANewCosmologicalModelforInertiaMoNDdoesnotperformwellontheseclusters.Ifitisusedthen,still,darkmatter,hastobeaddedaswellandtypically2or3timestheamountofvisiblematterneedstobeadded.SoitseemsthatevenMoNDneedsdarkmattertosaveitatthelargestscales.Soinsummary,MoNDhaspointedoutaveryinterestingpat-tern,butitisempiricalandnoreasonwasgivenforit,anditdoesnotworkforglobularclusters,becausetheirbehaviourdependsontheirinternalaccelerationsanditdoesnotworkforgalaxyclusters.Somethingmorefundamentalisneeded.
69June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04Chapter4ASolutionfromtheEdge:MiHsCThePhysicsoftheFluteIhaveplayedtheflutesinceIwas12andIamveryfondoftheinstrument.Apartfromthepleasantsoundwhichisuniquelypurewhenyouplayit,sincethesoundwavesareclosetobeingpuresinewaves,youfeelconnectedtoit,sincetoproduceasoundyouhavetovarytheairpressureyouapplyuntilyougetaresonancefromtheinstrument.Itbecomesasortofacousticinteractionbetweentheinstrumentandtheplayer.Thisisunlikethepianoforexample,onwhichyoupushkeyswithoutsomuchfeedback.Asaresult,onecouldplaythepianowithouttheabilitytohear,andthisiswhatBeethovendidafterhebecamedeaf.Isuspectyoucouldnotdothesamesoeasilywithaflutethoughyoucouldrelyonvibrationtransmittedthroughthebody.TheinterestingthingisthatthemodelforinertiaIamgoingtotalkaboutisverysimilartothephysicsoftheflute,whichmaybewhereIgotitfrom.Anyway,Icanusetheflutetointroducetheidea.Afluteisalongcylinder,completelyopenatoneend,andalsoopenattheotherviaamouthholealongitsside.Theinstrumentisplayedbyblowingoverthismouthhole.Sometimestheflowofairgoesovertheholeandsometimeitgoesobliquelyin,eventuallyanoscillationbetweenthesetwopatternsoccurs,awaveontheairjet,andifthepressurewavefromaparticularincursionofairintothefluteisabletogoatthespeedofsoundtotheotheropenendofthefluteandbackjustintimeforthenextincursion,thenthewavesreinforceeachother.Thisiscalledaresonanceanditmeansthatthewavesgrow,andthisishowthefluteproducessystematicvibrations(sound)intheairthatemanatefromtheopenend.57
70June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0458PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure1.(Upperpicture)Thefluteisopenatoneendandpartiallyopenattheother.(Middle)thevariationofpressurewithinthefluteforthefirstharmonic:thelongestwave.Thereislessvariationinpressureattheendssincetheairisconnectedtotheatmosphere,buttherecanbevariationinside.(Bottom)Themovementoftheairisatamaximumattheendswheretheairislessconstrainedbythecylindricalwalls.Ifyouwanttoproduceahigherpitchthenthewavesonthejetmustbeclosertogetherbecausethewavesofsoundintheinstrumentareclosertogether,theirwavelengthisshorter.Soyoumustblowtheairmorequicklyacrossthesidehole,sothatthewavespassacrossthisholeatjusttherightfrequencytomatchthesoundwavesinsidetheflute.Thepressureattheopenendsoftheflutemustbeatmosphericpressureandcannotvarybymuch,butinsidetheflutetheycanvary.ThesimplestarrangementlikethisisshowninFigure1.Figure1(upperpicture)isadrawingofmyflute.Thecurvejustbelowthatrepresentsthevariationinairpressureatvariouspointsalongitforthelongestwavepossibleinit.Thepressurevariesmostinthecentreandleastattheopenends,sinceattheendsairisalwaysavailablefromsurroundingareastofillinlowpressure,andhigherpressureaircandivergeintotheopenspace.Thesepointsofminimumvariationarecallednodes.Thecurvebelowthatshowsthe
71June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC59movementoftheair.Thisismaximumattheopenendssincetherearenowallstheretodampthemotion,andleastinthemiddle.Thewaveshowninthepictureisthelongestairpressurewavethatispossibleinthefluteandthismeansitisthelowestnotepos-sible,amiddleC.Thishasalwayscausedmeoccasionaldisappoint-mentsincemanynicetunes(dependingonthekeytheyarewrittenin)havebeenwrittenwiththeBbelowmiddleCinthem,sowhenItrytoplaymusicnotspecificallywrittenfortheflute,Ioftenhavetoskipanote.ThisisaniceanalogytothelongwavelengthcutoffofMilgromdiscussedabove.Justastheflutedoesnotallownoteswithawavelengthgreaterthanitslength,theobservableuniversedoesnotallowUnruhwavesgreaterthanitssize,butthefluteanalogyallowsustogoabitfurtherthanthat.IfwedraweitherthepressurewaveorthedisplacementwaveataninstantintimeitwouldlooklikeFigure1.Sothefundamental,thelowestnotehasawavelengththatistwicethelengthoftheflute.Thisisbecauseinordertomatchthenextwavethatappearsattheblowhole,thewavehastotravelfromoneendtotheotherandthenbackagain.Instrumentslikepanpipes,whichhaveaspecificlength,playjustthefundamentalandotherwavesthatfitintothecyclinder.Panpipesarearrangedsothattheyhavecylindersofdifferentlengthsotheplayerscanchosethelengthofthecylinderandthenotetheywanttoplay.Trombonistscanvarythelengthoftheircylindertovarythenote.Onthefluteyoucanvarythenoteintwoways.Thefirstistoblowharder,andbythisyoucanexcitethesecondharmonicorthethird,etc.ThiswouldmoveyouupanoctavefrommiddleCtotopCandthentheGabovethat.TheseharmonicsareshowninFigure2.Thesearethelongestwavesyoucanhavesothatthereisapres-surenodeattheopenendsoftheflute.ThelongestismiddleC,thenextoneistheCabovethat.ThenextoneisactuallyaG,thenhighCthenanEandsoon.Thisiscalledaharmonicseriesandifthefrequencyofthefirstnoteisf,thenthefrequenciesoftheothersare:2f,3f,4f,5f,etc.SohowdoyouplayanotebetweenmiddleCandupperC?Thereisaverynicetricktothis.Whatyoudoonthefluteisopenaholeatapointalongthefluteandsoenforceapressurenodeatthatpoint.
72June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0460PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure2.Pressurewavesinaflute.Fromtoptobottom,thefirst,second,thirdandfourthharmonics.Figure3.Howtoforceaparticularnoteontheflutebyopeningahole.Only,preferably,onewavecansatisfythisnode,soyoucanselectnotes(Wolfe,2012)asshowninFigure3.Tosummarise,onlysoundwaveswithwavelengthsshorterthantwiceitslengtharepossibleintheflute,justasinMilgrom’sbreak.However,intheflutetheconditionsareevenmorestringentinthatonlywavesthatfitexactlyintotheflute(ortwicethelength)areallowed,andallothersaredisallowed.Openingholesalongthelengthoftheflutemakestheconditionsevenmorestringentandfurther
73June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC61wavesaredisallowed.Thisnecessityforanexactfitofthewavewithinthesystem,isthebasisofthenewmodelofinertiapresentedinthisbook.Thisisaone-dimensionalexample,butIwillnowdiscussaclosertwo-dimensionalanalogyinvolvingparallelplatesandalsoshipsatsea.TheCasimirEffectHeisenberg’suncertaintyprinciplecanbewrittenas∆E×∆t=whichmeansthattheuncertaintyintheenergy(∆E)timestheuncertaintyintime(∆t)equalsaconstantwhichisPlanckconstantover2π().Soifweconsiderveryshorttimes(small∆t)thenalargeramountofenergy(∆E)isavailableforborrowingforthatshorttime.Asaresult,ifyoulookatavacuuminwhichthereisapparentlynomass-energy,youwon’tseeavacuumsincethevacuumcan“borrow”energyforashorttimetoproducewhatarecalledvirtualparticles.Thereisanotheruncertaintyrelation:∆p×∆x=andthissaysasimilarthing:thesmallerthescaleyoulookat(small∆x),themoreuncertainmomentumis(larger∆p)soparticleswithmomentumappearoutofnowhereonverysmallscales.Thisiscalledthezeropointfield,becausealthoughtheenergyissupposedtobeclassicallyatthezeropoint,therearestillparticles(alsoafield)causedbythisquantumuncertainty,likewavesonasea.AfterdeBroglie,weknowthateveryparticlecanalsobethoughtofasawave(afield),justasthesoundinafluteisbothawaveandanote(differentphysics,butthesamemaths).Ifwetaketheelectromagneticfieldthenhowcanwelimitthewavesasintheflute?Theansweristoputtwoflatconductingplatesclosetoeachotherandparallel.Thismeansthattheelectromagneticfieldmustbezeroatthetwoplatessincetheelectronsinthemetalmovearoundtocancelit,justasthesoundwaveintheflutemustbezero(haveanode)atthetwoopenendsasairmovestoequalisepressurethere.Justasinthefluteonlycertainwavelengthsareallowedsotherearefewerphotonsbetweentheplatesthanoutsidethem.Thismeansthatmorephotonsbangintotheplatesfromoutsidethanbangintothemfrominsideandsincephotonshavemomentumthereisanetforcepushingtheplatestogether.Thisiscalledthe
74June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0462PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure4.Twoparallelconductingplatessuppresszeropointfieldwavelengthsthatdonotexactlyfitwithintheplates.Wavelengthsoutsidetheplatesareunaffected,sothereisagreaterzeropointfieldoutsidetheplatesandsincetheparticlesofthisfieldhavemomentum,moreofthembangintotheplatesfromoutsidethanfrominside,pushingthemtogether.ThisCasimirforcewaspredictedbyCasimir(1948)andobservedbyLamoreaux(1997).Ithasbeenpredictedthatthereisasimilarforceduetooceanwavesthatpushestwocloseparallelshipstogetheratsea(seeBoersma,1996),butthisisextremelysmall.Ithasbeenmeasuredinthelaboratory.CasimireffectandtheCasimirforceafterHendrikCasimirwhofirstpredictedit(Casimir,1948).ItwasobservedjustaspredictedbyLamoreaux(1997).AnAsymmetricCasimirEffect(aCe)ItmayseemstrangethatthefirstpartofthenewmodelIamgoingtotalkaboutwasproposedinthelastpaperIpublishedonit(McCul-loch,2013),butconceptuallythispartcomesfirst.Wheredoesinertiacomefrom?WhatisthemechanismbywhichUnruhradiation(partofthezeropointfield)causesinertia?AmodelforinertiawasproposedbyHaischetal.(1994).Theyproposedthatoscillatingpartons(particles)withinanacceleratedobjectfeelamagneticLorentzforce,duetotheirinteractionwiththezeropointfield,thatopposestheaccelerationoftheobject.Theforcetheyderivedwas−Γwc2aF=(4.1)2πc2
75June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC63WhereΓistheAbraham–Lorentzdampingconstantofthepartonbeingoscillated,wcistheComptonscaleofthepartonbelowwhichtheoscillationsofthezeropointfieldhavenoeffectonit,isthereducedPlanck’sconstant,aistheaccelerationandcisthespeedoflight.Theirderivedformuladoeslookratherlikeinertia,sincetheforceisalwaysintheoppositedirectiontotheacceleration(thereisaminussign).However,theirderivationwascomplexandithasbeencriticisedonrelativisticandothermorebasicgroundsby,amongothers,Unruh,andLevin(2009).ThemainproblemIhavewiththismodelisthattheLorentzforcetheyusedtogenerateinertiaactsontheparticleswhentheyareoscillatingatveryhighfrequenciesandsincethefrequencycangoallthewayuptoinfinity,theyhadasimilarproblemtotheultravioletcatastrophewediscussedearlierandtheyhadtointroduceanarbitraryfrequencycutofftogetridofthehighfrequencies,andthatseemsunphysicalandisratherlikeatuningparameter:neveragoodsign.Iwillhereproposeasimplermodel.Asdiscussedabove,thezeropointfieldcanberegardedassomethingliketheoceanwithwavesandparticlesappearingallthetime,inpairstoconservemomentum.Hawking(1974)lookedattheproductionofthesetwinparticlesnearblackholesandfoundthattheeventhorizonsurroundingablackholecausedbyitsintensegravitywasabletoseparatesomeofthesepairsofparticles.HethuspredictedHawkingradiationandblackholeevaporation.ItwasthenproposedbyFulling(1973),Davies(1975)andUnruh(1976)thatasimilardynamiceventhorizonformsnearacceler-atedobjects.Consideraspacecraftthatisbeingacceleratedinthex-direction.Atacertaindistanceoppositetotheaccelerationvectorofthespacecraft,thereisasurfacebeyondwhichnoinformationcanevercatchupwiththespacecraft,thevolumebehindthissurfacecanneverbeobservedbythespacecraft(unlessthespacecraftturnsaroundofcourse!).ThissurfaceiscalledaRindlerhorizon.Justlikeablackholeeventhorizon,andjustasinthatcase,itwasproposedthatvirtualparticlesandradiationcanbesplitupbythisinformationboundaryandrealisedintotherealworld.Theresultisthatanacceleratedbodyseesthermalblackbodyradiation(awarmbackground)surroundingit,whereasanunacceleratedobject
76June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0464PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure5.Aschematicshowingaparticleacceleratingrightwards(theblackcircle).TheshadingshowstheRindlerhorizonclosebytoitsleft(atadistance2c/aaway)andthecosmichorizonfarawaytoitsright.TheUnruhradiationisdampedmoreontheleftsideproducinganasymmetricCasimireffect(theforceindicatedbythetwounequalopposingarrows)thatpushestheparticle(O)totheleftagainstitsaccelerations:amodelforinertia.inthesamespaceseesnoradiationatallandconcludesthatthebackgroundiscold(theheatiscausedbytheUnruheffect).Nowimagineaparticlebeingacceleratedtotheright,inthepos-itivex-direction(seetheschematicinFigure5).Anisotropic(fromalldirections)Unruhradiationappearsallarounditandradiationcanexertapressure.ThisistheprinciplebehindsolarsailingwherephotonsfromtheSunpushonathinsolarsail.NowontheoppositesideoftheparticletotheaccelerationvectorthereisformedaRindlerhorizonwhosedistanceawayisd=c2/a,wherecisthespeedoflightandaistheaccelerationoftheparticle.Figure5showsaparticlethatisbeingacceleratedtotheright.ARindlerhorizonappearsonitsleftside.IfwecalculatetheenergydensityoftheUnruhradiation,inthedirectionoftheacceleration,totheright,mostoftheUnruhwaveswillbeallowedsincetheonlyeventhorizonthereisthecosmiconefarawayattheHubbledistance(2.7×1026m),butontheoppositeside,totheleft,fewerwavesintheUnruhspectrumwillbeallowedbecausethedynamic(Rindler)eventhorizonismuchcloser,ata
77June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC65distanceofc2/a.Foranaccelerationof9.8m/s2thedistancetotheRindlerhorizonwouldbe1015m,oraboutatenthofalightyear.Sothemomentum/impactoftheUnruhradiationwillbelowerfromtheleft(withastronglydampingRindler-scaleCasimireffect)andgreaterfromtheright.Andthiswillpushtheobjectbackagainsttheappliedacceleration.ThisasymmetricCasimireffect(aCe)modelsinertiaintuitively.Thislookslikeinertia,butdothenumberswork?Thederiva-tionisinMcCulloch(2013)andsummarizedinAppendixDandthederivedequationfortheinertialforce(dFx)isasfollowsπ2hadFx=−(4.2)48clP(whenIfirstpublishedthisformulaImadeafactoroftwoerror,seeMcCulloch,2013,thatwaskindlypointedouttomebyDrJaumeGin´e,fromSpain).IntheformulahisPlanck’sconstant,aistheacceleration,cisthespeedoflightandlPisthesizeoftheparticle.IfweputinthePlancklengthforthesize,theinertialmassispredictedtobe2.75×10−8kg:about26%largerthanthePlanckmass.Thisderivationisonlyvalidforsingleparticles.Tocalculatetheinertialmassofacompoundobjectitwouldbenecessarytomultiplytheabovemassbythenumberofparticlespresent.Itisalsoonlyvalidforvelocitiesmuchslowerthanthatoflight,sinceotherwiseLorentzcontractionwouldalterthedimensionsoftheparticlesothatitwouldlookmorelikeapancake.Thisformulaforinertiaisconsiderablysimpler,inbothderiva-tionandfinalform,thantheformulaofHaischetal.(1994)andthemechanismisdifferent.InthearticleofHaischetal.(1994)theinertialprocesswasamagneticLorentzforceactingonparticlesoscil-latingatveryhighfrequencies,andtoavoidaninfiniteenergytheyhadtoimposeanarbitraryupperfrequencylimit.TheasymmetricCasimireffectproposedhereproducesanenergydifferencebetweenthetwosidesoftheparticle,sonocutoffisneeded.Howcanthismodelbetested?Well,anentirechapterofthebookwillbedevotedtopotentialtests,soIwon’tgointothatverydeeplyhere,butIhavesuggested(McCulloch,2008b,2013)thatitmaybepossibletomodifyinertiabycreatinganeventhorizonusingthemetamaterialsproposedbyPendryetal.(2006)andLeonhardt
78June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0466PhysicsfromtheEdge:ANewCosmologicalModelforInertia(2006).Theyhaveshownthatradiationofagivenwavelengthcanbebentaroundanobject(whichmustbesmallerthanthewavelength)usingametamaterial,makingthatobjectinvisibletoanobserveratthatwavelength.Itmaybepossibleinstead,tosetupametamate-rialtoreflectradiationsothatanartificialeventhorizonisformed.Thenaccordingtothemodeldiscussedhere,thiswilldampUnruhradiationononesideoftheobjectwhichwouldthenbeacceler-atedtowardsthehorizon.ItmaybesimplertousemetamaterialstodirectlydamptheUnruhradiationononesideoftheobject.ModifiedInertiabyaHubble-ScaleCasimirEffect(MiHsC)SoanasymmetricCasimireffectcanexplainstandardinertiaquitewell,butitwouldbenofunifwestoppedtheresinceweneedtopredictsomethingthatisdifferentfromthestandardmodeltostarttotestthenewideas.ThereisanotherhorizonIhaven’tconsideredyetinanydetailandthatistheHubble-scalehorizon.Letusseewhathappenswhenweaddthistothemix.Figure6(fromMcCulloch,2007)showstheenergyspectrumofUnruhradiationseenbyanacceleratedobjectonthey-axisandthewavelengthofthatradiationonthex-axis.OntheEarth,whereaccelerationsarehigh,oronahighly-acceleratedspacecraft,wewillseeshortUnruhwaveslikethoseontheleftandthebackgroundperceivedbytheobjectwillbewarm.TheprevioussectionshowedthatstandardinertiaiscausedbytheenergyintheUnruhradia-tionspectrum(theareaunderthecurve).Now,wewillconsidertheeffectoftheHubblehorizon.Thewavesthatfitintothishorizon,likethewavesthatfitintothefluteortheCasimirplates,areshowninFigure6bytheverticaldashedlinesandtheirwavelengthsareλ=2Θ/n,wheren=1,2,3,etc.Ontheleftoftheplottheallowedwavesareclosetogethersomostofthewavesforhighaccelerationsareallowed,theUnruhspectrumremainsmuchasbeforeandtheinertiaishardlyaffected.Incontrast,forabodywithalowaccelerationtheUnruhspec-trumwilllookliketheoneontheright,andwilllieonapartoftheplotwheretherearefewerallowedwavelengths.Intheplotthespectrumisonlysampledbyoneallowedwavelength(2Θ)soonlya
79June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC67Figure6.HowtheinertialmasscausedbytheasymmetricCasimireffect(areaunderthecurves)ismodified,andfilteredbytheHubble-scaleCasimireffect.verysmallproportionoftheUnruhspectrumwillremainandwecanexpecttheinertiatodecreaseinanewway.WhyshouldonlywavesthatfitexactlyintotwicetheHubblescalebeallowed?IoriginallyassumeditisduetoacosmicCasimireffect,butrecentlyIhavecometothinkofitphilosophically.Knowl-edgefromoutsidetheHubblehorizoncannotbeallowedin,soanypatternthatisbiggerthantheboundary,ordoesn’tfitexactlywithinit,wouldgiveobserversinsideaclueastowhatliesoutsidebyextrap-olation,andthiscannotbeallowed.Icallahorizonlikethisahard-horizon,sincenotonlydoesitnotallowinformationtopassthroughbymovement,italsomodifieswavesthatpassthroughittomakesuretheydon’tallowinformationthrougheitherviatheirpattern.ThismorephilosophicalapproachismathematicallyequivalenttoaHubble-scaleCasimireffectand,inmyview,canbeseendirectlyinthelow-lCosmicMicrowaveBackgroundanomaly(McCulloch,2014).Iwillexplainthisinmoredetaillater.
