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TheAxiomatizationofPhysics-Step1
ADerivationoftheLorentzTransformation
EugeneShubert
www.everythingimportant.org
updatedJanuary27,2007
Abstract
TheaimofTheAxiomatizationofPhysics–Step1istodefinephysicsand
thenaxiomatizeanincreasinglycomplexhierarchyofmathematicalmodelsof
spacetime.Webeginbybuildingtoymodelsofspacetimeandthenprogress
slowingsothathighschoolstudentswhohavemasteredalgebracanunderstand
thetheoryofrelativityeasily.Spacetimeinhighdimensionsbeginsbydefining
spaceandtimeinonespatialdimension.
1WhyAxiomatizePhysics?
Inthehistoryofphysics,ideasthatwereonceseentobefundamental,
general,andinescapablepartsofthetheoreticalframeworkaresometimes
laterseentobeconsequent,special,andbutonepossibilityamongmanyina
yetmoregeneraltheoreticalframework....Examplesaretheearth-centered
pictureofthesolarsystem,theNewtoniannotionoftime,theexactstatus
ofthelawsofthermodynamics,theEuclideanlawsofspatialgeometry,
andclassicaldeterminism.Inviewofthishistory,itisappropriatetoask
ofanycurrenttheory“whichideasaretrulyfundamentalandwhichare
‘excessbaggage’.”J.B.Hartle,ClassicalphysicsandHamiltonianquantum
mechanicsasrelicsoftheBigBang,PhysicaScriptaT36(1991),228-236.
“Thereductionistapproach—explainingphysicalphenomenainterms
ofsimple,mathematicallyprecise,quantities—hasbeenextraordinarily
successfulinalmostallareasofphysics.Itgoesagainsteverythingwehave
learnedaboutnaturetoproposeatheoryinwhichcomplicatedmacroscopic
objects,whoseprecisedefinitionmustultimatelybearbitrary,arefunda-
mentalquantities.”A.Kent,Againstmany-worldsinterpretations,Int.J.
Mod.Phys.5(1990),1745-1762.
“Agreatphysicaltheoryisnotmatureuntilithasbeenputinapre-
cisemathematicalform,anditisoftenonlyinsuchamatureformthatit
admitsclearanswerstoconceptualproblems.”A.S.Wightman,Hilbert’s
sixthproblem:mathematicaltreatmentoftheaxiomsofphysics,in:Proc.
Sympos.PureMath.,Vol.28,AMS,1976,pp.147-220.
1
2DavidHilbert’sPhilosophyofPhysics
Physicsshouldevolvefromasmallnumberofaxioms
“Ifgeometryistoserveasamodelforthetreatmentofphysicalaxioms,weshall
tryfirstbyasmallnumberofaxiomstoincludeaslargeaclassaspossibleofphysical
phenomena,andthenbyadjoiningnewaxiomstoarrivegraduallyatthemorespecial
theories....Themathematicianwillhavealsototakeaccountnotonlyofthosetheories
comingneartoreality,butalso,asingeometry,ofalllogicallypossibletheories.He
mustbealwaysalerttoobtainacompletesurveyofallconclusionsderivablefromthe
systemofaxiomsassumed.”DavidHilbert,InternationalCongressofMathematicians,
ParisFrance,1900. [1] .
Thedefinitionofphysics
Physicsisthemathematicalstudyofallconceivableuniverses.Auniverseisa
mathematicalmodelthatdescribesspacetime,matter,energyandtheirinteractions.
Thinkofeachmodeluniverseasfillingonepageintheatlasofallpossibleuniverses.
“Philosophyiswritteninthisgrandbook,theuniverse,...Butthebookcannotbe
understoodunlessonefirstlearnstocomprehendthelanguageandreadthecharacters
inwhichitiswritten.Itiswritteninthelanguageofmathematics.”—GalileoGalilei.
3TheCoordinatizationofSpace
Imaginetwoisolatedpoints.Twopointsdeterminealine.Thinkofonepointasbeing
onyourleft.Assignitthenumber0.Thinkoftheotherpointasbeingonyourright.
