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

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x Preface

gravitational ﬁelds that cannot be considered negligible. Section 4.2 describes

the philosophy espoused by Einstein for this purpose. Implementing this phi-

losophy, however, requires mathematical tools that played no role in the ﬁrst

three chapters so Section 4.3 provides a very detailed and elementary intro-

duction to just enough of this mathematical machinery to accomplish our

very modest goal. Thus supplied with a rudimentary grasp of manifolds,

Riemannian and Lorentzian metrics, geodesics and curvature we are in a

position to introduce, in Section 4.4, the Einstein ﬁeld equations (with cos-

mological constant Λ) and learn just a bit about one remarkable solution.

This is the so-called de Sitter universe dS and it is remarkable for a number

of reasons. It is a model of the universe as a whole, that is, a cosmological

model. Indeed, we will see that, depending on one’s choice of coordinates, it

can be viewed as representing an instance of any one of the three standard

Robertson-Walker models of relativistic cosmology. Taking Λ to be zero, dS

can be viewed as a model of the event world in the presence of a mass-energy

distribution due to a somewhat peculiar “ﬂuid” with positive density, but

negative pressure. On the other hand, if Λ is a positive constant, then dS

models an empty universe and, in this sense at least, is not unlike Minkowski

spacetime. The two have very diﬀerent properties, however, and one might be

tempted to dismiss dS as a mathematical curiosity were it not for the fact that

certain recent astronomical observations suggest that the expansion of our

universe is actually accelerating and that this weighs in on the side of the de

Sitter universe rather than the Minkowski universe. Thus, this ﬁnal chapter

is also something of an Epilogue to our story in which the torch is, perhaps,

passed to a new main character. Section 4.5 delves brieﬂy into a somewhat

more subtle diﬀerence between the Minkowski and de Sitter worlds that one

sees only “at inﬁnity.” Following Penrose [Pen

] we examine the asymptotic

structures of dS and M by constructing conformal embeddings of them into

the Einstein static universe. Penrose developed this technique to study mass-

less spinor ﬁeld equations such as the source-free Maxwell equations and the

Weyl neutrino equation with which we concluded Chapter 3.

The background required for an eﬀective reading of the ﬁrst three chap-

ters is a solid course in linear algebra and the usual supply of “mathematical

maturity.” In Chapter 4 we will require also some basic material from real

analysis such as the Inverse Function Theorem. For the two appendices we

must increment our demands upon the reader and assume some familiar-

ity with elementary point-set topology. Appendix A describes, in the spe-

cial case of Minkowski spacetime, a remarkable topology devised by Hawk-

ing, King and McCarthy [HKM] and based on ideas of Zeeman [Z

]whose

homeomorphisms are just compositions of translations, dilations and Lorentz

transformations. Only quite routine point-set topology is required, but the

construction of the homeomorphism group depends on Zeeman’s Theorem

from Section 1.6.

In Appendix B we elaborate upon the “essential 2-valuedness” of spinors

and its signiﬁcance in physics for describing, for example, the quantum

mechanical state of a spin 1/2 particle, such as an electron. Paul Dirac’s

Preface xi

ingenious “Scissors Problem” is used, as Dirac himself used it, to suggest, in

a more familiar context, the possibility that certain aspects of a physical sys-

tem’s state may be invariant under a rotation of the system through 720

◦

, but

not under a 360

◦

rotation. To fully appreciate such a phenomenon one must

see its reﬂection in the mathematics of the rotation group (the “conﬁguration

space” of the scissors). For this we brieﬂy review the notion of homotopy

for paths and the construction of the fundamental group. Noting that the

3-sphere S

is the universal cover for real projective 3-space RP

and

that RP

is homeomorphic to the rotation group SO(3) we show that

(SO(3))