80June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0468PhysicsfromtheEdge:ANewCosmologicalModelforInertiaHerewewillassumethattheunmodifiedinertialmassisjustthearea(A)underthecurvesasshowninFigure6:mi=A.Howdowetakeaccountofthelossofenergyinthedisallowedwavelengths?Well,atfirstItookaguessandassumeditwoulddecaylinearlywithwave-length,thenIgotscaredrealisingthatitwouldnotbeexactlylinear,andcalculateditwithaFortranprogram(arevieweralsopesteredmetodothis!).Itriedvaryingthepeakwavelengthandcountedthenumberofallowedwavelengthsremaininginthesubsampledcurvewherethespectralenergywasmorethan1%ofthepeakenergy.Itturnedoutthatitwasroughlylinearafterall.Anyway,wecanaddafactortotheformulatoaccountforthislineardecayofenergyasitissubsampledlessandlessefficientlyatlongerwavelengths,likethismi∝A×F(4.3)WhatisthefunctionF?Well,wecanassumeitlookslikeF=(λ/α)+βwhereαandβaretwoconstantstobefound.Weknowthatwhenthewavelengthiszero,thenFmustbeonebecausethecurveisperfectlysampledatshortUnruhwavelengthsandthereisnoenergyloss,soputtingthisassumptionthatwhenλ=0forF=1intotheequationweget:1=0/α+β.Soβ=1!WealsoknowthatwhenthewavelengthistwicetheHubblediameter(4Θ)thenFmustbezerosincewearejustabouttoexceedthelengthofthelongestUnruhwaveandthisisMilgrom’sbreak.Theonlywaytheaboveequationworkswiththisfact(0=4Θ/α+1)isifα=−4Θ.Sowehavenowλmmi∝A×1−(4.4)4ΘThebracketisthecorrectionweneedtoapplytotheunmodifiedinertialmass,whichistherestoftherighthandsidesowehaveanewequationforinertialmassλmmI=mi1−(4.5)4ΘIshouldmentionatthispointbrieflythatthisisaroughapprox-imation.Actually,thenumberofallowedwavelengthswilloscillateupanddownineverslowerintervalsasweincreasethewavelengthastheallowedwavelengthsresonatewiththeboundary,butthiswilldofornow.TheλmisthepeakwavelengthintheUnruhradiation
81June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC69spectrumwhichisgivenbyWien’slaw(Eq.(2.6))andifweputthisvalueforλintotheequationforinertialmasswegetβπ2c2mI=mg1−(4.6)aΘWecansimplifythisalittle,introducinganerrorofabout1%,ifwewrite2c2mI=mg1−(4.7)aΘThisistheformulawewillbeusingformostoftherestofthisbookanditdescribeshowthestandardinertiafromanasymmet-ricCasimireffect(McCulloch,2013)ismodifiedbyaHubble-scaleCasimireffect(McCulloch,2007).ThecompletemodelcanbecalledModifiedinertiabyaHubble-scaleCasimireffect(MiHsC)orsimplyquantisedinertia.ThemIfromMiHsCbehavesinasimilarwaytowhatwouldbeexpectedfromMoND.Figure7isagraphshowingtheaccelerationalongthex-axisfromtinyaccelerationsonthelefttonormalaccelera-tionsontherightandtheratiobetweentheinertialandgravitationalmassalongthey-axis.Fortheassumptionofanequivalenceprinciplethereisthestraightdashedlinemi=mg.ThepredictionofMoNDisshownbythedottedlineandagreeswiththeequivalenceprincipleuntiltheaccelerationisaslowasfittingparametera0(2×10−10m/s2)andtheninertialmassreduces.ThepredictedratiofromMiHsCisshownbythelongdashedlineanditapproximatestheequivalenceprincipleforhigh(normal)accelerations,butreducestheinertialmassinanewwayfortinyaccelerationssuchasthoseattheedgesofgalaxies.TheMiHsCcurveissmootherthanthatofMONDanddoesn’tneedafittingparameter.IshouldclarifythattheaccelerationintheMiHsCformula,isthatwithrespecttoothermassesassuggestedbyMachwhosaidthatonlythingsthatareobservableshouldbeputintoatheory.Forexample,foranobjectontheEarth,itsaccelerationwithrespecttotheEarthiszero,sinceitisattached.However,theEarthisspinning,sotheobjecthasanaccelerationrelativetoeverythingelseintheuniverse‘thefixedstars’whichislatitude-dependent.Iwillprovetheutilityofassumingthislateron.
82June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0470PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure7.Agraphshowingtheaccelerationalongthex-axisandtheratiobetweentheinertialandgravitationalmassalongthey-axis.Fortheassumptionofanequivalenceprinciplethereisthestraightdashedlinemi=mg.MoNDagreeswiththisuntiltheaccelerationisaslowasa0andinertialmasssuddenlyreduces.MiHsCapproximatestheequivalenceprincipleforhigh(normal)acceler-ations,butreducestheinertialmassinanewgradualwayfortinyaccelerations.Foranobjectinmid-latitudestheaccelerationwithrespecttothefixedstarsisabout0.03m/s2.Giventhisacceleration,theinertialmasswoulddecreasebelowthegravitationalmass(byMiHsC)by2×3000000002m=m1−=m(1−2.3×10−8)(4.8)Ii26i0.03×2.6×10Thisalossof2.3×10−8kgforeverykgofgravitationalmass:averysmalleffect,butexperimentshavebeendonetotesttheequivalenceprincipledowntorelativeaccelerationsof10−15m/s2,soitisimpor-tantthatbeforeweprogressIexplainwhytheseexperimentsdonotseetheeffectofMiHsC.Theseexperimentsareusuallydonewithatorsionbalance.Theymeasurethedifferentialattractionoftwoballsonacrossbarsus-pendedonawire,towardsdistantmassesliketheSun,bydetectingtinytwistsinthewire(seeGundlachetal.,2007).ThisisdonebyrotatingthetorsionbalancesothechangingangleoftheSunisappar-entintheresult.ThisislikeGalileo’sfamousexperimentinwhichhe
83June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch04ASolutionfromtheEdge:MiHsC71droppedtwoballsfromatower.TheheavierballismoreattractedtowardstheEarth(ithasmoregravitationalmass),butalsofindsithardertoaccelerate(ithasmoreinertialmass).Thisappliestotheballsinthetorsionbalanceintheir‘fall’towardsdistantmasses.Theseexperimentshaveindeedshownnotwistinthewire,soyoumightsaythatinordertomakesuchaclaimasIdoforMiHsCIwouldneedsomeballs.However,thechangeininertialmasspre-dictedbyMiHsCisindependentofmass,asyoucanseebyputtingEq.(4.7)intoNewton’ssecondlawandhisgravitylawlikethisGMmg2c2F=mia=2=amg1−(4.9)raΘwhichsimplifiestoGM2c2a=+(4.10)r2ΘImagineGalileodroppinghistwoballsofdifferentweightoffthetower.MiHsCpredictsthatthechangeininertialmassforbothballsisinthesameproportionatalltimes,intheaboveequationtheextraMiHsCaccelerationisindependentofthemass,sotheywillstilldroptogether,althoughMiHsCpredictsthattheywillbeacceleratedbytheEarth’sgravityanddropeversoslightlyfasterthanexpected(thenewtermontherighthandside).Similarly,inthetorsionbalanceexperimentstherewillbenotwistinthewire,asobserved.Figure8.WithanodtoDouglasAdams,apoorspermwhaleandabowlofpetuniasdebatethefutilityofknowingthatMiHsCincreasestheiraccelerationveryslightly.
84May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank
85June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05Chapter5EvidenceforMiHsCInthischapterIwillpresentcomparisonsofthepredictionsofMiHsCwithquiteafewobservedanomalies,astests.Nowsomeoftheseanomaliesarefairlysolid,likecosmicaccelerationandthegalaxyrotationproblemandsomearenotsosolid.Iamnotsayingbydiscussingthem,thatInecessarilybelievethatalltheanomaliesherearetrueones,somemaybeerrors,butIamsayingthattakentogethertheydoseemtobepointingtonewphysicsthatlooksratherlikeMiHsC.Notethatformostofthesepredictionsitwillbeassumedthatthestandardpartofinertia(modelledbytheasymmetricCasimireffectinChapter4)isunchanged,andtheHubble-scaleCasimireffectproducesaslightdeviation.CosmicAccelerationThefirstpredictionthatcanbegainedfromMiHsCcanbeobtainedjustasabovewherethisexpressionwasderived:GM2c2a=+(5.1)r2ΘNotethatthestandardNewtonianresultincludesjustthefirsttermontherighthandside,whereasMiHsCpredictstheexistenceofanewtermthatdependsonlyonthespeedoflightandtheHubblediameter.Thistermistinyandisabout6.7×10−10m/s2.Thisaccel-erationwouldtakeyoufromzeroto1m/sin317years,fromzeroto60milesperhourin8500years,andintriguinglyfromzerotothespeedoflightintheageoftheuniverse(13billionyears),soitisverysmall.However,thisextraaccelerationisimportantbecauseitdoesnotdependonthemassoftheobjectorthemassofthe73
86June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0574PhysicsfromtheEdge:ANewCosmologicalModelforInertiaobjectgravitationallyattractingit.Itstandsalone.Thismeansthatwhenanobject,forexample,leavestheSolarSystem,andtravelsintodeepspacewheretherearenomassestoexertagravitationalpull,theremuststillbethisresidualaccelerationandalso,asweshallseelater,whenspecialrelativityenhancesthemassofobjectsclosetothespeedoflight,thisaccelerationisunaffected.Wheredoestheextratermcomefrom?Whathappensisthatastheaccelerationofanobjectdecreasesasit,forexample,movesoutoftheSolarSystem,theUnruhwavesitseesandwhichmakeupitsinertialmass,getlonger,sothataccordingtoMiHsCagreaterproportionofthemdonotfitwithintheHubble-scaleandsoaredisallowedbytheHubble-scaleCasimireffect(seeEq.(4.7))sotheinertialmassdropsandtheobjectisthenmoreeasilyacceleratedbythesameexternalgravitationalforce(thegravitationalmassisunchanged).AtsomepointabalanceisreachedbetweentheeverincreasingrateoflossofinertiaduetoMiHsC,andtheaccelera-tionduetoconstantexternalforcescausedbythatlossofinertia.ThisbalancepointduetoMiHsCpredictsthatthereisaminimumaccelerationinnatureof2c2/Θ.TakingthistoanextremeyoucanseethatMiHsCdoesnotallowzeroaccelerationbecausethentheUnruhwavelengthswouldcertainlybelongerthanthecosmosandunobservable(sodisallowed,Figure1.MiHsCimpliesthatthereisaminimumaccelerationinnatureandsodisagreesveryslightlywithNewton’sfirstlaw.Thisdeviationissosmallthatitisverydifficulttodetect,and,unlikethefigurewouldnotevenimpacttheslowworldofsnails.Nevertheless,indeepspace,farfromanyforces,itcanbecomeimportant.
87June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC75ifweadopttheattitudeofMach,whichIdo).Thereforetheinertialmasswouldbezeroandaccelerationswouldbeinfinite.SoinMiHsC,astheaccelerationapproacheszero,inertiadisappearsslowlyandinfinitespeedsareavoided.Aswewilldiscusslater,thePioneercrafthavelefttheSolarSystemandareshowingjustthisanomalousacceleration,butbeforewecanbeencouragedtoomuchbythisitisimportanttopointoutthattheanomalousaccelerationofthePioneerscanalsobeexplainedbyananisotropicradiationofexcessheatfromthespacecraft(Tury-shevetal.,2012).AlthoughtheanalysisofTuryshevetal.mightbecorrect,itiscertainlynotconclusivebecausetheirmodelistoocomplextobepublishedinfull,hastwoadjustableparametersandcontainsover2000finiteelements.MiHsC,incontrast,predictsthecorrectresultwithoutanyadjustablefittingparametersandfromasimplephilosophy.Dowehaveanyotherexamplesofmassesmovinginfardeepspace?Yes,attheedgeoftheuniversethereareplentyofobjectscalledsupernovaewhicharethermonuclearexplosionsofstars(Fraser,ed.,2006,TheNewPhysics)andtheyarebrighter,emitmorelight,thanthegalaxiestheyexplodeinsotheyarevisibletousatgreatdistances,hencetheiruseascosmicmarkers.Perlmut-ter(1999)andRiess(1998)foundthatsupernovaeveryfaraway(athighredshifts)arefainterthanonewouldexpectifgravitywasslowingtheuniversalexpansiondownovertime.Ifthisgravitationalslowingwasoccurringthesupernovaewouldstillbebrightbecausetheywouldn’thavemovedasfarawayfromus.Whatcouldpossi-blyoffsetthegravitationalselfattractionoftheuniverse?Well,thestandardanswertothisisthatitis‘darkenergy’whichisspeedingtheuniversalexpansionup,andwhoseoriginisunknown,butaswehaveseenMiHsCpredictsthisaccelerationwithoutneedinganydarkenergy.So,howdoesthisworkintheframeworkofMiHsC?IwillnotassumetheBigBanginthisdiscussionbecauseinmyopinionitisbettertostayfocusedmoreonthingsthatcanbeasdirectlyseenaspossible.WhatIwilldoisconsideramodeloftheuniverseasitistoday,butputinmoreenergythaniscurrentlyseenintheformofrandommotionsandaccelerations.Slowlythisexcessenergyis
88June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0576PhysicsfromtheEdge:ANewCosmologicalModelforInertiaconvertedtoheatandtheobjectsintheuniverseslowdownandtheiraccelerationsreducetowardszero.ThismeansthattheUnruhwavesseenbyallthemassesintheuniversegetslowlylongerandlonger.However,inMiHsC,zeroaccelerationisnotallowedbecausetheUnruhwaveswouldthenbelargerthanthesizeoftheuniverse(theHubblescale)andthereforecannotexist(followingMach’sprinciple)sothelongerUnruhwavesdisappearinacloudoflogic,theinertialmassdissipatesanditbecomeseasiertoacceleratealltheobjectsintheuniverseandsotheyaccelerateagain.Asdiscussedabove,MiHsCpredictsthatabalanceisachievedaroundtheuniversalaccelerationof2c2/Θandweeventuallyfindthattheaccelerationofthecosmoscannotfallbelowthisvalue.IfitdidtheUnruhwaveswouldexceedthesizeofthecosmos,inertiawoulddisappearandtheaccelerationwouldincreaseagain.IneffectyoucouldarguethattheenergyfordarkenergyiscomingfromalossofUnruh-energycausedbytheHubblehorizon.Inmyopinionthisisaveryphysical,self-consistentandelegantwaytounderstandtheexistenceofthecosmicaccelerationandthereisawaytotestit.ThenewsecondtermintheequationdependsonthesizeofΘandincosmichistory,sincetheuniverseisexpandingthisscalehasincreasedintime,sothecosmicaccelerationmusthavebeenmuchgreaterinthepast.ThismayhelptoexplainthesocalledflatnessproblemwhichwaspointedoutbyDickeandPeebles(1979).Theproblemisthattheobservedmassofthecosmosisclosetothecriticalvaluethatdetermineswhethertheuniverseis‘closed’andwilleventuallyself-collapseor‘open’andwillcontinuetoexpand.Inotherwordsitis‘flat’.Guth(1981)proposedthattheuniversewashugelyinflatedearlyonsothatnowitismanytimeslargerthanthepartswecanseeandthisexplainsitsapparentflatness.MiHsCindeedpredictsthattheaccelerationmusthavebeenmuchgreaterinthepast,butIhavenotdonethedetailedcalculationsforthisyetandalsoItendtomistrustcosmologicalextrapolationslikethis,andprefertestsclosertohome.Thesecond,MiHsC,termmayalsodependonavariablespeedoflightc,buttheevidenceistooweaktodecideonthatatthemoment.
89June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC77TheLarge-ScaleCMBAnomalyIn1964ArnoPenziasandRobertWilsondiscoveredbackgroundradiationcomingfromalldirectionsinspaceandthisisnowcalledtheCosmicMicrowaveBackground(CMB)sinceitswavelengthisinthemicrowave(afewmicrometres).Thisradiationisjustwhatyouwouldexpectfromauniversewithanaveragetemperatureof2.725±0.002K,butitisnotquiteuniform.Thereisanasym-metrythatisthoughttobeassociatedwiththemovementofthelocalgroupofgalaxieswithrespecttothisradiation(aDopplershift),andwhenthisissubtractedouttherearesmallervariationsaswellwhichareaboutafewpartsin100,000ofthebackgroundtemperature.DatafromtheCosmicBackgroundExplorersatellite(COBE),thelaterWMAP(WilkinsonMicrowaveAnisotropyProbe)andmostrecentlythePlancksatellite(PlanckCollaboration,2013)showaverycloseagreementwiththeColdDarkMattermodel(ΛCDM)atmostscales,andparticularlythesmallscalesthosewithmultipoles(aparametergivenbythesymbolL)greaterthan40.NotethatΛrepresentsdarkenergy,anotherinvisiblecomponentofthemodel.ItwasshownbyHinshawetal.(1996)forCOBEandSpergeletal.(2003)forWMAPthatthereisanunexpectedlackofangulartwo-pointcorrelationonscalesbetween60and170◦inallwavebands.ThismeansthatonlargescalestheCMBradiationisunexpectedlyuniform,buttheseresultswerenotsignificant.TherecentresultsofthePlanckCollaboration(2013)areverysimilartothoseofWMAP(theyagreewithin3%)andthePlanckteamhaveagainfoundthatformultipoles(L)of:Llessthan40theCMBenergyisbetween5and10%lowerthanexpectedaccordingtothebestfitPlanckΛCDMmodel,andthisanomalyissignificanttothe2.5to3sigmalevel.ItturnsoutthattheHubble-scaleCasimireffectusedinMiHsCtosuppresslongUnruhwavesalsopredictsasuppressionoflargescalevariationsofthesamesizeasseenbyPlanck.AccordingtoMiHsCtheenergyintheCMBshouldbemodifiedasfollowsλmE=E1−(5.2)4Θ
90June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0578PhysicsfromtheEdge:ANewCosmologicalModelforInertiawhereλmisthepeakwavelengthoftheradiationandΘistheHubblediameter.TheCMBdataisusuallypresentedusingtheconceptofthemultipole.Thezerothmultipole,orL=0,isaconstantvalueacrossthesky,whichwouldrepresentawavemuchbiggerthanisallowedinMiHsC.ThefirstmultipoleorL=1representsthelongestwave-lengththatcanfitintothesky,orthecosmos,thishasawavelengthof2Θ.ThemultipolemomentcanbewrittenintermsoftheHubblediameterasfollows2ΘL=(5.3)λmSothatthecorrectionfromMiHsCis1E=E1−(5.4)2LSo,asusualwithMiHsC,forhighermultipolesorsmallercosmicscalesthereisnoeffectsincethesecondterminthisformulaisnegli-gible,butforlowmultipolesorlargescalesthereisaneffect:MiHsCsuppressesvariationsonalargescale.TheresultisshownonFigure2asthedashedline.ThesolidlineinFigure2showstheobservedangulartwo-pointcorrelationoftheCMBradiationasafunctionofthemultipolemoment(denotedL)asobservedbythePlanckCollaboration(2013).ThereisalargedecreaseinthepowerofthespectrumforL=2,and3,butthisisnotsignificantgiventhelargeerrorscausedbyalackofdataattheselargescales.Whatissignificant,to2.5to3standarddeviations,butnotapparentonthegraph,isthe5to10%reductioninpowerbelowthatexpectedfromtheΛCDMforallthemultipoles(L)whicharelessthan40.Thedottedlineshowsthepredictedspectrumusingthebest-fitΛCDMmodel.ThismodelfitstheCMBspectrumverywellforsmallerscales(L>40),butitoverestimatestheobservedenergyatthelargerscales,whereL<40,by5–10%.ThedatafromPlanckhasshownthatthisdifferenceissignificantatthe2.5to3sigmalevel.Thismeansthereisonlya0.27%chanceitisduetochanceandnotanerrorinthemodel,soitprobablyrepresentsaproblemwiththemodel.