Assignitthenumber1.Nowassumethatthereisawaytomeasuredistancebetween
points.Placethenumber2totherightof1thesamedistanceas0isfrom1.Imagine
thatprocedurebeingrepeatedindefinitelyforallthewholenumbers.Noticethatfor
everynumber,ifweadd1toit,wemovetotherightonestep.Logicallythen,forevery
number,ifwesubtract1fromit,wemovetotheleftonestep.That’showweassign
numberstotheleftof0.Here’sthepicture(it’suptoyoutoimaginetheintegers
extendingindefinitelyinbothdirections):
··· -9-8-7-6-5-4-3-2-10123456789···
Figure1
Inmathematics,fractionsarecalledrationalnumbers.Theyareeasytovisualize.To
understandwhataquotientoftwointegersmeansgeometrically,itsucestoconsider
thelinesegment[0,1].Thefraction1/2isthepointhalfwaybetween0and1.Halfway
meansthattherearetwoequaldistances.Thetwofractions1/3,2/3imply3equal
distances.Thedistancebetween0and1/3,1/3to2/3,and2/3to1areallequal.For
anynumbern,it’seasytoconceiveofn+2equallyspacedpointsonthelinesegment
[0,1].Thesepointsareassignedthelocationcoordinates0=0/(n+1),1/(n+1),
2/(n+1),3/(n+1),...(n+1)/(n+1)=1.
Intuitively,everypointonthelineisarbitrarilyclosetosomefraction.Weonly
needtomakethenumbernlargeenough.
2
Inancienttimes,therewerephilosopherswhobelievedthateverypointonanumber
linemayberepresentedasthequotientoftwointegers. [2] . ItwastheGreekphilosopher
HippasusofMetapontum,borncirca500B.C.inMagnaGraecia,whoprovedthat
presuppositionthattobefalse. [3] .
AconsequenceofthePythagore an theoremisthatthehypotenuseofarighttriangle
withequalsidesofunitlengthis
p
p
2.Itis
easytoprovethatthe p 2isnotaquotientoftwointegers.Numberslikethat,which
evidentlyexistbutarenotrational,aresaidtobeirrational.
Realnumbersareamildlyabstractgeneralizationofpracticalmeasurements.Real
numbersareeasytograspintuitively.Thefirststepistodefineconvergencefroma
mathematicaldefinitionofclosenessofpoints.Thentheessentialideaisremarkably
simple:Itispossibletogetarbitrarilyclosetoanypointonthenumberlinebyspe c-
ifyingaproperlybehavedinfinitesequenceofrationalnumbers.Forinstance,the
2
maybedefinedasthelimitoftheinfinitesequence1.4,1.41,1.414,1.4142,1.41421,
1.414213,1.4142135,...Noticethatthesquareofthesetermsisthesequence1.96,
1.9881,1.999396,1.99996164,1.9999899241,...whichisconvergingtowardthenumber
2.ItisastandardbeginningpointinmathtodefinerealnumbersintermsofCauchy
sequencesofrationalnumbers. [4] .
Itisoftenthecasewhererealnumbersarenotthoughtofasrepresentingpointson
alinebutonlyasanabstractsethavingalgebraicproperties,whereanytwoelements
ofthesetcanbeaddedtogetherormultiplied,andsoon.Whentherealnumberline
isinview,andthedistancebetweenpointsisimportant,thentherealnumberlineis
oftencalledaone-dimensionalEuclideanspace.AEuclideanspaceofonedimension
hasthefollowingdefinitionofdistancebetweenpoints:Ifrandsarerealnumbers,
thenthedistancebetweenrandsisdefinedbytheabsolutevalueoftheirnumerical
dierence.Inmathematicalparlance,d(r,s)=|r−s|.
p
4MyFirstToyUniverse
Littlechildrenknowintuitivelythatatinyarrowthatmovessteadilyalongacontinuum
ofnumbersisaclock.Wewilltakethenextstepupinageneralizationofthat.Canyou
easilyconceptualizetherebeingaclockateverypointintheuniverse?Marvelous!Our
beginningpointinderivingtheLorentztransformationistograsptheideathat,for
everypossiblespeed,thereisanimaginaryclockmovingthrougheachpointatevery
instant.Howcanweachievethis?Webeginbyconstructingthesimplestuniverse
imaginable.Thesymbolforitis 2 .Thisuniverseconsistsoftwoone-dimensional
Euclideanspaces.
