Wandering Towards a Goal: How can mindless mathematical ... · Wandering Towards a Goal: How can...

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

1

Theshortansweristhatmindlessmathematicallawscannot,andthereforethattheydonot,giverisetoaimsandintentions. Howisitthathumanhistorycanleadtoanessaycontestontheabovetheme?Andwhatdoesthatchoiceofthemetellusaboutourselvestoday? Ittellsusthatweareintellectuallystillyoungindeed—thatwestilldonotknowwhattherightquestionsare,though(evenso)wehavemademanywonderfulstridestowardeventualanswers.Forpleasedorememberthatitwasonlyabout50,000yearsago,inEastAfrica,thatwehumansevendevelopedspokenlanguage.ThathappenedwiththeappearanceoftheFOXP2gene,inonesmallgroupofhumans,givingthem(well,givingus—fortherestofthemillionorsopeoplethenonEarthleftnodescendants)theincrediblegiftofspeech:theabilitytosymbolicallyrepresentboththingsandideas. Theideasweretherealready,inchoate—forexample,peopleovertheageslookingatVenusinthewesternskyaftersunset!Iwasaguest,longago,inWesternSamoa,andIsaidtomyhost,"TapuiT'ear."IhadlearnedthatfromafriendwhohadvisitedWesternSamoayearsbefore.Myhostturned,andhelookedintothewesternsky:Venus!WhatisVenus? MarcusAureliusthoughtthatthestarsweregods—so,perhaps,didpre-FOXP2humanity. Wehumansarenotverysmart,forhowcanitbethatourmostbrilliantancientcivilizations—thoseofIndia,China,andsomeothers—failedtofindthemagickeytoourpresentintellectualexplosion:thatkeybeingmathematics(yes,mindlessmathematicallaws)mappingastoundinglytoourperceptionsofwhatwecalltheuniverse?Howcoulditbethat,insteadofbeingthefruitofthoseancientandmostgloriouscivilizations,thetrulymagicalandliberatingconnectionappearedandtookfireonlyamidstthechaosofhorribleprimitivemedievalEurope? Newtonmodestly(andcorrectly)saidthathestoodontheshouldersofgiants:yesindeedofGalileo,butNewtonreallywasreferringtothetag-teamofTychoBraheandJohannesKepler:thefantasticconjunctionofTycho,abuilderofgiantcelestialobservingdevices—and,more,theuserofsametopursueMarsrelentlessly—workingwithahumanmathematicalmachine,Kepler,amanofincalculableabilityandpersistence.KeplerworkeddirectlywithTycho,andafterTycho'sdeathusedTycho'sdatatopursueKepler'sown(andquitewrong)ideasaboutthemathematicalstructureofthesolarsystem.Keplerdiedwithoutunderstandingwhathistrueaccomplishmentwas—ittookNewtontoextractitfromKepler'sworkandtherebyopenupthephysicsoftoday:theadventofourwonderfulbutmindlessmathematicallaws.Above,Icriticizedthethemeofthiscontest.Butnowletmewithdrawthatcriticism,simplybecauseofthemarvelousopeningphraseofourtheme:"wanderingtowardsagoal."Therealizationofignoranceisthebeginningofwisdom.IhavementionedhowKepler,supremegenius,wanderedincorrectly,yetachievedgreatness;well,Newton,too,wanderedincorrectly,buthealso,likeKepler,achievedsuccessdespitethefalsityofhiscorebeliefs—forNewtonwasflabbergastedandrepelledbythesuccessofhisownlawof

