s. a. huggett and k. p. tod- an introduction to twistor theory

LONDON MATHEMATICAL SOCIETY STUDENT TEXTS

Managing editor: Dr C.M. Series, Mathematics InstituteUniversity of Warwick, Coventry CV4 7AL, United Kingdom

1 Introduction to combinators and k-calculus, J.R. HINDLEY & J.P. SELDIN

2 Building models by games, WILFRID HODGES

3 Local fields, J.W.S. CASSELS

4 An introduction to twistor theory: Second edition, S.A. HUGGETT & K.P. TOD

5 Introduction to general relativity, L.P. HUGHSTON & K.P. TOD

6 Lectures on stochastic analysis: diffusion theory, DANIEL W. STROOCK

7 The theory of evolution and dynamical systems, J. HOFBAUER & K. SIGMUND

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9 Automorphisms of surfaces after Nielsen and Thurston, A. CASSON & S. BLEILER

10 Nonstandard analysis and its applications, N. CUTLAND (ed)

11 Spacetime and singularities, G. NABER

12 Undergraduate algebraic geometry, MILES REID

13 An introduction to Hankel operators, J.R. PARTINGTON

14 Combinatorial group theory: a topological approach, DANIEL E. COHEN

15 Presentations of groups, D.L. JOHNSON

16 An introduction to noncommutative Noetherian rings, K.R. GOODEARL &

R.B. WARFIELD, JR.

17 Aspects of quantum field theory in curved spacetime, S.A. FULLING

18 Braids and coverings: selected topics, VAGN LUNDSGAARD HANSEN

19 Steps in commutative algebra, R.Y. SHARP

20 Communication theory, C.M. GOLDIE & R.G.E. PINCH

21 Representations of finite groups of Lie type, FRANQOIS DIGNE & JEAN MICHEL

22 Designs, graphs, codes, and their links, P.J. CAMERON & J.H. VAN LINT

23 Complex algebraic curves, FRANCES KIRWAN

24 Lectures on elliptic curves, J.W.S. CASSELS

25 Hyperbolic geometry, BIRGER IVERSEN

26 An Introduction to the theory of L-functions and Eisenstein series, H. HIDA

27 Hilbert Space: compact operators and the trace theorem, J.R. RETHERFORD

London Mathematical Society Student Texts 4

An Introductionto Twistor TheorySecond Edition

S. A. HuggettUniversity of Plymouth

K. P. TodUniversity of Oxford

CAMBRIDGEUNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESSCambridge, New York, Melbourne, Madrid, Cape Town, Singapore, SAo Paulo

Cambridge University PressThe Edinburgh Building, Cambridge C132 8RU, UK

Published in the United States of America by Cambridge University Press, New York

www.cambridge.orgInformation on this title: www.cambridge.org/9780521451574

© Cambridge University Press 1994

This publication is in copyright. Subject to statutory exceptionand to the provisions of relevant collective licensing agreements,no reproduction of any part may take place without the writtenpermission of Cambridge University Press.

First edition published 1985Second edition first published 1994

A catalogue record for this publication is available from the British Library

ISBN 978-0-521-45157-4 hardback

ISBN 978-0-521-45689-0 paperback

Transferred to digital printing 2008

To our parents

Contents

Preface

Preface to the second edition

ix

xi

1 Introduction 1

2 Review of Tensor Algebra and Calculus 5

3 Lorentzian Spinors at a Point 11

4 Spinor Fields 25

5 Compactified Minkowski Space 33

6 The Geometry of Null Congruences 45

7 The Geometry of Twistor Space 53

8 Solving the Zero Rest Mass Equations I 65

9 Sheaf Cohomology and Free Fields 71

10 Solving the Zero Rest Mass Equations II 91

11 The Twisted Photon and Yang-Mills Constructions 99

12 The Non-Linear Graviton 105

13 Penrose's Quasi-Local Momentum and AngularMomentum 119

14 Functionals on Zero Rest Mass Fields 137

Vii

viii CONTENTS

15 Further Developments and Conclusions 147

16 Hints, Solutions and Notes to the Exercises 153

Appendix: The GHP Equations 163

Bibliography 167

Index 175

Preface

This book is an introduction to Twistor Theory and modern geometricalapproaches to space-time structure at the graduate or advanced undergrad-uate level. The choice of material presented has evolved from graduate lec-tures given in London and Oxford and we have aimed to retain the informaltone of those lectures.

Topics covered include spinor algebrea and calculus; compactified Min-kowski space; the geometry of null congruences; the geometry of twistorspace; an informal account of sheaf cohomology sufficient to describe thetwistor solution of the zero rest mass equations; the active twistor con-structions which solve the self-dual Yang-Mills and Einstein equations; andPenrose's quasi-local-mass constructions. Exercises are included in the textand after most chapters.

The book will provide graduate students with an introduction to theliterature of twistor theory, presupposing some knowledge of special rela-tivity and differential geometry. It would also be of use for a short course onspace-time structure independently of twistor theory. The physicist couldbe introduced gently to some of the mathematics which has proved usefulin these areas, and the mathematician could be shown where sheaf coho-mology and complex manifold theory can be used in physics.

It is a pleasure to acknowledge many useful discussions, comments andcorrections from colleagues in London, Oxford and elsewhere. In particulara debt is due to Mike Eastwood, Andrew Hodges, Lane Hughston, TedNewman, David Robinson, George Sparling, Richard Ward, Ronny Wellsand Nick Woodhouse; we are grateful to Val Willoughby for typing themanuscript and dealing patiently with innumerable revisions; but above allwe are indebted for teaching and inspiration to Roger Penrose.

ix

Preface to the secondedition

In the eight years since the first edition of this book was published theliterature of twistor theory has increased very substantially. There havebeen several books published whose subject matter overlaps to a greater orlesser extent with ours: Ward and Wells (1990) contains a mathematicallymore thorough and rigorous account of sheaf cohomology and the machineryfrom complex analysis necessary for twistor theory than we have foundspace for; Baston and Eastwood (1989) is a mathematical account of the`Penrose transform' and its generalisation as part of representation theory;Bailey and Baston (1990) is a collection of commissioned reviews which goon in many directions from topics which we touch on more briefly; finally,Mason and Hughston (1990) follows on from Hughston and Ward (1979)by being a collection of articles which had previously appeared in TwistorNewsletter.

None of these precisely duplicates our book and so we have prepared anew edition taking account of some of the developments of the past eightyears. The principal changes for the second edition are as follows:

- chapter 9 has been rewritten with a slightly different emphasis. Ourintention is to provide a much clearer and more detailed motivation for thenotion of a sheaf, with readers new to the subject in mind. Then (as in thefirst edition) we go on to discuss the elements of Cech cohomology.

- chapter 13 has been extended to describe the present status of the originalquasi-local mass construction. There has been a lot of work in this area,and also a proliferation of other definitions of quasi-local mass. We describeone of them, that of Dougan and Mason, in detail.

- chapter 14 is new. It describes the cohomological framework for studyingthe kind of multi-linear functionals of zero rest mass fields which arise

xi

xii AN INTRODUCTION TO TWISTOR THEORY

naturally in twistor theory, and which provide a starting point for a twistorapproach to quantum field theory.

- chapter 15, the previous chapter 14, has been extended to describe thefurther `Further Developments' of the past eight years. These are impressive(even if some of twistor theory's original aims have not yet been fulfilled).Perhaps the browsing mathematical physicist could do worse than starthere!

- chapter 16 is new. Part of our aim in the first edition was to smugglein extra material disguised as exercises, some of which were perhaps un-reasonably difficult. With the inclusion of `Hints, Solutions and Notes tothe exercises' we continue this policy but with what we hope is a moreaccessible set of exercises.

The final major change is that the second edition has been completelyreset in LATEXand we are very grateful to Domonic Green for undertakingthis substantial labour.

Chapter 1

Introduction

Twistor Theory began as a subject in the late 1960's with the appearanceof Penrose's two papers (1967, 1968a). A more definitive statement ofits aims and accomplishments was `Twistor Theory: An Approach to theQuantisation of Fields and Space-time' (Penrose and MacCallum), whichappeared in 1973.

What is apparent from the start is the breadth of application whichRoger Penrose saw for it. In the decades since then the subject has grownin different direction to the extent that different traditions have emerged.

There is a purely mathematical strain which refers to the Penrose Trans-form and is interested- in its geometrical and complex analytic features,frequently in a positive definitve setting.

There is a quantum field theoretic strain concerned with elementaryparticles and their interactions in Minkowski space.

There is a modest point of view which simply holds that the theory isuseful for solving some non-linear equations and the object is to discoverwhich ones, and there is a full-blooded strain which holds that the repeatedoccurence and usefulness of complex numbers and complex analyticity tellsone something fundamental about the physical world.

The full-blooded strain of twistor philosophy may be seen in for example(Penrose 1975). One place where the diversity of twistor theory is manifestis the Twistor Newsletter, an informal publication produced by the Oxfordgroup about twice a year. The content of the first ten Newsletters waspublished as `Advances in Twistor Theory' (Hughston and Ward 1979)and later articles were collected in `Further Advances in Twistor Theory'(Mason and Hughston 1990).

In this book our aim is simply to give an introduction to the subjectand point the reader in the direction of these other possibilities. One thing

1

2 AN INTRODUCTION TO TWISTOR THEORY

we do want to emphasize is that a large part of the material falls under theheading of space-time geometry and indeed a course on that subject couldbe made out of the first six chapters.

The plan of the book is as follows.We begin in chapter 2 with a review of tensor algebra and calculus

as a reminder and to fix conventions. Lorentzian spinors at a point areintroduced in chapter 3 and spinor algebra is developed. Also the definitionof a complex manifold is given with projective spinors as a paradigm. Inchapter 4 we define spinor bundles and the spinor covariant derivative andintroduce various spinor differential equations, notably the zero rest massfree field equations and the twistor equation.

In attempting to define a Lie derivative of spinors, we are led in chap-ter 5 to a consideration of compactified Minkowski space and conformalinvariance.

The geometry of null geodesic congruences is discussed in terms ofspinors in chapter 6 and the connection between shear and complex ana-lyticity is noticed. We find shear-free congruences in terms of free analyticfunctions of three variables.

Twistors are introduced in chapter 7, first as spinor fields solving thetwistor equation and acted on by the conformal group. The geometri-cal correspondence between twistor space and complexified compactifiedMinkowski space is developed and other pictures of a twistor, as an a-planeor a Robinson congruence are given.

In chapter 8 we give the twistor contour integral solution of the zero restmass free field equations and in attempting to understand curious featuresof the solution we are led in chapter 9 to sheaf cohomology. After aninformal account of sheaf cohomology we return in chapter 10 to the zerorest mass equations and interpret the contour integrals cohomologically.

In chapters 11 and 12 we describe two `active' constructions where fieldequations in space-time are coded into deformations of complex structurein corresponding twistor spaces. These are the construction due to Wardfor solving the self-dual Yang-Mills equations and the construction due toPenrose for solving the self-dual Einstein equations.

An application of twistor theory in conventional general relativity isdiscussed in chapter 13. This is Penrose's proposal for a quasi-local momen-turnangular-momentum in an arbitrary curved space-time.

The sheaf cohomology of chapter 9 is extended in chapter 14, in whichwe show how to describe some spaces of multilinear functionals of zero restmass fields.

Finally in chapter 15 we briefly mention some other developments intwistor theory not mentioned elsewhere!

CHAPTER 1. INTRODUCTION 3

We have provided exercises throughout, both within the text and group-ed at the end of the chapters and have included a chapter (16) of hints,solutions and notes. Some of the exercises are just problems on the materialcovered but others are open-ended and intend to lead the reader into regionswhich we don't have the space to cover more fully. (Some of these exercisesare then referred to later in the text!)

Chapter 2

Review of Tensor Algebraand Calculus

This chapter is chiefly intended as a reminder and to fix conventions (whichlargely follow Penrose and Rindler 1984).

We shall be concerned with a real four-dimensional vector space V, itsdual V*and its complexification Vc = V ® C.