2 ! ··· -9-8-7-6-5-4-3-2-10123456789 ···
1 ···-9-8-7-6-5-4-3-2-10123456789···
Figure2
3
2.Ifweplacethishypotenuseonthenumberline
withoneendonzero0andtheotherendonthepositivenumbers,thenthat s econd
endpointwillspecifyap o intonthenumberlinewiththecoordinatevalue
 
Figure2isaninvitationtopicturethetwoabstractlines 1 and 2 aspristine,
frictionlessrulers. 1 and 2 layflatagainsteachother.Thereisnospacebetweenthe
rulers.Theonlyobjectsthatexistarethesetwononmaterialrulers.Inthisuniverse
wecall 2 ,timeandmotionalsoexistsandthearrowsinfigure2tellustoimagine 2
slidingon 1 ataconstantvelocity.
Ingeneral,itishelpfultoknowthedirectionofmotionforthetwolines.Because
thenumber1istotheleftofthenumber2onthenumberline,wewillassignameaning
totheindices1and2anddeclarethat 1 ismovingtotheleftof 2 .Thus,asthe
arrowsofthediagramindicate,allthenumbersi.e.pointsof 2 aremovingtotheright
of 1 .
Themostdistinctphysicalcharacteristicorpropertyoftheuniverse 2 isthata
uniquekindofmotionexists.Inadditiontothespatialstructureofthisuniverse,
thereisalsoaninterestingtimestructureaswell.Observethatthereisaverynatural
mathematicalclockthatexistsateverypointof 1 and 2 .Toseetheclockatthe
pointx 2 of 2 ,forinstance,imaginethatthepointx 2 isapointer(visualizeorl)
thatismovingalongthecontinuumofnumbers 1 .Bydefinition,that’saclock.For
theimaginaryclockatthepointx 1 of 1 ,likewise,imaginethatx 1 isapointerthatis
movingalongthecontinuumofnumbers 2 .
Itisimportanttoknowthedierencebetweenanarbitraryconventionandalawof
physics.Pleasenotethatallthemathematicalclocksof 2 assignadirectiontotime
asprogressingfromsmallertolargernumberswhereasallthemathematicalclockson
theline 1 runintheoppositedirection(i.e.,backwards).Thereisnophysicalreason
torejectthisoddity.It’sanarbitrarychoicethatweassignthedirectionoftimeasnot
countingdowntowardmorenegativenumbers.Wewilladoptthepopularconvention
thatallclocksshouldmeasuretimeascontinuallyprogressingtowardlargernumbers.
So,togetallourcoordinateclocksof 1 tobehavelikethemathematicalclocksof 2 ,
wewilldefinetheclocktimeatthegeneralpointx 1 of 1 mathematicallytobethe
negativeofwhatevernumberthearrowatx 1 ispointingto.
Itisnecessarytotranslatethemeaningofthesemathematicalclocks,whichwe’ve
defineduptillnowwithpictures,intothepreferred,universallanguageoffunctions.
Recallthedefinitionofafunction:
Inmathematics,afunctionrelateseachofitsinputstoexactlyoneoutput.Astandard
notationfortheoutputofthefunctionfwiththeinputxisf(x).
Intermsoffunctionsthen,t 1 =f 1 (x 1 ,x 2 )=−x 2 andt 2 =f 2 (x 1 ,x 2 )=x 1 .The
interpretationhasalreadybeengiven.Hereisarestatementthatmighthelptoclarify
themeaningofthemathematicalsymbols.Thefunctionsf 1 andf 2 specifyeverydetail
aboutthemathematicalclockson 1 and 2 ,respectively.Theinputforthesefunctions
isanyorderedpairofnumbers(x 1 ,x 2 )wherex 1 isapointon 1 andx 2 isapointon 2 .
Thesetuplesarepivotalforunderstanding 2 .Theorderedpair(x 1 ,x 2 )istheevent
wherex 1 meets(touches,fliesbyorpassesthrough)x 2 ,howeveryouwanttosayit.The
functionf 1 inputstheevent(x 1 ,x 2 )andassignsanoutputt 1 forthemathematicalclock
atx 1 .Theoutputsaystheeventhappensattimet 1 =−x 2 .Likewise,thefunction
f 2 inputsthesameevent(x 1 ,x 2 )andassignsanoutputt 2 forthemathematicalclock
atx 2 .Thisoutputforthemathematicalclockon 2 saystheeventhappensattime
t 2 =x 1 .
