A New Approach to Differential Geometry using Clifford's Geometric Algebra

471 Pages·2012·6.167 MB·English

by  John Snygg (auth.)

John Snygg
A New Approach
to Differential Geometry
Using Clifford’s Geometric
Algebra
JohnSnygg
433ProspectSt.
EastOrange,NJ07017
[email protected]
ISBN978-0-8176-8282-8 e-ISBN978-0-8176-8283-5
DOI10.1007/978-0-8176-8283-5
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011940217
MathematicsSubjectClassification(2010):11E88,15A66,53-XX,53-03
(cid:2)c SpringerScience+BusinessMedia,LLC2012
Allrightsreserved.Thisworkmaynotbetranslatedorcopiedinwholeorinpartwithoutthewritten
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,
NY10013, USA),except forbrief excerpts inconnection with reviews orscholarly analysis. Usein
connectionwithanyformofinformationstorageandretrieval,electronicadaptation,computersoftware,
orbysimilarordissimilarmethodologynowknownorhereafterdevelopedisforbidden.
Theuseinthispublicationoftradenames,trademarks,servicemarks,andsimilarterms,eveniftheyare
notidentifiedassuch,isnottobetakenasanexpressionofopinionastowhetherornottheyaresubject
toproprietaryrights.
Printedonacid-freepaper
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To PerttiLounesto
Preface
Thisbookwaswrittenwiththeintentionthatitbeusedasatextforanundergraduate
course.Theendresultisnotonlysuitableforanundergraduatecoursebutalsoideal
foramasterslevelcoursedirectedtowardfuturehighschoolmathteachers.Itisalso
appropriateforanyonewhowantstoacquainthimselforherselfwiththeusefulness
ofCliffordalgebra.Inthatcontext,instructorsteachingPh.D.studentsmaywantto
useitasasourcebook.
Mostintroductorybooksondifferentialgeometryarerestrictedtothreedimen-
sions. I use the notation used by geometersin n-dimensions. Admittedly,most of
myexamplesareinthreedimensions.HoweverinChap.3,Ipresentsomeaspectsof
thefour-dimensionaltheoryofspecialrelativity.Onecanpresentsomefunaspects
ofspecialrelativitywithoutmentionofsuchconceptsas“force”,“momentum”,and
“energy.”For example,addingspeeds possibly near the speed of lightresults in a
sumthatisalwayslessthanthespeedoflight.
The capstonetopic for this bookis Einstein’sgeneraltheoryof relativity.Here
again, knowledge of Newtonian physics is not a prerequisite. Because of the
geometric nature of Einstein’s theory, some interesting aspects can be presented
withoutknowledgeofNewtonianphysics.Inparticular,Idiscussthepossibilityof
twins aging at differentrates, the precession of Mercury,and the bendingof light
rayspassingneartheSunorsomeothermassivebody.
The only topic that does require some knowledge of Newtonian physics is
Huygen’sisochronouspendulumclock,andtherelevantsectionshouldbeconsid-
eredoptional.
Mystrategyforwritingthisbookhadthreesteps:
Step1: Stealasmanygoodpedagogicalideasfromasmanyauthorsaspossible.
Step2: ImproveonthemifIcould.
Step3: Choosetopicsthatarefunformeandfittogetherinacoherentmanner.
Frequently I was able to improve on the presentation of others using Clifford
algebra.
The selection of this book may introduce a hurdle for some instructors. It is
likely they will have to learn something new – Clifford algebra. Paradoxically,
vii
viii Preface
the use of Clifford algebra will make differential geometry more accessible to
students who have completed a course in linear algebra. That is because in this
book,Cliffordalgebrareplacesthemorecomplicatedandlesspowerfulformalism
ofdifferentialforms.Anyonewhoisfamiliarwiththeconceptofnon-commutative
matrixmultiplicationwillfinditeasytomastertheCliffordalgebrapresentedinthis
text.UsingCliffordalgebra,itbecomesunnecessarytodiscussmappingsbackand
forthbetweenthespaceoftangentvectorsandthespaceofdifferentialforms.With
Cliffordalgebra,everythingtakesplaceinonespace.