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

2

gravitation,whichwasnothingbutatinyformula,entirelylackinginthemechanismsthatNewtonsought. TheproblemthatNewtonhad,andthattoomanyofustodaystillhave,istheincorrectnotionthatmathematicsisjustatoolthatisusefulforexploringaphysicaluniverse.Butwhat"physical"meansisnotdefined—itstilltodayhasnomeaningfuldefinition.Acommonwordthatisusedis"real."Well,thereisno"real." Thatistheheartofmyessay.Theworldisnotreal—itreallyisnot.Theworldismental,notphysical.Physicalhasnomeaning.ArthurStanleyEddingtonknewthat,andSirJamesJeansknewthat.In1990IpublishedmyessayinNature,"Thementaluniverse,"anattempttoreviveEddingtonandJeans'pioneeringproposedintellectualbasisforustoproceedmoresuccessfullywithourinvestigationofthatwhichwecalltheuniverse. Whatdoesitmatterthattheworldisnotreal?Itmattersbecauseuntilonegraspsthe,well,reality,ofthatidea,onecannotmosteffectivelyworktowardsthegoal:ifyoudon'tknowtherules,youwillnotplaythegameverywell.Ofcoursemostpeopledonotknowtherules,yettheyneverthelessdoachievesomesuccesses.Thatisbecausemostotherpeoplehavethesameblindnessandthusgoalongwithwhatisinfactanaïveapproachtotheuniverse.Wewouldbemuchbetteroffshouldweadopta,well,again,realistic,approachtoourstudyoftheuniverseandofourselves.Weourselves,unliketheuniverse,arereal. Let'slookatasequenceof"casestudies"ofmytheme,whichIexpectwillconvertyou—startingwiththePythagoreantheorem,whichisthemostfundamentalkeyofall:

Noticethatthisproofofthatmostfamoustheoremdoesnotinvolveanymathematicsatall:wesimplyseethetruthofthetheorem,asapropertyofthatwhichwecalltheexternalworld.Myaimandintentionwiththisfigurewillgiverisetomathematicallaws—firstofall,tothe"planeordinary"Pythagoreantheorem,𝑑𝑠! = 𝑑𝑥! + 𝑑𝑦!. Thatequationisamentalhumanconstruct—itwasdevisedinItalyduringthefourteenthcenturytorepresentthealready-known,fromnature,Pythagoreantheorem.

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

3

Fastforwardnowtothenineteenthcentury,andtoWilliamRowanHamilton,thatsupremegeniusofphysicsandofmathematics,theinventorofquaternions(nowvectors),bywhichinventionHamiltonshowedthatthegreatGausswaswrongwhenGaussclaimed(in1830)that"wemustconfessinallhumilitythat,whilenumberisaproductofourmindalone,spacehasarealitybeyondthemindwhoseruleswecannotcompletelyprescribe."Well,Hamiltonwasindeedagenius—yesindeed;butgeniusdoeshaveitslimits,asweshallnowsee! ForHamiltontriedtoextendthePythagoreantheoremtoencompasstimeasafourthdimension:andour"supreme-genius-of-physics-and-mathematics"Hamiltonfailedutterlyinhisattempt—ahugeembarrassmentinretrospect,butonethatilluminatesthecoreweaknessofourhumanminds.Understandingthatilluminationofourweaknessisofimmensevalueintryingtoremovetheblindnesstodayexemplifiedinourcontesttheme. HowdoIknowthatHamiltontriedandfailed?Iknowitbecause(blesshim)Hamiltonwrotethefollowingpoemrevealinghisabjectfailure:

THE TETRACTYS by William Rowan Hamilton

Of high Mathesis, with her charm severe, Of line and number, was our theme; and we

Sought to behold her unborn progeny, And thrones reserved in Truth’s celestial sphere: While views, before attained, became more clear; And how the One of Time, of Space the Three,

Might, in the Chain of Symbol, girdled be: And when my eager and reverted ear

Caught some faint echoes of an ancient strain, Some shadowy outlines of old thoughts sublime,

Gently he smiled to see, revived again, In later age, and occidental clime, A dimly traced Pythagorean lore,

A westward floating, mystic dream of FOUR.IputthisforwardnottoshameHamilton'sghostbuttoencourageeveryreaderofthisessay:thegreatHamilton(andhewasindeedgreat)wassubjecttoanutterlynaïveprejudice:Hamiltoncouldnotbringhimself,inplaceof