91June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC79Figure2.TheobservedspectrumoftheCMB(thesolidline)formultipolesbelow40asderivedfromtherecentpaperbythePlanckCollaboration(2013).AlsoshownisthepredictionofthePlanckΛCDMbestfitmodel(thedottedline).InthereportbythePlanckCollaboration(2013)theyconcludedthatthereis:“atensionbetweenthebest-fitΛCDMmodelandthelow-Lspectrumintheformofapowerdeficitof5–10%atL<40,significantat2.5–3sigma.Wedonotelaboratefurtheronitscosmologicalimplications,butnotethatthisisourmostpuzzlingfindinginanotherwiseremarkablyconsistentdataset”(PlanckCollaboration,2013).ThedashedlineshowsthepredictionofaHubble-scaleCasimireffect,whichagreeswiththeobservations.ThedashedlineshowsthepredictionfromthebestfitΛCDMmodel,butwithacorrectionappliedusingtheHubble-scaleCasimireffectusingEq.(5.4).ThiscorrectionhasnoeffectonthesmallerscaleswhereL>40,butitdoesreducetheenergyfromtheΛCDMmodelforthelargestscales(L<40)by5.5%.Themodifiedpredic-tionnowagreeswiththePlanckobservationswhichshowedadeficitrelativetoΛCDMofbetween5to10%.Themeaningofthissuppressionissubtle:itisnotsimplyasuppressionofthelongestwavelengthsoftheCMBradiation,sincesuchwavelengthsarefartoolongtobemeasuredbyourinstruments,butisratherasuppressionofthelargescalevariationoftheradiationoverthecosmos.Asbrieflymentionedabove,thereisawaytounderstandthisthatusesandillustratesthephilosophyIhaveusedinMiHsC.TheHubblehorizonisahorizonbeyondwhichwecannotseeanythingin
92June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0580PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure3.HorizoncensorshipfortheCMBanomaly.OnlypatternsthatfitexactlywithintheHubblescale(withnodesattheboundaries)canbeallowed.IfallpatternswereallowedthentheywouldenableustoknowwhatliesbehindtheHubblehorizonandthiswoulddefeatthepurposeofthehorizon.principle(seeFigure3).Nowifthereisapatternofvariationthatfitsexactlywithinthisboundary(Figure3,seetheupperlineontheschematic)thenthisisallowedbecausethepatternisclosedattheboundaryanditdoesn’timplyanythingbeyondit.However,apatternthatdoesnotfitexactlywithintheHubblescale(seethelowerlineontheschematic)allowsustoguesswhatliesbeyondtheHubbleboundary,andthisshouldnotbeallowed.SowhatIamsuggestingisakindofhorizoncensorship.Thehorizonmakessurethatitremainsaboundaryforinformation,evenbyreachingintothecosmosanddisallowingpatternsthatmightallowustoguesswhatliesbehindit.Asatestofthisidea,theHubble-scaleCasimireffectpredictsthatthespectralpowershouldvaryinaparticularwayforverylowmultipoles.TheenergyshouldshowpeakswhenthewavesfitexactlywithintheHubblediameter,andtroughswhentheydonot.ItmaybepossibletosearchfortheseresonancesinthePlanckdataifitisavailableathigherspectralresolution.
93June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC81Figure4.Atypicaldiscgalaxy.Thefigureshowstheforcesonastarnearitsedge.Thegravitationalforce(dependentonthegravitationalmass)pullsthestartowardsthegalacticcentre,andtheinertialforce(dependentontheinertialmass)tendstokeepitgoinginastraightlineandthereforeoutwardsfromthegalacticcentre.GalaxyRotationwithoutDarkMatterThegalaxyrotationproblemisthebestknownanomalyinphysics.Toputitsimply:galaxiesshouldexplode,butdon’t.Thatis,thevisiblemassinthemshouldnotbeabletoholdontothestarsattheiredgessincethesearewhizzingaroundsofastthattheinertialforcesshouldpropelthemoutwards.Thisisusuallysolvedbyaddingdarkmattertothegalaxiesarbitrarilytoholdtheminbyforce.Instead,MiHsCsolvesthisproblembyreducingtheinertialmassofthestars.StartingwithNewton’sinertialsecondlawandhisgravitylawwegetGMmF=mia=(5.5)r2wheremiistheinertialmass,aistherotationalacceleration(v2/r)ofthestartowardsthegalacticcentre,Gisthegravitationalconstantwhichisjustaverysmallnumber,Misthegravitationalmasswithinaradiusr,andmisthegravitationalmassofthestaratthegalaxy’sedge.WecannowreplacetheinertialmassusingMiHsC:2c2GMmm1−a=(5.6)|A|Θr2
94June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0582PhysicsfromtheEdge:ANewCosmologicalModelforInertiaTheaccelerationAcannowbeseparatedintoameanpartthatisconstantintimethatissupposedtorepresentthesteadyrotationalaccelerationofthestararoundthecentreofmassofthegalaxy,thiscanbecalleda,andapartthatvarieswithtime,forexample,asthestaracceleratesintodensergalacticspiralarms,closetoneighbouringstars,throughglobularclustersorwhenitmovesoutsidetheplaneofthegalaxy’sdisc.Thisiscalleda(a-prime).So2c2GMmm1−a=(5.7)(|a|+|a|)Θr2Inordertosimplifythisformulaalittle,wecandividebymandmultiplyby|a|+|a|togive2c2GM(|a|+|a|)|a|+|a|−a=(5.8)Θr2Youmaythinkthatthishashelpedusnotatall,sincetheformulastilllooksaboutassimpleasaRubic’scube,butthereisanicelittlesimplificationwecannowmakeandthatinvolvesrememberingfromthelastsectionthatMiHsCpredictsthatthereisaminimumacceler-ationinnature,withnoexceptionsandthatthisisequalto2c2/Θ.Thisoccurs,torecap,becauseasaccelerationsreducetheUnruhwavesseenbyobjectsbecomesolargethatagreaterproportionofthemdonotfitintotheHubblescaleandtheyaredisallowed.InprincipletheycannotbeobservedandMachsaidthatifsomethingcannotbeobservedtheyshouldbeassumednottoexist.Nowattheveryedgeofagalaxy,themeanacceleration|a|becomeverysmall,butstilltheremustbeanaccelerationabove2c2/Θsothisaccel-erationmustbeprovidedbythelocalaccelerations|a|.Therefore|a|=2c2/Θandintheequationabovethesecondandthirdtermsonthelefthandsidecancel.So2GM(|a|+|a|)a=(5.9)r2Thislooksabitsimpler.Asbefore,wecansaythatattheedgeofthegalaxythemeanrotationalacceleration|a|ismuchsmallerthanthelocalacceleration|a|sowecanjustignorethe|a|ontheright
95June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC83handsidesinceitisdwarfedby|a|.So2GM|a|a=(5.10)r2Nowassumingacircularorbitwecansaythata=v2/rwherevistherotationspeedandristheradiussotheequationbecomesv4=GM|a|(5.11)Aswealreadydiscussed,theaccelerationaattheedgeofagalaxyhastobe2c2/ΘsinceMiHsCprecludesaccelerationslowerthanthatsoweneedtoreplaceawiththis:2GMc2v4=(5.12)ΘNowwehaveaformulathatpredictstheouterrotationspeedsofgalaxiesforcomparisonwiththoseobservedandithasnoadjustableparameters.ThisisacrucialpointsinceMiHsCthenhasonlyonechancetoberight,andcan’tbe‘tuned’,thevaluesareallknown.ThevalueofGhasbeendeterminedintheSolarSystemandontheEarthandiswellknown(6.6×10−11Nm2kg−2).TheMisthebaryonicmassofthegalaxy,sothisisthemasswecanseeandthisisdeterminedbylookingathowmuchlightiscomingoutofthegalaxyandassumingthattheamountoflightyougetisrelatedtothemassinthesamewaythattheamountoflightfromtheSunisrelatedtoitsmass,whichisknown.Thisiscalledthemasstolightratio.Thevalueofthespeedoflightisnowverywellknown.Thesizeoftheobservableuniverseisperhapstheleastwellknownoftheparameters,butitcanbederivedfromtheHubbleconstantwithanerrorof9%(Freedman,2001).Whatremainsistocomparethepredictiontotheobservations,soFigure5showstheresults.Thex-axisshowsthemassofthesystemlookedatinsolarmasses.ThemassoftheSunisonesolarmass,soavaluealongthisaxisof1.E+03means1000timesthemassoftheSun,sothesesystemsrangefrom1000solarmasses,whichtendtobedwarfgalaxiessuchasarefoundsurroundingtheMilkyWay,uptospiralanddiscgalaxiesinthemiddlerightuptogalaxyclustersupto1×1015timesthemassoftheSun.Thisisahugerange.The
96June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0584PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure5.Theobservedouterrotationalvelocityforcosmicstructuresrangingfromdwarfgalaxies,throughgasdiscsanddiscandspiralgalaxiesandouttogalaxyclustersfromMcGaughetal.(2009)(blackcircles,solidline)presentedincomparisonwiththepredictionsofMoND(dottedlines)andMiHsC(dashedline).BothMoNDandMiHsCcorrectlypredictthedatawithintheuncertain-tiesinboththedataandthemodels,butMiHsCdoesthiswithoutadjustableparameters.y-axis,theverticalone,showstherotationspeedofeachparticularsystem,andtheobservationisshownbythesolidline.Notethatthisaxisislogarithmic,whichisjustacomplicatedwayofsayingthatithasbeensqueezedsmallerintheverticaldirectionforthelargevaluesatthetoptogetthemallonthepage.TheobservationsarefromMcGaughetal.(2009).Thefirstthingtonoticeisthattherotationspeedgoesupasthecentralmassgoesup.Thisistobeexpected:asmorematterispresentinthecentreofthesystems,theycanpossessagreaterrotationalvelocityandstillremainstableagainsttheinertial(cen-trifugal)forcesthatwouldliketotearthemapart.ThedottedlineinthefigureshowsthepredictionofMoND(Mil-grom’sModifiedNewtonianDynamics)withtwodifferentvaluesfor
97June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC85itsadjustableparameter,a0:1.2×10−10m/s2and2.0×10−10m/s2.√Thisisdonebyusingtheformula:v4=GMa0.MoNDworksbestfortheintermediatemasssystems,andperformslesswellonthesmalldwarfgalaxiesandthehugegalaxyclusters.ThismightbeexpectedsinceMoNDwastuned,thatistheadjustableparameterinit:a0,wastunedtofittheintermediatespiralgalaxies.ThedashedlineshowsthepredictionofMiHsC.Thishasa20%errorbecauseofthe9%errorintheHubbleconstantmentionedaboveandafactorof2uncertaintyinthestellarlighttomassratio(assuggestedbyMcGaughetal.,2009).MiHsCoverestimatestherotationspeedoftheintermediatestructures,byupto50%,butthisisstillinagreementwiththedatagiventhelargeerrorbars.MiHsCismuchclosertotheobservationsfordwarfgalaxiesandgalaxyclusters.MoNDisfamouslyunabletomodelgalaxyclusters,butMiHsCcandothis.ItisalsoremarkablethatMiHsCperformssowellbecauseitdoesnothaveanyadjustableparametersandso,unlikeMoND,itcannotbetunedtofitthedata.Thisisanimportantadvantageforatheory.Itispossiblefora‘bad’theorytobetunedtofitthedatausinganadjustableparameter,whereas,ifatheoryfitswithoutanyadjustment,theneitheritisanexampleofverygoodluck,oritistellingussomethingaboutnature.TheFlybyAnomaliesSofar,Ihavemostlyfocusedoncosmologyandastrophysics,butitisbettertotesttheoriesonobservationsclosertohome,sincetheyaremorereliableandlessopentomultipleexplanations.SoIwillnowtalkabouttheflybyanomalies,showthatMiHsCisabletoexplainsomeaspectsofthem,butthattheresultsaremixed.FuelisexpensiveandheavytolaunchsointerplanetarymissionsweregivenahugeboostwhenMichaelMinovitch(1961)proposedthemethodnowcalledthegravityassist.Thefilmthatalwayshelpsmetounderstandthisis“BacktotheFuture”.InoneearlysceneMartyMcFlyisonhiswaytoschoolonhisskateboardandgrabsontoatruckwhichpullshimalongforawhile.Thatiswhathappenswithagravityassist.Say,youwanttosendaprobeintotheouter
98June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0586PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure6.AnEarthflybyassist.AspacecraftapproachesbehindtheEarthinitsorbitandispulledalongbytheEarth’sgravityandspeedsup,seethelengtheningdashes,withouttheneedforexpendingfuel.SolarSystem,toSaturn,butyoudon’thavethefuelnecessarytobreakoutoftheSun’sgravitywell.WhatyoucoulddoisfirethespaceprobesoitarrivesatJupiteratapositionbehinditinitsorbit,andthenJupiter’sgravitywillpullthespaceprobealongandspeeditup,justlikethetruckpulledMartyMcFlyalong.ThefirstspacecrafttousethistechniquewasMariner10,whichusedVenustoslowitselfdownsothatitcouldgointoorbitaroundMercury.VerycleverlythisisoftendoneusingtheEarthitself,thishashappenednowninetimes,andtheprogressofthespacecraftistrackedusinglaserranging.Todothistheyfirerepeatedlaserbeamsatthespacecraftandwaitforthereflection,andsincethespeedoflightisconstantinavacuum,theycanworkoutthedistancetothespacecraftasitchangesintimeandthereforealsothespeed.TheycanalsoworkoutthespeedbysendingsignalstothespacecraftandlookingattheDopplershiftinthereply.AntreasianandGuinn(1998)firstnoticedthatforthesespace-craftthespeedwithwhichtheylefttheEarthwassometimesslightlydifferentfromthatexpectedandthiswascalledtheflybyanomaly.ThedatafromthesevariousflybysisshowninTable1.Thefirstcolumnshowsthenameoftheflybycraft,columns2and3theincomingandclosestapproachvelocities,columns4and5theincom-ingandoutgoinglatitudes,column6thedistanceoftheclosestapproach(perihelion)andcolumn7showstheobservedanomalousspeedchange.
99June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC87Table1.Theflybyanomalies.Thenameofthespacecraft,geocentricvelocity◦atinfinityandatperihelion(km/s),theincomingandoutgoingdeclinations(),theperiheliondistance(km)andtheobservedanomaly(mm/s).v∞vperihelionLatinLatoutPeriheliondvFlybycraftkm/skm/sdegdegkmmm/sGalileo18.9513.74−12.5−34.273313.92±0.08Galileo28.8814.08−34.3−4.96681−4.6±0.08NEAR6.8512.739−20.8−72691113.46±0.13Cassini1619.026−12.9−57553−2Rosetta13.8610.517−2.8−34.383271.8±0.05Messenger4.0610.38931.4−31.987180.02±0.05Rosetta210.918.4117000.27Rosetta318.524.3152390.4EPOXI15.216.8219430EPOXI217.163.9498350EPOXI3305050Juno7000?IrememberwhenIsuddenlyfoundoutthatanewpaperhadjustbeenpublishedaboutallthis(Andersonetal.,2008).Idownloadeditthereandthenandreaditthroughduringmylunchbreak.Thepaperwasshort,butshowedthatthedatatheyhadatthetime(thefirstsixcolumnsinthetable)fittedaformulalikethisdv=3.099×10−6×v×(cosϕ−cosϕ)(5.13)∞inoutwherev∞isthehyperbolicexcessvelocitywhichisthevelocitythespacecraftwouldhavealongwayawayfromtheEarth(outsidethegravitywell),φinistheincidentlatitudeofthespacecraftandφoutisthelatitudeitexitedat.Theyhadnoticedthispatterninthedata:ifthespacecraftlefttheEarthatthesamelatitudethatitapproachedthentherewasnoanomaly,butifitslatitudechangedthentherewasananomaly.AgoodexampleofaflybywithoutananomalyisthesixthshowninTable1:theMessengerspacecraftwhichcameinatalatitudeof31.4degreesandleftat31.9degreesandshowedonlyatinyspeedanomaly.
100June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0588PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThenthereistheNEARflybywhichcameinat20degreessouthandleftat72degreessouthandhadthelargestspeedanomalyof14mm/s.IrememberreadingthispaperandfindingitbreathtakingandIwasupsetbecauseIcouldn’texplainitusingMiHsC.So,Ihadabathroombreakandthenwentovertoalectureroomtohaveathinkaboutit,andthereIrealisedthatIcouldexplainitwithMiHsCbythinkinginawayIhadalreadyconsidered,buthadn’thadthedatatopointouttheway.ItooktheunusualstepofconsideringalltheaccelerationsseenbytheNEARspacecraftonitsflyby.WecansimplifythingsbylookingatthespacecraftanditsinteractionwithjustonepieceofmasswithintheEarth(seeFigure7).OnitsequatorialapproachthespacecraftseestheaccelerationofthemassPawayfromittowardstheEarth’scentreasamutualacceleration(assumingthespacecraftisfarenoughawaythatitsownaccelerationrelativetothemassissmall).ByMiHsCthenthewavelengthoftheUnruhradiationisFigure7.SchematicshowinganEarthflybywherethespacecraftapproachesneartheequator(1)andleavesnearthepole(2).AlsoshownisanarbitrarymasswithintheEarth(P).
101June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC89shortandmanyofthewavesareallowedandtheinertialmassisclosetothegravitationalmass.Now,asthespacecraftleavesclosetothepole(seeposition2inthefigure)itnowdoesn’thaveasmuchmutualaccelerationwithrespecttothemasswithintheEarth(P)becausetheaccelerationvectorofthemassPisperpendiculartothelinejoiningthecraftandthemass,sothemutualaccelerationbetweenthecraftandthemassissmaller.SotheUnruhwavesarelongerandagreaterproportionofthemaredisallowedbytheHubble-scaleCasimireffect,soMiHsCdecreasestheinertialmass.Ifweassumethatmomentum(whichismasstimesvelocity)mustbeconserved,thenthespacecraft,sinceithaslostmass,mustgaininvelocity,soitspeedsup.Notealsothatthisdatataughtmesomething:theaccelerationstoconsiderwerenotthoserelativetoabsolutespace,butrather,themutualaccelerationsbetweenanobjectandotherobjectsneartoit.BeforethisIhadalwaysthoughtofaccelerationswithrespecttothebackground,butthiswasamovetoakindofthinkingclosertoErnestMach’s,asIfoundoutlater,anditwasnotmotivatedbyphilosophy,butbynecessity:tryingtogetthemodeltofitthedata.Now,althoughIhadahandwavingexplanationIstillneededtoshowwhethertheideapredictedtheactualnumberscorrectly,thisisthedifficultpart!AssumingtheconservationofmomentumforthecraftinthereferenceframeoftheEarthweget:m1v1=m2v2(5.14)wherethetermsare:theinitialandfinalmomentaofthecraft.WecannowreplacetheinertialmassesoftheunboundspacecraftusingtheMiHsCformulaforinertia:2c22c2mg1−v1=mg1−v2(5.15)|a1|Θ|a2|Θwheremgisthegravitationalmassofthecraft,ortheuncorrectedinertialmass.Somealgebraimpliesthat2c2v2v1v2−v1=dv=−(5.16)Θa2a1
102June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0590PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThisisthenew‘jump’inspeed(dv)predictedbyMiHsCbecauseitsmutualaccelerationwithrespecttoalltheseparatemasseswithintheEarthhasdecreasedduetoitspositionclosertotheEarth’sspinaxis.Nownormally,inNewtonianphysics,thesemyriadaccelerationsareforgotten,andthemassoftheEarthistreatedasifitwasallconcentratedatapointinthecentreoftheplanet.WithMiHsCinsteadweneedtoconsiderallthesemyriadaccelerationsandthisisoneofthemostdifficultthingstocommunicateaboutMiHsCandIamnotsureIhaveeversucceededinatalkorapaperingettingtheideaacrossclearly.TopicturealltheseinteractionsonecouldimaginealineconnectingthespacecraftwitheveryPlanckmasswithintheEarth.Onethenneedstosumuptheaccelerations(accelerationofthelengthsofthelines)foralltheseinteractionsanddividebytheirnumbertogettheaccelerationthatgoesintotheequationabove.ThisoftenremindsmeofagameIusedtoplaywithmysisterinthecar.Atnightweusedtoseethestreetlightsandthewaythediffractionofthelightthroughthewindowsmakesitseemasthoughthereisalightbeamextendingbetweenthestreetlightandthecar,andweimaginedthattheywerepullingthecaralong.ItisthesebeamsoflightIimaginewhenIthinkoftheflybysetuphere.Also,morerecentlyIgoregularlyforacousticscansoftheheartsinceIhaveabicuspidheartvalvethatleaksabit.Iusuallyhavetolieonthebedfacingawayfromthescreen,butbytwistingmyheadIcanseewhatisgoingonthereandtalktothetechnicianaboutit.Theheartvalveisoftenvisibleinblackandwhite,andthecomputershowsredandblueareas.Theyshowthevelocityofthebloodtowardsandawayfromthesonar.Whathappensis,thesonaremitsasoundwhichbouncesoffthematerial,backtothereceiverandfromthetimeintervalittakestocomeback,andknowingthespeedofsound,youcanworkoutthedistancetothematerialitbouncedoffandplotthesoundcomingbackfromaparticularlayerofthebodyonthescreen.YoucanalsolookattheDopplershiftofthesound.Ifthepitchofthesoundishigherwhenitbouncesback,thatmeanstheobject(probablyblood)ismovingtowardsthesonarandifitislower,the
103June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC91bloodismovingaway.Inthiswaythespeedofthefluidcanbeplottedincolour.Always,inlookingatmyheartonecanseeacolourfuljetofcolourwhichrepresentstheleakbackofbloodasmyheartvalvefailstopushitallthroughthevalve.Ialsothinkoftheflybysinthissortofway.Inthiscasethespacecraftseesaccelerations(notspeeds)ofthePlanckmassesintheEarthastheyspinaround.Thefasterthemutualaccelerationthemorecolourthespacecraftwouldsee,themoreUnruhwavesareallowedbyMiHsCandthereforetheinertialmassishigher,andclosertothegravitationalmass.Thisuseofmutualaccelerationsbetweentwomasses,isverysat-isfyingwhenyoulookatMach’spointthatyoushouldonlyincludeinatheorythingsthatyoucandirectlyobserve.Youcannotdetermineanaccelerationrelativetoabsolutespace,becauseyoucannotbesurehowthespace(whichisinvisible)isaccelerating,butyoucandeterminetheaccelerationbetweentwosolidbodies.NowIwon’tderivethewholethinghere(youcanseetheAppendixofMcCulloch,2008bifyouwanttoknow.Itisquiteinvolvedbecauseyouhavetoconsiderthefactthatthedensityoftheplanetisslightlygreateratthecentre),buttheaverageacceler-ationofthemasseswithintheEarthasseenbythespacecraft(thecolouredareas)isgivenby0.07ve2a=(5.17)Rwherethe0.07comesfromthevariationofdensityasyougotowardsthecore,veistheequatorialspinvelocityoftheEarthandRistheEarth’sradius.Thecomponentofthisaccelerationseenbyaflybyspacecraftatalatitudeofφistherefore0.07ve2a=cosϕ(5.18)RWecannowusethisexpressionfortheaccelerationsintheMiHsCformulafordvandassumethatv1=v2:2Rc2cosϕ1−cosϕ2dv=×v×(5.19)0.07ve2Θcosϕ1cosϕ2
104June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0592PhysicsfromtheEdge:ANewCosmologicalModelforInertiaWecannowsubstitutevaluesinhere.Ialwayslikeusingrealnum-bers.Itcouldalmostbethoughtofasasignthatyouaredoingsomethingreal.Intoomuchofmodernphysics,thenumbersareexcludedandpeopleset,forexamplec=1.Ifindthistobeabhor-rent,becauseitcutsthelinkbetweenthephysicsandtherealworld.Itmay,assomeargue,exposepatterninthetheorymoreclearly,butIthinkthismakesthephysicsthattheydoveryincestuous.Nodatacancomeinfromtherealworldthatway.So,Ilikenumbersandtheyare:R=6371km,c=3×108m/s,ve=465m/sandtheHubblediameterΘ=2.7×1026m,soweget−7cosϕ1−cosϕ2dv=2.8×10×v×(5.20)cosϕ1cosϕ2ThisformulaisverysimilartotheempiricalequationderivedbyAndersonetal.(2008)fromtheirflybydata,especiallyitsdepen-denceonthedifferenceinthecosineoftheincidentandoutgoinglatitude.Figure8showstheobservedflybyanomaliesinblackandthosepredictedbytheMiHsCformulaaboveingrey.ThereistheNEARflybywhichhasthelargestanomalyat14mm/s,andMiHsCpredicts10mm/s.Theonlyanomalyitdoesn’tpredictistheEPOXI-2and3anomalieswhichwerezeroandMiHsCpredictsjumpsof5mm/s.201510Observed5MiHsCgeocentric0Anomalousjump(mm/s)-5-10Figure8.Theflybyanomalies(inmm/s)bothobserved(black)andpredictedbyMiHsC(grey).MiHsCagreeswiththesignoftheanomalies,andwiththeNEARanomalywithinerror,whichisthelargestone,butitisfarfromperfect.TheEPOXIflybysoccurredabouttentimesfurtherawayfromtheEarththantheothersandIwouldnotexpectMiHsCtopredictthem,seethenextsectionontheTajmareffect.