∼

. One then sees Dirac’s demonstration as a sort of physi-

cal model of the two distinct homotopy classes of loops in SO(3). But there

is a great deal more to be learned here. By regarding the elements of SU

(Section 1.7) as unit quaternions we ﬁnd that, topologically, it is S

and

then recognize SU

and the restriction of the spinor map to it as a concrete

realization of the covering space for SO(3) that we just used to calculate

(SO(3)). One is then led naturally to SU

as a model for the “state space”

(as distinguished from the “conﬁguration space”) of the system described

in Dirac’s demonstration. Recalling our discussion of group representations

in Section 3.1 we ﬁnd that it is the representations of SU

, i.e., the spinor

representations of SO(3), that contain the physically signiﬁcant information

about the system. So it is with the quantum mechanical state of an electron,

but in this case one requires a relativistically invariant theory and so one

looks, not to SU

and the restriction of the spinor map to it, but to the full

spinor map which carries SL(2, C) onto the Lorentz group.

Lemmas, Propositions, Theorems and Corollaries are numbered sequen-

tially within each section so that “p.q.r” will refer to result #r in Section

#q of Chapter #p. Exercises and equations are numbered in the same way,

but with equation numbers enclosed in parentheses. There are 232 exercises

scattered throughout the text and no asterisks appear to designate those that

are used in the sequel; they are all used and must be worked conscientiously.

Finally, we shall make extensive use of the Einstein summation convention

according to which a repeated index, one subscript and one superscript, indi-

cates a sum over the range of values that the index can assume. For example,

if a and b are indices that range over 1, 2, 3, 4, then



a=1

= x

+ x



b=1

=Λ

+Λ

= η

+ η

+ ···+ η

and so on.

Gregory L. Naber

Acknowledgments

Some debts are easy to describe and acknowledge with gratitude. To the

Departments of Mathematics at the California State University, Chico, and

Drexel University, go my sincere thanks for the support, ﬁnancial and other-

wise, that they provided throughout the period during which the manuscript

was being written. On the other hand, my indebtedness to my wife, Debora,

is not so easily expressed in a few words. She saw to it that I had the time

and the peace to think and to write and she bore me patiently when one or

the other of these did not go well. She took upon herself what would have

been, for me, the onerous task of mastering the software required to produce

a beautiful typescript from my handwritten version and, while producing it,

held me to standards that I would surely have abandoned just to be done

with the thing. Let it be enough to say that the book would not exist were

it not for Debora. With love and gratitude, it is dedicated to her.

xiii

Contents

Preface ....................................................... vii

Acknowledgments ............................................ xiii

Introduction .................................................. 1

1 Geometrical Structure of M .............................. 7

1.1 Preliminaries........................................... 7

1.2 Minkowski Spacetime ................................... 9

1.3 The Lorentz Group ..................................... 15

1.4 Timelike Vectors and Curves ............................. 42

1.5 Spacelike Vectors ....................................... 55

1.6 Causality Relations ..................................... 58

1.7 Spin Transformations and the Lorentz Group .............. 68

1.8 Particles and Interactions ................................ 81

2 Skew-Symmetric Linear Transformations

d E

lectromagnetic Fields ............................... 93

2.1 Motivation via the Lorentz Law .......................... 93

2.2 Elementary Properties .................................. 95

2.3 Invariant Subspaces ..................................... 99

2.4 Canonical Forms ....................................... 105

2.5 The Energy-Momentum Transformation ................... 109

2.6 Motion in Constant Fields ............................... 113

2.7 Variable Electromagnetic Fields .......................... 117

3 The Theory of Spinors .................................... 135

3.1 Representations of the Lorentz Group ..................... 135

3.2 Spin Space ............................................ 153

3.3 Spinor Algebra ......................................... 160

3.4 Spinors and World Vectors............................... 169

xvi Contents

3.5 Bivectors and Null Flags ................................ 179

3.6 The Electromagnetic Field (Revisited) .................... 186

4 Prologue and Epilogue: The de Sitter Universe ........... 199

4.1 Introduction ........................................... 199

4.2 Gravitation ............................................ 199

4.3 Mathematical Machinery ................................ 202

4.4 The de Sitter Universe dS ............................... 249

4.5 Inﬁnity in Minkowski and de Sitter Spacetimes ............. 255

Appendix A Topologies For M ............................. 279

A.1 The Euclidean Topology ................................. 279

A.2 E-Co

ntinuous Timelike Curves ..

