Table Of Content

Moses Fayngold Special Relativity and Motions Faster than Light Author: (cid:2) Thisbookwascarefullyproduced.Never- MosesFayngold theless,authorandpublisherdonotwarrant DepartmentofPhysics, theinformationcontainedthereintobefree NewJerseyInstituteofTechnology,Newark. oferrors.Readersareadvisedtokeepin e-mail:[email protected] mindthatstatements,data,illustrations,pro- ceduraldetailsorotheritemsmayin- Illustrations: advertentlybeinaccurate. RolandWengenmayr,Frankfurt,Germany LibraryofCongressCardNo.appliedfor. CoverPicture: AlbertFayngold,NewYork,NY BritishLibraryCataloguing-in-Publication Data: 1stedition Acataloguerecordforthisbookisavailable fromtheBritishLibrary. DieDeutscheBibliothek–CIP-Cataloguing- in-PublicationData: Acataloguerecordforthispublicationis availablefromDieDeutscheBibliothek.  Wiley-VCHVerlagGmbH, Weinheim,2002 Allrightsreserved(includingthoseoftrans- lationintootherlanguages).Nopartofthis bookmaybereproducedinanyform–by photoprinting,microfilm,oranyother means–nortransmittedortranslatedintoa machinelanguagewithoutwrittenpermis- sionfromthepublishers.Registerednames, trademarks,etc.usedinthisbook,even whennotspecificallymarkedassuch,are nottobeconsideredunprotectedbylaw. PrintedintheFederalRepublicofGermany Printedonacid-freepaper Typesetting ProSatzUnger,Weinheim Printing betz-druckgmbh,Darmstadt Bookbinding Großbuchbinderei J.SchäfferGmbH&Co.KG,Grünstadt ISBN 3-527-40344-2 V Table of Contents Preface IX 1 Introduction 1 1.1 Relativity?Whatisitabout? 1 1.2 WeirdnessofLight 9 1.3 Asteamerinthestream 11 2 LightandRelativity 15 2.1 TheMichelsonexperiment 15 2.2 Thespeedoflightandtheprincipleofrelativity 19 2.3 “Obvious”doesnotalwaysmean“true”! 22 2.4 Lightdeterminessimultaneity 23 2.5 Light,times,anddistances 27 2.6 TheLorentztransformations 31 2.7 Therelativityofsimultaneity 34 2.8 Aproperlengthandapropertime 36 2.9 Minkowski’sworld 38 2.10 Whatishorizontal? 48 3 TheVelocities’Play 55 3.1 Theadditionofcollinearvelocities 55 3.2 Theadditionofarbitrarilydirectedvelocities 57 3.3 Thevelocities’play 58 4 RelativisticMechanicsofaPointMass 63 4.1 Relativistickinematics 63 4.2 Relativisticdynamics 66 5 ImaginaryParadoxes 72 5.1 Thethreeclocksparadox 72 5.2 Thedialogoftwoatoms 75 5.3 ThelongitudinalDopplereffect 82 5.4 Predicamentsofrelativistictrain 86 SpecialRelativityandMotionsFasterthanLight.MosesFayngold Copyright(cid:1)2002WILEY-VCHVerlagGmbH,Weinheim ISBN:3-527-40344-2 VI TableofContents 5.5 Dramaticstop 101 5.5.1 BrakinguniformlyinA 102 5.5.2 AcceleratinguniformlyinT 107 5.5.3 Non-uniformbraking 110 5.6 Thetwinparadox 113 5.7 Circumnavigationswithatomicclocks 123 5.8 Photonracesinacentrifuge 131 6 SuperluminalMotions 142 6.1 Velocity,information,signal 142 6.2 Thescissorseffect 143 6.3 Thewhirlingswords 144 6.4 Waltzinamagneticfield 145 6.5 Spiralingray 149 6.6 Starwargamesandneutronstars 153 6.7 Surprisesofthesurf 162 6.8 Thestoryofasuperluminalelectron 163 6.9 Whatdoweseeinthemirror? 167 6.10 Thestarrymerry-go-round 172 6.11 Weirddryspots,superluminalshadow,andexplodingquasars 174 6.12 Phaseandgroupvelocities 183 6.13 ThedeBrogliewaves 193 6.14 Whathappensatcrossingofrays? 