ℎ! = 𝑎! + 𝑏! + 𝑐! + 𝑡! towrite,andtoexploreℎ! = 𝑎! + 𝑏! + 𝑐! − 𝑡!Thelatter,ofcourse,isEinstein's1905TheoryofSpecialRelativity,complete.HadHamiltonexploredthat(inretrospectobvious)possibility,thenwithHamilton'smathematicalabilitythereisnoquestionthathewouldhaveanticipatedEinstein.Andindeed,EinsteinhimselfwasjustasblindaswasHamilton—itwasonlyEinstein'steacher,

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

4

HermannMinkowski,whofinallyrealizedexactlywhatEinstein'stwooriginal(andpatheticallynaïve)"postulates"actuallyimpliedabouttheinnernatureoftime. Carefulnow,dearreader:don'tmaketheterriblemistakeofacceptingtheresultbutmissingthepoint:wemustanalyzetheresult,indetail,tograspitshugeimplications—notfortheuniverse,butforourownminds. Geometryisahumaninvention:possiblyoriginallycreatedsimplytore-establishlandboundariesaftertheannualfloodingoftheNile?Thatearlymathematicswas,muchlater,expandedbyItalianmathematicianstoencompassalgebraicequations.Andnow,post-Einstein,abruptly,throughthemindofMinkowskiin1908,revealedtoallowhumanaccesstothesecretofthenatureoftimeitself,throughMinkowski'sℎ! = 𝑎! + 𝑏! + 𝑐! − 𝑡! Thatsingle,simple,equationcontainsthemostimportantphysicswehaveeverdiscovered.Wearelookingformeaninginthisessay,notjusttechnicalanswerstoquestions!Soletmepersist:ifweinsist,asweusuallydo,ontinourequationbeingexpressedinseconds,thenwemustmultiplytbysomenumberctoconvertittometers,soastohaveourequationbeconsistent.Butwhatisthenumericalvalueoftherequiredunits-conversionfactorc? Byexperiment(aclockflownathighspeedaroundtheworld,andthetimetaken,asmeasuredaboardthemovingplane,comparedwiththe(supposedly)sametimetaken,butasmeasuredwithanidenticalclockthathadbeenleftbehindonthetarmac)thenumericalvalueofcwasdeterminedtobeveryclosetoc=299,792,458meterspersecond—recognizedofcourseastheknownspeedoflight. So,today,weusuallychoosetowrite

h2=a2+b2+c2-(ct)2

or,ifwealsodecidetowrite,fortheseparationinspace-aloneoftwoevents(say,oftwosnapsofmyfingers,asIwaveanarmabout)

d2=a2+b2+c2thenweareabletomorecompactlywrite

h2=d2-c2t2

Thattiny“Pythagorean”equationis—complete—Einstein’stheoryofrelativity:allelse,includingE=mc2andtheatomicbomb,canbededucedfromit(aftermultiplyingthroughbym2)usingnothingbutthemagic—andIusethatwordadvisedly—ofourhuman-inventedalgebra.(Forthereader’sstudentsIprovidemyproofintheAPPENDIX.) Butwait—algebraisindeedahumaninvention—andindeed,isarecenthumaninvention. Whyshouldalgebra,whichwasinventedbyus,produceapredictionoftheatomicbomb?Yet,algebradoes!Wehavenoideawhy—butitisso.

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

5

Our“Pythagorean”equationnoweasilyshowsusthat"photonsoflight"arebuta,dareIsay,“whitelie”ofthephysicists(thoughstillausefultoolforstudentsandforengineers):forif,insteadofourtwoeventsbeingtwosnapsofourfingersatdifferenttimesandlocations,wechooseforourtwoevents:

