The Geometry of Minkowski Spacetime: An Introduction to the Mathematics of the Special Theory of Relativity

341 Pages·2012·2.45 MB·English

by  Gregory L. Naber (auth.)

Applied Mathematical Sciences
Volume 92
Editors
S.S.Antman
DepartmentofMathematics
and
InstituteforPhysicalScienceandTechnology
UniversityofMaryland
CollegePark,MD20742-4015
USA
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P.Holmes
DepartmentofMechanicalandAerospaceEngineering
PrincetonUniversity
215FineHall
Princeton,NJ08544
[email protected]
L.Sirovich
LaboratoryofAppliedMathematics
DepartmentofBiomathematicalSciences
MountSinaiSchoolofMedicine
NewYork,NY10029-6574
[email protected]
K.Sreenivasan
DepartmentofPhysics
NewYorkUniversity
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NewYorkCity,NY10012
[email protected]
Advisors
L.Greengard J.Keener J.Keller
R.Laubenbacher B.J.Matkowsky A.Mielke
C.S.Peskin A.Stevens A.Stuart
Forfurthervolumes:
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Gregory L. Naber
The Geometry of
Minkowski Spacetime
An Introduction to the Mathematics
of the Special Theory of Relativity
Second Edition
With 66 Illustrations
GregoryL.Naber
DepartmentofMathematics
DrexelUniversity
Korman Center
3141 Chestnut Street
Philadelphia, Pennsylvania
19104-2875
USA
[email protected]
ISBN978-1-4419-7837-0 e-ISBN978-1-4419-7838-7
DOI10.1007/978-1-4419-7838-7
SpringerNewYorkDordrechtHeidelbergLondon
LibraryofCongressControlNumber:2011942915
MathematicsSubjectClassification(2010):83A05,83-01
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For Debora