195 6.15 Themysteryofquantumtelecommunication 202 7 SlowLightandFastLight 209 7.1 Monitoringthespeedoflight 209 7.2 AdventuresoftheBump 213 7.3 Slowlight 217 7.4 Fastlight 219 8 TachyonsandTachyon-likeObjects 224 8.1 Superluminalmotionsandcausality 224 8.2 Thephysicsofimaginaryquantities 226 8.3 Thereversalofcausality 228 8.4 Onceagainthephysicsofimaginaryquantities 231 8.5 Tachyonsandtardyons 235 8.6 Tachyon–tardyoninteractions 245 8.7 Flickeringphantoms 251 8.8 Tobe,ornottobe? 258 8.9 Theyarenon-local! 265 8.10 CerenkovradiationbyatachyonandWimmel’sparadox 267 8.11 Howsymmetrybreaks 275 8.12 Paradoxesrevised 281 8.13 Laboratory-madetachyons 287 TableofContents VII References 296 Index 298 IX Preface ThisisabookaboutSpecialRelativity.Thepotentialreadermayaskwhyyetanother book needs to be written on this subject when so many have already covered this ground, including some classical early popularizations. There are four answers to thisquestion. First,thisbookisintendedtosupplementtheordinaryphysicstextsonSpecialRela- tivity.Theauthor’sgoalwastowriteabookthatwouldsatisfythedemandsofdiffer- entcategoriesofreader,suchascollegestudentsontheonehandandcollegeprofes- sorsteachingphysicsontheother.Tothisend,manysectionsarewrittenontwole- vels. The lower level uses an intuitive approach that will help undergraduates to graspqualitatively,fundamentalaspectsofrelativitytheory.Thehigherlevelcontains a rigorous analytical treatment of the same problems, providing graduate students andprofessionalphysicistswithagooddealofnovelmaterialanalyzedindepth.The readers may benefit from this approach. There are not many books having the de- scribedtwo-levelstructure(arareandoutstandingexampleisthemonographGravi- tationbyC.W.Misner,K.S.Thorne,andJ.A.Wheeler[1]). Second,thebookexploressomephenomenaanddelvesintosomeintriguingareas that fall outside the scope of the standard treatments. For instance, in the current bookmarketonrelativityonecanspota“hole”–anapparentlackofinformation(but forjustoneortwobooks[2])aboutfaster-than-lightphenomena.Oneofthepurposes ofthisbookistofillinthehole.Thecorrespondingchapters(Chap.6–8)aimtoeluci- dateareasrelatedtofaster-than-lightmotions,whichatfirstseemtocontradictrelativ- ity,butuponexaminationrevealtheconsistency,subtlety,anddepthofthetheory. Third,therehaveappearedrecentlyagooddealofnewtheoreticalstudiesandcorre- sponding experiments demonstrating superluminal propagation of light pulses, which,onthefaceofit,couldappeartoimplypossibleviolationofcausality.(Asimi- lar approach has been used to slow the light pulses dramatically and, finally, to “stop“ light by encoding information it carried, into the physical state of the med- ium.)Theseexperimentshavebeendescribedinthemostprestigiousjournals(see, forinstance,Refs.[3–6]),andhaveattractedmuchattentioninthephysicsandoptics communities.Thisbookdescribesthenewresultsatalevelaccessibletoanaudience withaminimalbackgroundinphysics(Chap.7).Itcontainsananalysisofasimpler versionofthistype ofexperiment[7–11],including apurely qualitative description, whichcanbeunderstoodbyanyinterestedpersonwithpracticallynomath. SpecialRelativityandMotionsFasterthanLight.MosesFayngold Copyright(cid:1)2002WILEY-VCHVerlagGmbH,Weinheim ISBN:3-527-40344-2 X Preface Fourth,thereexistsanother“gap”inavastpoolofbooks(andtextbooksespecially) onthespecialtheoryofrelativity:thesignificantlackofcoverageofacceleratedmo- tions. This has produced another long-standing and widespread misconception (even among professional physicists!) that the theory is restricted to inertial (uni- form)motionsofparticlesthat arenotsubjecttoexternalforces.Iwassurprisedto findeveninrecentlypublishedbooksstatementsthatthespecialtheoryofrelativity isincompletebecauseitcannotdescribeacceleratedmotionsofanykind. Nothingcanbefartherfromthetruththansuchstatements.Howcouldtheparticle acceleratorsthat areroutinely usedin high-energy physics have beendesignedand workproperlywithout thespecialtheoryofrelativity?Oneofthegoalsofthisbook istodispelthemyththatacceleratedmotionscannotbetreatedintheframeworkof the Special Relativity. The reader will find a standard treatment of accelerated motion in Chapter 4,which is devoted especially to the relativistic dynamics of a point mass. In Chapter 5 we describesubtle phenomena associatedwith accelerated motion of extended bodies (Sects. 5.4 and 5.5), and motions in rotating reference frames, including famous experiments with the atomic clocks flown around the Earth(seereferencesinChap.5,Sects.5.7and5.8).InChapter6thereaderwillfind adescriptionofthe rotationalmotionofarodandmotionofchargedparticlesina magnetic field (Sects. 6.3 and 6.4), and in Chapter 8 accelerated superluminal mo- tionisconsidered(Sects.8.10and8.12). Ratherthanbeingatextbookoramonograph,thebookisaself-consistentcollection ofselectedtopicsinSpecialRelativityandadjacentareas,whichareallarrangedina logicalsequence.Theyhavebeenselectedandarediscussedinsuchawayastopro- videtheabove-mentionedcategoriesofreaderswithinterestingmaterialforstudyor futurethought.Thebookprovidesnumerousexamplesofsomeofthemostparadox- ical-seemingaspectsofthetheory.Whatcancontributemoretotherealunderstand- ingofatheorythanresolvingitsparadoxes?ParaphrasingMartinGardner[3],“you have to know where and why opponents of Einstein go wrong, to know something aboutrelativitytheory.” ThefirstthreechapterscovertraditionaltopicssuchastheMichelson–Morleyex- periment,Lorentztransformations,etc. Afewchaptersdealwiththestrangeworldofsuperluminalvelocitiesandtachyons, and other topics hardly to be found elsewhere. Their investigation takes us to the boundariesof the permissiblein relativity theory, exploring the remote domains of superluminalphenomena,whileatthesametimeservingasthefoundationofadee- perunderstandingofEinstein’suniquecontributiontoscientificthought. Initiallytheappearanceofthetheoryofrelativity,withitsabsoluteinsistencethatno signalcarryinginformationcantravelfasterthanlightinavacuum,createdtheopi- nion among many that no superluminal motion of any kind was possible. In this bookagreatmanyphenomenaaredescribedinwhichsuperluminalmotionseems to appear or does appear. Such phenomena may occur in some astrophysical pro- cesses,inphysicallaboratories,andevenineverydaylife.Howeverincrediblesome ofthemmight seem,theyareallshowntobeinaccordancewithSpecialRelativity, since in an almost mysterious accord with the overriding dictates of the theory, subtledetailsalways conspiretoinsurethat noneofthesephenomena canbeused Preface XI forsignal transmissionfasterthan light in avacuum. And Special Relativity isjust thekindoftheoryfordescribingadequatelythiskindofmotion. A couple of decades ago there was a great controversy in the scientific literature about hypothetical superluminal particles – tachyons. After extensive