4
Becausemathematicalclocksexpresstimeintermsofrealnumbers,wearethenfree
toselectwhateverscalewedesirefortime.Ifaandbarepositiverealnumbers,then
t 1 7!t 1 /aandt 2 7!t 2 /bismerelychangingthetimescalesuchashourstominutes
orhourstoseconds.Sinceweaimatcreatingbeautifulimaginaryuniversesthathave
ahighdegreeofsymmetry,wewillinsistthata=b.Forthepurposeofnotingthe
similaritybetween 2 andrealworldphysics,wewillassignthesymbolµtorepresent
anappropriatetimescalingfactorandnotethatµhasdimensionsofvelocity.
Asexplained,t 1 =−x 2 /µandt 2 =x 1 /µareperfectlygooddefinitionsofclock
timefortheinfinitelymanyclocksof 1 and 2 ,respectively.Supposethatwewere
tonowreplacealltheseclockswithnewclockspointbypoint—theonlydierence
betweentheoldandnewclocksbeingthatforalltime,past,presentandfuture,the
newclocksdierfromtheoldbybeingsetaheadonehour,orbehindonehouror
byanyotherfixedamountoftime.Thent 1 7!t 1 +g 1 (x 1 )andt 2 7!t 2 +g 2 (x 2 ).
Consequently,themostgeneralsettingformathematicalclocksin 2 isgivenbythe
equationst 1 =−x 2 /µ+g 1 (x 1 )andt 2 =x 1 /µ+g 2 (x 2 ).Thefunctionsg i (x i ),i=1,2are
calledsynchronizationfunctions.
Wehavecreatedtheuniverse 2 withtheunderstandingthateverypointof 2 is
likethepointerofaclockthatmovesequaldistancesinequaltimes.Thismotionis
bestunderstoodbyfirstdefining“propervelocity”andthendoingafewcalculations.
Forthatweneedourtwofundamentalequations:
t 1 =−x 2 /µ+g 1 (x 1 ) (1)
t 2 =x 1 /µ+g 2 (x 2 ) (2)
Propervelocityisdefinedasfollows:
µ 12 = x 1
x 0 1 −x 1
t 0 2 −t 2
;µ 21 = x 2
t 1 =
x 0 2 −x 2
t 0 1 −t 1
(3)
Toillustratethemeaningoftheseequations,pickafixedpointx 2 from 2 ,astart
timet 2 andanendtimet 0 2 .Weunderstandthatthestationarypointx 2 from 2 is
steadilymovingthrough 1 .Accordingtoequation(2),thepointx 2 attimet 2 is
locatedatx 1 =µt 2 −µg 2 (x 2 )on 1 .Thesameequationstatesthatthepointx 2 ata
latertimet 0 2 willbeatx 0 1 wherex 0 1 =µt 0 2 −µg 2 (x 2 ).Weinsertthesevaluesintothe
definitionofµ 12 and,withalittlehighschoolalgebra,findthatµ 12 =µ.Theresult
forµ 12 issimilar,exceptthatwemustuseequation(1).Selectapointx 1 from 1 ,a
beginningtimet 1 andanendtimet 0 1 .Inthatelapsedtimet 0 1 −t 1 ,thepointx 1 ,which
isfixedon 1 ,willmovefromx 2 =−µt 1 +µg 1 (x 1 )tox 0 2 =−µt 0 1 +µg 1 (x 1 ).Plugging
thesevaluesintothedefinitionofµ 21 givesµ 21 =−µ.
Becauseallpointsoftheline 2 haveequalpropervelocities,andlikewiseforthe
line 1 ,weshallsaythatthepropervelocityof 2 withrespectto 1 isµ 12 andthat
thepropervelocityof 1 withrespectto 2 isµ 21 .Pleasenotethatµ 21 =−µ 12 and
thatµ 12 isapositivenumber.
Ibelievethatwehavenowcompletedourstudyof 2 .Tomakethetransitionfrom
thisuniversetothemoremathematicallychallengingoneseasier,Iwillpreemptively
stateafewdefinitionsandobservationssothatthereaderwillrecognizetheconnection.
5
t 2 =
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