Vectors or elements of V are written Va; covectors, elements of V*, arewritten Wa and the pairing

V xV* -* R is (V°,Wa)-aVaWa.

We follow the abstract index convention of Penrose (1968b) in whichthe index a on Va is simply an indication that the object is a vector, ratherthan one of a set of numbers.

Higher valence tensors are elements of tensor products of V with V* as,e.g. :

pal...arbl...b, EV ®...®V®V*(9 ...(9 V*r 3

Algebraic operations on tensors which we shall require are:

i) Contraction: pab...ede ...f pab...cae

...f

ii) Symmetrisation: p(a...b) _ rt Q pa(a)...Q(b)

where pa...b has r indices and the sum is over all permutations. For example

p(ab) = 1(pab + pba).

5


Figure 2.1. The null cone in V separating time-like from space-like.

iii) Skew or anti-symmetrisation: P[a...b] = T, Q (signa)P°(a)...a(b)

where sign o, is +1 according as a is an even or odd permutation. Forexample

p[ab] =2

(Pab - Pba).

A skew symmetric tensor of valence 2 will be referred to as a bivector.We shall suppose that V comes equipped with a Lorentzian metric, i.e. a

symmetric non-degenerate tensor r/ab which is equal to diag (1, -1, -1, -1)in an orthonormal frame.

Non-zero vectors in V are characterized as time-like, space-like or nullaccording as their `length' ulabV (Vb is positive, negative or zero. This givesrise to the characteristic picture of the null cone in V (see figure 2.1).

Removing the origin 0 disconnects the null cone, so that time-like andnull vectors can be further distinguished into two classes, one of which canbe labelled future-pointing and the other past-pointing.

The metric allows the identification of V and V*:

V->V*; Va--+ Va=?labVb

together with the inverse:

V*-+ V; Vb-+ Vb=rlbcVc

CHAPTER 2. REVIEW OF TENSOR ALGEBRA AND CALCULUS 7

where 77ab 77b. = - 6'.-Usually a choice of orientation for V is made, that is, a choice of skew

tensor Eabcd = Eabcd] with

EabcdEabcd = -24.

Then a right-handed orthonormal frame (T'' X b, Y°, Z') is one with

EabcdTaXbYcZd = 1.

With the aid of Eabcd and the metric we may define the dual, *Fab, of abivector Fab as:

*Fab = - 2 EabcdFcd

As a consequence of the signature of the metric, we find

**Fab = -Fab

Thus the eigenvalues of duality on bivectors are ±i and therefore the eigen-bivectors are necessarily complex.

The Lorentz group L = 0(1, 3) is the group of endomorphisms of Vpreserving 97ab:

Aab E L AabAcdr/ac = ?)bd

or, in matrix notation,

77 = AT77A where 77 = diag(1, -1, -1, -1).

Clearly det A = ±1. Lorentz transformations with negative determinantchange the orientation since

EabcdAa pAb yAc,.A ds = (detA)EPyrs.

Also a Lorentz transformation will interchange the future and past nullcones if A°° < 0. Thus L has four components which may be representedas

L=L+UL+ULT ULL

where ± indicates the sign of the determinant and ? indicates A°° > 0. Thefirst component L+ contains the identity and is referred to as the properorthochronous Lorentz group.

As examples of Lorentz transformations, we have

i) diag(- 1, 1, 1, 1) ELI : time-reflection

ii) diag(1, -1, 1, 1) E LT : space-reflection


cosh 0 0 sinh0 1 0 0

iii) E L+ : `boost' in the (03) plane.0 0 1 0

sinh 0 0 cosh

The way to calculate the dimension of the Lorentz group is to considerinfinitesimal Lorentz transformations: in matrix form

A=I+cS.

Then, to first order in e,

AT7A=r7 =>- ST?l+llS=0.

This gives ten conditions on the sixteen components of S, so that L issix-dimensional.

We assume the reader is familiar with the basic machinery of differentialgeometry, i.e. the definitions of a smooth manifold and its tangent andcotangent bundles as given for example in Hicks (1965) or Do Carmo (1976,1992).

We are thinking of the vector space V above as being the tangent spaceTPM = (TM)p at a point p of a real four-dimensional manifold M. Themetric on V comes from a metric on M, that is a smooth, valence 2, non-degenerate symmetric tensor field gab with signature -2. This allows thedefinition of the orthonormal frame bundle B of M and also determines aunique, torsion-free, metric-preserving connection, the Levi-Civita connec-tion, which we denote Va.

Commuted derivatives give rise to curvature via the Ricci identity:

(VaVb - VbVa)Vd = RabcdVC

(2.1)

which defines the Riemann tensor Rabod. The Ricci tensor Rab and Ricciscalar R are defined by

Rab =Racb' . R = gab Rab,

and the Einstein field equations of general relativity are

Rab - 2 R9ab = -Tab

with a suitable choice of units, where Tab is the stress-energy tensor ofmatter. The Riemann tensor satisfies the first Bianchi identity:

Ra[bcd] = 0 (2.2)

CHAPTER 2. REVIEW OF TENSOR ALGEBRA AND CALCULUS 9

which implies the interchange symmetry of the Riemann tensor and thesymmetry of the Ricci tensor:

Rabcd = Rcdab e Rab = Rba,

and also the second Bianchi identity (which is usually referred to as theBianchi identity):

V[aRbc]de = 0. (2.3)

Finally, the Lie derivative along a vector field Xa is defined by

i) LX f = XaVa f for functions f

ii) LxYa = XbVbYa -YbVbXa

iii) for arbitrary valence tensors, extend by the Leibniz rule.

The Lie derivative is actually part of the differential structure of M andis defined prior to the assumption of a metric. Put another way, the abovedefinition is independent of (symmetric) connection.

Our next step is to introduce spinors and develop the algebra and cal-culus of spinors by analogy with the tensor algebra and calculus.

Exercises 2

a) Given a tensor Rabcd with the symmetries of the Riemann tensor:

Rabcd = R[ab]cd = Rab[cd] ; Ra[bcd] = 0

define Sabed = Rc(ab)d.Show that Sabad has the symmetries of Rabcd but with skew-symmetris-

ers replaced by symmetrisers (i.e. square brackets replaced by round brack-ets). Find an expression for Rabcd in terms of Sab,d.

b) A bivector Fab is said to be simple if it can be written as the skew outerproduct of two vectors 2U[aVb]. Show that Fab simple EabcdFabFcd = 0

*FabFab = 0-

c) Show that Lx9ab = 2V(aXb), where Va is the Levi-Civita connectionpreserving the metric gab.

d) Show that Lx *Fab = *(LxFab) if Lx9ab = )gab for some function A. Avector field Xa with this property is known as a conformal Killing vector(see chapter 5).

e) If X a, Ya are two conformal Killing vectors show that LxY' is also.

Chapter 3

Lorentzian Spinors at aPoint

Given a vector Va E V with components (V°, V1, V2, V3) in some or-thonormal frame, we may define a Hermitian matrix *(V") by (Penrose1974; Penrose and Rindler 1984; Pirani 1965)

'T(Va) = VAA =

y°°

LV1°'

VOl'

' J

1

[

V° +V3 Vl 12V2V1l

2=VI - iV 2 V 0- V3

I -

(3.1)

The indices A, A' have the ranges 0, 1 and 0', 1' respectively. The sig-nificance of the prime will become apparent!

Clearly (3.1) gives a one-one correspondence between 2 x 2 Hermitianmatrices and elements of V. Further, the determinant of the matrix is halfthe length of the vector:

det *(Va) = 1nabVaVb.

If we multiply the matrix *(Va) on the left by an element of SL(2, C)(i.e. a 2 x 2 matrix with complex entries and unit determinant) and onthe right by its Hermitian conjugate:

VAA' VAA' = tABVBB'tA' B,

0to

where L t1 t1ilE SL(2, C) , and

to B' = TAB

11


then the result will be another Hermitian matrix and the determinant willbe unchanged. This process therefore defines a linear transformation onthe vector V' preserving its length, i.e. a Lorentz transformation:

Va-+Va=AabVb.

Thus we have a map SL(2, C) --+ L. The following properties of this mapmay be readily established:

i) it is a group homomorphism;

ii) it is into L+ ;

iii) the kernel consists of ±I in SL(2, C) , where I is the identity matrix.

Since SL(2, C) is also a six parameter group the map is necessarily ontoand so is a 2-1 isomorphism. In exercise 3a we establish that SL(2, C) issimply connected, so that this map exhibits SL(2, C) as the universal coverof L+.

The Lorentz transformations in L+ leaving invariant the time-like vec-tor in the chosen orthonormal tetrad evidently define a three-dimensionalrotation group, SO(3). The matrix To corresponding to the time-like vec-tor is proportional to the unit matrix, so that the SO(3) subgroup of L+is covered by an SU(2) subgroup of SL(2, C)

As an example of the map iY, the matrix

e2 0t

0 e- 2

determines a boost in the (03) plane while

0t

0 e-2-

determines a rotation through i in the (12) plane.

[A topological aside: a path from I to -I in SL(2, C) will correspondin L+ to a path beginning and ending at I, but which cannot be shrunk toa point. For example

sa2

e0 a0<A<27r

2 )

CHAPTER 3. LORENTZIAN SPINORS AT A POINT 13

corresponds to a family of rotations in L+ beginning with a zero rotationand ending with a 2-7r rotation, which is the identity in L+ . This pathcannot be shrunk to a point in L+ . However, if the path in SL(2, C) iscontinued on to return to I, say by extending the range of A to 0 < A < 4ir,then this path, which is just the previous path traversed twice, can beshrunk to a point in L+ . There are a number of ways of demonstratingthis fact, all essentially equivalent to the following (see e.g. Misner et al.1973): imagine a spherical lamp-shade suspended from the ceiling by a longflex and surrounded by notional concentric spheres attached to the flex.Rotate the lamp-shade about a vertical axis and suppose that this rotationis communicated to the concentric spheres so that each is rotated a littleless than the one within it, with the outermost one fixed in its originalposition. Then the various spheres define a set of rotations, i.e. a path inSO(3) from the identity to the rotation undergone by the lamp-shade. Ifthe lamp-shade is rotated through 21r then the flex has a single twist init and this twist cannot be removed by rearranging the spheres. However,if the lamp-shade is rotated through 4ir, then the flex has two twists in itand a suitable rearrangement of the spheres will undo this. Essentially onesimply takes a loop of flex and passes it below the lamp-shade, but thisinvolves moving all the spheres and shrinking the original path to a point.

Clearly the same argument applies if the lamp-shade is actually con-nected by any number of strings to different places on the walls, ceiling andfloor when the disentangling is more impressive!]

Now we return to the matrices T(V") and remark that if the vectorVa is null, then the rank of $(Va) drops to one. Thus it may be writtenas the outer product of a complex two-dimensional vector and its complexconjugate:

VAA' = I V00: Vol' 1 1 aoa a0al 1 A-A'L Vlo V11, ala , alas, J = a a (3.2)

We are therefore led to consider a complex two-dimensional vector spaceS with elements aA on which SL(2, C) acts. This is spin-space and theelements are spinors. The complex conjugate vector space S = S' haselements QA' and we also have the two dual spaces S*, S'* with elements`YA, 6A'

We may develop the spinor algebra by analogy with the previous section.Higher valence spinors are elements of tensor products:

40 ...BA'...C'C...DB'...F' E S®...®S®S'®...®S'®S*®...®S*®S'*®...®S'*

We adopt the convention that the relative order of primed and unprimed


indices among the upper or among the lower indices is unimportant, i.e.

"PAB'C' = "DB'AC' = 4 B'C'A

There is an operation of conjugation between S and S': aA E S de-fines aA' in S' by aA' = aA. This extends to higher valence spinors, e.g.aABC' E S ® S ®S' defines aA'B'C E S' ®S' ® S by

aA'B'C = aABC' so that e.g. 501111 = a011'

A spinor with equal numbers of primed and unprimed indices, sayQABC'D', is Hermitian if

aA'B'CD = aA'B'CD so that e.g. ao'o'oi = aooo'1' = ao'o'oi

There is a 1-1 correspondence between real tensors of valence n andHermitian spinors with n primed and n unprimed indices. For n = 1, thisis the correspondence in (3.2): V = Hermitian part of S ® S.