Preface
It is the intention of this monograph to provide an introduction to the spe-
cial theory of relativity that is mathematically rigorousand yet spells out in
considerable detail the physical significance of the mathematics. Particular
care has been exercised in keeping clear the distinction between a physi-
cal phenomenon and the mathematical model which purports to describe
that phenomenon so that, at any given point, it should be clear whether we
are doing mathematics or appealing to physical arguments to interpret the
mathematics.
The Introduction is an attempt to motivate, by way of a beautiful theo-
rem of Zeeman [Z1], our underlying model of the “event world.” This model
consists ofa 4-dimensionalrealvectorspace onwhich is defined a nondegen-
erate, symmetric, bilinear form of index one (Minkowski spacetime) and its
associated group of orthogonal transformations (the Lorentz group).
The first five sections of Chapter 1 contain the basic geometrical infor-
mation about this model including preliminary material on indefinite inner
product spaces in general, elementary properties of spacelike, timelike and
nullvectors,timeorientation,propertimeparametrizationoftimelikecurves,
theReversedSchwartzandTriangleInequalities,Robb’sTheoremonmeasur-
ing proper spatial separationwith clocks and the decompositionof a general
Lorentztransformationinto aproductoftworotationsandaspecialLorentz
transformation. In these sections one will also find the usual kinematic dis-
cussions of time dilation, the relativity of simultaneity, length contraction,
the addition of velocities formula and hyperbolic motion as well as the con-
struction of 2-dimensional Minkowski diagrams and, somewhat reluctantly,
an assortment of the obligatory “paradoxes.”
Section 6 of Chapter 1 contains the definitions of the causal and chrono-
logical precedence relations and a detailed proof of Zeeman’s extraordinary
theorem characterizing causal automorphisms as compositions T ◦ K ◦ L,
whereT isatranslation,Kisadilation,andLisanorthochronousorthogonal
vii
viii Preface
transformation. The proof is somewhat involved,but the result itself is used
only in the Introduction (for purposes of motivation) and in Appendix A to
construct the homeomorphism group of the path topology.
Section 1.7 is built upon the one-to-one correspondence between vectors
in Minkowskispacetime and 2×2 complex Hermitian matrices and contains
a detailed construction of the spinor map (the two-to-one homomorphism of
SL(2,C) onto the Lorentz group). We show that the fractional linear trans-
formation of the “celestial sphere” determined by an element A of SL(2,C)
has the same effect on past null directions as the Lorentz transformation
corresponding to A under the spinor map. Immediate consequences include
Penrose’s Theorem [Pen1] on the apparent shape of a relativistically mov-
ing sphere, the existence of invariantnull directions for an arbitrary Lorentz
transformation, and the fact that a general Lorentz transformation is com-
pletelydeterminedbyitseffectonanythreedistinctpastnulldirections.The
material in this section is required only in Chapter 3 and Appendix B.
InSection1.8(whichis independent ofSections 1.6and1.7)weintroduce
intoourmodeltheadditionalelementofworldmomentumformaterialparti-
clesandphotonsanditsconservationinwhatarecalledcontactinteractions.
Withthisonecanderivemostofthewell-knownresultsofrelativisticparticle
mechanics and we include a sampler (the Doppler effect, the aberration for-
mula, the nonconservation of proper mass in a decay reaction, the Compton
effect and the formulas relevant to inelastic collisions).
Chapter 2 intr(cid:2)oduces charg(cid:3)ed particles and uses the classical Lorentz
World Force Law FU = mdU as motivation for describing an electromag-
e dτ
netic field at a point in Minkowski spacetime as a linear transformation F
whosejobitistotellachargedparticlewithworldvelocityU passingthrough
that point what change in world momentum it should expect to experience
due to the presence of the field. Such a linear transformation is necessarily
skew-symmetric with respect to the Lorentz inner product and Sections 2.2,
2.3and2.4analyzethealgebraicstructureoftheseinsomedetail.Theessen-
tial distinction between regular and null skew-symmetric linear transforma-
(cid:3) (cid:3) (cid:3) (cid:3)
tionsisdescribedfirstintermsofthephysicalinvariantsE·B and|B|2−|E|2
of the electromagnetic field (which arise as coefficients in the characteristic
equationofF)andthenintermsoftheexistenceofinvariantsubspaces.This
material culminates in the existence of canonical forms for both regular and
nullfieldsthatareparticularlyusefulforcalculations,e.g.,ofeigenvaluesand
principal null directions.
Section 2.5 introduces the energy-momentum transformation for an arbi-
trary skew-symmetric linear transformationand calculates its matrix entries
intermsoftheclassicalenergydensity,Poynting3-vectorandMaxwellstress
tensor.ItsprincipalnulldirectionsaredeterminedandtheDominantEnergy
Condition is proved.
In Section 2.6, the Lorentz World Force equation is solved for charged
particles moving in constant electromagnetic fields, while variable fields are
introducedinSection2.7.Herewedescribetheskew-symmetricbilinearform
Preface ix
(bivector)associatedwiththelineartransformationrepresentingthefieldand
useitanditsdualtowritedownMaxwell’s(source-free)equations.Assample
solutionstoMaxwell’sequationsweconsidertheCoulombfield,thefieldofa
uniformly moving charge, and a rather complete discussion of simple, plane
electromagnetic waves.
Chapter3is anelementaryintroductionto the algebraictheoryofspinors
in Minkowski spacetime. The rather lengthy motivational Section 3.1 traces
the emergence of the spinor concept from the general notion of a (finite di-
mensional)grouprepresentation.Section3.2 contains the abstractdefinition
of spin space and introduces spinors as complex-valuedmultilinear function-
als on spin space. The Levi-Civita spinor (cid:2) and the elementary operations
ofspinor algebra(type changing,sums,components,outerproducts,(skew-)
symmetrization, etc.) are treated in Section 3.3.
In Section 3.4 we introduce the Infeld-van der Waerden symbols (essen-
tially,normalizedPaulispinmatrices)andusethem,togetherwiththespinor
mapfromSection1.7,todefinenaturalspinorequivalentsforvectorsandcov-
ectorsinMinkowskispacetime.Thespinorequivalentofafuture-directednull
vectorisshowntobeexpressibleastheouterproductofaspinvectorandits
conjugate.Reversingtheprocedureleadstotheexistenceofafuture-directed
null “flagpole” for an arbitrary nonzero spin vector.
SpinorequivalentsforbilinearformsareconstructedinSection3.5withthe
skew-symmetricforms(bivectors)playingaparticularlyprominantrole.With
these we can give a detailed construction of the geometrical representation
“up to sign” of a nonzero spin vector as a null flag (due to Penrose). The
sign ambiguity in this representation intimates the “essential 2-valuedness”
of spinors which we discuss in some detail in Appendix B.
Chapter3culminateswithareturntotheelectromagneticfield.Weintro-
duce the electromagnetic spinor φAB associated with a skew-symmetric lin-
eartransformationF andfindthatitcanbedecomposedintoasymmetrized
outerproductofspinvectorsαandβ.The flagpolesofthesespinvectorsare
eigenvectorsfortheelectromagneticfieldtransformation,i.e.,theydetermine
its principal null directions. The solution to the eigenvalue problem for φAB
yieldstwoelegantspinorversionsofthe“Petrovtype”classificationtheorems
ofChapter2.Specifically,weprovethataskew-symmetriclineartransforma-
tionF onM isnullifandonlyifλ=0istheonlyeigenvalueoftheassociated
electromagnetic spinor φAB and that this, in turn, is the case if and only if
theassociatedspinvectorsαandβ arelinearlydependent.Nextwefindthat
the energy-momentumtransformationhas a beautifully simple spinor equiv-
alent and use it to give another proof of the Dominant Energy Condition.
Finally, we derive the elegantspinor form of Maxwell’s equations and briefly
discuss its generalizations to massless free field equations for arbitrary spin
1n particles.
2
Chapter 4, which is new to this second edition, is intended to serve
two purposes. The first is to provide a gentle Prologue to the steps one
must take to move beyond special relativity and adapt to the presence of