....

..................... 280

A.3 The Path Topology ..................................... 284

Appendix B Spinorial Objects .............................. 293

B.1 Introduction ........................................... 293

B.2 The Spinning Electron and Dirac’s Demonstration .......... 294

B.3 Homotopy in the Rotation and Lorentz Groups............. 296

References .................................................... 307

Symbols ...................................................... 311

Index ......................................................... 317

Introduction

All beginnings are obscure. Inasmuch as the mathematician operates

with his conceptions along strict and formal lines, he, above all, must

be reminded from time to time that the origins of things lie in greater

depths than those to which his methods enable him to descend.

Hermann Weyl, Space, Time, Matter

Minkowski spacetime is generally regarded as the appropriate arena within

which to formulate those laws of physics that do not refer speciﬁcally to

gravitational phenomena. We would like to spend a moment here at the

outset brieﬂy examining some of the circumstances which give rise to this

belief.

We shall adopt the point of view that the basic problem of science in gen-

eral is the description of “events” which occur in the physical universe and

the analysis of relationships between these events. We use the term “event,”

however, in the idealized sense of a “point-event,” that is, a physical occur-

rence which has no spatial extension and no duration in time. One might

picture, for example, an instantaneous collision or explosion or an “instant”

in the history of some (point) material particle or photon (to be thought

of as a “particle of light”). In this way the existence of a material particle

or photon can be represented by a continuous sequence of events called its

“worldline.” We begin then with an abstract set M whoseelementswecall

“events.” We shall provide M with a mathematical structure which reﬂects

certain simple facts of human experience as well as some rather nontrivial

results of experimental physics.

Events are “observed” and we will be particularly interested in a certain

class of observers (called “admissible”) and the means they employ to describe

events. Since it is in the nature of our perceptual apparatus that we identify

events by their “location in space and time” we must specify the means by

which an observer is to accomplish this in order to be deemed “admissible.”

, : An Introduction

, Applied Mathematical Sciences 92,

G.L. Naber The Geometry of Minkowski Spacetime to the Mathematics 1

of the Special Theory of Relativity

DOI 10.1007/978-1-4419-7838-7_ , © Springer Science+Business Media, LLC 20120

2 Introduction

Each admissible observer presides over a 3-dimensional, right-handed,

Cartesian spatial coordinate system based on an agreed unit of length

and relative to which photons propagate rectilinearly in any direction.

A few remarks are in order. First, the expression “presides over” is not

to be taken too literally. An observer is in no sense ubiquitous. Indeed, we

generally picture the observer as just another material particle residing at

the origin of his spatial coordinate system; any information regarding events

which occur at other locations must be communicated to him by means we

will consider shortly. Second, the restriction on the propagation of photons

is a real restriction. The term “straight line” has meaning only relative to a

given spatial coordinate system and if, in one such system, light does indeed

travel along straight lines, then it certainly will not in another system which,

say, rotates relative to the ﬁrst. Notice, however, that this assumption does

not preclude the possibility that two admissible coordinate systems are in

relative motion. We shall denote the spatial coordinate systems of observers

O,... by Σ(x

Σ(ˆx

, ˆx

),....

We take it as a fact of human experience that each observer has an innate,

intuitive sense of temporal order which applies to events which he experiences

directly, i.e., to events on his worldline. This sense, however, is not quantita-

tive; there is no precise, reliable sense of “equality” for “time intervals.” We

remedy this situation by giving him a watch.