The operations of symmetrisation and anti-symmetrisation are definedfor spinors just as for tensors. However, the fact that S is two-dimensionalmeans that any skewing over more than two indices gives zero, and that upto complex multiples there is a unique non-zero skew two-index spinor. Wemake a choice of one such and call it CAB. Its complex conjugate is writtenEA'B' = TA'B', omitting the bar for brevity.

Now for tAB E SL(2, C) ,

tABtCDEBD = (det t)eAC = 6AC-

Thus the chosen CAB is preserved by SL(2, C) much as the metric 91abis preserved by the Lorentz group L. All spinors are `null' with respect toCAB in that EABaAaB = 0 for all aA, and conversely if CABaAI3B = 0 fornon-zero aA and flB then they are proportional.

CAB allows the identification of S with S* via

aA -a aA = aBCBA

and converselyaA-+ aA=EABaB

where CABECB = SCA. Care is needed with these expressions because CABis skew. Thus, for example,

aA = aBEBA = -aBCAB-

The mnemonic is `adjacent indices - descending to the right'. In thesame fashion, CAIB, and EA'B' identify 9 with S'*. If we choose CAB so


that Eol = 1 in the basis of S implicitly given by (3.1) and (3.2) then theHermitian spinor EABCAIBI, which corresponds to some real tensor, in factcorresponds to the metric flab. To see this, we first observe that it defines asymmetric, non-degenerate tensor. Next, for any null vector Va = aAaA'we have

CABCA'B'aAaA'aBaB' = (eABaAaB)(eA1B1aA'aB') = 0

so that it is certainly correct up to proportionality. Finally, for the scale,

if to = tAA' is the unit time like vector with T(ta) _ 11

1 0

Jthen

EABEA'B,tAA'tBB' = e01e0'1't00't11' + e10e1,0,t11't00' = 1.

Since S is two-dimensional, the analogue of a tetrad is a dyad (oA, tA)and we can say that this is normalised if eABOAtB = OBtB = 1. A nor-malised dyad defines a tetrad in V by

la = oAOA'; na = tAIA'; ma = oAjA'; m = LA-5A'. (3.3)

Here la and na are real future-pointing null vectors and ma is a complex nullvector. With the identification of EABEA'B' with flab we may also calculate

1lablanb = 1 = -1Jabmam

and all other scalar products are zero. Further, the tetrad is right-handed.A tetrad with these properties is referred to as a null tetrad and evidentlydetermines a normalised dyad up to a sign ambiguity, i.e. (oA,1A) is notdistinguished from (-o A, -t A). Note that, for a normalised dyad,

OAtB - tAOB = CAB (3.4)

This a particular case of a more general formula. We observe thatEA[BECD], being skew on three indices, necessarily vanishes. Writing thisout in full, rearranging terms, and raising some indices we find

EABECD = SAC5BD - 6AD8BC. (3.5)

Now suppose that a given spinor is skew on a pair of indices, say

4'...CD... = 4...[CD]...

where the dots denote some other collection of indices. Then multiplyingby the identity (3.5) we find

CABECD...CD... = 2...AB...


1'D ...AB... = 1 CABD...CC...

Thus a skew pair of indices can be removed as an CAB with a contraction. Inthis sense, only symmetric spinors count, that is, any spinor can be writtenin terms of symmetric spinors and the special spinors CAB, CA'B' and theirinverses.

A further simplification is that symmetric spinors factorise in the sensethat a symmetric valence n spinor (PA ... B can be written as a symmetrisedouter product a(A.../3B) of n valence one spinors. To see this, choose a dyadand consider components; define eA = (1, x) and form 6B. Thisis a polynomial of degree n in x and so, since S is complex, it factorises. Thelinear factors are of the form /3AeA etc. whence 4A...B = a(A.... 8B).The individual factors are unordered and defined only up to scale. Eachfactor defines a null vector via (3.2) and these are known as the principalnull directions (p.n.d.s) of 4A...B

As an example of these properties of spinors, we consider the spinorequivalent of a valence 2 tensor,

Tab = TAA'BB' = TABA'B'

On the unprimed index pair we have

TABA'B' = T(AB)A'B' +T[AB]A'B'1= T(AB)A'B' + 1 EABTCCA'B'

and now on the primed pair

= T(AB)(A'B') + 1 CA,B,T(AB)C'C' + 1 CABTCC(A'B')2 2

1+ 1CABEA'B'TCC'CC(3.6)

Here the first term is a Hermitian spinor and corresponds to the sym-metric trace-free part of Tab, the fourth term is a real multiple of the metricand so must be the trace 4gabTe,'. The middle two terms give a Hermitianspinor when taken together. Further if A and B and also A' and B' areinterchanged, each of the two middle terms changes sign. Thus these twogive the skew part T[ab] of Tab. We deduce that if Fab is a real bivector then

Fab = 4ABEA'B' +TA'B'EAB

where (DAB is symmetric.


The spinor 4'AB factorises as a(A)3B) and the principal null directionsare related to eigenvectors of Fab (exercise 3b).

The spinor equivalent of Eabcd is found by a similar analysis to be

Eabed = i(EACEBDEA'D'EB'C' - EADEBCEA'C'EB'D')

so that

Eabcd =i(SACSBDSA,D'SB C' - SADSBCSA,C'SB,D').

In this form it is easy to see the effect of dualising the bivector Fab (3.7):

Fab = 'DABEA'B' +4A'B'EAB

*Fab = 1 EabcdFed = -24'ABEA'B' + 2T A'B EAB,

The eigenspaces of dualising are now apparent:

if Wab = 4'ABEA'B' then *Wab = -iWab (3.8)

if Wab = 'A'B'EAB then *Wab = iWab. (3.9)

The first case is referred to as an anti-self-dual bivector and the secondas a self-dual bivector. As anticipated, the eigen-objects are complex.

We may exploit this discussion of bivectors to give a geometrical inter-pretation of a spinor. Firstly, a spinor oA defines a null vector la = OAOA'.However, this ignores the phase of oA, i.e. oA and e'OoA define the samenull vector. To include the information of the phase we define the bivector

Fab = OAOBEA'B' + OA'OB'CAB. (3.10)

By exercise 2b, this bivector is simple. If we introduce a spinor eA to forma normalised dyad with OA then

Fab = OAOB (OA' 6B' - GA'OB') + OA'OB' (OAGB - GAOB )

lamb - lbma + lamb - lbma

in terms of the null tetrad (3.3) defined by the dyad, so

Fab = 2l[aXb]

where Xa = ma + ma.Now Xa is space-like and orthogonal to la. A different choice of 1,A, say

bA = tA + AOA, adds a multiple of la to Xa. Thus oA, as well as definingthe null vector 1a, defines a `null flag' or two-plane element containing the'flag-pole' la (see figure 3.1).


Figure 3.1. The null flag.

If we set ma = Xa + iYa, then a change of phase o° ei0oA takesto -+ a-ietA and so ma -+ e2iema. This rotates ma in the plane of Xaand Ya by 20. Thus as the spinor oA rotates by 0, the flag-pole rotates by20. As anticipated, there is no way of distinguishing geometrically between±OA.

We end this chapter with a consideration of `spinors up to scale,' i.e.the set of one-dimensional subspaces of the two-dimensional space S'. Thisis the one-dimensional complex projective space P1 (or CP1 if we wish todistinguish it from RP1). We may describe it as follows: a point of P1 isa proportionality class of primed spinors, an equivalence class [7ro,, iris] ofpairs of complex numbers (7ro,, 7r1,) under the equivalence relation

(7ro,, 7rl,) - (A7ro,, A7r1,) ; A E C*,

with 7ro, and 7r1, not both zero. We may therefore distinguish two openneighbourhoods which cover P1:

Uo = {[7ro,, 7rl,] : 7ro, # 0}; U1 = {[7ro,, 7r1,] : ir1, # 0}.

On Uo, the quantity C = is a good coordinate, i.e. it maps Uo one-to-0

one and onto the complex plane. On U1, i1= is a good coordinate andin the overlap Uo fl U1i71=-1

This exhibits P1 as a complex manifold (see e.g. Morrow and Kodaira1971; Field 1982). The definition is: a paracompact Hausdorff topologicalspace X is a complex manifold if

a) X has an open cover {Ui}iEI with coordinate functions fi : Ui --+ Cl;

b) on each non-empty intersection Ui fl Uj, the `transition' functionsfj o fi 1 are holomorphic from C' to Cn.


Figure 3.2. Stereographic projection.

The coordinates C and 77 can be given a more geometrical interpretationby means of stereographic projection. P1 is the space of spinors regardlessof scale and phase, and this is the space of null directions or the projectivenull cone at a point in Minkowski space. Clearly this is topologically asphere, the celestial sphere. If we represent it as the unit sphere in R3, thenstereographic projection is the map from the north pole to the equatorialplane as in figure 3.2.

This map takes the point p with polar coordinates (8, cp) to the point p'with

x + iy = cot 0

and maps the whole sphere with the exception of the north pole onto theArgand plane of C. To get the north pole we project from the south poleand the point with polar coordinates (8, cp) goes to

Here the minus sign is because the equatorial plane when seen from belowhas the opposite orientation.

Thus Uo is the sphere minus the north pole, Ul is the sphere minus thesouth pole and in the overlap, 77 = (-1.

The isomorphism of SL(2, C) with L+ can be seen directly with the aidof these considerations.

The Minkowski metric in spherical polar coordinates is

ds2 = dt2 - dr2 - r2 (d82 + sine 8dcp2 )

and the future null cone N+ of the origin is just the surface t = r. The


Figure 3.3. Two cuts of the future null cone.

degenerate metric on N+ is therefore

ds2 = -r2 (d02 + sin 2 0dcp2) (3.11)

where 0 and cp label the null geodesics generating N+ and r is an affineparameter along them. A particular section or cut of N+ is given by spec-ifying r as a function of 0 and cp (see figure 3.3).

It is clear from (3.11) that the map down the generators from Sl to S2is conformal. The cut E : r = 1 is invariantly defined as the set of nullvectors l° with talc = 1 where to is the unit vector defining the time axisand the intrinsic metric of E is that of a unit sphere. If we apply a Lorentztransformation to ta, say

to - to =A a bt'

then

E-+ t:tala=1and clearly t must also have the unit sphere metric (see figure 3.4).

The map down the generators from t to E is conformal so a Lorentztransformation defines an element of the group C(2) of conformal transfor-mations of the unit sphere. If Aab is a rotation then E coincides with Eand the corresponding element of C(2) is a rotation.

Evidently, this is a group homomorphism and is one-one. To go furtherwe need to know more about C(2).

In terms of the coordinate C, the unit sphere metric is

ds2 = - (1 + -)2 dCdC. (3.12)


Figure 3.4.

Now we know from the theory of one complex variable (and it is clearfrom (3.12) - see also exercise 3h) that a conformal map in two dimensionsmust be holomorphic (or anti-holomorphic, in which case it is orientation-reversing). Thus a conformal transformation of the sphere must be a glob-ally defined holomorphic transformation:

S S=fm77 -4 = 9(r1)

with , = C-' where finite.Intuitively, we may find these as follows: the function f (C) must have a

simple zero at the transformed south pole and a simple pole at the trans-formed north pole, so

f (C) = S + d h(S)

where h(C) has no zeroes or poles. Therefore h(C) must be constant byLiouville's theorem and

f(C) =cC +

d. (3.13)

Without loss of generality we may take ad - be = 1. Thus the conformalgroup C(2) is the group of Mobius transformations (3.13) which is clearlyisomorphic to SL(2, C) /{±I} and we must have

SL(2, C) - C(2) = L.

