醫(yī)療糾紛專職人民調(diào)解員招聘報名表T

Simple,Interesting,andUnappreciatedFactsaboutRelativisticAccelerationJohnMallinckrodtCalPolyPomona2003AAPTWinterMeetingAustinTexas江西不孕不育醫(yī)院

15January200312003AAPTWinterMeeting,AustinQuestionDoesrelativityallowanobjectto“accelerateasarigidbody”?Answer:Yes.

“Simply”applyanappropriateamountofforcetoeverypieceoftheobject.15January200322003AAPTWinterMeeting,AustinFollowupIfanobjectisacceleratingasarigidbody,howdoestheaccelerationofits“front”endcomparetothatofits“rear”end?

a)Obviously,theywouldhavetobethesamesothatallpointsmovewiththesamespeedatalltimes. b)Obviously,pointsnearerthe“frontend”wouldhavetohavesmalleraccelerationssothattheobjectLorentzcontractsproperly. c)Bothoftheabove.(Ifnot,whynot?)15January200332003AAPTWinterMeeting,AustinOutlineofTalkAnalyzeasimpleworldlinethatwillturnouttobethatofapointobjectundergoingconstantproperacceleration.Summarizeitsgeometriccharacteristicsonaspacetimediagram.Considerpairsofseparatedparticlesthatstartfromrest,maintainconstantproperaccelerations,andeithera)

haveidenticalaccelerations,orb)

maintaintheirproperseparation.Extendtheanalysistocontinuousbodies.Findaconstraintonthelengthofarigidlyacceleratingobjectandunderstanditsconnectiontotheexistenceofaneventhorizonforrigidlyacceleratingreferenceframes.Considerthebehaviorofclocksonarigidly,butotherwisearbitrarilymovingobject.Watchasimulationofarodthatacceleratesfromresttoamaximumspeedanddeceleratesbacktorest.Summarizemainpoints.15January200342003AAPTWinterMeeting,AustinClaims,Disclaims,

andAcknowledgementsIbelievethismaterialisaccessible,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.)15January200352003AAPTWinterMeeting,AustinClassicalFeaturesoftheMotionofInterestConsiderthemotiondescribedinaninertialframebyInunitswherec=1andwiths=“vertexdistance”=constant.

Usingnothingmorethanthedefinitionsofvandaitiseasytoshowthat

15January200362003AAPTWinterMeeting,AustinRelativisticFeaturesoftheMotionofInterest1Howmuchdoesthemovingobject“feel”itsspeedchangewhenitsspeedobservedintheinertialframechangesfromvtov+dv?(Lorentzvelocityaddition)Howmuchtimeelapsesintheframeoftheobjectduringatimedtintheinertialframe?(Timedilation)Thus,theproper(“felt”)accelerationisgivenbyThatis,theproperaccelerationisconstantandinverselyproportionaltothevertexdistance.15January200372003AAPTWinterMeeting,AustinRelativisticFeaturesoftheMotionofInterest2Howdoesthepropertimechangealongthetrajectory?Notethatthisformulafortheelapsedpropertimedependsonlyonthevelocityasmeasuredintheinertialframe(whichismonotonicallychanging)andthevertexdistanceandthat

itisdirectlyproportionaltothevertexdistance.WithtP(v=0)=0,wecanintegratetofind15January200382003AAPTWinterMeeting,AustinGeometricConsequencesAneventhorizonexistsandpreventsanycausalconnectiontoeventsonthe“darkside”ofthathorizon.Theworldlineforanobjectundergoingconstantproperaccelerationisahyperbolathatcorrespondstothelocusofalleventshavingaconstantspacelikeseparationfrom“theorigin.”

(x2-t2=2=constant)Theproperaccelerationissimplytheinverseofthatseparation.(ap=1/Astheobjectaccelerates,the“l(fā)ineofinstantaneoussimultaneity”alwayspassesthrough“theorigin.”

(dx/dt=t/x)15January200392003AAPTWinterMeeting,AustinGeneralizationto

ArbitraryWorldlinesTheinstantaneousposition,velocity,andaccelerationofanarbitrarilymovingobjectassociatesitwithaunique“instantaneousconstant(proper)accelerationworldline.”15January2003102003AAPTWinterMeeting,AustinTwoObjectswithIdenticalConstantAccelerationEachobjectmovesalongahyperbolicpathhavingitsownasymptoticlightconeTheseparationisconstantintheinertialframeAandBdisagreeonmattersofsimultaneityInfact,BmightevensaythatA’strajectoryistime-reversedexceptforthefactthat……thatportionofA’sworldlineishiddenbehindB’seventhorizon15January2003112003AAPTWinterMeeting,AustinTwoObjectswithIdenticalAsymptoticLightConesSincethevertexdistancesaredifferent,soaretheaccelerationsThe“front”object,B,hasasmallerproperaccelerationthanthe“rear”object,AAandBagreeatalltimesonmattersofsimultaneityAandBagreeatalltimesontheircommonvelocityAandBagreethattheirproperseparationisconstantAandBagreethatB’sclockrunsfasterindirectproportiontotheirrespectivevertexdistances.15January2003122003AAPTWinterMeeting,AustinARigidlyAcceleratingRodthatFlashesSynchronouslyConsiderthesequenceofeventsinboththeinertialandacceleratingframes.Notethat,evenwithintheframeoftherod,flashingsynchronouslyisnotthesameasflashingatadefinitetimeintervalbecausetheclocksrunatdifferentrates.15January2003132003AAPTWinterMeeting,AustinExtensiontotheDarkSide

oftheEventHorizonConsiderafamilyofhyper-bolicworldlinessharingthesamefocus(vertex).Anyspacelikelinethroughthefocusisa“l(fā)ineofinstan-taneoussimultaneity”andintersectsallworldlinesatpositionsofidenticalslope(velocity).Positiveaccelerationsontheright,negativeontheleft.Whycan’tarigidbodystraddlethevertex?15January2003142003AAPTWinterMeeting,AustinAcceleratingasaRigidBodyuptoaFinalSpeedHowdowegetaLorentzcontractedrodintheinertialframe?Mustacceleratetoauniformvelocityintheinertialframe.Requirestherearendtostopacceleratingbeforethefront.Clocksarenotsynchronizedinthemovingframe.Notesubsequentpenetrationoftheformer“eventhorizon.”15January2003152003AAPTWinterMeeting,AustinGeneralized

RigidBodyMotionUnderuniformaccelerationtheworldlinesofthefrontandrearareidenticalbutshifted.ThebodydoesnotLorentzcontract.Under“rigidbodyacceleration”thebody’smotionisarbitraryaslongastheaccelerationofthe“frontend”neverexceeds1/L.Theworldlineforthe“frontend”isalwayslesscurved(smallera)thanthatofthe“rearend.”Clocksreturntosynchronizationwhenevertheyreturntothevelocityatwhichtheyweresynchronized.15January2003162003AAPTWinterMeeting,AustinSimulationHereisanInteractivePhysics?simulationoftheendpointsofarigidbody(ofadjustablelength)whoserearendaccelerateswithconstantproperaccelerationfromresttoan(adjustable)maximumvelocityandthenbacktorestwiththeoppositeproperacceleration.15January2003172003AAPTWinterMeeting,AustinSummaryofMainPointsUniformlyacceleratingthevariouspartsofabodyincreasestheirproperseparationsanddevelopstensilestresseswithinthebody.Bodiescanacceleraterigidly,butonlybyacceleratingnonuniformly.Theworldlinesofdifferentpositions(asobservedinaninertialframe)followafamilyofhyperboliccurvessharingthesamefocus(ver