Each admissible observer is provided with an ideal standard clock based

on an agreed unit of time with which to provide a quantitative temporal

order to the events on his worldline.

Notice that thus far we have assumed only that an observer can assign a

time to each event on his worldline.Inorderforanobservertobeableto

assign times to arbitrary events we must specify a procedure for the place-

ment and synchronization of clocks throughout his spatial coordinate system.

One possibility is simply to mass-produce clocks at the origin, synchronize

them and then move them to various other points throughout the coordinate

system. However, it has been found that moving clocks about has a most

undesirable eﬀect upon them. Two identical and very accurate atomic clocks

are manufactured in New York and synchronized. One is placed aboard a

passenger jet and ﬂown around the world. Upon returning to New York it is

found that the two clocks, although they still “tick” at the same rate, are no

longer synchronized. The travelling clock lags behind its stay-at-home twin.

Strange, indeed, but it is a fact and we shall come to understand the reason

for it shortly.

To avoid this diﬃculty we shall ask our admissible observers to build their

clocks at the origins of their coordinate systems, transport them to the de-

sired locations, set them down and return to the master clock at the origin.

We assume that each observer has stationed an assistant at the location of

Introduction 3

each transported clock. Now our observer must “communicate” with each

assistant, telling him the time at which his clock should be set in order that

it be sychronized with the clock at the origin. As a means of communication

we select a signal which seems, among all the possible choices, to be least

susceptible to annoying ﬂuctuations in reliability, i.e., light signals. To per-

suade the reader that this is an appropriate choice we shall record some of

the experimentally documented properties of light signals, but ﬁrst, a little

experiment. From his location at the origin O an observer O emits a light

signal at the instant his clock reads t

. The signal is reﬂected back to him

at a point P and arrives again at O at the instant t

. Assuming there is no

delay at P when the signal is bounced back, O will calculate the speed of

the signal to be distance (O, P)/

−t

). This technique for measuring the

speed of light we call the Fizeau procedure in honor of the gentleman who

ﬁrst carried it out with care (notice that we must bounce the signal back to

O since we do not yet have a clock at P that is synchronized with that at O).

For each admissible observer the speed of light in vacuo as determined

by the Fizeau procedure is independent of when the experiment is per-

formed, the arrangement of the apparatus (i.e., the choice of P), the

frequency (energy) of the signal and, moreover, has the same numer-

ical value c (approximately 3.0 × 10

meters per second) for all such

observers.

Here we have the conclusions of numerous experiments performed over the

years, most notably those ﬁrst performed by Michelson-Morley and Kennedy-

Thorndike (see Ex. 33 and Ex. 34 of [TW] for a discussion of these exper-

imen

ts).

The r

esults may seem odd. Why is a photon so unlike an electron

whose speed certainly will not have the same numerical value for two ob-

servers in relative motion? Nevertheless, they are incontestable facts of na-

ture and we must deal with them. We shall exploit these rather remarkable

properties of light signals immediately by asking all of our observers to mul-

tiply each of their time readings by the constant c and thereby measure time

in units of distance (light travel time, e.g., “one meter of time” is the amount

of time required by a light signal to travel one meter in vacuo). With these

units all speeds are dimensionless and c = 1. Such time readings for observers

O,... will be designated x

(= ct), ˆx

(= c

t),....

Now we provide each of our observers with a system of synchronized clocks

in the following way: At each point P of his spatial coordinate system place

a clock identical to that at the origin. At some time x

at O emit a spherical

electromagnetic wave (photons in all directions). As the wavefront encounters

P set the clock placed there at time x

+distance(O, P)andsetitticking,

thus synchronized with the clock at the origin.

At this point each of our observers O,

O,... has established a frame of

reference S(x

S(ˆx

, ˆx

),.... A useful intuitive visualiza-

tion of such a reference frame is as a latticework of spatial coordinate lines

with, at each lattice point, a clock and an assistant whose task it is to record