Naber G.L. The Geometry of Minkowski Spacetime. An Introduction to the Mathematics of the Special Theory of Relativity

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Applied Mathematical Sciences

Vo l u m e 9 2

Editors

S.S. Antman

Department of Mathematics

and

Institute for Physical Science and Technology

University of Maryland

College Park, MD 20742-4015

USA

ssa@math.umd.edu

P. Ho lm es

Department

of Mechanical and Aerospace Engineering

Princeton Univ

ersity

215 Fine Hall

Princeton, NJ 08544

pholmes@math.princeton.edu

L. Sirovich

Laboratory

of Applied Mathematics

Department of Bi

omathematical S

ciences

Mount Sinai School of Medicine

New York, NY 10029-6574

lsirovich@rockefeller.edu

K. Sreenivasan

Depart

ment of Physic

New

York University

70 Washington Square South

New York City, NY 10012

katepalli.sreenivasan@nyu.edu

Advisors

Greengard J. Keener J

. K

eller

R. Laubenbacher B.J. Matkowsky A. Mielke

C.S. Peskin A. Stevens A. Stuart

For further volumes:

http://www.springer.com/series/34

Gregory L. Naber

The Geometry of

Minkowski Spacetime

An Introduction to the Mathematics

of the Special Theory of Relativity

Second Edition

With 66 Illustrations

ISBN 978-1-4419-7837-0 e-ISBN 978-1-4419-7838-7

DOI 10.1007/978-1-4419-7838-7

Springer New York Dordrecht Heidelberg London

Library of Congress Control Number: 2011942915

Mathematics Subject Classiﬁcation (2010): 83A05, 83-01

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gln22@drexel.edu

Gregory L. Naber

Department of Mathematics

Drexel University

USA

Korman Center

Philadelphia, Pennsylvania

3141 Chestnut Street

19104-2875

For Debora

Preface

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 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 [Z

], our underlying model of the “event world.” This model

consists of a 4-dimensional real vector space on which 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

null vectors, time orientation, proper time parametrization of timelike curves,

the Reversed Schwartz and Triangle Inequalities, Robb’s Theorem on measur-

ing proper spatial separation with clocks and the decomposition of a general

Lorentz transformation into a product of two rotations and a special Lorentz

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,

where T is a translation, K is a dilation, and L is an orthochronous orthogonal

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 Minkowski spacetime 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 [Pen

] on the apparent shape of a relativistically mov-

ing sphere, the existence of invariant null directions for an arbitrary Lorentz

transformation, and the fact that a general Lorentz transformation is com-

pletely determined by its eﬀect on any three distinct past null directions. The

material 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 introduce

into 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 particle

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 introduces charged particles and uses the classical Lorentz

World Force Law



FU =

dτ



as motivation for describing an electromag-

netic ﬁeld at a point in Minkowski spacetime as a linear transformation F

whose job it is to tell a charged particle with world velocity U passing through

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,

2.3 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 ﬁrst in terms of the physical invariants



and |



−|



of the electromagnetic ﬁeld (which arise as coeﬃcients in the characteristic

equation of F ) and then in terms of the existence of invariant subspaces. This

material culminates in the existence of canonical forms for both regular and

null ﬁelds that are particularly useful for calculations, e.g., of eigenvalues and

principal null directions.

Section 2.5 introduces the energy-momentum transformation for an arbi-

trary skew-symmetric linear transformation and calculates its matrix entries

in terms of the classical energy density, Poynting 3-vector and Maxwell stress

tensor. Its principal null directions are determined and the Dominant Energy

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

introduced in Section 2.7. Here we describe the skew-symmetric bilinear form

Preface ix

(bivector) associated with the linear transformation representing the ﬁeld and

use it and its dual to write down Maxwell’s (source-free) equations. As sample

solutions to Maxwell’s equations we consider the Coulomb ﬁeld, the ﬁeld of a

uniformly moving charge, and a rather complete discussion of simple, plane

electromagnetic waves.

Chapter 3 is an elementary introduction to the algebraic theory of spinors

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) group representation. Section 3.2 contains the abstract deﬁnition

of spin space and introduces spinors as complex-valued multilinear function-

als on spin space. The Levi-Civita spinor  and the elementary operations

of 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 spinor

map from Section 1.7, to deﬁne natural spinor equivalents for vectors and cov-

ectors in Minkowski spacetime. The spinor equivalent of a future-directed null

vector is shown to be expressible as the outer product of a spin vector and its

conjugate. Reversing the procedure leads to the existence of a future-directed

null “ﬂagpole” for an arbitrary nonzero spin vector.

Spinor equivalents for bilinear forms are constructed in Section 3.5 with the

skew-symmetric forms (bivectors) playing a particularly prominant role. 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.

Chapter 3 culminates with a return to the electromagnetic ﬁeld. We intro-

duce the electromagnetic spinor φ

associated with a skew-symmetric lin-

ear transformation F and ﬁnd that it can be decomposed into a symmetrized

outer product of spin vectors α and β. The ﬂagpoles of these spin vectors are

eigenvectors for the electromagnetic ﬁeld transformation, i.e., they determine

its principal null directions. The solution to the eigenvalue problem for φ

yields two elegant spinor versions of the “Petrov type” classiﬁcation theorems

of Chapter 2. Speciﬁcally, we prove that a skew-symmetric linear transforma-

tion F on M is null if and only if λ = 0 is the only eigenvalue of the associated

electromagnetic spinor φ

and that this, in turn, is the case if and only if

the associated spin vectors α and β are linearly dependent. Next we ﬁnd that

the energy-momentum transformation has a beautifully simple spinor equiv-

alent and use it to give another proof of the Dominant Energy Condition.

Finally, we derive the elegant spinor form of Maxwell’s equations and brieﬂy

discuss its generalizations to massless free ﬁeld equations for arbitrary spin

n particles.

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