This relationship was pointed out in Penrose (1955,1956) `The VisualAppearance of a Moving Sphere'. A sphere S intersects a circular cone


of null geodesics on an observer's past null cone, i.e. S is represented asa circular disc on the celestial sphere. If the sphere moves rapidly pastthe observer, what is its appearance? (We know that it suffers a 'Lorentzcontraction' but does it appear flattened?) To answer this question, applya Lorentz transfomation, that is a Mobius transformation, to the celestialsphere. Since this is conformal it takes circles to circles and hence discs todiscs, so that the sphere will continue to appear circular!

Exercises 3

a) Show that SL(2, C) is simply connected.

b) Show that if the real bivector Fab corresponds to the symmetric spinor4AB = a(AI3B), then aAaA, and QA,3A' are eigenvectors of Fab.

c) A null bivector Fab is one for which the principal null directions coincide,i.e. 4AB = aAaB. Show that a bivector Fab is null if

FabFab = Fab *Fab = 0.

Deduce that a null bivector is simple (exercise 2b).

d) If Rabcd has the symmetries of the Riemann tensor, show that it has aspinor decomposition of the form

Rabcd = TABCDEA'B'EC'D' + TA'B'C'D'EABECD

+ 4ABC'D'EA'B'ECD + 4A'B'CDEABEC'D'

+ 2A(EACEBDEA'B'EC'D' + EABECDEA'D'EB'C')

where T ABCD is symmetric, <bABA'B' is symmetric on each pair of indicesand is Hermitian, and A is real. By taking contractions of this, find thespinor equivalents of the Weyl tensor, the Ricci tensor and the Ricci scalar.

e) CP' (respectively RP') is defined as the set of one-dimensional sub-spaces of an n+1-dimensional complex (respectively real) vector space.Show that CP' is a complex manifold by constructing an appropriate coverand coordinates.

In a similar way, one defines a Grassmann manifold G(k, n) as the setof k-dimensional subspaces of an n-dimensional vector space.

f) Recalling thatt=

e 2 01E SL(2, C)0 e 2


defines a boost in the (03) plane, show that the effect of a boost on thecelestial sphere is to move points towards the direction of motion.

g) This exercise is concerned with (complex) line bundles over CP'. Sucha bundle, say L, may be defined in terms of the cover {Uo, U1} as follows:there exist local trivialisations

ti:piL - UixC

(where p2L means L restricted to Ui), and a transition relation between thefibre coordinates pi

Yo = fol(C)µ1

where the transition function fol is holomorphic on Uo n U1.A section of L is defined by a pair of holomorphic functions

µo = so(C), t ti = si(c)

related by the transition function on Uo n Ui.We may define a line bundle L over CP1 by associating to a point

[7rA1] in CP1 the corresponding one-dimensional subspace of C2. ThenL C CP1 X C2 and the projection L -+ CP1 is

{[7rA,], A7rA, : A E C} -+ [7rA,].

Show that in this case we may take fol = C. This particular L is known asH-1 or O(-1). In a similar way we may define H-' or 0(-n) by

{A 7rAB1 : A E C} -> [7rA']n

Show that now fol = Cn. By analogy, we define Hn by fol = C-n. Showthat sections of Hk are defined by functions F(irA') homogeneous of degreek in the sense that

F(A7rA,) = AkF(7rA,).

Evidently this construction produces line bundles Hn or 0(n) on any com-plex projective space (Wells 1980).

h) An alternative approach to defining a complex manifold is in terms ofextra structure on a real smooth manifold (Chern 1967).

If M is a smooth manifold with tangent bundle TM, form TMc, thecomplexification ((TMc)p = (TM) (9 C). What is required is a way ofrecognising vectors in (TMc)p which are to be regarded as holomorphic.


A complex structure J on a real vector space V is a tensor Jab withJabJbc = -6aC. In Vc, J has eigenspaces with eigenvalues ±i. The holo-morphic or type (1,0) vectors are defined to be the +i eigenspace and theanti-holomorphic vectors are the -i eigenspace.

An almost complex structure on M is a field of complex structures onTM. The almost complex structure is said to be integrable if the differentdefinitions of holomorphic at each point fit together. This means that if Vaand Ua are type (1,0) vector fields then their commutator must be also, i.e.

(Jab_i6ab)Va = 0 = (Jab-i6ab)Ua (Jab-i6ab)(VCVQUa-U0VcVa) = 0.

Show that this is equivalent to

U"VbO[aJb]' = 0

or8[ae Jb] dOcJde = 0

and that, for a two-dimensional manifold M, this condition is vacuous.The Newlander-Nirenberg Theorem then asserts that complex coordi-

nates can be found for M so that M becomes a complex manifold in theearlier sense.

Show that, for an orientable two-dimensional manifold M, having aconformal structure is equivalent to having an almost complex structure.(The point is that type (1,0) vectors will be null.)

Chapter 4

Spinor Fields

By analogy with vector fields, we wish to define a spinor field on a manifoldM as a section of a suitable bundle S over M. In chapter 3, we saw thata spinor dyad at a point determines a null tetrad which is a right-handedtetrad containing future-pointing null vectors. To define S therefore themanifold M must be orientable (so that a global choice of orientation ispossible) and time-orientable (so that a global choice of future-pointing ispossible). This means that the orthonormal frame bundle B of M can bereduced to an L+ bundle, i.e. that we can choose the transition matricesfor B to be in L+ .

Now a dyad determines a null tetrad but, conversely, the tetrad deter-mines a dyad only up to sign. Thus, having reduced B to an L+ bundle, wemust find S as a double cover of B. For this to be possible, M must satisfycertain topological restrictions which can be illustrated diagrammatically(see below) but which we shall first discuss in the language of Cech coho-mology (Bott and Tu 1982). Here the sheaves are constant (Z2) and thereader wishing to follow this argument but unfamiliar with the terminologymay want to refer first to chapter 9.

We begin with the question of orientability. We take a locally finiteopen cover {Ui}iEI of M, and a choice of orthonormal frame fi over U.On the non-empty intersections Ui fl Uj, the frames fi and fj are relatedby a Lorentz transformation Pij:

fjPij = A.

(Throughout this discussion, we suspend the summation convention!) Thetransformations Pi j define the orthonormal frame bundle B and must sat-

25


isfy

Pji = PPj1

PijPkiPjk = I if Ui n Uj n Uk # 0.

We define Tij = detPij.Then Ti., is an assignment of ±1 to every non-empty intersection Ui n Uj

and Tij = Tji. Thus Tij defines a one cochain T E C1(M; Z2). FurtherTi2TkjTjk = 1 so that T is actually a cocycle, r E Z1(M; Z2).

A change of orientation of some of the fi corresponds to a zero cochainw E C°(M; Z2) as follows: wi = 1 if the orientation of fi is unchanged,wi = -1 if the orientation of fi is changed. This in turn modifies the Pijand hence the Tij according to

Tij -4 WiTijWj

i.e. T changes by a coboundary.If the class [T] of r in Hl(M; Z2) is trivial then r is a coboundary,

Tij = wiwj. Now using the cochain w we can modify the frames fi so thatall the Tij become one. This is then a choice of orientation - thus themanifold M is orientable if the class [T] in H1(M; Z2) is trivial.

A similar argument shows that M is time-orientable (i.e. that a consis-tent choice of one part of the null cone of each point as the future null coneis possible) if another class is trivial.

Suppose this is done, so that M is orientable and time-orientable andthe matrices Pij are all in L+ . Now to construct the spin bundle S, wechoose an SL(2, C) matrix aij which is one of the two inverse images of Pij(the other being -ajj). These can evidently be chosen to satisfy

aji = o 1

but on the non-empty triple intersections we shall have

QijQkiQjk = zijkI

where I is the 2 x 2 unit matrix and zijk = ±1. To be able to construct Swe need to be able to choose the aij so that all the Zijk are +1. Again, zijkdefines a cochain z, now in C2(M; Z2), which again is actually a cocycle.We may define a one-cochain wij by changing the choice of Qij : wij = -1if we take the choice -Qij, otherwise wij = 1. This changes zijk by acoboundary:

zijk ' zijkTijTki1Tjk

CHAPTER 4. SPINOR FIELDS 27

Figure 4.1. Trouble arises if the path IF can be shrunk to zero over a surfaceE not wholly in the fibre containing I'.

The class [z] of zijk in H2(M; Z2) is trivial if z equals a coboundary, say

zijk = TijTki1Tjk.

In this case we may use the cocycle Tij to change the signs of the vij:

a --4 Tijaij.

Now the new aij satisfy

aij = o,.,,' and OrijOrkiO'jk

and can be used to build the bundle S. The class [z] in H2(M; Z2) is thusthe topological obstruction to building S. If [z] is trivial there will be in-equivalent bundles S parametrised by one-cocycles with trivial coboundary,i.e. by elements of H1(M; Z2).

These topological considerations may be shown in diagrammatic formas in figure 4.1: the difficulty is that a path in the fibre of the frame bundlenot homotopic to zero in that fibre may be homotopic to zero if deformedin the whole frame bundle.

For IF we imagine a rotation of a frame, at a point p, through 2ir. Fromthe discussion in chapter 3, we know that IF is not homotopic to zero withoutmoving away from p. However, it is possible, and an example is providedby exercise 4a, that if the frame is carried along paths in M beginning andending at p then this rotation may be undone.


We shall suppose that these pathologies are absent and that the bundleS has been constructed. Local sections of S are unprimed spinor fields,7rA(x). We will also have the dual, conjugate, and conjugate dual bundlesS*, S', S'*.

Again from the discussion in chapter 3, a spinor field defines a null anti-self-dual (a.s.d.) bivector, with an unavoidable sign ambiguity. The Levi-Civita connection Va of M can be used to define the covariant derivativeof null a.s.d. bivectors, and so will extend uniquely to define a connectionVAA' on the spin bundles if we demand

DaEBC = 0 = Da6B'C'. (4.1)

The discussion of spinor calculus parallels the review of tensor calculusin chapter 2.

The commutator of derivatives on a spinor will introduce curvature. Wedefine the operator DAB by

VaVb - VbVa = EA'B'DAB + EABOA'B' (4.2)

so thatAAB = OC,(AVB)C,. (4.3)

Note that this operator is linear over functions in that

DAB(.faCl3D') = I(aCAAOD' +I3D'DABaC)

for any function f and pair of spinors ac, /3D'. Thus we must have

AABaC = VE'(AOB)El aC = XABCDaD (4.4)

DA'B'aC = DE(A,DEB,)aC = (4.5)

for some curvature quantities XABCD and 4'A'B'CD. From (4.1) and (4.3),these must have the symmetry properties

XABCD = X(AB)(CD)'A'B'CD = 4(A'B')(CD)

Further, from the Ricci identity of chapter 2 applied to an a.s.d. bivector

DABa(AI3B) = 0

for any a(A,QB). (This follows from the symmetry of the Ricci tensor whichin turn follows from (2.2).) Thus

XABBD = 3AEAD


for some A. We may therefore decompose XABCD as

XABCD = T ABCD - 2AED(AEB)C (4.6)

where 'ABCD is totally symmetric. To relate the three curvature quantities1 ABCD, 4'A'B'CD and A obtained in this way to the Riemann tensor, wehave from (4.3), (4.4) and (4.5):

DAB(aCI3C') = I3C'XABCDaD +aCTABC'D'ID'

= -(XABCDEC'D' +4IABC'D'ECD)aDI3D

= -RABCC'DD'aD;F

where Rabcd = RABCC'DD'EA'B' + complex conjugate. Comparing thiswith the spinor decomposition of the Riemann tensor found in exercise 3dwe find

Cabcd = TABCDEA'B'EC'D' + complex conjugate (4.7)

Rab- 4R9ab = -24ABA'B' (4.8)

R = 24A (4.9)

for the Weyl tensor, trace-free Ricci tensor and Ricci scalar. In particular,this means that (DABA'B' is Hermitian and A is real.

Note that the Einstein vacuum field equations are just

'ABA'B' = 0 = A.

The (second) Bianchi identity (2.3) translates to

VA/ ATABCD = V (BB'4)CD)A'B'

VAA'4)ABA'B' + 3VBB'A = 0.