105June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC93AsIamalwaysremindingmystudents,whendecidingwhetherornotatheoryagreeswiththedataitisveryimportanttoconsiderwhatuncertaintiesthereareinthetheoryandwhatuncertaintiesareinthedata.Ifthetwoagreewithintheseerrorbarsthenthetheorycanbesaidtobesupportedbythedata.WithMiHsC,anerrorisalwaysincurredbyourinaccurateknowledgeoftheHubblescale(Θ)givingusanerrorof9%.Inthisparticularcasethereisalsoanerrorincurredbecauseofasimplifiedgeometryusedinthederivations,forexampleIhavesofarneglectedthegeocentricaccelerationsbetweenthespacecraftandthecentreoftheEarth.Maybeifyouhavethetimeandinclinationyoucandoabetterjob.In2009IwasinvitedtoattendaflybyworkshopattheInter-nationalSpaceScienceInstituteinBern,Switzerland.ThereImetoneofthediscoversoftheflybyanomaly,DrJohnCampbell,andhegavemetheEPOXIdatapoints,whichallshowedazeroanomaly.ThatwasfineforoneofthembecauseMiHsCpredictedaverysmallanomaly,butitwasnotfinefortheEPOXI-2andEPOXI-3anoma-lies,sincetheywerezeroandMiHsCpredictedaspeedjumpofabout5mm/s.Again,thisupsetmequitealotatfirst,and,ashappenswithmeabouttwiceaweek,IbegantofearthatIwaschasingfairiesdownthegardenpath,buttheinterestingthingherewasthattheperigeeoftheEPOXIflybyswasabouttentimesfurtherawayfromtheEarththanalltheotherflybys.WhatifthereisadistancedependenceofMiHsC?Irealisedthatthereisadistancedependence,notbythinkingphilosophicallyaboutMach’sprinciple,butratherbylookingatanotherexperimentalanomaly.ThistimemoredowntoEarth,thoughnolesscontroversial:theTajmareffect.Progress(andhopefullythisdoesrepresentprogress)neverliesdownasafepath.TheTajmarEffectWhenyouarelookingatananomalyinspaceyoucan’tchangetheconditionstotestyourhypothesis,youhavenocontrol,soyoucanneverbeconclusive.Incontrast,alaboratoryexperimentgivesonesomecontrol.
106June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0594PhysicsfromtheEdge:ANewCosmologicalModelforInertiaOnesuchexperimentwasthatofMartinTajmar.WhereastheflybyanomalieswereananomalousmotionofspacecraftwhichwerechangingtheirorientationandthereforetheiraccelerationwithrespecttomatterinthespinningEarth,theTajmareffectisananomalousmotionofagyroscopeclosetoaringwhichchangesitsrotationalaccelerationintime(forthedetailsoftheexperimentseeTajmaretal.,2007,2008,2009).MartinTajmarandhisteamusedringsmadeupofvariousmate-rials(niobium,aluminium,stainlesssteelandTeflon)whichwereabout7.5cminradius,soaboutthesizeandshapeofsmallcircularFrisbeesandtheycooledtheminacryostatdownto5K(−268◦C).Theyplacedaccelerometers(deviceswhichmeasureacceleration)closetotherings(about5,10and23cmaway)andmadesuretheywerenotinanyfrictionalcontactwiththerings.Thentheyspantheringsbothclockwiseandanti-clockwiseandamazinglytheyfoundthattheaccelerometersmeasuredanaccelerationalmostasiftheywereinslightcontactwiththering(butthiswasnotso).Thisaccelerationwasverysmall:itwasonlyoneintenbillionththeaccelerationoftheringitself,butitcouldbedetected.Theylaterusedmoresensitivelasergyroscopesandobtainedthesameresult.AstheynotedthisisalittleliketheLense–Thirringeffectthatispredictedbygeneralrelativity,wherearotatingbodypullsaroundspace-timewithit,soobjectsneartoalargerotatingbodyareexpectedtospinaroundwiththatbody.ThiseffectwasapparentlymeasuredbyGravityProbeB.However,whatTajmaretal.sawisabout20ordersofmagnitude(or100,000,000,000,000,000,000times)largerthantheLense–Thirringeffectexpectedfromsuchasmallring.TheexperimentalsetupisshowninschematicformbyFigure9.Theringwashorizontalandthethreelasergyroscopeswereplaced5,10and23cmawaydirectlyabovethering’sedge.Wecansimplifythemathematicsofthisnicelybyforgettingforamomentthatwehaveaspinandjustlookinginthevicinityofthegyroscopeandbelowit.Here,theringissimplymovinginastraightlinebelowthegyroscope,thecurveissmall,so,justaswedidfortheflybys,wecanwriteaconservationoflinearmomentumequation
107June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC95msFixedstarsrsrRmRRingFigure9.AschematicoftheTajmarexperiment.Thefigureshowstherotatingring(bottom),thelasergyroscope(middle)andthefixedstarsabove.likethis:mg1vgr1=mg2vgr2(5.21)Thelefthandsideisthemomentumofthegyroscopewithrespecttothering(hencethesubscript“gr”,thegisforgyroscope,therisforring)alongthedirectionofthering’sspinbeforetheringspins,andtherighthandsideisthemomentumafterwards.Thisjustsaysthatweareassumingthatthemomentumdoesn’tchange.Ifwetrytheusualtrickofreplacingtheinertialmasses(the“m”s)withtheMiHsCinertialmassweget2c22c2vgr11−=vgr21−(5.22)|ag1|Θ|ag2|ΘAndrearrangingthisweget2c2vgr2vgr1vgr2−vgr1=−(5.23)Θ|ag2||ag1|
108June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0596PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThisformulaisthesameastheoneshownabovefortheflybyanoma-lies.InthatcasetheinitialandfinalaccelerationsseenwerethemutualaccelerationsbetweenthespacecraftandthespinningEarth.Inthiscaseitisthemutualaccelerationsbetweenthemassinthegyroscopeandallthesurroundingmatter(includingthatinthering)thatneedtobeconsidered.Firstofall,thewholeexperimentisdoneinacryostatcooleddownto5K,sothismeansthatthetemperature-dependentthermalaccelerationsofnearbyatomsshouldbeminimised.WecanalsosaythattheaccelerationsbetweenatomsinthegyroscopeandatomsinthesolidEartharesmallbecausetheexperimentissolidlyfixedtotheEarth’ssurface.Whataccelerationsdoesthegyroscope(whichisassumedtobeinthesameplaneastheground)see?Well,itisfixedtoaspinningplanetsofromitspointofviewthefixedstarsintheskyarespin-ningaroundandthisisamutualacceleration,onethatMachwouldappreciate.Thefixedstarsare,onaverage,veryfaraway,buttheircombinedmassishuge.Theapparentspinrate(mutualacceleration)ofthefixedstarsdependsonthelatitudeyouareat.Ifyouareattheequatorthenthespiniszero,thefixedstarsjustseemtopassacrosstheskylinearly.Ifyouareatthepolethenthestarsappeartorotate(spin)aroundPolaris,thePolestar.Ifyouareinthemid-latitudesthereisanintermediateamountofspin.Mybackgroundisinphysics,butalsoinoceanography,andphysicaloceanographersknowthisdependenceverywell(consideringthespininthesameplaneastheground),itiscalledtheCoriolisaccelerationanddependsonthelatitude.TheCoriolisaccelerationisbestunderstoodbythinkingaboutamissilelaunchedfromtheequator,sayfromEquador.Therota-tionoftheequatorbeneaththemissilewaszero(aroundalocallyverticalaxis),sothemissilealsohaszerorotation.Asthemissilemovesfurthernorthorsouthitismovingintoareasthathavenon-zerorotation.Inthenorthernhemispherethemissilewouldseethegroundbeneathitstarttospinanticlockwise.Apersonstandingonthegroundlookingupatthemissilewouldseethemissilespinning
109June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC97clockwise,andasittravelsalongitwouldappeartoturntotheright.Inthesouthernhemisphereeverythingwouldbereversed.AtSiebersdorfinAustria,whereTajmaretal.conductedtheirexperimentthelatitudeis48◦N,and,asanyphysicaloceanographerknows,theCoriolisaccelerationisfv,wherefistheCoriolisparam-eter(0.0001/sinmid-latitudes)andvisthespinvelocityoftheEarthatthatlatitude(whichis311m/s).Thisgivesanapparentacceler-ationwithrespecttothefixedstarsof0.0311m/s2.Thisisalsothespinaccelerationoftheexperimentwithrespecttothestars(sinceitisontheEarthspinninganticlockwiseinthenorthernhemisphere)butsinceonlythemutualaccelerationisimportant,theyarethesame.So,intheaboveformulatheinitialmutualaccelerationseenbythegyroscopeisag1=0.0311m/s2.WhentheringwassuddenlyrotatedbyTajmar,therewasanaccelerationofaR=33rad/s2=2.5m/s2,butinordertoapplyMiHsCtothiscasestudythereisaproblem.Whatistherelativeimportanceforthefixedstarsandtheringfortheinertialmassofthegyroscope?Aretheyequallyimportant?Surelynot!Surelylargerandclosermassesshouldbemoreimportant?WhatIchosetoassumewasthattheimportanceofanothermassfortheinertialmassofthefirstisproportionaltothemassoverthedistancesquared,byanalogywithgravity.So,tocalculateanaveragemutualacceleration(ag2)foruseintheMiHsCformulaforinertiaweneedtowritedowntheaccelerationoftheheavybutdistantstarsandthelightbutclosering,asfollows:msa+mRars2gsr2grRag2=msmR(5.24)rs2+r2Rwheremsandrsarethemassandtheaveragedistanceawayofthefixedstars,andmRandrRarethemassanddistanceawayofthering.Wecannowputvaluesinforthesequantities.Themassoftheobservableuniverseisnotaveryrobustfigure,butthishasbeenestimatedbycountingstarstobecloseto2.4×1052kg.Theaver-agedistanceawayofthesestarscanbeestimatedtobetheHubbledistancewhichis2.7×1026m(rs=2c/H,whereHistheHubble
110June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch0598PhysicsfromtheEdge:ANewCosmologicalModelforInertiaconstantfromFreedman,2001).Themassofthe(steel)ringcanbeestimatedfromthedensityofsteel(8000kg/m3)andthedimensionsofthering.Ithadacircumferenceof2π×0.075m,aheightof0.015mandawidthof0.006m.Thedistancefromthelowestgyroscopetothering,was0.0533m.Usingthesevalueswehave0.33ags+118agrag2=(5.25)118Wecanthereforeforgetallabouttheagsvaluesinceitissmallcom-paredwiththeothertermintheformula,andsimplifythisalittleag2=agr(5.26)Sothefinalaccelerationissimplytheaccelerationofthering,andthefixedstarsareirrelevant(buttheyarenotirrelevantbeforetheringisspunbecausetheyareresponsiblefortheinitiallylowinertialmass).Wecannowsubstitutethevaluesforag1andag2intotheconservationofmomentumformula:2c2vgr2vgr1vgr2−vgr1=−(5.27)Θ|agr||ags|Thefirstterminthebracketshastheaccelerationofthegyroscopewithrespecttotheringonthebottom,andthisisaprettylargequantitywhentheringisspinningsofast,sothismeansthatthistermisdividedbyabignumber,andsoisverysmallandwecanforgetit.Sowegetleftwith2c2vgr1vgr2−vgr1=−(5.28)Θ|ags|Nowwecandifferentiatethischangeinvelocity(dv)withtimetofindthechangeinacceleration(da)2c2agrda=−(5.29)Θ|ags|Ifweswaptheagr,theaccelerationofthegyroscopewithrespecttothering,withtheaccelerationoftheringwithrespecttothe
111June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC99gyroscope(arg)thenwehavetomultiplyby−1,so2c2argda=(5.30)Θ|ags|Thisequationpredictsthattherotationalaccelerationofthegyro-scopewillbeinthesamedirectionasthatofthering,justasobservedbyTajmaretal.intheirexperimentandwillbeproportionaltotheratiooftheaccelerationoftheringbelowittotheaccelerationofthefixedstarsaboveit.Ifweassumefirstthattheaccelerationofthegyroscopewithrespecttothefixedstarsisconstantthenthisgivesusanaccelerationof−10arg−8da=6.6×10=2.1×10arg(5.31)0.0311Tocomparethispredictionwiththedata,wealsoneedtoknowhowuncertainitis.Itisknownthatthereisa9%uncertaintyintheHubbleconstant,whichleadstoa9%uncertaintyinΘ,and9%ofthepredictionisabout0.2.Sothepredictionisda=(2.1±0.2)×10−8×am/s2(5.32)rgThisishappilyinagreementwiththeaccelerationratioseenbyTajmaretal.(2009)whichwas(3±1.2)×10−8argm/s2.AllthedetailscanbefoundinMcCulloch(2011a).ThatisthemeatypartofthedescriptionofthewaythatMiHsCpredictstheresultsoftheTajmarexperiment.MiHsCpredictstheresultverywell,anditdoesthiswithoutadjustableparameters.Thetrickistothinkaboutthingsinaslightlydifferentway:inertiahereisindeedduetomatterthere,asMachspeculated,butyouhavetoaccountforthingsproperly.Nonetheless,thereareproblemswiththeexperiment.ForastarttheTajmarexperimenthasneverbeenfullyreproduced,notevenbyTajmar,althoughmaybethisisnotsurprisingsinceitwasdoneprettyrecently,andrequiresasophisticatedsetup,andTajmarhimselfhasbeenmovingaroundtheworld,fromVienna,toSouthKorea,toGermany.Maybeitwillbeperformedagainsoon.