Event#1,aphotonoflightiscreatedonthesunandheadstowardEarth;and,

Event#2,8minuteslater,thatphotonarrivesatEarth,andisabsorbedinaretina theninthatcasedisthedistancefromthesuntotheEarth,andtisthetimetakenbythephotoningoingfromthesuntoyoureye. Now,ofcoursedistanceisequaltovelocitytimestime:d=vt.Substitutingthattinynewequationintoourredrelativityequationgives

h2=v2t2-c2t2

wherehistheseparationinspace-and-time(thatis,inouruniverse)ofthetwoevents:thebirthofthephotononthesun,andtheextinctionofthatsamephoton,ingeneratinganelectricalsignalintheeyeofsomepersononEarth. Wehavereachedthecrux:thespeedvofaphotonoflightis,byexperiment,cmeterspersecond.Settingv=cshowsthattheseparationinouruniversebetweenthetwoeventsofaphoton’sbirth,andofthatsamephoton’sextinction,is

c2t2-c2t2=h2=0

Soinspacetime—thatis,inouruniverse—thereisnoseparationatallbetweenthetwoevents:thusoursupposedphotoncanhavenoexistence.BetweenitsemissiononthesunanditsreceptiononEarth,thephotonismostdefinitelynot"outthereandonitsway."

“All these fifty years of conscious brooding have brought me no nearer to the answer to the question, 'What are light quanta?' Nowadays every Tom, Dick and Harry thinks he knows it, but he is mistaken.” (Albert Einstein, 1954) Thatisabigpartofwhywegettheodditiesofthewell-knownquantum-mechanicaldouble-slitexperiment:theresimplyarenophotons—thereareonlyresults:whicharethemeasurementsthemselves. Butwait,there'smore! Supposetwoobservers(onemovingwithvelocityv1andtheothermovingwithvelocityv2)witnessthosesametwoevents:byhypothesishisinvariant,so

v12t12-c2t12=h2=v22t22-c2t22orv12t12-c2t12=h2=0-c2t22ifthesecondobserverhasv2=zeroorv12t12-c2t12=-c2t22

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

6

Looksharplyatthatlastequation! Thesquareofanynumberisalwayspositive.Sobecauseoftheminussign,therighthandsideoftheequationisnecessarilyanegativenumber.Butifv1isfasterthanlight(thatis,ifv1isgreaterthanc)thelefthandsideofthatequationwouldbeapositivenumber,whichcannotbeequaltoanegativenumber.Soourequationassertsthatvelocitiesgreaterthanthatoflightsimplycannotoccur:ifanyonesaysthattheycan,thatwouldbeanalogoustosomeone—absurdly—announcingthattheycangonorthofthenorthpole. Ourequationproducedtheatomicbomb—ourequationisknowntobetrue:soweconcludethatvelocitiesfasterthanlightarenotjustnotpossible,theyareinconceivable—the4-dimensionalgeometryofspacetimesimplydoesnotpermitit—theresimplyisnofaster. Solightisdiscoveredtogoasfastasitispossibletogo.Asdogravitonsandgluons. Ifphotonsdonotexist(andwehavejustseenthattheydonot)neitherinfactdoevenelectronsoranyotherparticles(andthishasalsobeenconfirmedbyexperimentaltests).Sotheuniversedoesnotexist,exceptinourmindsasexperiences.Yes,yourmindisreal,andyes,theobservationsarereal:buttheobservationsarenotobservationsofanything. Goodphysicistsareperfectlywellawareofthis,butitisasmind-bogglingtothemasitistoanyone,andthereisnoclearconclusionastowhatallofthisreallymeans:arewebutdreamsinthemindofGod?Whoknows?Sophysicistsgenerallysticktoonly—atleastinpublic—discussingafakeuniverseofthings—forwhymakeyourselfsoundlikeafool,evenifyouknowthatyouarenot?Tomakeprogressintheirphysics,allphysicistsknowthattheymuststicktotheirequations,becausetheyhavelearnedthatonlybymeansofthoseequationsdotheygetanswersthatturnouttobecorrectwhencheckedbyexperiment. IfthePythagoreantheoremisworkedoutnotinaflatspacetimebutinacurvedspacetime,itsalgebraicformchanges,ofcourse.Andifweapplythesamemagicthatwelearnedtoadoptfortime(simplychangingtheone+signtoa-sign)wequicklygetEinstein’stheoryofGeneralRelativity. Curvedspacetime—asjustslightlypatchedupbyEinsteinsoastoensureconservationofenergy—wasfoundin1915toyieldNewton’sLawofUniversalGravitationasitsfirstapproximation(butonlyifafactorof8πwasstuckin—for,still,nounderstoodreason)and,wonderfully,Einstein'snewtheorypredictedmanysince-confirmedeffects(includingthemostgloriousrecentlydetectedgravitationalwaves)neitherpredictedbynorexplicablewithNewton’soriginalgravitation:another,andhuge,successforAlbertEinstein. Relativity—eveninitsgeneralform—isasimpletheory.Itisperhapsverystrange,butitiscertainlytrue:true—bytheacidtestofmanyexperiments.Andithelpsmakecleartousthattheuniverseisamentalmystery.