The fact that Clifford algebra (otherwise knownas “geometric algebra”)is not
deeply embedded in our current curriculum is an accident of history. William
Kingdon Clifford wrote two papers on the topic shortly before his early death
in 1879 at the age of 33. Although Clifford was recognized worldwide as one
of England’s most distinguished mathematicians, he chose to have the first paper
publishedinwhatmusthavebeenaveryobscurejournalatthetime.Quitepossibly
itwasagestureofsupportfortheeffortsofJamesJosephSylvestertoestablishthe
first American graduate program in mathematics at Johns Hopkins University.As
partofhisendeavors,SylvesterfoundedtheAmericanJournalofMathematicsand
Clifford’sfirstpaperonwhatisnowknownasCliffordalgebraappearedinthevery
firstvolumeofthatjournal.
Thesecondpaperwaspublishedafterhisdeathinunfinishedformaspartofhis
collected papers. Both of these papers were ignored and soon forgotten. As late
as 1923, math historian David Eugene Smith discussed Clifford’s achievements
without mentioning “geometric algebra” (Smith, David Eugene 1923). In 1928,
P.A.M.DiracreinventedCliffordalgebratoformulatehisequationfortheelectron.
Thisequationenabledhimtopredictthediscoveryofthepositronin1931.
In 1946 and 1958, Marcel Riesz published some results on Clifford algebra
thatstimulatedDavidHestenestoinvestigatethesubject.In1966,DavidHestenes
publishedathinvolumeentitledSpace-timeAlgebra(Hestenes1966).And18years
later, with his student Garret Sobezyk, he wrote a more extensive book entitled
Clifford Algebra to Geometric Calculus – A Unified Language for Mathematics
andPhysics(HestenesandSobczyk1984).Sincethen,extensiveresearchhasbeen
carriedoutinCliffordalgebrawithamultitudeofapplications.
Had Clifford lived longer, “geometric algebra” would probably have become
mainstreammathematicsnearthebeginningofthetwentiethcentury.Inthedecades
following Clifford’s death, a battle broke out between those who wanted to use
quaternions to do physics and geometry and those who wanted to use vectors.
Quaternions were superior for dealing with rotations, but they are useless in
dimensions higher than three or four without grafting on some extra structure.
Eventuallyvectorswonout.
Sincethestructureofbothquaternionsandvectorsarecontainedintheformalism
ofCliffordalgebra,the debatewouldhavetaken a differentdirectionhadClifford
lived longer. While alive, Clifford was an articulate spokesman and his writing
for popular consumption still gets published from time to time. Had Clifford
participated in the quaternion–vector debate, “geometric algebra” would have
receivedmoreseriousconsideration.
Preface ix
The advantage that quaternions have for dealing with rotations in three
dimensions can be generalized to higher dimensions using Clifford algebra. This
is important for dealing with the most important feature of a surface in any
dimension–namelyitscurvature.
Suppose you were able to walk from the North Pole along a curve of constant
longitudetotheequator,thenwalkeastalongtheequatorfor37ı andfinallyreturn
totheNorthPolealonganothercurveofconstantlongitude.Inaddition,supposeat
the start of your trek, you picked up a spear, pointed it in the south direction and
thenavoidedanyrotationofthespearwithrespecttothesurfaceoftheearthduring
yourlongjourney.Ifyouwerecareful,thespearwouldremainpointedsouthduring
theentiretrip.However,onyourreturntotheNorthPole,youwoulddiscoverthat
yourspearhadundergonea37ı rotationfromitsinitialposition.Thisrotationisa
measureofthecurvatureoftheEarth’ssurface.