112June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05100PhysicsfromtheEdge:ANewCosmologicalModelforInertiaInlightoftheabove,anespeciallyinterestingexperimentwouldbeanattempttodotheexperimentinthesouthernhemisphere,sinceMiHsCthenpredictsthatthegyroscopewouldstillfollowtheringinitsrotation,butthatinthesouthitwouldbetheanticlockwiserotationthatwouldbeslightlylarger.ThisisbecausethespinoftheEarth(ofthestars,actually)shouldproduceaparityviolation.So,peopleinNewZealand,Australia,SouthernAfricaandSouthAmerica,etc.:pleasehaveago!SomeguysinNewZealand(Grahametal.,2008)didattemptit,andtheparityviolationintheirresultsdoesindeedlookreversed,butunfortunatelytheirdatahaderrorbarsmuchlargerthantheresults,sotheresultmustbediscounted(Grahametal.,2008).WhenIhavebeenthroughaboutofmathematicsIalwaysliketorecapthingsintuitively,sincethatiswherealltheoriesmusteven-tuallybeconsidered,soheregoes.Beforeitssurroundingsarecooledthegyroscopeseesallkindsoflocalaccelerationsduetotheheatofthemoleculesneartoit.Theyarevibratingandaccelerating,sothegyroscopeseesUnruhwavesthatarerelativelyshortcomparedtotheHubblescale,soveryfewofthemaredisallowedbytheHubble-scaleCasimireffectandthegyro’sinertialmassisnormal.Thenthesurroundingsarecooleddownto5Kinthecryostatandsuddenlymostofthesethermalaccelerationscease.Thegyro-scopeisalsonotacceleratingwithrespecttotheEarth,atleastatfirst,sotheonlyaccelerationitseesiswithrespecttothefixedstars.ThegyroscopeisonthespinningEarth,soithasamutualaccelerationwithrespecttothefixedstars,butthisissmall,sotheUnruhwavesitseesarelongernowandagreaterproportionofthemaredisallowedbytheHubble-scaleCasimireffectbecausetheydon’tfitwithintheHubblescale.Theinertialmassofthegyro-scopedecreasesinthiscasebyabout2×10−8kgforeverykgofmassbecauseofMiHsC,sopresumablyitmusthaverotatedslightlywithrespecttothefixedstarsatthispoint,butthiswasnotmonitoredbyTajmar.SuddenlytheringacceleratesandthegyroscopeseestheUnruhwavesproducedbythatmutualacceleration.Theseareshort,since
113June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC101theaccelerationislarge,andsofewerofthemaredisallowedbyMiHsCandtheinertialmassofthegyroscopegoesup.Nowhowdoesthesystemrepresentedbythegyroscopeandtheringcon-serveitsmomentum(masstimesvelocity)?Theinertialmassofthegyroscope(m)hasjustgoneup,soitsvelocityhastogodown,inthereferenceframeofthering.Theonlywaythiscanhappenisifthegyroscopestartstorotatewiththering,sotheirmutualvelocityisreduced.Afurthercomplicationisthatwhenthegyro-scopestartstorotatewithrespecttothefixedstars,thisgener-atesmoremutualaccelerationsthatcauseaparityviolationsinceaclockwiserotationincreasesagsandananticlockwiserotationdecreasesags.SomethingIdidnotconsiderinmypaper,isthatwhenitstartstomove,thegyroscopealsostartstoacceleratewithrespecttothematterinthesolidEarth,sothisshouldincreaseaccelerationsandincreasetheinertialmassstillfurther.Thiscomplicationdeservesanotherpaper,butIhavenotyethadtimetogointodetailslikethis.Theartofscientificrevolution,ifthisiswhatthisis,istofocusonthingsthatotherswillregardasbeingconclusiveproofs.Easiersaidthandone!Thereareplentyofembarrassmentswhenoneistryingtodosomethingnew,andoneofthesewasbroughthometome,actuallyinthetownwhereEinsteinhashisannusmirabilis:BerneinSwitzer-land.WhenIhadjustpublishedmypapersonMiHsCandtheflybyanomalyIwasinvitedtoattendaweek-longworkshopontheflybyanomaly.Iwasinvitedbackayearlaterandthisworkshopwasalsoorgan-isedbyProfessorClausLammerzahlofBremerhaven,whoisprob-ablyoneofthemostimpressivemenIhavemetinmylife:huge,German,professorial,etc.,andIthinkIdiddisappointhimonthisoccasion.ThiswasbecausethenightbeforeIwastogivemytalkonMiHsCtotheworkshop,IwasgoingthroughmyproofthatMiHsCpredictstheTajmareffect(inmyfirstpaperonit)andatabout10pmInoticedahugeflawinthemathematicsinmypublishedpaper!Ihadusedcompletelythewrongreferenceframe.Ifyouarecurious,youcanreadthefirstpaperIwroteonthis—pleasedo
114June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05102PhysicsfromtheEdge:ANewCosmologicalModelforInertianot!—andeventuallyyouwillrealisethat,althoughtheintuitiveideaisperfectlysound,Iambasicallytalkingabsolutemathematicalrubbishbytheendofit.Thenextday,Ionlyhadhalfatalktogive:thefirstpartabouttheflybyanomalyandIthensatdownwithstillanotherhourtogo.Prof.Lammerzahlcameovertoaskmepolitelywheretherestofmytalkwas,andIhadtotellhimthatwasit.Itwasextremelyembar-rassingforme,andmaybeIshouldjusthavepresentedtheflawedderivationandaskedforhelp,butIdidn’t.Furthermore,thatnightInoticed(youcancheckthesethingsonthearxiv)thatlotsofpeoplewerereadingmyflawedpaper!(Presumablytheworkshopattendeesandtheircolleaguesbackhome.)FormonthsIhadbeendreamingaboutpeopleactuallyreadingmypaper,andfewpeoplehadnoticedit.AssoonasIrealisedthatthepaperwasflawed,everyonewassuddenlyreadingitandpresumablyrealisingtheflawandwritingmeoffforevermore.Laterthatyearmypaperactuallywonanawardfromthejour-nalitwaspublishedin,anddespiteitsflawsIdothinkitdeservesit,becausealthoughitisflawedmathematically,Istillthinkitisgroundbreakingandintuitivelycorrect.ItisinterestingthatwhenIfirstsubmittedittothejournalthemathswasrightandonlygotscrewedupasIlosttrackduringthereviewprocess.Anyway,IoncetoldProf.HorneattheUniversityofStAndrewsaboutmyerrorandhemadeacommentthatcheeredmeupalot:“Don’tworry:thesearehardthingstothinkabout.”Ithinkthatisabsolutelyright,althoughhedidlatergoontosaythat“mymathswasalittlesloppy”,andIthinkthatisrightalso.Infact,theproofofmysloppinesshasbeenpublished!Nevertheless,donotletthisputyouoffMiHsC,sinceIdeviseditintuitively,whichIusuallyseemtogetright.Themathematicsalwayscomeslater,andalthoughImakefrequentmistakes,Ikeepgoingobsessivelytillthemathematicsfitsmyintuition.Iwroteanotherpapertocorrectthefirstone,andthatalsogotanaward.ThewayIintroducedtheTajmareffectatthebeginningofthischapterwasbysayingitwas“theflybyanomalywritsmall”andthat
115June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC103itwouldofferasolutiontotheerroneouspredictionoftheEPOXI-2andEPOXI-3flybys.Ifyouremembertheseflybysshowednojumpatall,butMiHsCpredictedalargeoneof5mm/sorso.Ialsosaidthattheseflybysweremuchfurtherawaythanalltheothers,andifyoulookatEq.(5.24)aboveyouwillseethattheinfluenceofanobjectontheinertialmassofanotherone,inMiHsCdependsonthesquareofthedistancebetweenthem.ThisissomethingIhaveaddedtoMiHsCsincethinkingabouttheEPOXIflybysandtheTajmareffect.Now,youmaysayIamjust‘fitting’thetheorytothedata,butthisideamakesperfectsense:effectsoftendecreasewithdistance,andIhavestilldoneitwithoutadjustableparameters.ThefitIhavesuggested(to1/r2)isextremelysimpleandisanalogoustothegravitationaleffect.WhatthisideameansfortheEPOXIflybysisthattheEPOXIspacecraftwassofarawayfromtheEarth(about10timesfurtherawaythanalltheothers)soitsaccelerationwithrespecttothespinningEarthwaslessimportant,initsversionofEq.(5.24),andits,relativelyconstant,accelerationwithrespecttothefixedstarsandtheotherplanetswasmoreimportant,sotherewasmuchlessofajumpinspeed.ThisdistancedependencealsoprovidesawaytotestMiHsCusingaTajmarexperiment.Itpredictsthattheeffectshoulddecreasewithdistanceaway.Forexampleat1metre,5metres,20metresand56metersawayfromtheringtheeffectofMiHsCshouldbe0.03%,0.8%,11.5%and50%smaller.Thedecreaseat20metersshouldbedetectable,but,admittedly,cryostatsthislongarehardtofind.Maybetheexperimentcouldbedoneinspace?MiHsCalsopredictsthattheanomalouseffectshoulddisappearveryquicklyoutsidethecryostat,becauseofthelargerthermalaccelerationsthere.ThereisanotherdeepermeaningtoallthisthatrelatesittoMach’sprinciple.Ifyouremember,MachconsideredNewton’sbucketexperimentandclaimedthatthewaterinthebucketwasbecomingcurved,notbecauseitwasacceleratingrelativetoabsolutespaceasNewtonhadclaimed,whichisathoughtthingonly,butratherthatitwastheaccelerationsrelativetosurroundingmatterthatwereimportant,specificallyaccelerationsrelativetothefixedstars.Mach
116June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05104PhysicsfromtheEdge:ANewCosmologicalModelforInertiawonderedwhatwouldhappenif,insteadofrotatingthebucket,onecouldrotatethefixedstarsinsteadandhesaidthatno-onewascompetenttosayhowthiswouldturnout(Bradley,1971).Wouldthesurfaceofthewatercurveornot?ThinkingaboutthisusingMiHsCmakesalotmoresenseandenablesustopredictwhatwouldhappenifthefixedstarswererotated.MiHsCpredictsthatifthebucketitselfisrotatedtheinertialmassofthewaterwillundergoaslightincreaseasitsuddenlyrotatesrelativetothefixedstars,justlikethegyrodiscussedabove,andthisgainininertialmasswillslowthewaterdowninitsrotation.Anotherwayofsayingthisisthatanew(tiny)MiHsCrotationwillappearcontrarytotherotationofthebucket.Now,inthecasewherethebucketisstaticbutthefixedstarsaresuddenlyrotated,thenagaintheinertialmassofthewaterwillslightlyincreasefollowingMiHsCandthehighermutualaccelerationandtoconservemomentum,thewaterwillrotate(slightly)withthefixedstars,andthisisinthesamedirectionasbefore:contrarytothemotionofthebucket.Thisallsoundsveryphilosophical,butIhavenowdiscussedsev-eralobservationsthatfitthepredictionsofMiHsC.Withtheflybyanomaliestheresultswerenotveryimpressive,butasIsaidinthatsection,Ihavenotyetbeenabletomodelalltheaccelerationspresentasthespacecraftflyby.Intheflybycasetheobservedjumpsinthespeedofthespacecraftcanperhapsbeexplainedasvariationsoftheinertialmassofthecraftbecauseoftheacceleration(spin)ofthematterinthenearbyspinningEarth.WiththeTajmareffecttheinertialmassofthegyroscope,asapparentinitsmotion,isaffectedbyboththenearbyspinoftheringandthespinofthedistantstars.Theseexperimentsshowthatincold/low-accelerationenvironmentsMach’sprinciple(inertiahereisduetomassthere)mayhavebeenseen.ThePioneerAnomalyBeforeweleavethetestsofMiHsCthatIhavepublished,andtalkaboutpossiblefutureones,Iwouldliketotalkabouttheanomalythat,forme,startedallthis:thePioneeranomaly.First,Ishall
117June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC105talkaboutthePioneermission,basedoninformationgiveninthecomprehensivesummaryofAndersonetal.(2002).On2ndMarch1972andthe5thApril1973Pioneer10and11werelaunchedfromCapeCanaveral.ThesespacecraftusedgravityassistsatJupiterandSaturntogivethemtheboostinspeedneces-sarytoleavethegravitywelloftheSolarSystem.ThecrucialcharacteristicofthePioneerspacecraftisthattheywerespinstabilised(spinning),ratherlikebullets,andthiswasdoneasitisforbulletssothattheywouldbemorelikelytotravelinastraightline,withouttheneedformessyrocketcoursecorrec-tions.Atlaunchtheyweretravellingat4.28and7.8revolutionsperminuterespectively.ThismeansthatforthePioneerspace-craftveryfeworbitalcorrections,andenginefirings,wereneededandsotheaccelerationsofthecraftcouldbedeterminedtoanaccuracyof10−10m/s2.IncontrasttheVoyagerspacecraftwerenotspinstabilisedandrequiredfrequentrocketpropelledcoursecorrec-tionstheresultsofwhicharedifficulttopredict,sothereisalotofnoiseinthetrajectorydataandaccelerationscannotbedeter-minedsoaccurately.So,toquoteAndersonetal.(2002)directly:“thePioneerspacecraftrepresentanidealsystemtoperformpre-cisioncelestialmechanicsexperiments”.Thiswaspartlydeliberate,sinceitwasfeltthatsmallperturbationsinthespacecrafttrajec-torywouldrevealthepresenceofsmallbodiesintheouterSolarSystem.Thebestexperimentsarethosewhereyoucanreducethenumberofeffectsactingonasystemdowntoone,soyoucanlearneverythingaboutthatoneeffect.ThatiswhythePioneercraftmakesuchabeautifulexperiment:becausetheyareinspacetheeffectsactingonthemareminimised.AwayfromtheSunthereisjustgravity......well,notquite!Oneofthemainproblemsearlyoninthiscelestialmechanicsexperimentwassolarradiationpressure.PhotonsstreamingoutfromtheSuncarrymomentum,andbangintothespacecraftpushingitandthisdidhaveadetectableeffectonthetrajectorywithin20AU,butitgraduallyreducedsothatbythetimethecrafthadpassedtheorbitofSaturnitcouldbeneglected.Then,oddlyenough,Andersonetal.(1998,2002)foundasystematicextraaccelerationpushingthePioneercraftbacktowardstheSunthatwas
118June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05106PhysicsfromtheEdge:ANewCosmologicalModelforInertia(8.09±0.2)×10−10m/s2forPioneer10and(8.56±0.15)×10−10m/s2forPioneer11.TheanalysispublishedbyAndersonetal.(2002)showedthattheanomalycouldnotbeduetosolarradiationpushingonthecraft,thesolarwindpushingit(bothinthewrongdirection),theeffectofthesolarcoronaontheradiowavesusedtocommunicatewiththecraft,electromagneticLorentzforcesactingonachargedspacecraft,thegravityoftheKuiperbelt,thedriftofclocksonboardthespace-craft,ortheeffectoftheantennaeusedbytheDeepSpaceNetwork“drooping”duetogravity(!).Alsoconsideredwastheradiobeamreactionforce.WhenthePioneerspacecraftbeamasignal,aradiowave,attheEarth,thereisasmallrecoilanditwasshownthatthisisalsotoosmallandinthewrongdirectiontoexplaintheanomaly(itwasabout1.1×10−10m/s2).ProbablythemostdebatedexplanationofthePioneeranomalyistheworkleadbyTuryshevetal.(2012)thatthePioneeranomalyisduetoheatradiationfromtheRadio-ThermalGenerators(RTGs)bouncingoffthespacecraftantennaeandpushingitsunwards.TheyandthefolkattheUSPlanetarySocietyhavedoneanexcellentjobraisingfundstorecoverallthePioneerdataanddigitisingitintoamodernformat.Thisworkisessential,sincegoodobservationsarecrucial.Ialsoappreciatealltheworktheyhaveputintoanalysingthisdata,butIhavesomecriticismsoftheirthermalexplanation:(1)Astheysayintheir2011paper(onpage4)thePioneerdataistoonoisytoprovewhetherthereisadecaywithtimeintheanomalyornot,andathermalexplanationcan’tbesupportedwithoutaprovendecaysincetheRTGsoutputdeclinessignifi-cantlywithtime.(2)Thehalf-lifeofthedecaythatbestfitstheirthermalmodelis28.8or36.9yearswhereasthehalf-lifeoftheplutoniumonboardis87years.(3)TheirpredictedanomalyisatitslargestintheinnerSolarSys-temwheretherewasnoPioneeranomaly.Theyhavegotaroundthisbyproposingthattherewasanexactlycancellingpushbecausethesunwardsideofthecraftwaswarmerduetosunlight,butlookingattheirFig.2fromtheirlatterpaperthisdoesnot
119June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC107exactlycanceltheonboardthermaleffect,soIguesstheyhadtoadjustthemomentumfluxfromphotonsclosetotheSunthatwasoriginallyassumedbyAndersonetal.(1998).(4)Andersonetal.(2002)saidthatsincemostoftheheatfromtheRTGswasradiationfromfinswhoseflatsurfaceswerenotpointingattheantenna,onlytheirnarrowedges,only4Wofpower,couldhavehittheantenna,leadingtoamaximumaccel-erationofonly0.55×10−10m/s2.(5)SincethepowerradiationQisproportionaltothefourthpowerofthetemperature,Iwouldliketoseethetemperatureerrorstheyget,sinceanyerrorsfromthissourcewouldbehugelymagnified.(6)Moregenerally,Ialwaysfinditdifficulttoacceptapaperwhenaverycomplexandunrevealedprocess(over3000finiteelementswereusedintheirthermalmodel,notdetailedinthepaper)andwithtwofittingparameters,isusedtogettoapreviouslyknownanswer,andnoexperimentissuggestedthatmightunambigu-ouslytestitagainstrivals.Idislikedarkmatterforsimilarandmoreextremereasons.(7)Thedetailsoftheirmodelhavenotbeenpublishedsotheirmodelisnotfalsifiable.TomakeitfalsifiableIwouldliketoseethempresentasimplifiedorderofmagnitudecalculationsootherscanreproducewhattheyhavedonewiththethermalmodel,onpaper.Havingsaidallthis,Icannotprovethattheyarewrong,asIsaidtheirresearchisunfalsifiablesincetheyhavenotpublishedthedetails.Isimplysuspectthattheyare,sincetheyseemtobework-ingwiththesamekindofattitudethatmostofmoderntheoreticalphysicsis:thatphysicsiscomplete,andsoifthedatadoesn’tagreewiththetheoryyoucanmanipulatethedatabyinventinginvisiblematter,oruseincreasinglytortuousmethodstomakethedatafitthetheory.IamsureTuryshevetal.havetheirheartsintherightplaces,butthisisanattitudethatisendemictoallmoderntheoret-icalphysicsanditneedstobefixed,possiblyatthephysicsdegreelevel,sincepassingexamsencouragesaparrotmentality.Thisisnotasbadinphysicsasinsay,history,wherefactsarelearnedparrotfashion,butwhatislearnedinphysicsisthemodel,andexamstest
120June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05108PhysicsfromtheEdge:ANewCosmologicalModelforInertiawhetherstudentsusethemodelproperly.Whatexamsdonotdoistestthestudentbygivingthemrawdataandaskingthemtoinventanewmodel.Anyway,backtothePioneeranomaly.ItisveryeasytoexplainitwithMiHsC,since,asshownabovetheequationofmotionforMiHsCisasfollows:GM2c2GM−102a=+=+(6.7±0.7)×10m/s(5.33)r2Θr2Theobservedaccelerationswas(8.74±1.33)×10−10m/s2(anaverageforbothofthecraft,fromAndersonetal.,2002)andsoMiHsCagreeswithit.NotethatMiHsChasnoadjustableparameters,isaverysimplemodelandalsomakessensephysically.Toexplainintuitively:whenthePioneercraftwerelaunchedtheyalreadyhadtheanomalousaccelerationspredictedbyMiHsC,sinceitissimplyaddedontotheequationofmotiontogiveanextraaccel-erationforeveryobjectnomatterwhatitsstateofmotionis.Ini-tially,becausealltheotheraccelerationsweresolarge,thisextraaccelerationwashiddeninthenoise,butasthePioneertrajectorymovedoutintotheSolarSystemtheotherforcesactingbecamesmallerandthenoisevanished,sothattheMiHsCaccelerationstoodout.TheideahereisthatMiHsCactstodecreasetheinertialmassofthelowaccelerationspacecraftslightlybelowthegravitationalmass,sothegravitationalforceactingonthecraftfromtheSunisthesame,butsincethecraft’sinertialmassesareslightlyless,theycanrespondslightlymoretotheSun’sgravitybyacceleratingmorethanexpectedtowardstheSun,astheydid.However,iftheinertialmassofeverythingattheedgeoftheSolarSystemislowerthanthegravitationalmasssuchthatthereisalwaysananomalousaccelerationtowardstheSun,thenwhatabouttheplanets?TheyorbitatthesamedistancefromtheSunwherethePioneercraftshowedupananomaly.Theanomalyexistsfurtherin,butisobscuredbyotherthings.Whydon’ttheplanetsshowanomalies?Theanswertothiscouldbethattheslightlylowerinertialmassoftheplanetscanbehiddenbyerrorsinourestimatesoftheirgravitationalmasses.
121June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch05EvidenceforMiHsC109Asthischapterhasshown,MiHsChasbeendevelopedbylookingatcontroversialandmessyanomaliesfromtherealworld.Despitethis,orratherbecauseofit,MiHsCdoessuccessfullypredicttheverywellobservedanomaliesofcosmicaccelerationandgalaxyrotation.Theremainingproblemisthatother,lesselegant,theoriescanalsoexplainthese,soamoreunambiguoustestisneeded.Inthenextchaptersomewillbeproposed.