Thereisabiggermysterythanrelativityorevenquantummechanics:themysteryofthehumanmind—ofitsstrengthsandofitsfoibles.ThehumanracehasproducedsuchasRamanujan,instinctivelysuperblymathematical.Thatsuggeststhatevolutioncouldresult

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

7

ininstinctivemathematicalbrillianceinall,induecourse.Andthankfullythehumanraceneednotpayanypriceforthat:thebrilliantphysicistArthurStanleyEddington,forexample,wasbothmathematicallysupremelygifted,andyetwasalsoasuperblyrounded,albeitimperfect,humanbeing. ButevenEddingtonwasunabletothinkupthatmagicminussign—theminussignthatcantodayeasilyrevealrelativitycorrectlyeventoelementaryschoolstudents.Butforthatmatter,aswehaveseen,neitherdidEinsteinhimself!Instead,Einstein,in1905,putforwardtwoclumsypostulates,postulatesthatwerenotmathematical,butratherweresentencesinGerman.Yes,thosesentencesdidwork,despitetheirclumsiness,butthreeyearslater,Einstein’sformerprofessor,HermannMinkowski,finallydidcomeupwiththePythagoreanminussignthatabruptlyrevealedEinstein’srelativitytheorytobemostbeautifulandnatural,ratherthanfreakishandarbitrary. Sowhataboutquantummechanics? Inmyexperience,thevastmajorityofwell-educatedpeoplewhoareacquaintedwithquantummechanicsbelieveittobedeeplymysterious.Indeed,Irecallarevealingtalk,beforeascientificaudience,byRogerPenrose.Itispowerfultoreceivetheideasofluminariesinperson,ratherthanjustfromtheirwritings,becauseinperson,theysaythingsthattheydonot,incoldblood,write. Rogerwasemotionalonthesubjectofquantummechanics! Now,thisessayofminesummarizesmypersonaljourneyoversomedecades. TeachingphysicshadgraduallyrevealedtomewhatIhavesummarizedabove,andso,happily,itfinallydawnedonmethatquantummechanicstoo,mustsurelybenatural,andmustbepurelymentalinitsorigins. So,onesummeratLosAlamos,IsatdownandIsaidtomyself,supposeIjustask,“whatisanobservation?”Anyobservation,ofanything?Theanswerclearlywas,thatwhencloselyexamined,anyobservationis“anumber.”Thatwasthelaunchingpointformypaper“Quantummechanicsmadetransparent.”AsIcomposedmypaper,quantummechanicsappeared,tomygreatpleasure,automatically,simplyfromconsideringnumbers,Noether’stheorem,andnothingelse.Quantummechanicswasnatural,andquantummechanicswasinevitable! Subsequent(andmuchmoresophisticated)abinitioderivationsofquantummechanicshavecometoexist:Shapiro(2008). Soletmefinallysumup:wanderingTowardsaGoal:Howcanmindlessmathematicallawsgiverisetoaimsandintention:badquestion!Lawscannotgiverisetoanythingatall:lawsarequantitativeabstractmathematicalconclusionsthatcanonlybedrawnfromobservationsandthedisciplinedpowerofourminds.Absentobservations,mindscouldonlyhypothesizeconceivablelaws.Absentobservation,mindhasnothingtosearchforanypossiblelaws.Sotheentirepremiseofthiscontestishopelesslymuddled.Butwhyisthat?