discussion it was decided by the overwhelming majority of physicists that tachyons cannot exist sincetheirexistencewouldbringaboutviolationsincausality,plungingtheUniverse into unresolvable paradoxes, by changing the past. There are numerous papers which argue that the kind of tachyon hypothesized in the early discussionscannot exist (seethe referencesin Chap.8). Yet the readerof this bookwill find a description of real tachyon-like objectsthat can be “manufactured“ in the laboratory. They possessakindofduality,whichallowsonetorepresentatachyon-likeobjectaseither asuperluminalorsubliminalobject,dependingonwhatphysicalquantitiesarecho- senforitsdescription. Manyofthesetopicsarehardlytobefoundelsewhere,andsomeofthemhavesofar onlybeenpublishedinafewhighlyspecializedprofessionaljournals.Inthisrespect thisbookshouldbeauniquesourceofinformationforbroadcategoriesofreaders. As already mentioned above,the bookis intended to satisfy also the demands of thosereaderswithaminimal backgroundinmath.Theywillfindinmanydescrip- tions an easy part showing the inner core of a phenomenon, its physical picture. Thesereaderscanstopatthispoint–theyhavegraspedthemainidea. Forthebetterprepared,aftertheyhavebeenmadecapableofseeingtherathercompli- catedfeaturesinvolved,therefollowsaquantitativedescriptionwiththeequationsand other details. Many of the examples discussed are unusual and thought provoking; theyoftenstartasunsolvableparadoxes,tobe,afterafewunexpectedturns,finallyre- solved.OnecanfindanexampleofsuchanapproachinChapter5,Section5.4. Anotherexampleofthisapproachcanbefoundinthediscussionofphaseandgroup velocities(Chap.6,Sect.6.12).Theyarediscussedonthreedifferentlevels.Thefirst – intuitive – gives a pictorial representation of the phenomenon using a simple model. This will help the beginnerwith no mathat allto grasp the relationship be- tweenthetwovelocities.Thenthesamerelationshipisobtainedgraphically.Finally, itisobtainedbyanalyzingthesuperpositionoftwowavefunctions.Thelasttwole- velsare appropriate foreverybodyfamiliar with college math. The firstonemay be goodfortwoextremecategoriesofreader:theleastpreparedattheonepole,andthe mostsophisticated(e.g.collegeprofessors)attheother.Theformermayfinditgood tolearn,whilethelattermayfinditgoodtoteach. Insummary,thebookcanbeusedassupplementaryreadingforcollegestudents takingcoursesinphysics.Highschoolandcollegeteacherscanuseitasapoolofex- amplesforclassdiscussion.Further,becauseitcontainsmuchnewmaterialbeyond standardcollegeprograms,itmaybeofinterestforallthosecuriousaboutthework- ingsofNature.Amathematicalbackgroundontheundergraduatelevelwillbehelp- ful in understanding quantitative details. More advanced readers can find in the bookmuchthought-provokingmaterial,andprofessionalphysicists,whileskipping thetopicsthatarefamiliartothem,orwrittenontheelementarylevel,maywellfind somenewinsightsthereorseeaprobleminafreshlight. XIII Acknowledgements IamgratefultoBorisBolotowsky,JulianIvanchenko,andGregoryMatloff,whoen- couraged me to keep on working on the book on its earlier stages. Stephen Rosen and Leo Silber helped me with their comments and good advise. Slawomir Piatek spentmuchofhistimediscussingwithmeafewsamplechapters,andIusedhisin- sightfulremarksintherevisedversionofthetext.YuryAbramianinfarawayArme- nia helped me in my searches for a few