ThecomponentsoftheRiemanntensor,usedtomeasurecurvature,aresomewhat
abstract in the usual formalism. Using Clifford algebra, the components of the
Riemann tensor can be interpreted as components of an infinitesimal rotation
operatorthatindicateswhathappenswhenavectoris“paralleltransported”around
aninfinitesimalloopinacurvedspace.
Inmanycoursesondifferentialgeometry,theGauss–BonnetFormulaisthecap-
stoneresult.ExploitingthepowerofCliffordalgebra,aproofappearsslightlyless
thanhalfwaythroughthisbook.Ifoptionalinterveninghistoricaldigressionswere
eliminated,theproofoftheGauss–BonnetFormulawouldappearonapproximately
p.115.
This should leave time to cover other topics that interest the instructor or the
instructor’sstudents.Ihopethatinstructorsendeavortocoverenoughofthetheory
of general relativity to discuss the precession of Mercury. The general theory of
relativityisessentiallygeometricinnature.
Whatevertopicsarechosen,Ihopepeoplehavefun.
Acknowledgments
Anauthorwouldliketohavetheillusionthatheorshehasaccomplishedsomething
withouttheaidofothers.However,alittleintrospection,atleastinmycase,makes
such an illusion evaporate. To begin with as I prepared to write various sections
ofthis book,I read andrereadthe relevantsectionsin DirkJ. Struik’sLecturesin
ClassicalDifferentialGeometry2ndEd.(1988),andBarrettO’Neill’sDifferential
Geometry 2nd Ed. (1997). These two texts are restricted to three dimensions.
For higher dimensions, I devoted many hours to Johan C.H. Gerretsen’s Lectures
on Tensor Calculus and Differential Geometry (1962). For my chapter on non-
Euclideangeometry,IfoundalotofveryusefulideasinGeometry1stEd.(1999)
byDavidA.Brannan,MatthewF.Esplen,andJeremyJ.Gray.FromtimetotimeI
usedothergeometrybooks,whichappearinmybibliography.
Inaddition,IhavebenefitedfromthosewhohavedevelopedthetoolsofClifford
algebra.PerhapsthemostrelevantbookisCliffordAlgebratoGeometricCalculus
(1984)byDavidHestenesandGarretSobczyk.Itshouldbenotedthatothershave
demonstratedtheusefulnessofCliffordalgebraatafairlyelementarylevel.These
include Bernard Jancewicz with his Clifford Algebras in Electrodynamics (1988),
Pertti Lounesto with his Clifford Algebras and Spinors (1997), and William E.
BayliswithhisElectrodynamics–AModernGeometricApproach(1999).
As I was writing the first draft, I found myself frequently contacting Frank
Morgan by e-mail and pestering him with questions. As I neared completion of
the first draft, I was able to get Hieu Duc Nguyen to go over much of the text.
He suggested that I give prospective teachers better guidance on how to use the
text. This motivated me to reorganize the book so that the later chapters became
moreindependentofoneanother.Thisgivesprospectiveteachersmorefreedomto
designtheirowncourse.(AtHieu DucNguyen’surging,I also addeda sectionof
theIntroductionaddressedtoteachers).IamalsogratefultoPatrickGirardforhis
comments.AsI wasmakingsomefinishingtouchestothemanuscript,I gotsome
valuablecommentsfromG.StaceyStaples.
I also got substantial assistance in my quest to include some historical back-
ground.IusuallystartedasearchontheInternetthatinvariablyledtoitemsposted
by John J. O’Connor and Edmund F. Robertson at the School of Mathematics
xi
xii Acknowledgments
and Statistics at the University of St. Andrews, Scotland. Due to the ephemeral
nature of some web sites, I have been hesitant to cite their work. A few of their
entriesarementionedinmybibliographybutmoreoftenthannot,Iexploitedtheir
bibliographiestofindoutmoreaboutthemathematiciansIwantedtodiscuss.