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

8

Itisclearlybecauseofyearningonthepartofthosehumanspirits—thoseminds—whose“aimsandintentions”gaverisetothecontestinthefirstplace! Goodforthosespirits!Theyshallberewarded: Icouldproposeafollow-upessaycontest:Howcanaimsandintentiongiverisetomathematicallaws?Iwouldwinthatcontest—indeed,I’llhappilyshowhowI'dwinitrightthisminute:itisclearthataimsandintentions—consciousspirits—thatdonotgiverisetomathematicallawswillbe‘aimsandintentions’—thatis,minds—thatonlywanderinchoatein—literal—darkness.Onlyaimsandintentionsthatgiverisetomathematicallawswillfindthemselvesinapositionwheretheycan,forexample,launchintellectualcontests.Keplerwasthereforewell-motivatedindeedinlookingforsymmetries—hewassimplylookingforthewrongsymmetriesinthewrongplaces.ItwasEmmyNoetherwhofoundforusthenecessarysymmetries—thosesymmetriesthatsnatchcoherenceoutofchaosandthattherebyallowthehumanmindandthatalso,necessarily,requirethattherebeavastandemptyuniverse—withnopurposeotherthantosimplyprovidethestageonwhichwehumansdelightfullypranceanddance.References:Henry,RichardConn,"Quantummechanicsmadetransparent,"1990,AmericanJournalof

Physics,58,1087,1990Henry,RichardConn,"Thementaluniverse,"2005,Nature,436,29,2005Henry,RichardConn,citedin:https://en.wikipedia.org/wiki/Kretschmann_scalarHenry,RichardConn,additionalhelpforyou:http://henry.pha.jhu.edu/trivial.pdfShapiro,Moshe,"Derivationoftherelativistic'proper-time'quantumevolutionequations

fromcanonicalinvariance,2008,J.PhysicsA:Mathematical&Theoretical,41,175303

(Alltheweblinkswereverifiedon2017March07Tuesday:whichwasmy77thbirthday!)

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

9

Appendix: to show that E = mc2 Manyreadersofthisessay,Iexpect,willbeteachers,whomightappreciateatransparentderivationofthatmostfamousformulaE=mc2fromtheequallyfamoustheoremofPythagoras(sotrivially—butsoconsequentially—extendedbyMinkowski'simprovingonAlbertEinstein)—

h2=d2-c2t2wherehistheseparation(inspacebyd;intimebyt)ofanytwoevents,suchasthespaceship(redheart)leavingthesun,andarrivingatEarth,ofmyessay.[OfcourseIignoredgravitycompletely(Ihopeyounoticedthat)—Ichosethatgeographyjusttoestablishaneasilyvisualizedandacceptedgreatdistance.]Hereisthattrip:d=dxisthedistancefromthesuntoEarth,t=dtthetimethetriptakes,andtheprimedcoordinatesystemisthatofyourmovingsisterredheart,theunprimedcoordinatesystembeingthatofEarth.

𝑑ℎ! = 𝑑𝑥! − 𝑐!𝑑𝑡! = 𝑑𝑥!! − 𝑐!𝑑𝑡!"Thatequalityistheonlyhypothesisofspecialrelativity.Thehypothesisisthatdhisinvariant:thatdhisthesameforyouonEarth,asitisforyourmovingsister,redheart.Sotheclaimisthat(slightlyreorganizingourequation)

𝑐!𝑑𝑡!" − 𝑑𝑥!! = 𝑐!𝑑𝑡! − 𝑑𝑥!Ourtwochoseneventsareredheartsnappingherfingersassheleavesthesun,andredheartsnappingherfingersonceagainasshearrivesatEarth.WeonEarthhavestrategicallylocatedtwocarefullysynchronizedclocks,oneonEarth,theotheroperatedbyagraduatestudentlocatedatthesun.dt′isthedurationofherjourneyasmeasuredbyherwristwatch,whiledtisthesametimeintervalasmeasuredbyus,stationary.Snaptest:studythetwodiagramsagain,andtellmethenumericalvalueofdx′.