122May2,201314:6BC:8831-ProbabilityandStatisticalTheoryPST˙wsThispageintentionallyleftblank
123June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06Chapter6FutureExperimentalTestsofMiHsCOneshouldalwaysbeabletotesttheories.Iwouldgofurtherandsaythatifyouinventatheorythatcannotbetested,youshouldforgetthetheory.ThiswasthebasicprincipleofscienceoutlinedbyFrancisBaconinthe17thcentury.ItinspiredtheRoyalSociety,whichhadasitsmotto“Nulliusinverba”whichmeans“Wordsmeannothing”,or“Takeno-one’swordforit”andNewtonwhowasamemberoftheRoyalSocietywasverygoodattestinghistheories.ThisattitudeledtothreecenturiesofquickscientificadvancesfromNewtontothe1970s,butunfortunately,inrecenttimes,per-hapsbecauseofthestellarinfluenceofthelaterEinstein,whoputhisfaithmoreandmoreintomathematicsdespitealackofsuccess,theBaconianattitudehasdiedoutinmainstream‘theoretical’physics.(ItisstillaliveatplaceslikeCERN).TheoristsseemtobecopyingthefailedlaterEinstein,nottheearliersuccessfulone.Thishasgivenrisetoallsortsofuntestabletheoriessuchasstringtheory,super-symmetry,loopquantumgravity,darkmatterandmillionsofpoundsoffundinghasbeenpumpedintotheblackholeoftheseuntestabletheories.Idonotwishtomakethesamemistake,sothissectionisdevotedtodiscussingexperimentsthatmightbeabletotestMiHsC.OverthepastfewyearsIhavethoughtofmanydifferentexperiments,butsinceIamnotanexperimentalistmyselfandhaveverylittleexperienceofit,Ihavebeenunabletoproperlyjudgethem,anditmaybethatyouareinabetterpositionto.ASpinningDiscwithaWeightAboveInscienceaselsewhere,tomakeprogressitisimportanttoallowyourpassion,orcuriosity,todirectyou,andtorelyonyourown111
124June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06112PhysicsfromtheEdge:ANewCosmologicalModelforInertiathinking,andnotbeputoffbythegroupfeelingofthecrowd,sincethecollectiveintelligenceofcrowdsisusuallyclosetozero.Havingsaidthis,thereisadangerinfreethinkinglikethis,sincehumanscanmakeerrors,andalsoifyoudofreethinkthecrowddoesn’tlikeyouverymuch,butIthinkitisworthtakingarisknowandthen,sincenewdiscoveriesarenevermadebycommit-tees.Nowherewerethedangersoffreethinkingintoday’sclimatemoreexposedthaninthecaseofEugenePodkletnov’s“antigravity”experiment.In1991EugenePodkletnovandhisteam,wereworkingatTampereUniversityofTechnologyinFinland.Theywerelevitat-ingsupercooledsuperconductingringsabout12inchesindiameterusingstrongACmagneticfieldsandspinningthemat5000rpm.OneofPodkletnov’sco-workershappenedtobesmoking,whenheshouldn’thavebeen—somuchfortherules—,andtheynoticedthatthesmokeroseupinacolumnabovethespinningdiscs(Cook,2001).Whentheyinvestigatedthisbysuspendingmagneticandnon-magneticmassesabovethediscstheynoticedthattheylostafractionofapercentoftheirweight,andifthespinofthediscsincreasedsodidtheeffect,uptoamaximumweightlossof2%.Theeffectwasnotduetoamomentumtransferfrommovingairsincethesechangespersistedwhenthemasseswereencasedinglass,andthephenomenonwasindependentofthemasses’composition.Itwasnotamagneticeffectsinceitpersistedwhenametalscreenwasplacedbetweenthemassandthedisc.Aschematicisshownoverleaf.AfterPodkletnovpublishedtheseresultsinareputablejournal,PhysicaC,andafterhehadhadanotherpaperonitacceptedbytheJournalofPhysicsD:AppliedPhysics,anewspaperreportwrittenbyRobertMatthewsandIanSampleappearedintheGuardiansayingthatPodkletnovhaddiscoveredakindof“antigravity”.Forsomereasontherewasahugefussaboutthisword,andsuddenlyPodklet-novfoundhimselfsackedbyhisuniversity.Itseemsthatfindinganeffectthatcannotbeexplainedbymainstreamphysicsisacapitalcrimeinphysicscircles.Itshouldnotbe.Itshouldbeappreciatedthattheexperimentmightbeanerror,butalsoitmightbeagen-uineanomaly:awaytolearnmorethanweknownowabouttheworld.Peopleshouldbeencouragedtotakeexperimentalriskslike
125June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC113thisratherthanbeingbeatenaroundtheheadiftheysomuchaschal-lengeacceptedwisdom.Withouttheserisks,progressisimpossible.IhavebeenfascinatedbythePodkletnovresultseversinceabout2002,whenafriendsentmeanonlinearticleaboutit,andIfinallypublishedapaperin2011,testingMiHsConthePodkletnovresults.Ihadbeenwarnedbysomephysicistfriendsnottodothis,sincetheysaidIwouldbelabelledasacrank,butIpublisheditanyway,sinceIamnotpaidbyanyphysicsdepartmentsoIcandowhatIwant.ThishasthedisadvantagethatIamonthebreadline,butithastheadvantagethatIdon’thavetolistentothecrowd.TherehavebeensomeconsequencesofmepublishingsomethingmentioningPodkletnovinit.Forexample,myuniversityreceivedoneletterofcomplaintfromthepublicsayingthatIshouldnotbeallowedtoresearchexperimentslikethis,andmydepartmenthead,ProfessorNeilJames,aftercheckingmeout,wroteabrilliantletterbacktothischapsayingthatauniversityresearchershouldhavethefreedomtoinvestigatewhatevertheywant.Icouldhavekissedhim,butdidn’t!Italsohadtheconsequencethatmypapersarenowputinto“generalphysics”onthearXiv,asortofnaughtyspotforerrantresearchers,butitdoesn’treallybotherme(althoughatthemomentmylatestpaperhasbeenheldbythearXivforseveralmonthssinceFigure1.SchematicshowingthesetupofthePodkletnovexperiment.Asuperconductingdiscwasplacedinacryostat,cooledto77K,levitatedandspun.Atestmasssuspendedabovethecryostatlost0.06–2%ofitsweight.
126June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06114PhysicsfromtheEdge:ANewCosmologicalModelforInertiatheycan’tclassifyit,whereasitwaspublishedinthejournalwithinthreeweeks).Luckily,journalreviewersarestillacceptingmypapers.Myleapoffthedeependalsohadamoreimportant,tome,positiveeffectthatforexample,theLifeboatFoundationinvitedmetojointhem,andIdid.Theyarededicatedtosavingmankindbyhelpingus(amongotherthings)coloniseotherworlds,andthislatterisacauseIamproudtohelpthemwith.Thereisalessonhere:takerisksandyoumayembarrassyourselfintheeyesoftheignorant,makemistakesandprobablylosemoneytoo,butontheupside,atleastyouwillbenoticedbyinterestingpeople.Anyway,theresultsIgotfromcomparingPodkletnov’sresultswithMiHsCweremixed.Intuitively,andthereasonIlookedatit,wasthatIcouldseehowMiHsCmightapplytothisexperiment.Thereisamasssuspendednearacoldenvironment,sotheaccelerationsitseesaredueonlytoitsmotionwithrespecttothefixedstars,andthisissmall,soinitiallyUnruhwavesaredisallowedandtheinertialmassreducesbelowthegravitationalmass.ThensuddenlyyouswitchonanACmagneticfieldwhichhastheeffectofvibratingthesuperconductor.Thesuspendedmassnowseeslargermutualaccelerationsandgainsinertialmass.CalculatingthisaccelerationinthesamewayaswasdonefortheTajmareffect(Eq.(5.30)),showsthatthevibrationofthediscshould,viaMiHsC,causeanincreaseintheinertialmassofthetestmassandthereforeitshouldbecomelesssensitivetotheaccelerationofgravity.Specifically,giventheaccelerationofthediscbytheACmagneticfield,MiHsCpredictsthetestmassshouldaccelerateupwards(againstgravity)by0.03%ofg.TheweightlossobservedbyPodkletnovwas0.06%ofg.However,whenPodkletnovspunhisdiscstheeffectincreaseduptobetween0.3%and0.6%oftheweight,andIhavenotbeenabletoexplainthisusingMiHsC(itmaybepossible,butIdon’tknowenoughabouttheaccelerationscausedbyasuperconductingdiscthatisbothvibratingandspinning).TheTajmarexperiments,doneovertenyearslater,weresimilartothePodkletnovexperiment,butusedspinningnormalmetalringsratherthanlevitatedsuperconductingdiscs.TherotatingringofTaj-marispreferableinsomewayssinceasimplerotationinsteadofacom-plexvibrationprovidesa‘known’accelerationthatcanbepluggedinto
127June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC115MiHsCandthepredictedmasslosscanbetested.ThedisadvantageoftheTajmarexperimentwasthattheaccelerationswereverysmallsotheeffectneededasensitivelasergyroscopetodetectit.TheexperimentIwouldliketosuggestcombinesthebestofbothexperiments.ATajmar-typerotatableringisplacedinacryostat,andaPodkletnov-typemassissuspendedaboveandoutsidethecryo-stat,isolatedfromanyfrictionalconnectiontothering.Whenthecryostatiscooled,thechangeintheweightofthemassduetothecoolingisnoted.ThismaynotbeduetoMiHsC.Mostofthischangewillbeduetochangesinbuoyancy,airflowduetothehugechangeintemperature,butoncethisisdone,anysubsequenteffectsshouldbeduetothering’sspin.ToseetheeffectofMiHsC,weneedthentospintheringatafewthousandrpm.Tocalculatethechangeinacceleration(da)thiswillcauseforthesuspendedmasswecanusethesameformulathatweusedfortheTajmarexperiment:2c2amrda=(6.1)Θamswhereamristheaccelerationofthesuspendedmasswithrespecttotheringafteritspins(thespin-accelerationofthering)andamsistheoriginalaccelerationofthemasswithrespecttothefixedstarsbecausetheEarthisspinning(thesizeofthiswilldependonthelatitudetheexperimentwasdoneat).Wecanworkouttheaccel-erationoftheringasa=v2/r,wherevistherotationspeedoftheedgeoftheringandrisitsradius,ora=4π2R2r/3600,whereRistherotationrateinrpm.TheresultsofapplyingMiHsCtoaringofradius5cm(Eq.(6.1))areshowninTable1.(Theformulaisda=7.24×10−12×R2r/ams.)Sincethisaccelerationisanattemptbythering+masssystemtoconservemomentum,thechangeduetoMiHsCwillbemeasuredasanextraaccelerationupwards,andsoalossofweight.Thisanoma-louseffectshouldincreaseastheaccelerationoftheringincreases,andtowardsthepoleastheaccelerationrelativetothefixedstars(ams)decreases.TheexperimentshouldalsobeconclusivesinceifMiHsCpredictscorrectlythechangeinweightduetodifferentaccelerationsofthe
128June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06116PhysicsfromtheEdge:ANewCosmologicalModelforInertiaTable1.ThelossofweightpredictedbyMiHsCforatestmasssuspendedoverasupercooledringofradius5cmwithaspinrateasshownincolumn1,atthelatitudeofPlymouth,UK.SpinrateAccelerationofringAccelerationofbobAccelerationofbob22Rpmm/sm/s%ofg300049350.000150.001610,00054,8000.00150.018100,000547,5560.1761.76753,994311,541,8089.8100ring,differentdirectionsofspin,differentweightsofthering,itwillbehardtoarguewiththeconclusions.MiHsCpredictsthattheeffectseenintheexperimentshouldbemaximisedwithawiderring,afasterspinandahigherlatitude(ordoingtheexperimentinaframethatfollowstheEarth’srotation).Anotherpossibleeffectinthisexperimentwillbethechangeinthecentrifugalforceonthesuspendedmass.Astheinertialmassincreasesthisshouldincreasebyafactorof10−8,butthegravitationalforcedownwardsshouldnotincrease.Thereforethemassshouldtendtomoveupwards.Sincethecentrifugalforceupwardsattheequatorisabout1/365thofthegravitationalforce,thenetchangeintheweightofthesuspendedmassshouldbe5.3×10−8×1/365=1.4×10−10kg.Thiseffectissmaller,butitwouldbeinterestingtoconsiderwhatmighthappeniftheinertialmasscouldbeboostedinthiswayby365times—anobjectattheequatorwouldliftoff!OneodditynoticedbyPodkletnovandhisteamwasthattheanomalytheysawextendedupwardsapparentlyinacolumn,sincetheyalsomeasuredweightlossesforsuspendedmassesinthefloorabovetheircryostat.Thereisawaytoexplainthisusingtheexten-sionofMiHsCusinglocalhorizonsthatIpublishedin2013(McCul-loch,2013).Sincetheaccelerationofanobjectisrelatedtothedis-tancetoitsRindlerhorizona=c2/dRonecanwriteEq.(6.1)intermsofRindlerhorizons,likethis2c2doldda=(6.2)Θdnew
129June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC117Insteadoflookingatchangesinlocalaccelerations,thisequationshowsthattheanomalousaccelerationscanbethoughtofasbeingduetochangesintheclosenessofRindlerhorizons.IfthedistancetotheRindlerhorizonhalves,thenananomalousaccelerationappearsthatistwiceaslargeastheminimumaccelerationofMiHsC.InthecaseofthePodkletnovexperiment,thesuddenaccelerationofthediscpullstheRindlerhorizoninfrominitially3×1018mto9×1011mandthecloserhorizonwouldexertagreaterpullupwardsonanyobjectsclosetothedisc.ThisemphasisonRindlerhorizonsalsopointsoutanalternativemethodofeffectingdynamics.Rindlerhorizonsareboundariesofknowledge,soinformationfrombehindthemshouldnotbeavail-abletotheacceleratedobjects.Howaboutgravitationalinforma-tionthen?CouldyouhideagravitationalsourcebehindaRindlerhorizon?Forexample,Eq.(6.2)canberearrangedtopredicttherotationrate(R,inrpm)ofaring(radius0.1m)thatisrequiredtobringtheRindlerhorizonseenbymaterialinitcloserthantheSun.UsingknownvaluesforthespeedoflightandthedistancetotheSun(1.5×1011m)thisrotationrateis900c2R==23,422rpm(6.3)π2rdsThegravitationalattractionoftheSunis0.006m/s2or0.06%ofg.SincetheSunisabouthalfadegreewideintheskyonlyanareaof1/720thoftheringwouldfeelthedisappearanceoftheSunsotheactualaccelerationwouldbelowerthanthis.HowabouttheEarth’sgravity?Couldweblockthisout?ThedistancetothecentreoftheEarthis6300kmsotheaccelerationneededwouldbec2a==1.43×1010m/s2(6.4)6300000Thisisahugeaccelerationandprobablynotachievablebyspin-ningaring.However,Nano-ElectromechanicalSystems(NEMS)havenanopendulumsthatdoproduceaccelerationsofthissize.Ifoneweretoweighonesuchpendulumwhileitwasstillandthenagainwhen
130June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06118PhysicsfromtheEdge:ANewCosmologicalModelforInertiathependulumwasacceleratingthenthependulumshouldnotbeabletoseetheEarth’sgravity,foratleastpartofitsswing,andthiscouldbedetectable.Hugeaccelerationscouldalsobemadebyacceleratingelec-tronsinsuperconductorsandthendeceleratingtheminthevertical(downwards)directionusingordinaryconductorsphysicallyattachedbelow.ThisisanotherwayofthinkingaboutthePodkletnovexper-imentwhichhadjustthissetupandmayexplainthe‘gravityshield-ing’thattheysaw.Athirdwaytogetthesehugeaccelerationswouldbetouseelec-tronsacceleratingaroundgoldnanotips,asdonebyBeversluisetal.(2003)andcommentedonbySmolyaninov(2008).Theaccelerationoftheelectronsoverthecurvedsurfaceofthenanotipproducesanaccelerationofc2/r=9×1022m/s2.IfawholearrayofthesenanotipswassetupthenmaybetheywouldgenerateaRindlerhorizonthatwouldaffectnearbytestmasses.AfinalthoughtbeforeweleavethePodkletnoveffect.Whenthediscisspuntheinertialmassofthetestmassincreasesbyonlyaverytinyamount:about10−8kgperkg.Howthendoesthisproduceanaccelerationupwardsthatis0.06%ofg?WecanseethisusingathoughtexperimentshowninFigure2.WehaveadiscinfreefallFigure2.Thoughtexperiment.Adiscisinfreefall(left).Nowsliceithorizon-tallyandspinthetwohalvesrelativetoeachotherpreservingtheoverallangularmomentum.Theinertialmassofthetwohalvesincreasesandtodothisyouneedtoovercomeaforce.Forcemustbeconserved,andaforceappearsexternaltothesystemofjustthesamesize.
131June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC119ontheleft.Nowsliceithorizontallywithalaserintotwopartsandspintheminoppositedirectionssotheyaremutuallyaccelerating.MiHsCsaysthattheinertialmassgoesupbyafractiondm:dm2c2=(6.5)maΘwhereaisthespinacceleration.Tokeepthismutualaccelerationgoingdespitetheextrainertialmass,aforcehastobeaddedwhichisequaltoF=dm×a.Nowthisforce(F)cannotbeproducedoutofthinair.Forceshouldbeconserved,soaforcemustappearexternaltothesystemtobalancethisinternalforceF=F=dm×a=ma(6.6)extintwhereaistheanomalousexternalaccelerationproducedbytheexternalforce.Thereforedma=a(6.7)mInthePodkletnovexperimentaisthemutualaccelerationofthediscandtestmasswhich(duetotheACmagneticfieldandthespin)iscloseto105m/s2.Sotheanomalousaccelerationfora1kgmassis10−8a∼×105=10−3m/s2(6.8)1Thisisanattemptatadeeperunderstandingofwhatisgoingoninthiscase.TheCERNAcceleratorandRadioWavesOneoftheuniquecharacteristicsofwaves,andUnruhwavestoo,isthattheycaneitherconstructivelyordestructivelyinterferewitheachother.Figure3illustratesthis.Iftherearetwowavesofequalamplitudeorheight,onewiththecharacteristicsofwaveAandonelikewaveB,andtheymeetup,thenwherethereisapeakinwaveAthereisatroughinwaveB,theyaresaidtobeoutofphase,andtheycancelout.Thefinalresultisnowaveatall.Ontheotherhand,ifwaveAmeetswaveBsuchthatwherethereisapeakinwaveAthereisalsoapeakinwaveB,thentheywillconstructivelyinterferewith
132June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06120PhysicsfromtheEdge:ANewCosmologicalModelforInertiaFigure3.SchematicshowingthedestructiveinterferenceoftwowavesoutofphaseAandB(above),andtheconstructiveinterferenceoftwowavesinphase(below).eachotherandtheresultingwavewillbetwiceasbig.IsitpossibletointerferewithUnruhwaves?Itisdifficultiftheyareasbigasthegalaxysincewedonotknowyethowtocreatewavesthatlong,electromagneticorotherwise.ThewavelengthoftheUnruhradiationdependsonthespeedoflight(c)andacceleration(a)as4βπ2c27.11×1017λ==(6.9)aaSoanobjectacceleratedat0.0311m/s2wouldexperienceUnruhwavesofwavelength2.3×1019m.Anobjectacceleratedat9.8m/s2wouldseewavelengthsofabout7.3×1016m.ThetrickistoproduceanaccelerationsolargethattheUnruhwavelengthsseenareshortenoughtobemanipulatedbypresenttechnology.Therearesomeveryhighlyacceleratedsystemsthatmightfitthebill.OnethatIconsideredasanidealandcheapexperimentforawhilewassonoluminescence(describedaccessiblyinCrum,1994)whereinacollapsingbubblewithinafluidundergoeshugeaccel-erations,andoddlyemitslightbyamechanismsthatisnotwellunderstood.ItmaybethatthislightisUnruhradiationproducedbythehighacceleration,atwavelengthsshortenoughforustosee
133June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC121(Schwinger,1992),butthissystemdoesnotmakeaverygoodexper-imentbecausewaterabsorbsmostoftheradiationemittedmakingitdifficulttodrawfirmconclusionsandtheactualmaximumaccel-erationisnotwellknown.AsdiscussedintheprevioussectionNEMSandnanotipsalsocanproducehugeaccelerations.AnothermanmadesystemthatproduceshugeaccelerationsistheparticleacceleratoratCERN(EuropeanOrganizationforNuclearResearch)nearGenevaontheFrench–Swissborder.Hereparticlesareroutinelyacceleratedaroundthe1kmdiameterringatspeedsclosetothespeedoflight.Thismeansthattheaccelerationsproducedarev2(3×108)2142a===1.8×10m/s(6.10)r500PuttingthisaccelerationintoEq.(6.9),theUnruhwavelengthsseenbytheseparticlesshouldbe7.11×1017λ==3950m=3.95km(6.11)1.8×1014Electromagneticwavesofthiswavelengthcanbeproducedbyourpresenttechnology.Thesearelongradiowaves.Soapossibleexperi-mentwouldbetofireradiowavesofthiswavelengthattheparticlesbeingacceleratedatCERNtoseeifwecaninterfereormodifytheirinertialmass.Alsosincetheparticlesaretravellingclosetothespeedoflight,specialrelativitywouldhavetobetakenintoaccount.Forexample,iftheparticlewastravellingat0.9c(0.9timesthespeedoflight)thentheUnruhwavesseenbyitwouldbelengthenedinthedirectionoftravelfrom3.95kmtoabout8km,soradiowavesofthiswavelengthwouldinsteadhavetobeapplied.IfMiHsCisright,thentheresultshouldbeavariationintheinertialmassoftheparticle,whichmaybedetectablefromchangesinitstrajectoryorspeed.ThesechangeswouldberathersensitivetothephasedifferencebetweentheUnruhwavesandtheappliedelectromagneticones.
