ɷnaber g l the geometry of minkowski
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Applied Mathematical SciencesVolume 92
EditorsS.S. AntmanDepartment of MathematicsandInstitute for Physical Science and TechnologyUniversity of MarylandCollege Park, MD 20742-4015USAssa@math.umd.edu
P. HolmesDepartment of Mechanical and Aerospace EngineeringPrinceton University215 Fine HallPrinceton, NJ firstname.lastname@example.org
L. SirovichLaboratory of Applied MathematicsDepartment of Biomathematical SciencesMount Sinai School of MedicineNew York, NY email@example.com
K. SreenivasanDepartment of PhysicsNew York University70 Washington Square SouthNew York City, NY firstname.lastname@example.org
AdvisorsL. Greengard J. Keener J. KellerR. Laubenbacher B.J. Matkowsky A. MielkeC.S. Peskin A. Stevens A. Stuart
For further volumes:http://www.springer.com/series/34
Gregory L. Naber
The Geometry ofMinkowski Spacetime
An Introduction to the Mathematicsof the Special Theory of Relativity
With 66 Illustrations
ISBN 978-1-4419-7837-0 e-ISBN 978-1-4419-7838-7DOI 10.1007/978-1-4419-7838-7Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011942915
Mathematics Subject Classication (2010): 83A05, 83-01
All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if they arenot identied as such, is not to be taken as an expression of opinion as to whether or not they are subjectto proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
Springer Science+Business Media, LLC 2012
Gregory L. NaberDepartment of MathematicsDrexel University
Philadelphia, Pennsylvania3141 Chestnut Street
It is the intention of this monograph to provide an introduction to the spe-cial theory of relativity that is mathematically rigorous and yet spells out inconsiderable detail the physical signicance of the mathematics. Particularcare has been exercised in keeping clear the distinction between a physi-cal phenomenon and the mathematical model which purports to describethat phenomenon so that, at any given point, it should be clear whether weare doing mathematics or appealing to physical arguments to interpret themathematics.
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 modelconsists of a 4-dimensional real vector space on which is dened a nondegen-erate, symmetric, bilinear form of index one (Minkowski spacetime) and itsassociated group of orthogonal transformations (the Lorentz group).
The rst ve sections of Chapter 1 contain the basic geometrical infor-mation about this model including preliminary material on indenite innerproduct spaces in general, elementary properties of spacelike, timelike andnull vectors, time orientation, proper time parametrization of timelike curves,the Reversed Schwartz and Triangle Inequalities, Robbs Theorem on measur-ing proper spatial separation with clocks and the decomposition of a generalLorentz transformation into a product of two rotations and a special Lorentztransformation. In these sections one will also nd 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 denitions of the causal and chrono-logical precedence relations and a detailed proof of Zeemans extraordinarytheorem characterizing causal automorphisms as compositions T K L,where T is a translation,K is a dilation, and L is an orthochronous orthogonal
transformation. The proof is somewhat involved, but the result itself is usedonly in the Introduction (for purposes of motivation) and in Appendix A toconstruct the homeomorphism group of the path topology.
Section 1.7 is built upon the one-to-one correspondence between vectorsin Minkowski spacetime and 2 2 complex Hermitian matrices and containsa detailed construction of the spinor map (the two-to-one homomorphism ofSL(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 eect on past null directions as the Lorentz transformationcorresponding to A under the spinor map. Immediate consequences includePenroses Theorem [Pen1] on the apparent shape of a relativistically mov-ing sphere, the existence of invariant null directions for an arbitrary Lorentztransformation, and the fact that a general Lorentz transformation is com-pletely determined by its eect on any three distinct past null directions. Thematerial in this section is required only in Chapter 3 and Appendix B.
In Section 1.8 (which is independent of Sections 1.6 and 1.7) we introduceinto our model the additional element of world momentum for material parti-cles and photons and its conservation in what are called contact interactions.With this one can derive most of the well-known results of relativistic particlemechanics and we include a sampler (the Doppler eect, the aberration for-mula, the nonconservation of proper mass in a decay reaction, the Comptoneect and the formulas relevant to inelastic collisions).
Chapter 2 introduces charged particles and uses the classical LorentzWorld Force Law
(FU = me
)as motivation for describing an electromag-
netic eld at a point in Minkowski spacetime as a linear transformation Fwhose job it is to tell a charged particle with world velocity U passing throughthat point what change in world momentum it should expect to experiencedue to the presence of the eld. Such a linear transformation is necessarilyskew-symmetric with respect to the Lorentz inner product and Sections 2.2,2.3 and 2.4 analyze the algebraic structure of these in some detail. The essen-tial distinction between regular and null skew-symmetric linear transforma-tions is described rst in terms of the physical invariants
B and |
E|2of the electromagnetic eld (which arise as coecients in the characteristicequation of F ) and then in terms of the existence of invariant subspaces. Thismaterial culminates in the existence of canonical forms for both regular andnull elds that are particularly useful for calculations, e.g., of eigenvalues andprincipal null directions.
Section 2.5 introduces the energy-momentum transformation for an arbi-trary skew-symmetric linear transformation and calculates its matrix entriesin terms of the classical energy density, Poynting 3-vector and Maxwell stresstensor. Its principal null directions are determined and the Dominant EnergyCondition is proved.
In Section 2.6, the Lorentz World Force equation is solved for chargedparticles moving in constant electromagnetic elds, while variable elds areintroduced in Section 2.7. Here we describe the skew-symmetric bilinear form
(bivector) associated with the linear transformation representing the eld anduse it and its dual to write down Maxwells (source-free) equations. As samplesolutions to Maxwells equations we consider the Coulomb eld, the eld of auniformly moving charge, and a rather complete discussion of simple, planeelectromagnetic waves.
Chapter 3 is an elementary introduction to the algebraic theory of spinorsin Minkowski spacetime. The rather lengthy motivational Section 3.1 tracesthe emergence of the spinor concept from the general notion of a (nite di-mensional) group representation. Section 3.2 contains the abstract denitionof spin space and introduces spinors as complex-valued multilinear function-als on spin space. The Levi-Civita spinor and the elementary operationsof spinor algebra (type changing, sums, components, outer products, (skew-)symmetrization, etc.) are treated in Section 3.3.
In Section 3.4 we introduce the Infeld-van der Waerden symbols (essen-tially, normalized Pauli spin matrices) and use them, together with the spinormap from Section 1.7, to dene natural spinor equivalents for vectors and cov-ectors in Minkowski spacetime. The spinor equivalent of a future-directed nullvector is shown to be expressible as the outer product of a spin vector and itsconjugate. Reversing the procedure leads to the existence of a future-directednull agpole for an arbitrary nonzero spin vector.
Spinor equivalents for bilinear forms are constructed in Section 3.5 with theskew-symmetric forms (bivectors) playing a particularly prominant role. Withthese we can give a detailed construction of the geometrical representationup to sign of a nonzero spin vector as a null ag (due to Penrose). Thesign ambiguity in this representation intimates the essential 2-valuednessof spinors which we discuss in some detail in Appendix B.
Chapter 3 culminates with a return to the electromagnetic eld. We intro-duce the electromagnetic spinor AB associated with a skew-symmetric lin-ear transformation F and nd that it can be decomposed into a symmetrizedouter product of spin vectors and . The agpoles of these spin vectors