Wandering Towards a Goal: How can mindless mathematical laws give rise to aims and intention? Richard Conn Henry

10

Right!dx′=0becauseitisthedistancebetweenthetwofinger-snapsasmeasured,inherownreferenceframe,byyoursister.Shecarriesthatreferenceframewithherasshemoves.Thetwoeventstakeplaceattheidenticallocationinherreferenceframe.Nowwelaunchonabunchofalgebra.Itisnothingbutalgebra:noadditionalphysics,ofanykind.(ButhowdoesMotherNatureknowalgebra?Noonehasanyidea,butknowitMotherNaturedoes!)Becauseinredheart‘srestframedx′=0,wedecidetore-labeldt′asdτ,andso 𝑐!𝑑𝜏! = 𝑐!𝑑𝑡! − 𝑑𝑥!.Herrocket’sspeedis𝑣 = 𝑑𝑥/𝑑𝑡asmeasuredbyusonEarth[and𝑢 = 𝑑𝑥/𝑑𝜏asmeasuredbyredheartherself—theverysamedistanceofcourse,but(asweshallbringout)aquitedifferenttimeinterval].Sowerearrange:

𝑐! !!!"

!= 𝑐! − 𝑣!,so

𝑑𝑡𝑑𝜏 =

1

1− 𝑣!

𝑐!

≡ 𝛾 = 1−𝑣!

𝑐!

!!!= 1+

12𝑣!

𝑐! +⋯

Thatlaststepisabinomialexpansion(inventedbyNewton).Sincenormallyvisverymuchsmallerthanc,allhigherorderterms(+⋯)canbeignored.Thefirstpartofourequationdemonstratestimedilation.Iftherocketspeedv(measuredfromEarth)iszero,ourequationsaysthat𝑑𝑡 = 𝑑𝜏andthetimeaboardthe(stationary)shipisthesameastimeonEarth.Butasvapproachesc,thecoefficientofdτapproachesinfinity,andsodτ,thetriptimeasmeasuredbyredheart,approacheszero.Timeslowsdownasoneapproachesthespeedoflight.Andtimedoesnotpassatallforaphoton!Photonsareabsorbedandceasetoexistthesamemomentthattheyarecreated,accordingtotheirownwrist-watches!Nowlet’sreturntoourequation 𝑚!𝑐!𝑑𝜏! = 𝑚!𝑐!𝑑𝑡! −𝑚!𝑑𝑥!whereIhavenowmultipliedbothsidesofourequationbym2(afterall,weseekE=mc2,som,themassofeitheryoursister,orofSisplusherrocket,hastobeintroducedsomehow,doesn’tit?)Fromthat,

𝑚!𝑐! = 𝑚!𝑐! !"!"

!−𝑚! !"

!"

!or𝑚!𝑐! = 𝑚𝑐𝛾 ! −𝑚𝑢!

Noticethatitisthevelocityuthatappearsinthisequation.Nowre-writethisinthisform:

𝑚!𝑐! = 𝑚𝑐 1+12𝑣!

𝑐! +⋯!

−𝑚!𝑢!

𝑚!𝑐! = !!𝑚𝑐! + !

!𝑚𝑣! +⋯

!− 𝑚𝑢 !or,ifwe,now,abandonNewtonandwe

redefinemomentumtobep=muandenergytobe𝐸 = 𝑚𝑐! + !!𝑚𝑣! +⋯then

𝐸! = 𝑐𝑝 ! + 𝑚𝑐! !

Ifredheartisnotmovingatall,hermomentumpiszero,andwehave,atlast:𝐸 = 𝑚𝑐!