134June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06122PhysicsfromtheEdge:ANewCosmologicalModelforInertiaNEMSAccelerationsandRadioWavesAcoupleofyearsagoaDr.DavideIannuzzi,fromHolland,readmypaperproposingtheCERNexperimentthatIhavejustdiscussed,andwrotetomesuggestinganexperimentusingtheNano-ElectricalMechanicalSystemsthathewasusing.Helaterdecidednottopur-suethisresearchbecauseMiHsC’sdisagreementwiththeequivalenceprinciplewas‘alittlescary’buthissuggestionwasagoodone.TheseNEMSaretinynano-scalependulums,andtheyapparentlyexperi-enceaccelerationsashighas4×1011m/s2.SotheUnruhradiationtheyseeshouldhavearelativelyshortwavelengthasfollows:4βπ2c22λ==1.78×106m/s(6.12)aRadiowavesofthislength,ExtremelyLowFrequency(ELF),canbegenerated,andaregeneratedroutinelybyPIGs(PipelineInspectionGauges).Soitiscertainlypossiblethatradiowavesofthiswave-lengthcouldbefiredattheNEMS.GiventheirwavenaturetheappliedwavesmayenhanceorcanceltheexistingUnruhwavesseenbythependulum(constructiveordestructiveinterference)thereforeboostingorreducingitsinertia.Therefore,themotionofthepen-dulummaybeaffectedinanirregularmanner.Withmorecareitshouldbepossibletotunetheappliedradiationtoproducemoreregularresults.Also,theeffectwillbeasymmetricsincetheappliedradiowaveswillhaveadifferentapparentwavelengthdependingontherelativedirectionofthependulum’sswing.Toestimatethestrengthofradiowavesweneedtoapply,weneedtostartfromthemassofatypicalgardenvarietyNEMS,whichis2×10−16kg(Meyeretal.,2005).SinceE=mc2thisinertialmassisequivalentto18joulesofenergy.Thisenergyappearsinthepen-dulum’sacceleratedframeofreference,asUnruhwaves,butcannotbeseenbyanoutsideobserver.Soifwecandirect18joulesofradiowaveenergyatthependulumwecanpotentiallydampenorenhanceitsinertialmass.Thefocusingofradiowavescouldbeachievedusingaparabolicdish,orperhapsametamaterial.OnesuchmetamaterialisdiscussedinEhrenbergetal.(2012).Thisoneisabletofocusmicrowaveradiowaves,sowouldbenouseforwaveskilometerslong,
135June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC123butitisabletofocusthewavesontoasmallvolumethesizeofamoleculeandthisfieldisrapidlydeveloping.DeviationsfromtheStefan–BoltzmannLawPartofMiHsCistheassumptionthatUnruhradiationofverylowtemperatureisweakerthanexpectedbecauseagreaterproportionofthelongerUnruhwavesaredisallowedbytheHubble-scaleCasimireffect.Itislogicaltoextendthistothetemperatureofanyobject,sotheradiationemittedbyaverycoldobjectshouldbeslightlylessthanexpectedfromtheStefan–Boltzmannlaw.Thecoldesttemperatureachievedsofaris100pK(picokelvin)attheHelsinkiUniversityofTechnology(Knuutila,2000).UsingWien’slaw,anobjectthiscoldwouldhaveapeakradiatingwavelengthofλm=3×107m,sobyanalogytoEq.(5.2),theenergyoftheblackbodyradiationspectrum(E)wouldbemodifiedtoE,asfollowsλm−20E=E1−=E(1−2.7×10)J(6.13)4ΘIdon’tknowifvariationsinradiatedenergyassmallasthiscanbedetected.DiscretisedVariationsinInertialMassOneoftheassumptionsofMiHsCistheHubble-scaleCasimireffect,andIhavemodelledthisrathersimplybyassumingthatthepropor-tionofallowedwavelengthsdecreaseslinearlyasthepeakwavelengthoftheUnruhspectrumincreasestowardstheHubblescale.Thisiscorrect,onaverage,butifyoulookcloser,itistoocrudeamodel.Forcertainaccelerations,thewavesatthepeakoftheUnruhradiationspectrum(themostcommonwaves)willfitexactlywithintheHubblescale,sotheinertialmasswillbegreateratthisacceleration.Atotheraccelerations,thewavesatthepeak,willnotfitexactlywithintheHubblescaleandtheinertialmasswillbelowerforthisacceleration.Thisimpliesthat,forexample,asyouconsidergreaterandgreaterradiiinagalaxyyouwillpassthroughsomeradiiwithstarsattheresonantaccelerationsandthesewillhavegreateriner-tialmass,sowillbelessattractedbythegalacticcentreandwill
136June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06124PhysicsfromtheEdge:ANewCosmologicalModelforInertiamoveoutwards,andsomestarsinbetweenwillhavelesserinertialmasses,sotheywillbemoregreatlyattractedbythegalacticcentreandwillmovein.Theconsequencesofthisarenoteasytopredictwithoutrunningafullgalaxymodel(somethingIwouldliketodo),butonepossibilityisthatthestarsingalaxiesshouldshowsometendencytocongregateintoconcentriccircles.Thistendencywillbemostapparentattheedgesofgalaxieswherethenumberofallowedwavelengthsisfewer,sothedifferencebetweenaresonantcase,andanon-resonantcasewillbegreater.DropTowerExperimentAttheUniversityofBremeninGermanytheyhaveadroptowerthathasafallheightof110metres.Whenyoudropanobjectdownit,thenaccordingtoMiHsCitshouldfallwithanaccelerationofGM2c2a=+(6.14)r2ΘThisimpliesthatwhenyoudropsomethingitshouldarriveatthebottomalittleearlierthanexpectedifMiHsCisright.Thediffer-enceindistancetravelledinequaltimeiss=1at2,whichforan2extraaccelerationof6.7×10−10m/s−2andafalltimeof4.74sis7.5nanometres.Ifyoudroppedtwoobjectsdownthetower,asinGalileo’sfamousexperiment,thenMiHsCpredictsthattheyshouldbothfallatthesameratesinceterm2ontherighthandsideofEq.(6.14)isindependentofmass,butbothshouldfallslightlyfasterthanexpected.GPSSatelliteTrajectoriesGPS(GlobalPositioningSystem)satellitesareidealtestobjectsfornewdynamicslikeMiHsCsincetheyorbitat20,000kmwheretheatmosphereisthinsotheirtrajectoriesarelessaffectedbyfrictionwhichisdifficulttomodelandsofundamentalanomaliesareeasiertosee.InMiHsCtheinertialmassofabodyisaffectedbythemutualaccelerationbetweenitandnearbymatter.ThismeansthatasGPSsatellitestravelnearertothepole,andtheEarth’sspinaxis,themutualaccelerationsbetweenthemandthemassoftheEarthare
137June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06FutureExperimentalTestsofMiHsC125lowersotheirinertiashoulddecreaseandtheyshouldspeedupbyafewmm/s(asimilarspeed-uphasalreadybeenseeninflybyspace-craft,asdiscussedinChapter5).Also,infreefall,MiHsCreducestheinertiaofobjectsveryslightly,sothesatellitesshouldfallintheirorbitslightlymorethanexpected(givenairdrag)givinganextraEarthwardaccelerationof6.7×10−10m/s2.ThisimpliesthattheGPSsatelliteshoulddropby1metreforevery12hoursoftheirorbit.Ananomalyofjustthissize(10×10−10m/s2)hasrecentlybeenseenbyZiebartetal.(2007)butitisdifficulttoprovethatitisnotamundane(frictional)effect.SummaryofExperimentsSuggestedTable2liststhevariouspossibleexperimentsthatIhavesuggestedtotestMiHsCwithanattempttoquantifywithnumbers(1,2,and3)Table2.Thistableisalistofthevariouspossibleexperimentsthathavebeensuggestedandanattemptedquantitativemeasureofhowconclusive(column2),howpracticallyeasy(column3)andhowcheap(column4)itwouldbeforeachofthetests.Column5showstheoverallscoreforalloftheproposedtestsandshowsthatthediscwithaweightaboveitwouldbemostconclusive,easiestandcheapest,asfarasIcantell.ExperimentConclusiveEasyCheapTotalProblemDiscwithweight3339Frictionalabovecouplingeffects?GPSsatelliteTest2338MightbeduetoothereffectsDropTowerTest3227MightnotbedetectableRedotheTajmar2226HardtoduplicateexperimentexactlyRedothePodkletnov2226HardtoduplicateexperimentexactlyStefan–Boltzmann,2226Needtemperatureslow-Tof200pK!Galacticresonant2226MightbeduetoringsothereffectsNEMS&radiowaves1225EMeffectstoconsiderCERN&radiowaves2114Expensive,EMeffects
138June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch06126PhysicsfromtheEdge:ANewCosmologicalModelforInertiahowconclusive,easyandcheaptheywouldbe.Forexample,theexperimentwiththerotatingdiscandtheweightaboveitwouldberelativelyconclusive,andgetsamarkof3,becausetheexperimentcanbebroughttoanequilibriumandthenthedisccanbespunatdifferentratesandindifferentdirectionsprovidinganicesetofinde-pendentdatathatcanbecomparedwiththepredictionsofMiHsC.Ontheotherhand,theexperimentwithCERNparticles,thediscandNEMSwithappliedradiowaveswouldnotbesoconclusivesinceallsortsofelectromagneticeffectswouldhavetobecalculatedtoo.Youmayverywellhaveabetterexperimentalsuggestion,andIwouldencourageyoutotryit.Pleaseletmeknowhowitgoes.
139June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch07Chapter7MiHsCandFasterThanLightTravelTheconservationofmomentumcombinedwithspecialrelativitysug-geststhatanobject’sinertialmassdependsonitsvelocityvasm0mi=(7.1)v21−c2wherem0istherestmass.Byplayingaroundwiththisequationyoucanshowthatwhenthevelocityofanobjectissmallcomparedtothespeedoflight(vc)theinertialmassisclosetotherestmassm0,butwhenthespeedvapproachesthespeedoflight(c)theinertialmassmiapproachesinfinityandthereforefurtherincreasesofspeedareimpossible.Thiseffecthasbeenverifiedinparticleacceleratorsandtherearealsologicalreasonsforit(therearerestrictionsfromcausalitytoo).ThislimitstherangeofinterstellartravelwithinthelifetimeofthoseleftbehindonEarth.Assuggestedinthisbook,MiHsCalsoaffectsinertialmass.Com-biningspecialrelativity(Eq.(7.1))andMiHsC(Eq.(4.7))producesthisexpressionfortheinertialmassmi2c2(1−)m=maΘ(7.2)i0v21(1−2)2cwherem0istherestmass.UsingNewton’ssecondlawgivestheaccelerationv21F(1−2)2a==Fc(7.3)m2c2im0(1−aΘ)Multiplyingbothsidesbythedenominatorandrearranginggives1Fv222c2a=1−+(7.4)m0c2Θ127
140June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch07128PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThefirsttermontherighthandsideistheusualonefromspecialrelativitythatstatesthatifv=c,thennomatterhowlargeaforce(F)isappliedtoanobjectitwillnotaccelerate.ThenewresultfromMiHsCisthesecondterm,whichstatesthatevenwhenv=ctherewillalwaysbeaminimumaccelerationof:2c2/Θ∼6.7×10−10m/s2eveniftheforceapplied(F)iszero.Toexplainintuitively:asaspacecraft’sspeedapproachesc,specialrelativitypredictsthatitsinertiaincreasesandtheaccelerationfallstowardszero,butMiHsCsaysthatinertiadependsontheexistenceofUnruhwavesandatlowaccelerationsthesebecometoolongtofitwithintheHubblediam-eter,sowhereasspecialrelativitypredictsthattheinertiaincreasestoinfinityatspeedc,MiHsCpredictsthatasthishappens,Unruh-inertiadissipates.Theresultisthataresidualaccelerationremains(Eq.(7.4),term2).ThispredictionissupportedindirectlybytheobservationsofPerlmutteretal.(1999)andReissetal.(1998)whoobservedthissamevalueofaccelerationforthedistantstarswhicharetravel-lingawayatspeedsrelativetousofclosetoc.Thisaccelerationhasbeenattributedtoarbitrary‘darkenergy’butitispredictedbyMiHsC.Thissuggestsaparadox:astarneartheobservableuniverse’sedgeismovingatspeedc,amomentlater,becauseofcosmicacceler-ation,itismovingfasterthanc.Thisiscontrarytospecialrelativityalone,butnotwhenMiHsCisalsoconsidered(Eq.(7.4)).TheminimumaccelerationpredictedbyMiHsCistiny:fromrest,itwouldtake14billionyearstoreachthespeedoflight(Itisinter-estingthattheempiricalaccelerationcoefficientneededbyMoND(Milgrom,1983)isofasimilarsizetothatpredictedbyMiHsC,andasMilgromnoted:wouldproducethespeedoflightintheageoftheuniverse,13.7billionyears).Equation7.4impliesthatthewaytoboostthisforce-independentMiHsCaccelerationandachievefasterthanlight(FTL)travelinashortertimeisnottoincreaseFinthefirsttermwhichwillrunupagainsttheconstraintsofspecialrelativity,butinsteadreduceΘ(theHubblescale)inthesecondterm.
141June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch07MiHsCandFasterThanLightTravel129FTLTest1:Man-MadeEventHorizonsOfcourse,shrinkingtheobservableuniverseislikelytoremainimpos-sibleforsometime,butitmaybepossibletocreateasmaller‘effec-tive’Hubblescale.IsuggestedinthepapersMcCulloch(2008a,2013)thatonewaytoachievethiswouldbetousethemetamaterialsrecentlydevisedbyPendryetal.(2006)andLeonhardt(2006).Theyhavedemonstratedthatelectromagneticradiation(whichformspartoftheUnruhradiation)canbebentaroundanobject,whichmustbesmallerthanthewavelength,usingametamaterial(aspeciallydesignedmetalstructure),makingthatobjectinvisibletoanobserveratthatwavelength(cloaking).ByinsteadbendingUnruhradiation(orjusttheEM-componentofUnruhradiation)backtowardsaspacecraftitmaybepossibletocreateaneventhorizonsimilartotheoneassumedbyMiHsCtoexistattheedgeoftheobservableuniverse,thesizeofwhichistheΘinEq.(7.4).ThevalueofΘisusually2.7×1026m,butforaspacecraftsurroundedbyacarefullyarrangedmetamaterialshell,Θcouldbethediameteroftheshell,makingtheMiHsC-accelerationpredictedbyEq.(7.4)verymuchlarger.Thespeedoflightcouldthenbeachievedandpassedatamuchgreateracceleration.Forexample,ifacraftisacceleratedat9.8m/s2thewavelengthofUnruhradiationwouldbe4π2βc2λ=∼7.3×1016m(7.5)aIfthespacecraftwassurroundedbyametamaterialthatbentthiswavelengtharoundit,thenaccordingtoMiHsC,theobject’sinertialmasswouldreduce.PendryandWood(2007)havedevisedmetama-terialsthatareabletocontrolverylowelectromagneticfrequenciesincludingwavelengthsaslargeasthatabove(bendingotherUnruhfieldswouldbeharder).FTLTest2:ParticleAcceleratorsTheeffectsofMiHsChavenotbeenobservedinparticleaccel-eratorswhichaccelerateparticlestoclosetothespeedoflight.Thisisbecausetheseparticlestravelalongcirculartrajectories
142June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch07130PhysicsfromtheEdge:ANewCosmologicalModelforInertiaandarethereforehighlyaccelerated,makingMiHsC’seffectslessapparent.ThiscanbeshownquantitativelyusingthecombinedMiHsC+relativitymodelforinertia(Eq.(7.2)).SubstitutingtheaccelerationofaparticlearoundtheCERNparticleaccelerator:a=v2/r,whereristheradiusoftheacceleratorandvisthevelocity,intoEq.(7.2),weget(1−2c2r)v2Θmi=m0v21(7.6)(1−2)2cReplacingalltheknownconstantswithvalues,so:c=3×108m/s,r=4km(fortheLargeHadronCollideratCERN),andΘ=2.7×1026m,givesm0(1−2.6×10−6/v2)mi=√(7.7)1−1.1×10−17×v2122v2vUsingabinomialseriesforthedenominator:(1−c2)∼1−2c2···sothatapproximatelym0(1−(2.6×10−6/v2))mi∼(7.8)1−(5.5×10−18v2)ThechangeintheinertialmassfromspecialrelativityandMiHsCcannowbefound.Whenv=0.9ctheeffectofMiHsCis22ordersofmagnitudesmallerthanthechangeduetorelativity,andwhenvishigherstill,theeffectofMiHsCdecreasesevenfurther.ThereforeitwouldbeextremelydifficulttodetecttheeffectsofMiHsCinaparticleaccelerator.However,MiHsCpredictsthatmeasureableeffects,andFTL,mightbeachievedforsystemswithhighlinearvelocitywithverylowaccelerationsaroundthem,suchasinmanyoftheexamplesdis-cussedinChapter5.Twootherpossibleexamplesnotyetmentionedarecosmicraysenteringtheatmosphereandgalacticaxialjets(itispossiblethatM87hasanaxialjetthatactuallymovesfasterthanlight,andnotjustapparently).
143June16,201417:119inx6inPhysicsfromtheEdgeb1771-ch07MiHsCandFasterThanLightTravel131FTLTest3:GalacticJetsMiHsCpredictsthatjustastheEarthflybycraftareboostedwhentheyexitalongtheEarth’spolaraxis,objectsmayloseinertialmassandbemoreeasilyacceleratedalonggalacticaxes.GalacticjetshavebeenknownforsomeyearsandBirettaetal.(1999)observedtheM87axialjetandcalculatedtheapparentspeedofrecognisable‘knots’oflightwithinthejet,takingaccountoftheestimateddistancetoM87.Theyfoundanapparentspeedof6c.ItwasshownbyRees(1966)thatthisapparentsuperluminalspeedisenhancedbyanopticalillusioncausedbyspecialrelativity.FromRees(1966)theapparentspeed(vapp)ofarelativisticobjectmovingatanangleθtotheobserverdependsonitsrealspeed(v)asvsinθvapp=(7.9)1−cosθAccordingtoBirettaetal.(1999)themostlikelyangleoftheM87jettoourlineofsightis64.5◦,andtheyalsosaidthatbecauseoftheobservedshapeoftheknots:“placingthejetwithin20◦ofthelineofsightpresentsseveralchallenges”.Table1showstheassumedangle(column1)andtherealveloc-ityimpliedbytheobservedapparentvelocityofvapp=6c,usingEq.(7.9).InordertoproducerealvelocitieslessthanthespeedoflightfortheM87jet,itisnecessarytoassumeunrealisticanglesoflessthan20◦.Thisofcourseisacontroversialarea,andestimatesofthejet’sangleorthedistancetoM87maychange,butitispossiblethatthisisevidencefortheFTLspeedsthatMiHsCimpliescanbeachieved.Table1.TheassumedangletothelineofsightoftheM87jetandtheimpliedabsolutespeed.AssumedangletolineofsightAbsolutevelocity◦643.7c◦301.6c◦201.06c
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145June16,201417:119inx6inPhysicsfromtheEdgeb1771-conConclusionsItisobvioustomethatarecastingofphysicsisneeded.Thiscanbeseenconceptuallybysayingthatquantummechanicsisincom-patiblewithgeneralrelativity,sooneofthem,atleast,iswrong.Ontheobservationalside,anomaliesarepilingup,andsomeofthem,liketheanomalousdynamicsofglobularclusters,cannothopetobeexplainedbydarkmatter.Yet,thereisstillahugestigmainphysicsagainstalternativethe-ories.Ifyoulike,theecosystemhastoolittlediversitytobehealthy.Itseemsthatphysiciststodayconfrontedwithafascinatinganomalywillproposeinvisiblematterorinvisibledimensions,anything‘hid-den’ratherthanchangingEinstein’stheories.Thisisacenturiesoldfailsafeofmankind’s:whensomethingisnotunderstooditisattributedtoinvisibleentities,likePtolemy’sepicycles,Descartes’vortices,darkmatterordarkenergy.Thesetendtostickaroundforalongtimebecausetheyaresoflexible,thereforehardtodisprove.InthisbookIhavepresentedanalternativetheorythatdoesn’trelyonover-complexinvisibleentities,butreliessimplyonUnruhradiation(whichmayormaynothavebeenobserved,butisatleastobservable).ThistheorymovesawayfromtheparticlesandsmoothfieldsusedbyNewtonandEinstein,andisbasedonawayofthinkingthatuseshorizonsandinformationwhichmodifytheUnruhradia-tion.Ihavesuggestedthatwhenobjectsaccelerate,Rindlerhorizonsformintheoppositedirectiontotheiracceleration,suppressingtheUnruhradiationinthatdirectionandopposingtheacceleration.Thiscorrectlymodelsinertia.IhavediscussedhowforverylowaccelerationseventhefaroffHubblehorizonstartstocanceltheUnruhradiation,sothatinertia133
146June16,201417:119inx6inPhysicsfromtheEdgeb1771-con134PhysicsfromtheEdge:ANewCosmologicalModelforInertiadecreasesinanewwayfortheselowaccelerations,andthisdiffer-encecanexplaintheobservedanomalousgalaxyrotation,cosmicacceleration,thelow-LCMBanomaly,maybetheflybyanomalies,theTajmareffectandpossiblythePodkletnovresultsandsimilarresultsobtainedbyPoherandModanese.Finally,Ihavetriedtosuggestunambiguoustests.Probablythebesttestwouldbetoproduceanaccelerationsolarge(byacceleratingspinningdiscs)thatitproducesUnruhwavesandtheinertialmassanddynamicsofanearbymassaredetectablyaffectedbyit.AnotherroutewouldbetotrytodirectlyinterferewiththeUnruhwavesthatmakeuptheinertialmassusingelectromagneticwaves.Ihaveshownhowthisnewtheorysuggestsanewwaytolaunchthingsintospaceandthatitsuggestsverytentativelythatfasterthanlighttravelmaybepossible.Despitewhatwearetoldatuniversity,physicsisnotsetinstone,butisaworkinprogress.Hopefullythesenewideaswillencouragesomeofyoutotestordevelopthemfurther.Iftheyareprovenrightbyexperiment,then,notonlywillwehavefascinatingnewinsightsintoourcosmos,butalsonewwaystotravelthroughit.