references in Russian scientific literature. MyeldersonAlbertmadethefrontcoverofthebook.RolandWengenmayr,inanex- tensivecollaboration,whichIfoundveryrewarding,turnedAlbert’sandmineinitial crude sketches into line drawings, and then created in his illustrations a series of characters,which,inmyopinion,perfectlymatchthetext. MyspecialthankstomyyoungersonVadimforhisvicious,butconstructivecriti- cismofthefirstdraftsofthemanuscriptandforhisinvaluabletechnicalhelp;and also to David Green for his time and angelic patience in translating my version of the English language into English (any remaining linguistic and other errors that mighthavesurvivedandslippedintothefinaltextaretobeblamedentirelyonme). IwishtothanktheconsultingeditorEdmundImmergutforhisprofessionalgui- danceinfindingthemostappropriatepublisherforthisbook. IenjoyedworkingwithVeraPalmer,thepublishingeditoratWilew-VCH,and,on thelateststages,withMelanieRohnandPeterBielintheintensivecopy-editingpro- cess. IamdeeplygratefultomywifeSophiewhodidallinherpowertosavememore timeforwriting. SpecialRelativityandMotionsFasterthanLight.MosesFayngold Copyright(cid:1)2002WILEY-VCHVerlagGmbH,Weinheim ISBN:3-527-40344-2 1 1 Introduction 1.1 Relativity?Whatisitabout? OneofthecornerstonesoftheSpecialTheoryofRelativityisthePrincipleofRelativ- ity.Agoodstartingpointfordiscussingitmaybeabattlefield.Soimagineabattle- fieldwithdeadlybulletswhistlingaroundandletmeaskaquestion:couldyoucatch suchabulletwithyourbarehands? Thelikelyansweris:“NotI.You’dbettertrytodoityourself!” Whichimplies:that’simpossible. Irememberthat,asaschoolboy,Ihadgivenpreciselythesameanswertothisques- tion.ButthenIreadastoryaboutapilotinWorldWarIwhohadinoneofhisflight missionsnoticedastrangeobjectmovingalongsidetheplane,rightnearthecockpit. Thecockpitscouldeasilybeopenedinthosetimes,sothepilotjuststretchedouthis armandgrabbedtheobject.Hesawthatwhathehadcaughtwas…abullet.Ithad beenfiredathisplaneandwasatthefinalstageofitsflightwhenitcaughtupwith theplaneandwascaughtitself. Thestoryshowsthatyoureallycancatchaflyingbullet.Nowadays,havingspace- ships,onecan,inprinciple,catchaballisticmissile.Assumingunlimitedtechnolo- gical development,we do not see anything that would prevent us from “catching” anyobjectbycatchingupwithit–beitasolid,aliquid,orajetofplasma–nomat- terhowfastitismoving.Ifanaturalobjecthadbeenacceleratedtoacertainspeed, thenahumanbeing,whoisalsoanaturalobject,can(although,perhaps,ataslower rate)beaccelerateduptothesamespeed. Weseethatthevelocityofanobjectisasortof “flexible”characteristic.Thebullet thatisperceivedbyaground-basedobservertobemovingappearstobeatresttothe pilot.Wewillcallsuchquantitiesobserver-dependent,orrelative. Notallofthephysicalquantitiesarerelative.Someofthemareobserver-independent, orabsolute.Forexample,thepilotmayhavenoticedthatthebullethehadcaughtwas madeofleadandcoatedwithsteel,andthemassratioofleadandsteelinitis24:1. Thisproperty ofthebulletisabsolutebecauseitistrueforanyoneindependently of one’sstateofmotion.Thegunnerwhohadfiredthebulletwillagreewiththepiloton theratio24:1characterizingitscomposition.Buthewilldisagreeonitsvelocity.Hewill holdthatthebulletmoveswithhighspeedwhereasitisobviouslyatrestforthepilot. SpecialRelativityandMotionsFasterthanLight.MosesFayngold Copyright(cid:1)2002WILEY-VCHVerlagGmbH,Weinheim ISBN:3-527-40344-2