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Simple,Interesting,andUnappreciatedFactsaboutRelativisticAccelerationJohnMallinckrodtCalPolyPomona2003AAPTWinterMeetingAustinTexas江西不孕不育医院Doesrelativityallowanobjectto“accelerateasarigidbody”?Answer:Yes.“Simply”applyanappropriateamountofforcetoeverypieceoftheobject.15January20032003AAPTWinterMeeting,Austin3FollowupIfanobjectisacceleratingasarigidbody,howdoestheaccelerationofits“front”endcomparetothatofits“rear”end?a)Obviously,theywouldhavetobethesamesothatallpointsmovewiththesamespeedatalltimes.b)Obviously,pointsnearerthe“frontend”wouldhavetohavesmalleraccelerationssothattheobjectLorentzcontractsproperly.c)Bothoftheabove.(Ifnot,whynot?)15January20032003AAPTWinterMeeting,Austin4OutlineofTalk1.Analyzeasimpleworldlinethatwillturnouttobethatofapointobjectundergoingconstantproperacceleration.2.Summarizeitsgeometriccharacteristicsonaspacetimediagram.3.Considerpairsofseparatedparticlesthatstartfromrest,maintainconstantproperaccelerations,andeithera)haveidenticalaccelerations,orb)maintaintheirproperseparation.4.Extendtheanalysistocontinuousbodies.5.Findaconstraintonthelengthofarigidlyacceleratingobjectandunderstanditsconnectiontotheexistenceofaneventhorizonforrigidlyacceleratingreferenceframes.6.Considerthebehaviorofclocksonarigidly,butotherwisearbitrarilymovingobject.7.Watchasimulationofarodthatacceleratesfromresttoamaximumspeedanddeceleratesbacktorest.8.Summarizemainpoints.15January20032003AAPTWinterMeeting,Austin5Claims,Disclaims,andAcknowledgementsIbelievethismaterialisaccessible,surprising,uncontroversial,butneverthelessnotwellknown.Ontheotherhand,therearecloselyrelatedquestionsthatIfeellesscompetenttodiscuss.Afew(incomplete)references:Hamilton(AJP,46,83)DeslogeandPhilpott(AJP,55,252)Desloge(AJP,57,598)Desloge(AJP,57,1121)Nikolic(AJP,67,1007)TaylorandWheeler“SpacetimePhysics”Mould,“BasicRelativity”(EspeciallyChapter8).(IfI’mlucky,theywon’tcomeup.)15January20032003AAPTWinterMeeting,Austin6ClassicalFeaturesoftheMotionofInterestConsiderthemotiondescribedinaninertialframebyInunitswherec=1andwiths=“vertexdistance”=constant.Usingnothingmorethanthedefinitionsofvandaitiseasytoshowthat15January20032003AAPTWinterMeeting,Austin7RelativisticFeaturesoftheMotionofInterest1Howmuchdoesthemovingobject“feel”itsspeedchangewhenitsspeedobservedintheinertialframechangesfromvtov+dv?(Lorentzvelocityaddition)Howmuchtimeelapsesintheframeoftheobjectduringatimedtintheinertialframe?(Timedilation)Thus,theproper(“felt”)accelerationisgivenbyThatis,theproperaccelerationisconstantandinverselyproportionaltothevertexdistance.15January20032003AAPTWinterMeeting,Austin8RelativisticFeaturesoftheMotionofInterest2Howdoesthepropertimechangealongthetrajectory?Notethatthisformulafortheelapsedpropertimedependsonlyonthevelocityasmeasuredintheinertialframe(whichismonotonicallychanging)andthevertexdistanceandthatitisdirectlyproportionaltothevertexdistance.WithtP(v=0)=0,wecanintegratetofind15January20032003AAPTWinterMeeting,Austin9GeometricConsequencesAneventhorizonexistsandpreventsanycausalconnectiontoeventsonthe“darkside”ofthathorizon.Theworldlineforanobjectundergoingconstantproperaccelerationisahyperbolathatcorrespondstothelocusofalleventshavingaconstantspacelikeseparationfrom“theorigin.”(x2-t2=s2=constant)Theproperaccelerationissimplytheinverseofthatseparation.(ap=1/sAstheobjectaccelerates,the“lineofinstantaneoussimultaneity”alwayspassesthrough“theorigin.”(dx/dt=t/x)15January20032003AAPTWinterMeeting,Austin10GeneralizationtoArbitraryWorldlinesTheinstantaneousposition,velocity,andaccelerationofanarbitrarilymovingobjectassociatesitwithaunique“instantaneousconstant(proper)accelerationworldline.”15January20032003AAPTWinterMeeting,Austin11TwoObjectswithIdenticalConstantAccelerationEachobjectmovesalongahyperbolicpathhavingitsownasymptoticlightconeTheseparationisconstantintheinertialframeAandBdisagreeonmattersofsimultaneityInfact,BmightevensaythatA’strajectoryistime-reversedexceptforthefactthat……thatportionofA’sworldlineishiddenbehindB’seventhorizon15January20032003AAPTWinterMeeting,Austin12TwoObjectswithIdenticalAsymptoticLightConesSincethevertexdistancesaredifferent,soaretheaccelerationsThe“front”object,B,hasasmallerproperaccelerationthanthe“rear”object,AAandBagreeatalltimesonmattersofsimultaneityAandBagreeatalltimesontheircommonvelocityAandBagreethattheirproperseparationisconstantAandBagreethatB’sclockrunsfasterindirectproportiontotheirrespectivevertexdistances.15January20032003AAPTWinterMeeting,Austin13ARigidlyAcceleratingRodthatFlashesSynchronouslyConsiderthesequenceofeventsinboththeinertialandacceleratingframes.Notethat,evenwithintheframeoftherod,flashingsynchronouslyisnotthesameasflashingatadefinitetimeintervalbecausetheclocksrunatdifferentrates.15January20032003AAPTWinterMeeting,Austin14ExtensiontotheDarkSideoftheEventHorizonConsiderafamilyofhyper-bolicworldlinessharingthesamefocus(vertex).Anyspacelikelinethroughthefocusisa“lineofinstan-taneoussimultaneity”an