147June16,201417:119inx6inPhysicsfromtheEdgeb1771-appaAppendixABriefDiscussionofPapersonMiHsCThisisaquickchronologicalsummaryofallthepapersIhavewrittensofaronMiHsC(asof2013).2007.IassumedthatinertiawascausedbyUnruhradiation,andsubjecttoaHubble-scaleCasimireffectsothatsomeUnruhwavesaredisallowedbecausetheydon’tfitwithintheHubblescale.Thisleadstoanewlossofinertiaforlowaccelerations.Iappliedthismodel(calledMiHsC)tothetrajectoriesofthePioneerspace-craftandshowedthatthelossofinertialeadstoanextraSunwardaccelerationequaltothePioneeranomaly.Irememberthedelightfulcommentsofthereviewerofthispaperwhowasamusedbymyuseoftheword“forecast”insteadofprediction(IworkedattheUKMete-orologicalOfficeatthetime)andsaidsomethinglike:“Idon’tquitebelievehissolution,butit’smoreplausiblethanothersthathavebeenpublished,so...”SubsequentworkbyTuryshevetal.(2012)hasproposedthatthePioneeranomalycouldbeduetoananisotropicradiationofheat,butthemodeltheyuseiscomplexwithadjustableparametersandthereisnodecayintheanomalywithtimetobackathermalmodel.Mon.Not.Roy.Astron.Soc.,376,338–342,2007.2008a.Theflybyanomaliesareanomalouschangesofafewmm/sinthespeedofspacecraftflyingbytheEarth.InthispaperItriedtomodelthembysayingthatwhenthecraftpassthroughazonewherethenetaccelerationislowtheyloseinertialmassbyMiHsCandspeedupbymomentumconservation.Ispentthebetterpartofayearmodellingtrajectoriesinmysparetime,anditdidnotworkbecauseIdidnotyetconsidermutualaccelerations.However,inthispaper,IalsosuggestedcontrollinginertiabybendingUnruhwavesusingmetamaterials.J.Brit.Interplanet.Soc.,61,373–378,2008.2008b.ThispaperwasinspiredbyobservationsofAndersonetal.(2008)thatshowedthattheflybyanomalywaslargewhen135
148June16,201417:119inx6inPhysicsfromtheEdgeb1771-appa136PhysicsfromtheEdge:ANewCosmologicalModelforInertiathespacecraftcametowardstheEarthattheequatorandleftatthepole.WhenIdownloadedthepaperitupsetmebecauseIcouldn’texplainit,butthenIrealisedwithjoythatIcouldmodelitusingMiHsCifIconsideredthe‘mutual’accelerationsbetweenmasses,sincethemutualaccelerationbetweenaspacecraftandmassesinthespinningEarthislowerclosertothespinaxis.MiHsCthenpredictsthecraft’sinertiaislowernearthepoleandtoconservemomentumthecraftspeedsup.Thismodelstheflybyanomaliesfairlywellwith-outadjustableparameters,butnotperfectly.Mon.Not.Roy.Astron.Soc.Lett.,389(1),L57–60,2008.2010a.InthispaperIappliedMiHsCtotheobservationsofTajmaretal.(2007)whonoticedanunexplainedaccelerationofaccelerometersclosetorotatingrings.Itooktheideaofmutualaccel-erationsfurtherandconsideredtheinertialmassoftheaccelerome-tertobedependent(viaMiHsC)onnotonlyitsaccelerationwithrespecttothespinningringbuttothefixedstarstoo(withanodtoErnestMach).TheideawassoundbutImessedupthemaths.IrealisedmyerrorthenightbeforeIwasduetogiveanimportanttalkonitinBerne!Ihadtowriteanotherpapertocorrectit(see2011a).Europhys.Lett.,89,19001.2010b.ThiswasamoredetailedlookatthepredictionbyMiHsCthatsinceUnruhwaveslengthenasaccelerationsreduce,andbecausetheUnruhwavescannotinprinciplebeobservediftheyaregreaterthantheHubblescale,theremustthenbeaminimumaccelerationallowedinnature.Ishowedthatthisisclosetotheobservedcosmicaccelerationthatisusuallyattributedtoarbitrary‘darkenergy’.MiHsCalsopredictstheobservedminimummassfordiscgalaxiesseenbyMcGaughetal.(2009).InthispaperIalsosug-gestedmodifyingtheinertialmassofanobjectbyinterferingwithUnruhradiationusingEMradiation.Europhys.Lett.,90,29001.2011a.Icorrectedmymathematicalmistake(in2010a)andMiHsCworkedwellbutdidn’tfitoneofTajmar’sresults.WhenIemailedTajmarhetoldmethatparticularresultwasduetoawrongsteppermotor,soIwasecstatic.ThepredictionofMiHsCisthatwhentheringacceleratestheaccelerometergainsinertialmassandhastomovewiththeringtoconservetheoverallmomentumofthesystem.MiHsCpredictstheresultsverywell.Thispaperand
149June16,201417:119inx6inPhysicsfromtheEdgeb1771-appaBriefDiscussionofPapersonMiHsC1372010awon“BestofYear”awardsfromtheEPLjournal.Europhys.Lett.,95,39002.2011b.ThiswasmyattempttoexplaintheweightlossseenbyPodkletnovwhenhevibratedandspanasuperconductingdiscbelowvarioustestmasses.MiHsCprovidedapossibleexplanation,butnotacompleteoneandIcouldn’tgofurtherbecauseIhadnowaytoknowwhattheaccelerations/vibrationsofthediscwerewhenitwasspun.Thispaperonacontroversialexperimentledtomebeingconsignedtogen-phonthearXivandledtoacoupleofcriticallettersbeingsenttomyuniversityfaculty,butthengreatjoyastheheadofmyschoolwroteanemailsupportingmyacademicfreedom.PhysicsProcedia,20,134–139.2012.ImusthavesubmittednearlysixdifferentpapersseveraltimeseachoverfouryearstryingtomodeladiscgalaxywithMiHsCwithdifferentmethods.WitheachrejectionItriedagainandmymethodbecamesimplertilleventually,therewasnothingforthereviewerstorejectiton.MiHsCpredictstherotationspeedsofdwarf,discandgalaxyclusterswithintheerrorbarswithoutanyadjustableparametersandmostcrucially:withoutdarkmatter.Ihaveyettomodelagalaxyindetailthough.Astrophys.SpaceSci.,342(2),575–578.2013.InallthepapersaboveIusedaHubble-scaleCasimireffecttomodel‘deviations’fromstandardinertia,andjustassumedstan-dardinertia.InthispaperIproposedthatstandardinertiaisduetoaRindler-scaleCasimireffect.Asanobjectaccelerates,say,totheright,aRindlerhorizonformstoitsleftsinceinformationfur-therawaycannevercatchup.ARindler-scaleCasimireffectthensuppressesUnruhwavesontheleft,sothattheobjectfeelsmoreUnruhradiationpressurefromtheright.Thispressurepushesitbackagainstitsacceleration:anelegantmodelforinertiathatneedsnoadjustableparameters.Thismodelalsorepresentsanewwayofthinkingaboutmotionandenergyintermsofhorizonsandinforma-tion.Europhys.Lett.,101,59001.
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151June16,201417:119inx6inPhysicsfromtheEdgeb1771-appbAppendixBHawkingRadiationToarriveveryroughlyatHawking’s(1974)resultwecanapplytheuncertaintyprincipletoaphotonemittedfromtheeventhorizonofablackholeasfollows∆p∆x∼The∆pistheuncertaintyinmomentumofthephotonandthe∆xistheuncertaintyinposition.Heisenbergstatedthattheirproductwasequaltoh-barwhichisPlanck’sconstantdividedby2π.Sincetheoutwardsphoton’scompanionhasdisappearedintotheblackholesomewhere,theuncertaintyinpositionisclosetotheradiusoftheblackholeofmassMwhichis2GMR=c2Puttingthisintotheuncertaintyprinciplefor∆xwegetc2∆p∼2GMSincetheenergyE=pcthisgivesc3∆E=2GMWhenenergyisinarandomthermalform,itcanbewrittenintermsoftemperatureasE=kT.Hence,c3T=2GMkThisistheHawkingtemperatureofablackholeandwhenthisisderivedproperlyan8πappearsinthedenominator.139
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153June16,201417:119inx6inPhysicsfromtheEdgeb1771-appcAppendixCUnruhRadiationWecannowdoasimilaranalysistoHawking’sfortheRindlerhorizoncausedbyanobject’sacceleration(thiswasdonebyUnruhinamorerigorousway).Heisenberg’suncertaintyprincipleformomentum(p)andposition(x)is∆p∆x∼Theuncertaintyinspace(∆x)ofanobjectwithanaccelerationaisthendeterminedbythedistancetoitsRindlerhorizon,c2/a:c2∆p∼aSoa∆p∼c2SinceE=pca∆E∼cIftheenergyisthermalisedsoE=kTthena∆T∼ckUnruh(1976)derivedthisformulafirst,afterHawkinghaddonehisworkontheblackholeandtheonlydifferencewasthathehadanextra2πinthedenominator,givinga∆T≈2πckWhatthismeansisthatasanobject’sacceleration(a)increasesthebackgroundtemperatureitmeasures(T)alsoincreases.ThisiscalledUnruhradiationanditoriginatesfromtheRindlerhorizon.141
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155June16,201417:119inx6inPhysicsfromtheEdgeb1771-appdAppendixDAsymmetricCasimirEffectThisisaderivationofamodelforinertiausinganasymmetricCasimireffect(forthecompletederivation,seeMcCulloch,2013).Asingleparticleisconsideredforsimplicity.ItisacceleratingtotherightasinFigure5inChapter4,reproducedbelow.Theradiationpressure(force)onanysmallsurfacearea(A)exposedtoisotropic(directionindependent)radiationisuAF=3Figure1.Aschematicshowingaparticleacceleratingtotheright(theblackcircle).TheshadingshowstheRindlerhorizonclosebytoitsleft(atadistance2c/aaway)andthecosmichorizonfarawaytoitsright.TheUnruhradiationisdampedmoreontheleftsideproducinganasymmetricCasimireffect(theforceindicatedbythetwounequalopposingarrows)thatpushestheparticle(O)totheleftagainstitsaccelerations:amodelforinertia.143
156June16,201417:119inx6inPhysicsfromtheEdgeb1771-appd144PhysicsfromtheEdge:ANewCosmologicalModelforInertiaThethreecomesfromthefactthattherearethreedirectionspossible(x,yandz)andsothetotalenergyissplitbetweenthem.TheustandsfortheUnruhradiationenergydensity,Aisthesurfaceareaoftheparticleinterceptingthisradiationandthisisassumedtobeasmallpartofthesurfaceareaofthewholeparticle.Nowweneedthenetdifferencebetweentheforcesfromtherightandtheleftsowedrawalinethroughtheparticleatanarbitraryangleθ(thedashedline)andcalculatetheforcesalongthisline(seethearrowsinthefigure)andtakeitscomponentalongthex-axisbymultiplyingbycosθ.uleftAcosθurightAcosθdFx=−33TheenergycontentoftheUnruhradiationcomingfromtheright(thesecondtermontherighthandside)issubjecttotheusualHubble-scaleCasimireffect,souright=u(1−λ/4Θ),whereuistheunmodifiedenergyandλisthepeakwavelengthoftheUnruhradiationspectrum.Incontrast,theenergyoftheUnruhradiationcomingfromtheleft(thefirsttermontherighthandside)issubjecttoaRindler-scaleCasimireffectandthishorizonismuchcloser,atadistanceofc2/acosθ(whereacosθisthecomponentoftheaccel-erationinthedirectionofθ)sothatuleft=u(1−λ/4(c2/acosθ)).Thedifference,thenetforceinthex-direction,isuAcosθλacosθλdFx=1−−1+34c24ΘWecancancelthe1−1inthebrackets,sothissimplifiestouλAcosθ1acosθdFx=−34Θ4c2Wecannowintegratethecontributionfromallthepossibleangles.Todothisweintegratewithintheplaneofthepage,fromtheangleθ=0toπ/2(90◦),thendoublethistogettheresultforthewholex–yplaneandthenintegratethiscircularily180◦orπaroundthex-axis
157June16,201417:119inx6inPhysicsfromtheEdgeb1771-appdAsymmetricCasimirEffect145(aroundtheangleφ)tocalculatethetotalforce:uλAππ/2cosθacos2θdFx=2×−dθdφ3004Θ4c2ππ/22uλAsinθaθasin2θdFx=−−dφ304Θ8c216c20whichis2uλA1πadFx=π×−.34Θ16c2ThefirstterminthebracketsistheMiHsCcorrectiontowhatwenormallyexpectfrominertia.Itistinycomparedtothesecondtermunlesstheacceleration,a,isverysmallandthesecondtermbecomessmall.Ifweassumeanormal(terrestrial)accelerationthenwecanneglectthisfirstterm,soπ2uλAadFx=−24c2SinceuistheenergywithinavolumeV,sou=E/V=hc/λV,thenπ2hAadFx=−24cVThisisthepredictedforceononesphericalparticle,duetotheasymmetricCasimireffect.ItmeansthattheRindlerhorizonthatappearsintheacceleratedparticle’sreferenceframedampsUnruhradiationinthedirectionoppositetoitsacceleration,sotheradiationthereisunabletobalancethemomentumimpartedbytheradiationhittingitfromtheotherdirection.Theresultisanetforcecountertotheacceleration(theminussign)andthislooksjustlikeinertia.TheratioA/V=area/volumecouldbesimplifiedbychoosingadistancescale(x)smallerthantheparticlesothatA/V=x2/x3=1/x.UsingthePlanckdistance(lP=1.616×10−35m)asxfor
158June16,201417:119inx6inPhysicsfromtheEdgeb1771-appd146PhysicsfromtheEdge:ANewCosmologicalModelforInertiaexample,thentheequationabovebecomesπ2hadFx=−24clPAndtheinertialmassforthePlanckscaleparticleis5.5×10−8kgwhichisabouttwicethePlanckmass.ItwaslaterkindlypointedouttomebyJ.Gin´ethatIhadmadeafactorof2errorandthe24inthisresultshouldbe48.Thismakesthepredictionbetter:only26%greaterthanthePlanckmass.Apapertocorrectthis(Gin´eandMcCulloch,2014)isbeingreviewedatthetimeofwritingthisbook.
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163June16,201417:119inx6inPhysicsfromtheEdgeb1771-indexIndexabsolutespace,11darkmatter,detection(lackof),47accelerationsontheEarth,41deBroglie,wave-particleduality,33accelerations,needtoconsiderallinDopplershift,24–25MiHsC,88–89droptowerexperiment,124Adams,D.,18allowedUnruhwavelengthsinEinstein,A.,specialrelativity,18MiHsC,66embarrassmentinBerne,101Aristotle,1Empedocles,16artificialeventhorizonsmadewithequivalenceprinciple,5,9,25,70metamaterials,129escapevelocity,34asymmetricCasimireffect,62–66eventhorizons,63experiments,totestMiHsC,111Berkeley,B.G.,11extrapolationoverscalesblackbodyradiation,30(difficultyof),48blackhole,35,37Boltzmann’sconstant,value,32fallingstoneargument,3–4Borro,G.,3–4FitzGerald,G.,18flatnessproblem,76Campbell,J.,93flute,physicsof,57Casimireffect,61flybyanomalies,85–93centripetal‘force’,41flybyworkshop,93controllableexperiments,advantageframedraggingeffectofgeneralof,93relativity,29cooling,cryostat,needfor,100fundamentalnoteinmusic,59Copernicus,N.,6Coriolisforce,96galacticjets,MiHsCand,131cosmicacceleration,73–76,128galaxyrotationcurve,typical,49CosmicMicrowaveBackground,galaxyrotationandMiHsC,81–85CMB,36Galilei,G.,1curvedspace,26generalrelativity,26geodeticeffectofgeneralrelativity,28darknessofthenightsky,24globularclusters:crucialdarkenergy,75experiments?,52,54darkmatter,43–47GPSsatellitesandrelativity,28darkmatter,problemswith,45–47GPSsatellitesandMiHsC,124darkmatter,tooflexible,46gravitationalconstant,G,8151
164June16,201417:119inx6inPhysicsfromtheEdgeb1771-index152PhysicsfromtheEdge:ANewCosmologicalModelforInertiagravitationalforce,Newton’slaw,41McFly,Marty,85gravitationalmassdefinition,9metamaterials,inertialdampingwith,gravitationalwaves,2966gravityassist,85MichelsonandMorleyexperiment,17GravityProbeB,28Milgrom’sbreak,andproblemswith,Greekscience,152MiHsC(ModifiedinertiaduetoaHafeleandKeatingclockexperiment,Hubble-scaleCasimireffect),6622MiHsCderivation,66–69hailstormdebate,3,46MiHsCmodifiedinertiaformula,69Haischetal.’smodel,62MiHsC’sbehaviouratdifferenthardhorizon(horizoncensorship),67accelerations,70harmonicseries,59MiHsC’sequationofmotion,71Hawkingradiation,36,63MiHsCandequivalence,71Hawkingtemperature,36MiHsC,testsof,73–126heartscans,90MiHsC,intuitiveexplanation,74horizoncensorship,79–80MiHsC,cosmicacceleration,73–76HulseandTaylor’sbinarypulsartestMiHsC,low-lCMBanomaly,77–80ofGR,29MiHsC,galaxyrotation,81–85hypothesisnonfingo,Newton,9MiHsC,lackofadjustableparameters,83importanceofanomaliesinscience,40MiHsC,flybyanomaly,85–93inertialmassdefinition,8,9,11MiHsC,Tajmareffect,93–104inertiahereisduetomassthere,MiHsC,andNewton’spail,103Mach,11MiHsC,andPioneeranomaly,inertialresonanceinMiHsC,123104–109InternationalSpaceStation,1MiHsC,andPodkletnoveffect,InternationalSpaceScienceInstitute,112–114Berne,Switzerland,93MiHsC,levitation,114–119interferingwithinertialmass,MiHsC,interferewithinertia119–123electromagnetically,119–123MiHsC,summaryofproposedtests,largescale(low-l)CMBanomaly,12677–80MiHsC,andfasterthanlighttravel,lawofinertia,4127–131Leibniz,G.,11MoND(ModifiedNewtonianlowestnoteontheflute,59Dynamics),47–54Lorentz,H.,18MoND,adjustableparametera0,Lorentztransformations,1949–50MoND,inertialversion,50Mach,E.,11MoND,problemwith,54–56Mach’srefutationofabsolutespace,11–12Newton,I.,6Mach’sattitudetoscience,13,53,91Newton’slawsofmotion,8Mannheim’sconformalgravity,47Newton’sgravitylaw,8Maxwell,J.C.,17Newton’spailexperiment,9–11,103
165June16,201417:119inx6inPhysicsfromtheEdgeb1771-indexIndex153nodes,inwaves,58specialrelativity,18Nordstrom,G.,26speedoflight,16speedoflightlimit,23,127Oort,J.,44spinningdiscexperiment,111orbitalmotionisabalance,41steadystatetheoryofFredHoyle,orbitalvelocities,4235streetlightcargame,90particleaccelerators,MiHsCin,129subsamplingtheUnruhspectrum,66photoelectriceffect,32supernovae,75Pioneerspacecraft,75,104Pioneeranomaly,75,104–109Tajmareffect,93–104Pioneeranomaly,thermaltestability,oftheoriesisessential,111explanation,106thoughtexperimentPlanck,M.,quantumpioneer,31followingalightbeam,17Planck’sconstant,valueof,32trainandlightning,20Podkletnov’sexperiment,partiallaserclock,21agreementwithMiHsC,112–114starattheHubbleedge,24laserinalift,26quantisedinertia(MiHsC),66ff.torsionbalances,nullMiHsCresults,70realityiswhatyoucanobserve,22referenceframe,6Unruhradiationrelativisticinertialmass,23,127(Fulling–Davies–Unruh),38resonance,57Unruhradiation,observationof?,38Rindler(dynamic)horizon,38,63untestability,recentfashionfor,111Roemer,O.,16Uranus’orbit,discoveryofNeptune,Rubin,V.,discoveredthegalaxy45rotationproblem,45Virialtheorem,44Scarpaetal.,brilliantsignpostpaper,51–52wave-particleduality,33Scott,D.,4Wien’slaw,32simultaneity,20snailsandminimumaccelerations,74zeropointfield,ZPF,61SpaceXDragonspacecraft,1Zwicky,F.,proposeddarkmatter,43
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