In vacuum, this reduces to the vacuum Bianchi identity:

DA'ATABCD = 0. (4.10)

Since the Weyl spinor 'ABCD is totally symmetric, it will factorise

TABCD = a(A)3B'YCSD)

The factors define the four principal null directions (p.n.d.s) of the Weyltensor (Pirani 1965; Penrose 1968b). The Petrov-Pirani-Penrose classifi-cation of Weyl tensors is based on a consideration of coincidences among


(1,1,1,1)

(2,1,1) (2,2)

(3,1)

Figure 4.2. The PPP classification.

the p.n.d.s. These may all be distinct, when the Weyl tensor is said tobe algebraically general or Type 1 (symbolically (1,1,1,1,)) or may coincidein various ways, when the Weyl tensor is said to be algebraically special.The possible coincidences may be represented as (2,1,1), (2,2), (3,1) or (4),which are known respectively as Types 2, D, 3, and N, or the Weyl tensormay vanish (-). The various cases may be arranged in a table of increasingspecialisation as in figure 4.2.

We shall end this section with some examples of spinor field equations.An electromagnetic field is represented by a bivector Fab. The source-

free Maxwell equations on Fab are just

V[aFbc] = 0; VaFab = 0.

If we introduce the a.s.d. part of Fab :

Wab = Fab + i *Fab

then we can write these compactly as

VaWab=0. (4.11)

In spinor terms Wab = WABEA'B' for a symmetric spinor PAB and then(4.11) is just

v A'AVAB = 0. (4.12)

A linearised solution of Einstein's vacuum equations in Minkowski spaceis represented by a linearised Weyl spinor satisfying the vacuum Bianchi


identity, i.e. by a symmetric spinor field 'PABCD satisfying (4.10):

A,AcPABCD = 0. (4.13)

Both (4.12) and (4.13) are examples of a general class of field equations -the zero rest mass free field equations or z.r.m. equations for brevity. Thefield is a symmetric spinor WAB...C and the field equation is

VA'AcPAB...C=O- (4.14)

If the spinor has 2s indices, this represents a physical field of spin or helicitys. (Strictly speaking the field has positive helicity +IsI if it has positive fre-quency and negative helicity -Is I if it has negative frequency. Here positivefrequency means that the Fourier transform of the field has only positivefrequency components. We shall deal with this point more fully later on.)Thus the Maxwell field has spin 1 and the (linearised) gravitational fieldhas spin 2. The spin .1 member of the set is the Dirac or massless neutrinoequation:

VA'AVA=0. (4.15)

The other type of equation which we shall be discussing in some detailis the twistor equation:

DA'(AWB) = 0 (4.16)

and its higher valence generalisations:

VA (AWB...C) = 0 (4.17)

where wB. C is symmetric. For the moment we simply observe that, if(4.16) is satisfied then

DAI A W B = -iCABIrA, (4.18)

for some other spinor field irA', so that (4.16) and (4.18) are equivalent.

Exercises 4

a) Consider the total space of one of the bundles Hk of exercise 3g as a realfour-dimensional manifold. Show that if k is odd, then spinors cannot bedefined on W.

b) A Maxwell field is null if the corresponding bivector Fa.b is a null bivector.If the spinor representation of a null Maxwell field is c°AB = oAOB showthat Maxwell's equation implies

0A0BV AA'OB = 0


and that cOAB = f (x)cpAB also defines a Maxwell field if

OAVAA'f = 0.

What are the analogous statements for a linearised Type N Weyl spinorVABCD = OAOBOCOD?

c) Show that, if w-:1 are solutions of (4.16) for i = 1, ... , n then w(,A wB)

satifies (4.17).

d) If VA...B is a spin 2s solution of (4.14) and (wA, 7rA,) satisfies (4.18) showthat

XA...B = (PA...BCwC

wA...C = wA'DAA,WB...C - i(s + 1)7(AcpB...C)

are respectively spin (2s - 1) and spin (2s + 1) solutions of (4.14). Theseprocesses are referred to as spin-lowering and spin-raising (Penrose 1975;Hughston and Ward 1979).

e) If WAB is a solution of

DA'(AWBC) = 0

so that VAA'wBC = -i6A(BPC)A' for some Pa, show that pa is a Killingvector and further that VaPb is a self-dual bivector.

Chapter 5

Compactified MinkowskiSpace

A natural question to consider with spinor calculus is how to define the Liederivative of a spinor aA along a vector field Xa. If we had a definitionsatisfying the Leibniz property then necessarily the Lie derivative of thespinor CAB, LXCAB, being skew would have to be a multiple of CAB

LX EAB = AEAB.

However, this would entail

1X9ab = (A + X)9ab = k9ab (5.1)

Thus we could hope to define LxaA only for vectors Xa satisfying (5.1).These are conformal Killing vectors and correspond to infinitesimal confor-mal isometrics, that is maps P of M to itself with

P*(9ab) = k9ab

In flat space, M, these transformations define the conformal group C(1, 3)and we now turn to a consideration of this group.

By exercise 2c, and redefining k, equation (5.1) can be written

VaXb + VbXa = kgab (5.2)

from which, by taking the trace,

k= 1VaXa.

33


We shall find the general solution of (5.2). Differentiating again we have

VaObXc + VaVcXb = Dakgbc

Taking the cyclic sum (abc) + (bca) - (cab) gives

DaVbXe =2

(Vakgbe - Vckgab + Vbkgac)

Differentiating again gives

VaObVcXd = 1 (gcdVaVbk - gbcVaVdk +9bdVaVck)

Now since we are in flat space and there is no curvature, the skew part ofthis equation on (ab) must be zero:

9d[bVa]Ock - 9c[bOa]Odk = 0.

From the trace on (cb):

2V aOdk + 9adDk = 0

where k = gbcVbVck. From the trace of this on (ad):

k = 0therefore VaVbk = 0,

and so k = 2A + 4Baxa

for constant A and Ba. Thus by (5.4)

VaObXc = 2Ba9bc - 2Bc9ab + 2Bbgac

Integrating once

ObXX = 2Baxagbc - 2Bcxb + 2Bbxc + Agbc + Mbc

for constant Mbc. Substituting into (5.2) gives Mb, = -Mcb. Integratingagain finally gives

Xc = Pc + Mbcxb + Axc + (2Bbxbxc - Bcxbxb) (5.5)

for constant P. The general conformal Killing vector therefore depends on15 parameters. Ten of these, Pa and Mab, correspond to the translationsand rotations of the Poincare group; A defines a dilation and the remainingfour, Ba, define what are known as special conformal transformations. To

CHAPTER 5. COMPACTIFIED MINKOWSKI SPACE 35

understand these last we consider the integral curves of the vector field Xaobtained from (5.5) by setting Pa, Mab and A to zero but retaining Ba.These integral curves are the solutions of

dxa2Bbxbxa - Baxbxb.

ds

By a straightforward integration we find

xa(s) =xa(O) - sBaxb(O)xb(O)

(5.6)1 - 2sBbxb(O) + s2BbBbxC(0)xe(0).

From this we see at once that the vector field X' is incomplete onMinkowski space; that is we shall reach infinite Xa at finite parametervalues s where the denominator in (5.6) vanishes. To give a proper dis-cussion of the conformal group, we need to extend Minkowski space byadding some points at infinity. The special conformal transformations willthen interchange the points at infinity with some finite points of M. Weshall describe the points at infinity in two different ways. For the first weconsider the 0(2,4) null cone (Penrose and Rindler 1986). That is we con-sider a six-dimensional real manifold with a flat metric of signature -2. Incoordinates T, V, W, X, Y, Z the metric is

ds2 = (dT)2 + (dV)2 - (dW)2 - (dX)2 - (dY)2 - (dZ)2. (5.7)

The null cone N is simply given by

(T)2 + (V)2 - (W)2 - (X)2 - (Y)2 - (Z)2 = 0

and we wish to consider PN, the space of generators of N. (For a laterapplication we remark that PN is a quadric Q in RP5.)

Clearly PN is a compact four-dimensional manifold. To investigate itstopology, we may intersect N with a five-sphere S5:

(T)2 + (V)2 + (W)2 + (X)2 + (Y)2 + (Z)2 = 2. (5.9)

This will intersect each generator twice. Solving (5.8) and (5.9) simultane-ously we see that

(T)2 + (V)2 = 1

(W)2 + (X)2 + (Y)2 + (Z)2 = 1.

The intersection of N and S5 is therefore S' x S3 topologically. This isa double cover of PN; to obtain PN we must identify antipodal points of


the S1 and of the S3 simultaneously. However this is again topologicallyS1 x S3 (exercise!).

By the same considerations as in chapter 3, any two cuts of N havemetrics which are conformally related, so that PN has a conformal metric.

The group 0(2,4) is the group preserving the quadratic form (5.7) andso maps PN to itself preserving the conformal metric. Since both I and-I in 0(2,4) give the identity map on PN, 0(2,4) is a double cover of theconformal group of the conformal metric on PN. To make the connectionwith Minkowski space we observe that the map

Xa -+ (X0, 2(1 -XbXb), -2(1 + XbXb), X1, X2, X3) (5.10)

embeds Minkowsi space in the null cone N. In fact the image of Minkowskispace is the intersection of N with the hyperplane:

V-W=1. (5.11)

Substituting (5.11) into (5.7), we see that the embedding is actually iso-metric.

On any generator of N with V - W 0 0, we can find a point satisfying(5.11) and hence a point of M. Thus M is identified with a subset ofPN, and the conformal metric of M agrees with the conformal metric ofPN. PN is therefore a compactification MC of M, such that the conformalmetric of M extends to the whole compactification. The points in MCnot in M correspond to the generators of PN with V - W = 0. This isthe intersection of N with a null hyperplane through the origin. All suchhyperplanes are equivalent under 0(2,4) so to see what these extra pointsrepresent, we consider the null hyperplane

V+W=0.

From (5.10) we see that the points of M corresponding to generators of Nwhich lie in this hyperplane are just the null cone of the origin. Thus MCconsists of M with an extra null cone added at infinity. Conversely, M'is a compact manifold with conformal structure and if we remove the nullcone of one point then what is left is conformally related to M.

We have also seen by this construction that the group 0(2,4) is 2-1isomorphic to the conformal group C(1, 3) of Minkowski space, and thatthe null cones of points in MC are represented by the intersection of N withhyperplanes through the origin. An element of 0(2,4) leaving invariant theplane V - W = 0 will leave infinity invariant in M' and so will act only onM. This corresponds to a Poincar6 transformation or dilation. A reflectionin the plane V = 0 defines an inversion in M' which interchanges the null


cone at infinity and the null cone of the origin. To see this, we write A forxbxb and square brackets for the homogeneous coordinates on PN. Thenthe transformation is

[x°,2

(1 - A), - 2 (1 + A), x', x2, x3]

-4 [x°, -2(1 - A), -2(1 +O), x1, x2, x3]

x0 1 1 1 1 x1 x2 x3[0, 2(1 - 0), -2(1+ A- A 01

[°, 2(1 - A), -Z(1 + A), -;6 1, x2, x3]

= xaxa.where xa = o and ,&We may now interpret the special conformal transformations: they con-

sist of the sequence invert-translate-inverta a a a

xa -+x -->x -sBa-4 x- s$A A 1 - 2s(B.x) + s2B2A

where the translation is by -sBa.

There is another way to describe the points at infinity in MI whichwe shall now consider. The metric of Minkowski space written in terms ofspherical polar coordinates is

ds2 = dt2 - dr2 - r2(d02 + sin2 9dcp2).

If we introduce in- and out-going null coordinates by

(t + r), u= 1 (t - r)v = 172 72

this takes the form

ds2 = 2dudv - 2 (v - u)2(d02 + sine 9dcp2)

where the range of coordinates is -oo < u < v < +oo. We aim to add onthe boundary points at u, v = ±oo. To do this, set u = tan p, v = tan q so

ds2 = 2 sect p sect q(4dpdq + sin 2 (q - p) (d92 + sin 2 9dcp2)) .