At a CliffordalgebraconferenceheldatCookeville,Tennesseein 2002,Sergiu
Vacarudescribedto me some of hispersonalexperienceswith DimitriIvanenkov.
Thissetmeoffonagrandadventuretodiscoveralittle bitaboutdoingphysicsin
StalinistRussia.AlongthewayIgothelpfromGennadyGorelik,VitalyGinzburg,
SashaRozenberg,andEngelbertSchu¨cking.
Engelbert Schu¨cking shared several anecdotes along with a strong sense of
history. He also translated a number of German passages that were too difficult
for me. I was also able to elicit some commentsfrom Joseph W. Dauben,a noted
historianofscience.IfoundmyselfatoddswithGeorgeSalibaononeaspectofthe
impactofIslamicastronomersontheworkofCopernicus.NonethelessIwouldlike
tothinkthatmostofwhatIwroteonIslamicscienceisareflectionofhisviews.
IbenefitedfromthetechnicalsupportstaffatMacKichanSoftwareinsupportof
theirScientificWorkPlaceprogram.AndIamalsogratefultoAlanBellforkeeping
mycomputerworkingatsomecriticaltimes.
I wish to thank my daughter, Suzanne,who used her strong editorial skills to
clarifythehistoricalsections.Iamalsogratefultomyson,Spencer,whogaveme
some useful advice on some of my drawings.My brotherCharles reviewed a few
sectionsandmadesomeusefulcomments.
In1996,IwasforcedintoearlyretirementwhenUpsalaCollege(myemployer)
inEastOrange,NJwentbankrupt.Inthiscircumstance,IamgratefultotheRutgers
MathDepartmentforgrantingmestatusasa“visitingscholar.”Thishasgivenme
valuablelibraryprivilegesinallthevariousRutgerslibrarybranchesspreadoutover
thestateofNewJersey.IamalsoindebtedtothereferencelibrariansintheNewark
branchoftheRutgerslibrarysystem whosomehowmadesenseoutofincomplete
referencesandlocatedneededsources.
EastOrange,USA JohnSnygg
Contents
1 Introduction................................................................. 1
2 CliffordAlgebrainEuclidean3-Space................................... 3
2.1 Reflections,Rotations,andQuaternionsinE3 ..................... 3
2.1.1 UsingSquareMatricestoRepresentVectors.............. 3
2.1.2 1-Vectors,2-Vectors,3-Vectors,andCliffordNumbers .. 5
2.1.3 ReflectionandRotationOperators......................... 6
2.1.4 Quaternions................................................. 8
2.2 The4(cid:2) PeriodicityoftheRotationOperator ....................... 13
2.3 *ThePointGroupsfortheRegularPolyhedrons ................... 15
2.4 *E´lieCartan1869–1951............................................. 23
2.5 *SuggestedReading.................................................. 25
3 CliffordAlgebrainMinkowski4-Space ................................. 27
3.1 ASmallDoseofSpecialRelativity ................................. 27
3.2 *AlbertEinstein1879–1955......................................... 37
3.3 *SuggestedReading.................................................. 45
4 CliffordAlgebrainFlatn-Space.......................................... 47
4.1 CliffordAlgebra...................................................... 47
4.2 TheScalarProductandMetricTensor.............................. 53
4.3 TheExteriorProductforp-Vectors.................................. 61
4.4 SomeUsefulFormulas............................................... 65
4.5 Gram–SchmidtFormulas ............................................ 68
4.6 *TheQibla(Kibla)Problem......................................... 69
4.7 *MathematicsofArabSpeakingMuslims.......................... 74
4.7.1 *GreekScienceandMathematicsinAlexandria.......... 74
4.7.2 *Hypatia..................................................... 77
4.7.3 *TheRiseofIslamandtheHouseofWisdom............ 79
4.7.4 *TheImpactofAl-Ghaza¯l¯ı................................. 86
xiii