The range of coordinates is now -Z < p < q < E.. The points at infinitycorrespond to the places where the first factor is infinite. If we defineanother metric 62 by

d§2 = S22ds2; Q2 = 2 cost pcost q


Figure 5.1. Minkowski space on the Einstein Static Universe, representingRxS3byRxS1.

so that d§2 = 4dpdq + sin2(q - p)(d62 + sin2Odcp2) then this metric isconformally related to the Minkowski metric where 0 0 but is perfectlyfinite on a larger manifold. To see what the larger manifold is, set

p= 1 (T-R), q= 1(T+R)2 2

to findds2 = dT2 - dR2 - sine R(d82 + sin 2 OdO2 ).

This is the metric on the manifold R x S3 corresponding to the EinsteinStatic Universe (Penrose 1965b; Hawking and Ellis 1973). The region cor-responding to the Minkowski space is indicated cpt on figure 5.1. The lightcone of the point i- with coordinates T = -7r, R = 0 is the null surface1-. This refocusses at the antipodal point i° at T = 0, R = 7r. The lightcone of i° is Z+ refocussing at i+, T = 7r, R = 0. These surfaces thereforebound the compactified Minkowski space as a manifold with boundary. Inthe vicinity of i- (i+), the Minkowski space M has the character of an in-terior future (past) light cone, while in the vicinity of i°, M is the exterior


i-

Figure 5.2. Compactified Minkowski space showing the neighbourhood of1.

of the light cone as in figure 5.2. Therefore to make M°, the compactificartion of M discussed earlier, we identify opposite generators of Z- and Z+,thereby identifying the points i-, i° and i+ as one point I. In this picturethe Sl x S3 topology is also manifest; a null geodesic is now topologicallySl since its past endpoint on Z- is identified with its future endpoint onZ+, while a space-like hypersurface has the topology of S3, the one-pointcompactification (by Z) of R3.

Finally we remark the terms future/past null infinity for Z-/I+ re-spectively since all null geodesics have a past endpoint on 1 and a futureendpoint on Z+; future/past time-like infinity for i+/i- since all time-likegeodesics start from i- and end at i+; and space-like infinity for i° (Penrose1965b).

We were led to the conformal group by a desire to define a Lie derivativefor spinors. We may now give this definition. If Xa is any conformal Killingvector then the derivative of Xa must take the form

VaXb =Fab +k

2gab (5.12)

where k = 2 VbX b and Fab is skew-symmetric so that

Fab = WABEA'B' + PA'B'EAB (5.13)


for a symmetric spinor cpAB . We define

kCXaA = XbVbaA - cpABaB - 4aA. (5.14)

We need to check that for a null vector Va = aA/3A' the Lie derivativesatisfies

GXVa = aAGX'8A' + a 'L xc

which is a straightforward calculation. From (5.14) and the Leibniz rule wededuce

LXI3A = XbVbfA + cpABI3B +4#A-

Given this definition we may seek those parts of spinor calculus whichhave a simple behaviour under Lie differentiation along conformal Killingvectors, as this will indicate a simple behaviour under conformal transfor-mation.

The first example is the z.r.m. equations. For suppose ')A...B is asolution of the spin s z.r.m. equation:

VA'A0A...B = 0

and consider

1XA...B = GX' 'A...B + 1 kOA...B

= XCVC A...B + 2Stp(ACOB...D)C +S -F 1

2k'tyA...B-

From (5.13) and (5.5) we have

V AA'cOBC = 1 CA(BkC)A'

and from (5.15) and (5.16) it easily follows that

vAA'XA...B = 0

(5.15)

(5.16)

Thus the conformal group acts on solutions of the z.r.m. equations accord-ing to (5.15).

As another example, we consider the twistor equation. If

l3A = LXaA = XEVEaA - cpABaB - 2aA (5.17)


then

VA'(AQB) = i A'C'O(BC,aA) + kVA'(AaB) - ccBCVA'(CaA)4

- WACV A,(CaB)

so that 13A will satisfy the twistor equation if aA does.

Earlier in this chapter, we encountered the notion of conformally reseal-ing a metric gab. That is, we introduced a metric gab by

gab = Q2gab (5.18)

While transformations from the conformal group may have this effect ongab, we are here concerned with something different. Conformal resealingmust change the connection: the Levi-Civita connection Va preserving gabevidently differs from Va preserving gab by terms involving derivatives ofQ. Acting on a covector field Va we find (Penrose and MacCallum 1973)

DaVb = DaVb - TaVb - TbVa +gabecdTcVd (5.19)

where Ta = Va In Q. For the effect on spinors, we define conformal resealingby

EAB = 2EAB; EA'B' = QEA'B'

Then it is readily established that

DAA'aB = VAA'aB -TBA'aA (5.20)

so that alsoDAA'QB = VAA'13B +SABTCA'fC.

Now it is a sort of meta-theorem that Poincare invariant theories with asimple or `good' transformation under conformal resealing have a simple or`good' transformation under the conformal group and vice-versa. (There isa source of confusion in that both properties may be referred as `conformalinvariance'.)

As examples, we consider the effect of conformal resealing on the z.r.m.equations and the twistor equation.

For the first we require that, under (5.18), a z.r.m. field VA...B trans-forms as

OA...B - OA...B = 1WA...B. (5.21)

Then from (5.20) we deduce

DA,AOA...B = Q-2VA,AVA...B


so that solutions of the z.r.m. equations go to solutions under conformalresealing.

For the second, the correct transformation is

WA = WA

so that

V AA,WB = V AA,WB + SABTCA,WC

andDA'(AWB) = SZ-1VA'(AWB).

We also deduce the transformation

7WA, = 7fA, +iTAA,WA

(5.22)

(5.23)

where xA, is as in (4.18). From this it follows that the real scalar E definedby

E=W4ifA+WA1rA, (5.24)

is invariant under conformal resealing. This result will be significant inchapter 7.

Exercises 5

a) There is an alternative way of defining I in terms of causal properties ofM. Begin by defining a causal curve ry as one whose tangent is everwheretime-like or null and the past J- (S) of a set S as the set of all points p E Mwhich can be connected to S by future-pointing causal curves.

A causal curve when extended indefinitely into the future is held todefine a point of causal infinity. Two curves define the same point if theyhave the same pasts. Show that all time-like geodesics in M define a singlepoint at infinity, i+, in this sense. Show that two null geodesics in Mdefine the same point at infinity if they lie in a null hyperplane. Thus inthis picture 1+ appears as the set of all null hyperplanes. (For more oncausal relations see e.g. Penrose 1972; Hawking and Ellis 1973.)

b) Define Pab = -2

(Rab -g

R9ab) = 4'ABA'B' - ACABEA'B' From thetransformation (5.20) of the connection under conformal resealing, deducethe transformation of the curvature:

PABCD = TABCD

Pab = Pab - VaTb + TaTb - 29abTcTc

(5.25)

(5.26)

where Ta = Va In a


c) Show that under a special conformal transformation (5.6) the metrictransforms like (5.18) with

S2 = (1 - 2sBbxb(O) + s2BbBbx'(O)xc(O))-1.

d) With E defined by (5.24) and (WA, 7rA') satisfying the twistor equation,show directly that LxE = 0 for any conformal Killing vector X.

e) De Sitter space is a four-dimensional space of constant curvature arisingas the hyperboloid

T2-W2-X2-Y2-Z2=-1in the five-dimensional space with metric

ds2=dT2-dW2-dX2-dY2-dZ2

(Hawking & Ellis 1973). By introducing coordinates

T = sinht,W = cosh tsinxsinOsincp,X = cosh t cos x,Y = cosh tsinxcosO,Z = cosh t sin x sin 8 cos cp,

show that de Sitter space is conformal to part of the Einstein Static Uni-verse (Penrose 1965b). In particular this means that de Sitter space isconformally flat.

Chapter 6

The Geometry of NullCongruences

In this chapter, we discuss the geometry of null geodesic congruences interms of spinors. We encounter the important notion of shear and itsconnection with complex analyticity which is one of the underlying themesof twistor theory.

A null geodesic congruence I' in a region U of a space-time M is a setof null geodesics, one through each point of U. The tangent vectors to thegeodesics of IF define a field of null vectors la up to scale and so a spinorfield oA again up to scale. If we write D for laVa, the geodesic equation is

Dla = kla for some function k

or equivalentlyOADOA = 0. (6.1)

We may fix the scale of OA along each geodesic by demanding

DOA = 0 (6.2)

which corresponds to parallel propagation of both the flag-pole and flag-plane of OA.

To discuss the relative motion of nearby geodesics we consider Jacobifields. A Jacobi field 77' along a particular null geodesic 'y of r is simply aLie-dragged vector:

Gdy/a = 0 or Dqa = 77bvbla. (6.3)

45


From (6.3) and (6.2) we deduce the equation of geodesic deviation:

D2rfa = (6.4)

which relates the relative acceleration of geodesics neighbouring y to thecurvature tensor.

From (6.2) and (6.3) we see that ?jala is constant along y and so willalways be zero if it is zero at one point of y. The vanishing of rlala meansthat 77a lies in the null hypersurface E containing la so that the nearbygeodesic represented by 77a is `abreast' of y (rather than earlier or later).

Next we introduce a spinor to with

OAGA = 1; DtA = 0. (6.5)

This implies that the complex null vector ma =SAGA'

is parallelly propa-gated along y.

Now ma and ma span a two-plane element orthogonal to y and lying inE; call this the 'screen-space', S. If is lies in S then

r/a = zma + zm, z E C (6.6)

and we may think of S as the Argand plane of z. Substituting (6.6) into(6.3), contracting with ma and changing the sign leads to

Dz = -pz - vx (6.7)

where

PU

= ma-bVbla = OA-MbVbOA

mambVbla = OAmbVbOA-

Thus the relative motion of neighbouring geodesics is described by the twocomplex functions p and v. To disentangle their effects, we consider theprogress of a small circle C in S, i.e. we follow a small circular tube of nullgeodesics near -y (Pirani 1965).

Suppose C is given initially by

z=ee"v; 0<V<2ir.

For a small affine parameter distance SA along y, (6.7) gives

Sz = -(pz + oz)6A.

We consider first the case

v=0, p=Re0.

CHAPTER 6. THE GEOMETRY OF NULL CONGRUENCES 47

(a) (b)

Figure 6.1. The solid circle is changed (a) by convergence and rotation and(b) by shear.

Then the circle (6.8) becomes

z + Sz = f '(1 - Rea*0SA).

This represents another circle C'. The radius is diminished by the factor(1 - R cos -+/iS)) and the circle is rotated by an amount proportional toR sin &SA. We may write symbolically p = convergence + i x rotation.

Now supposep = 0, v = Jule2i,k

so thatz + Sz = e(ea`' -

To see what this represents, set

z + Sz = e$'o (x + iy)x + iy =

x = e(1- IuISA) 0)

y = e(1 + luISA) sin(cp - 0).

This is an ellipse. The major and minor axes are e(1 ± lvlSA) and theinclination is r/ = 2 arg u. Note also that the area is unchanged to thisorder. The term for this behaviour is shear: v is the complex shear (seefigure 6.1).

This much of the discussion follows from (6.3). If we substitute (6.6) in(6.4) we find

D2z = -Rabcdlamblc?7d= -4I oz - 4ooz (6.10)

48

where

AN INTRODUCTION TO TWISTOR THEORY

To = Rabcdlamblcmd = TABCDOAOBOCOD (6.11)

4,ABA1B,OAOBOA'OB' _ -1 Rablalb. (6.12)

From (6.7) and (6.10) we obtain the propagation equations for p and v :

Dp = p2 + as + 'Poo (6.13)

DQ = (p+P)o-+`yoo. (6.14)

These are known as the Sachs equations (Sachs 1961; Penrose 1968b) andallow an understanding of the effects of curvature on the congruence. From(6.13) we see that

D(p-P)=(p-P)(p+P)since, by (6.12), 4oo is real.

This means that if there is no rotation at one point on y, none will everappear. We shall see in exercise 6b that the geometrical significance of norotation is that the geodesics of the congruence lie in null hypersurfaces.From now on we shall suppose that p is real.

Next we remark that (Poo is related by the Einstein equations to theenergy-momentum tensor Tab of the matter content of space-time:

4>oo = 1Tablalb

For most `normal' matter, too will be positive since it represents thelocal matter density measured by an observer whose world line is tangentto la. Of course, since la is null, no observer can have such a world line, butit is enough that the matter density be positive for a sequence of time-likevectors tending to la.

Thus if the geodesic y passes through a region of Ricci curvature rep-resenting such normal matter, then by (6.13) p will increase, i.e. matterintroduces convergence.

Finally, the effect of To, which is a component of the Weyl tensor, is tointroduce shear or, in the language of optics, astigmatic focussing. This inturn enters into (6.13) and causes convergence (Penrose 1966).

Returning to the screen space S, we observe that the relative motionof geodesics as recorded in S is holomorphic iff the shear vanishes. Wemay consider a related question: given a null hypersurface E ruled by nullgeodesics tangent to la, when is there a conformal metric on the space ofgenerators? This metric would be of the form

r/ab = 2m(amb)


and to be a conformal metric on the space of generators its Lie derivativealong the generators must be proportional to itself, modulo terms tangentto the generators. That is, we require

Glm(amb) = km(amb) + 1(a Vb) (6.15)

for some scalar k and vector Va. (6.15) is equivalent to

mambL1m(amb) = 0

which, by the definition of a, is just v = 0.Thus the (two-dimensional) space of generators of a null hypersurface

E has a conformal metric if the generators of E are shear-free. From thediscussion in chapter 3, we know that a special case of such a E is the lightcone of a point in Minkowski space.

Also from exercise 3h, for a two-dimensional manifold, having a confor-mal metric is equivalent to having a complex structure. Thus we have tworelated examples of the interplay between shear and complex analyticity.

To provide a third, we consider the problem of constructing shear-freecongruences, that is, null geodesic congruences for which or vanishes.

In conformally curved space, we see at once that this will be difficult.If v is to vanish then from (6.14) so must To which means by (6.11) thatOA must be a principal null direction. At any point of space-time there areonly four p.n.d.s so the possible shear-free congruences are few. In fact thesituation is worse in that, by the Goldberg-Sachs theorem (Pirani 1965), anull geodesic congruence in vacuum is shear-free if it is a repeated p.n.d. ofthe Weyl tensor. Thus, among vacuum space-times only the algebraicallyspecial ones admit shear-free congruences and they have just one each,except for Type D which has two.

We therefore confine our attention to flat space.From (6.1) and (6.7) the conditions for a congruence to be geodesic and

shear-free (g.s.f.) are

O Dog - o o OB'VBB'QA = 0v = 0A0BIB' 0BB10A = 0.

These conditions may be combined to give the g.s.f. condition as

OAOBVBB,OA = 0- (6.16)

We remark that the condition in this form is independent of the scaling ofoA, and also that it is holomorphic in the components of OA, i.e. there are


no appearances of SAC. To solve (6.16) we choose a constant normaliseddyad (aA,/3A) and coordinatise oA as

1

oA=A(aA-LQA)sothat L= 00 _-001 0°

(6.17)

Then (6.16) reduces to the two equations

VOA,L-LV1A,L=0; A'=0',1'. (6.18)

We write these equations out in Minkowski coordinates

xAA' - [ C v ] VG L x±y t-z Jas

(6.19)

L- =0; L =0.au ac

By the method of characteristics, the general analytic solution of theseequations is given implicitly in terms of an arbitrary analytic function ofthree variables:

F(Lu + , L( + v, L) = 0. (6.20)

This can be written more symmetrically, taking account of (6.17), in termsof an arbitrary analytic function of four variables which is homogeneous ofsome degree

f (AW0, AW1, AW2AWs) = Akf(Wo Wi Wz Ws)

as

f (-ixA'BOB, oB) = 0 (6.21)

whereF(a, b, c) = f (-ia, -ib, c, 1).

So the general analytic g.s.f. congruence in flat space is defined by anarbitrary homogeneous holomorphic function of four variables. We shallfind a geometrical interpretation of this result, which is known as the KerrTheorem (Penrose 1967), in the next section.

Exercises 6

a) A vector field V° is said to be hypersurface-orthogonal (h.s.o.) if it isorthogonal to a family of hypersurfaces, or equivalently if it is proportionalto a gradient. Show that a necessary and sufficient condition for this is

VaVbV] = 0.


b) Show that the null vector field la is h.s.o. if In is geodesic and p = p.

c) If t° is the tangent vector to the world-line -y of the origin of coordinatesin Minkowski space, show that the g.s.f. congruence obtained from thefunction

I = tA,AXA BOAOB

consists of the generators of the in- and out-going null cones springing from'V.

Chapter 7

The Geometry of TwistorSpace

Our first definition of a twistor is as a spinor field QA(x) in Minkowskispace M satisfying the twistor equation

VA,(ASZB) = 0 (7.1)

or equivalentlyV AA'QB = -i6AB7rA (7.2)

for some other spinor field 7rA' .

Our first task is therefore to solve this equation. Since there is nocurvature we have

VA(A,VB,)AQB = 0 = VA,(AVA'B)QC.

Substituting from (7.2) we have at once that

V AA17FBI = 0

and so (7.2) can be integrated immediately:

QA = WA - ixAA' 7rA, (7.3)

where the constant spinor wA is a constant of integration.Thus twistor space T, the vector space of solutions to (7.1), is a four--

dimensional complex vector space and may be coordinatised with respectto a choice of origin, by a pair of spinors,

(WA, 7rA,) = Z`x ; a = 0, 1, 2, 3. (7.4)

53


We have seen already that the conformal group acts linearly on solutionsof (7.1) via

Lx QA =

so that also

LX7rA,

-iXAB'7rB, - (PABc2B - 4kQA

_ caA'B'7rB, + 4k7rA,

GX (QAWA +QA' WAI ) = 0.

We may now interpret this as saying that the conformal group acts linearlyon T and preserves the pseudo-Hermitian inner product

r = QA7rA + S2A'1rA, = WATA + WA'7rA' (7.5)

where we have used (7.3). Furthermore, no conformal transformation leavesall of twistor space invariant. (7.5) may be written

r = Z0Z + Z1Z3 + ZOZZ + Z1Z3 (7.6)

= E«a,Z«ZP,

where 716' E T is the complex conjugate of ZO E T. Since the inner productE is non-degenerate, we may use it to identify T with T* so that primedtwistor indices never appear. We therefore view complex conjugation as amap to T* rather than T:

Z« = (wA, 7rA,) - Z« = (WA,wA').

From the CAB and CA'B' on Minkowski space, we define a totally skewtwistor C«p.yb by

C«A.y6Z«Z/ Z)'Z6 = CABWAWBCA1 B17rA'7rB' (7.7)i k l [i k 1]

It is easy to see that this is also invariant under the conformal group.The signature of E is zero, so that the endomorphisms of T preserving

E constitute the group U(2, 2). If we restrict to endomorphisms preservingC«pyb, this is reduced to SU(2,2) which has fifteen parameters. Above wesaw that C(1, 3) acts linearly on T preserving E. Since C(1, 3) also hasfifteen parameters these groups must be locally isomorphic.

We shall see their relation more precisely below.This will give us a sec-ond definition of twistor space as the representation space of the spinorrepresentation for SU(2, 2).

CHAPTER 7. THE GEOMETRY OF TWISTOR SPACE 55

For a third definition, we consider for a particular field (7.3) the pointsof (complexified) Minkowksi space at which the field vanishes. These satisfy

WA=ixAA1rA'. (7.8)

Evidently, we must allow xa to be complex to find solutions of this ingeneral. By complexified Minkowski space, written CM, we mean just C4with the usual Minkowski metric:

(z0)2 _ (z1)2 _ (z2)2 - (z3)2

(in particular, not a Hermitian metric).The general solution of (7.8) is given in terms of a particular solution

x0A' byXAA' = x0A' + AAA' (7.9)

where AA is an arbitrary spinor.This defines a 2-plane in CM, every tangent of which has the form

)`AirA' for varying AA. Thus every tangent is null and any two are orthogo-nal. Further the tangent bivector is self-dual. Such a totally null 2-plane isreferred to as an a-plane. (A /3-plane is a totally null 2-plane with anti-self-dual bivector and arises as the zeroes of a solution of the complex conjugatetwistor equation, i.e. from a dual twistor.)

Note that the a-plane itself is defined by the proportionality class [Z']of the twistor Z'. If we define the projective twistor space PT as theprojective version of T, then the points of PT correspond to a-planes.Likewise the points of PT* correspond to /3-planes. The extra informationin T is the choice of scale for the spinor IrA, associated to a particulara-plane. PT and PT* are examples of CP3 (see exercise 3e).

In general, an a-plane will have no real point. If one does, say x0 A' in(7.9) is real, then from (7.8)

WATA = ix0 A TA7A'

so that from (7.5), E vanishes. The corresponding twistor is said to be null.Conversely, if E is zero then

WATA = ia, a E R.

Define

then

XAA' I WAWo = a

wA = ix0 A IrA'


PT+

PN

PT-

Figure 7.1. The null geodesics yyx and 'Yy meeting at p correspond to nulltwistors X and Y lying on the projective line L.

and xoA' is a real point on the a-plane. (If a happens to be zero a slightlydifferent construction of a real point on the a-plane is necessary.) Thusan a-plane contains a real point if the corresponding twistor is null. Thea-plane will then contain the whole null geodesic:

xAA = xo A + ryA7rA , r E R.

We may divide T and PT into three parts respectively T+, T- and N orPT+, PT- and PN according as E is positive, negative or zero. Then wehave seen that PN corresponds to unscaled real null geodesics in M. Weproceed to extend this correspondence between geometry in M or CM andgeometry in T or PT (figure 7.1).

First we ask: given two null twistors X1, Y' representing null geodesicsin M, what is the condition for the null geodesics to meet?

We have

Xa = (iXOA'XA,,XA'); Ya = (ixAA'YA',YA')

with corresponding null geodesics

yXxAA'=xpA'+AXAXA'. xAA'=xAA'+µYAYA,

If these intersect at say x2A' then

x2 A' = xo A' + AXAXA' = xAA' +PYAYA (7.10)


for some real A and it, so that

x2A'YAXA, = X A'YAXA, = XAA'YAXA'

and

X xY ' = i(x0 `4'YAXA' - xAA'YAXA') = 0-O (7.11)

Conversely, given (7.11) we deduce that

xoAA'_ xAA' _ -AXAXA' +pYAYA'

for some real A and p, provided XA,YA' 0. If XA,YA' is zero, so that'Yx and yy are parallel, then from (7.11) we deduce instead that

xpA' - xAA' _ AXAXA' + CAXA' + A'XA

for some CA and real A. If CA is zero, then ryx and ryY coincide, while if (Ais non-zero then ryx and -yy are two generators of a null hypersurface andconsequently they intersect at I.

Thus, in all cases, the null geodesics ryx and rye- will intersect if X aYais zero.

Since also X a Xa and YcYa are zero this means that any linear com-bination

Za = (X' + 77Ya; C, 77 E C

will be null and will define a null geodesic which meets the previous two.All these geodesics therefore define the null cone of a point in M: the

null cone of a point in M is a two-dimensional subspace of T lying entirelyin N or a projective line Lp = P1 in PT lying in PN. The equation of Lpis again

WA = ixAA 7.A

where we now think of xAA' as fixed (the coordinate of p) and (w A, 7rA') asvarying.

The points of PT lying on Lp correspond to the various projectivespinors at p, i.e. to the projective null cone or celestial sphere of p.

In M, two points p and q are null separated if there is a null geodesicconnecting them. Translated to PT, this means if the lines Lp and Lq inter-sect, since the point of intersection represents the connecting null geodesic.

The points of compactified Minkowski space MC not included in M makeup the null cone of the point at infinity, I. There is therefore a special linein PN, also written I, and I is represented by all the lines which intersectthis one. In coordinates (w, 7rA, ), the line I has equation 7rA' = 0.'1


We now consider the lines in PT which don't lie entirely in PN. Anarbitrary line may be represented in terms of two points X', Y' on it bythe bivector

Pap = XaYp - YaXp = 2X [aYp]. (7.12)

Evidently the proportionality class of Pap depends only on the line and noton the choice of points on the line. In the space of proportionality classesof all possible bivectors, which is a projective space P5, the lines of PT arethe simple ones. By exercise 2b, these are just the solutions of the equation

eap76PapPry6 = 0. (7.13)

(7.13) is the equation of a quadric in P5. Thus the space of lines in PTis represented by a compact complex manifold Q4 in P5; this is knownas the Klein representation. (In the language of exercise 3e, Q4 is theGrassmannian G(2,4) of two planes in the four-dimensional complex vectorspace T.) It also follows from exercise 2b that if Pap and Qap are bothsimple then the corresponding lines in PT will intersect if

eap76Pa'Q-y6 = 0. (7.14)

We define a conformal metric on Q4 by saying that two points are nullseparated if the corresponding lines intersect in PT. To see that this is aquadratic condition, consider a particular point Pap on Q4. Then (7.14) forall possible Qap defines the tangent plane to Q4 at P. The intersectionof this tangent plane with Q4 is therefore a cone, and this is the null coneof Pap.

With this conformal metric, Q4 is CM', that is complexified compacti-fied Minkowski space.

To locate the real points we need to use the Hermitian structure on PT.We recall that a line L defined a bivector Pap as the outer product of anytwo points on it. We could equally have used the bivector Pap defined asthe outer product of any two planes through L. This is the dual description

1Pap = 1 fap.y5P76

Also, given a line Pap, we can define a complex conjugate line Pap usingthe Hermitian structure. We may therefore define a line as real if

Pap =126ap.y6Pry6. (7.15)

It is then an exercise to show that the line Pap is real if all the points Zaon Pap are null.


(a) (b)

Figure 7.2. The tangent plane at P to Q4i (a), is represented in PT, (b),by the lines LQ intersecting Lp. Infinity is represented by the line I in PN.

The space of projective bivectors satisfying (7.15) is a real projectivespace RP5 and the simple ones give a real quadric RQ4. This is the realcompactified Minkowski space MC of chapter 5, and in suitable coordinates,(7.13) is (5.8).

The group SU(2, 2) acts on PT and so on RQ4. Since both groups havefifteen parameters, this gives a local isomorphism of SU(2, 2) and O(2, 4).However, minus the identity in SU(2, 2) gives the identity in O(2, 4). Also,as we have seen, minus the identity in 0(2,4) gives the identity in C(1, 3),so we have a chain of isomorphisms

SU(2,2) 2-i 0(2,4) - C(1, 3).

The infinitesimal generators of SU(2, 2) may be written as vector fields

iEa pZ,6 (9

on T, where the matrices Eap are Hermitian and trace-free:

Eap = Epa, Eaa = 0.

In coordinates,

EapZpaaA a

BaB a AB'E BW +P B'

A+B ' WA B

&J AC9 &rA


B1C7E iB,A,+A'

where the vectors pAA' and BAA' are real and

EAB = 'PAB + AEAB, WAB = 'P(AB) , A E R.

The transformations leaving the line I invariant are those with BAA'

zero. These are the translations Pa, the rotations'pAB and the dilation A.Thus Bn defines the special conformal transformations.

The trace part of Ea p would be the vector field

A,. (7.16)Y Za

8Z49

a = wAaWA + 7rA1

This is the homogeneity or Euler operator on T. A homogeneous functionf(Z') has

Tf = nfwhere n is the homogeneity. (Such an f should be thought of as a sectionof the bundle 0(n) discussed in exercise 3g.)

In particular, a geometrical object w on T is defined on PT if it isLie-dragged along T:

GTw = 0. (7.17)

In practice, one can tell by inspection whether (7.17) holds, since the re-quirement is just that w have homogeneity zero.

We conclude this chapter with a discussion of the Kerr Theorem whichleads to a fourth picture of what a twistor is.

The zero set of an analytic function f (Za) of the twistor variable is welldefined on PT provided f is homogeneous of some degree. Locally the zeroset is a hypersurface which will intersect PN in a three-dimensional set K.Then K defines a congruence of null geodesics in M. The content of theKerr Theorem is that this congruence is shear-free and conversely that allanalytic g.s.f. congruences arise in this way. To see this, recall that a pointon N is a twistor Za = (w A, 9rA,) with WA = iXAA7rA, for some real pointXAA' and that Za defines the null geodesic y through xAA' in the directionof 7rA7rA'. Thus given f (Z"), we define K by

f (ZxAA WA,, WA,) = 0

to obtain 7rA, as a function of xAA'As an example, the simplest case is to take for f a linear function

f = AaZ°`. (7.18)


Figure 7.3. The Kerr theorem.

The resulting congruence, known as a Robinson congruence (Penrose 1967),will then give a picture of the (dual) twistor Aa. The calculation is straight-forward, though a little messy. If A. = (AA,AA) we have from (7.18)

(iAAxAA + AA )7r,q' = 0

so that 7rA' = k(iAAxAA'+AA'). Choosing coordinates as in (6.19), supposealso

A' A'AA=oAi A =-abthen 7rA' = ik(x - iy, ia + t - z). For convenience of visualisation, we choosea surface E of constant t and project the null vector ld = WA7rA' into E:

l6 = l6 - tatblb.

Now we seek the integral curves of l° in E. This requires the solution of

dx°

ds= la

<ds - 2kC(iat - z) (7.19)

dz _ds

k(CC - a2 - (t - z)2) (7.20)


where C = x + iy. If we choose the parameter s to be proper distance then

(7.21)

The solution of (7.19) with (7.21) may be written in terms of the arbitrarycomplex constant of integration a as

Qi k+aC+aC--2a2=0Q2 aC-aC-2ia(t-z)=0.

If we set a = tan 6ez`* these become

Q1 = x2 + y2 + (t - z)2 + tax tan 0 cos cp - 2ay tan O sin cp - a2 = 0(7.22)

2

2aQ2 = (t - z) - x tan 9 sin cp - y tan 9 cos cp = 0. (7.23)

If we eliminate cp and set p2 = x2 + y2 we obtain

(p-asec9)2+(t-z)2 =a2tan20. (7.24)

Now (7.22) defines a sphere with centre (-a tan 0 cos cc, a tan 0 sin cp, t) and(7.23) defines a plane through (0, 0, t). Thus the integral curves are circles.These circles, for varying cp but constant 0, lie on (7.24) which defines atorus. These are not the `obvious' circles obtained by intersecting the toruswith horizontal and vertical planes. Instead they twist round the torus andeach links all the others. The sense of the twisting depends on the sign of

a=-1AaAa.

For varying 0, (7.24) defines a family of coaxial tori obtained by rotating aset of coaxial circles round the z-axis. The degenerate torus 9 = 0 is justthe circle p = a, t = z. All the circles link through this degenerate torus.

The Robinson congruence is reconstructed by attaching torches tangentto all the circles on all the tori. These are switched on and the whole systemis moved up the z-axis at the speed of light!

In the limit of zero a, i.e. when A,, is a null vector, the congruencedegenerates in the spatial picture to the set of all radial rays from the pointx = y = t - z = 0. Thus, in space-time, the congruence is defined by all thenull geodesics meeting the particular null geodesic defined by Ate. For anon-null twistor, the geodesics just fail to meet. Instead they twist aroundthe degenerate torus, which moves along at the speed of light.

Exercises 7


a) Verify that the signature of E in (7.7) is zero.

b) If are two solutions of (7.1), show that aAA' is a conformalKilling vector.

c) Show that the line L corresponding to the point za = xa -iya in CM liesentirely in PT+ iff ya is time-like and future pointing. The correspondingpart of CM, written CM+, is known as the future tube.

d) A line L in PT is represented by a simple bivector P. Show that Xalies on L if X[aPO'r] = 0.

e) Verify that the simple bivector P'O is real in the sense of (7.15) if allthe points Za on the corresponding line are null.

f) There is another way of describing the relationship between projectivetwistor space PT and compactified, complexified Minkowski space CM'which will be of use at the beginning of chapter 10. Regarding T as afour-dimensional complex vector space define

F1 = {L1 : L1 is a one-dimensional subspace of T}F2 = {L2 : L2 is a two-dimensional subspace of T}

F = F1,2 = {(L1, L2) : L1 and L2 as above with L1 a subspace of L2}.

Now define projections It and v to obtain the double fibration

F1,2

F1 F2

Show that F1 can be identified with PT, F2 with CM', and F1,2 with theprimed spin bundle of CM'. Show also that

tc(v-1(CM)) = PT - I.

g) Show that two a-planes always intersect but that the a-plane defined byAc, meets the /3-plane defined by Ba if A0Ba = 0. In this case show thatthe intersection is a whole null geodesic.

h) Since de Sitter space is conformally flat, it will also be represented bypart of the quadric Q4 with the same definition of the conformal metric.To fix the conformal scale, pick a point Iap of RP5 not on the quadric.


The polar plane of this point is represented by I«p = 2 E«p7al7a Now forany two points Xap,Yap on the quadric consider the two-plane defined byXap,Yap and Iap. This meets Q4 in a projective line which meets Iap intwo more points, say Pap and Q. Now define the distance d from Xapto Yap by

d= 1 log (x, y;; p, q) I I

where (x, y; p, q) is the cross-ratio of Xa13,Yap, Pap and Q. Show thatthis agrees with the definition of de Sitter space in exercise 5e provided Iapis `inside' Q.

In the limit as Iap moves onto Q4i this defines the metric of Minkowskispace. Thus de Sitter space is distinguished twistorially from Minkowskispace by having a non-simple infinity twistor P0.

Chapter 8

Solving the Zero RestMass Equations I

In chapter 6 we encountered the g.s.f. condition and showed how it wassolved by holomorphic, homogeneous twistor functions. We shall now seethat the zero rest mass free field equations can likewise be solved by twistorfunctions, though the solution is more elaborate. The solution raises anumber of questions of interpretation which will be answerable with theaid of sheaf cohomology in the following chapter.

We proceed by means of an example: consider the twistor function

1 (8.1)f(Za) =(AaZa)(BpZQ)

where A. = (AA, AA'), Ba = (BA, BA') are two constant dual twistors.We aim to calculate a z.r.m. field at the point xAA' in CM. Restrict

Za to the line Lx so that

AaZa =B0Za =

(iAAxAA' +AA'

)9rA' = aA'7rA, (8.2)

(iBAxAA' +BA'

)lrA' = QA'*7rA'

and consider the contour integral

W(x) = 2-7ri (aA' 7rA' ) ('3B (8.3)

This is well defined on P1 since the integrand has total homogeneity zero.Further, there will exist a contour around which to do the integral provided

65


the two poles are distinct, i.e. provided

a'4'8A, U.

We may coordinatise P1 in this case by

when (8.3) becomes

7rA/ = aA' + Z/3A'

W(X) _ 1 dz27ri (aA'fA,)z

1aA'fA.

From (8.2), if we suppose that the two dual twistors Aa, Ba meet in theline L., we find

aA/0A, = 1 AABA(x - y)2

where (x - y)2 = (xa - ya)(xa - ya). Thus

(8.5)cp(x) = AABA(x2

- y)2'

It is easy to see, by a direct calculation, that this is a solution of the waveequation. However, it is actually easier to see that, from the definition, itmust be a solution of the wave equation. If we use the symbol px in frontof a twistor expression to mean that the subsequent twistors are restrictedto the line L, then the integral (8.3) can be written

(8.6)Ax) = 27rijPXf(Z-)7rC,d7rC'

1 j f(ixAA'7rA1, 7rA')7rC,d7rC'.21ri

Now

So

O(P

8x'a2 cP

OXAA'OXBB'

O _ Of

BxAA' px f - 27rA,Px awA . (8.7)

21 f P., Ow 27rA'7rC,d7rC'

2ri i(-1)px aas2

C,

CHAPTER 8. SOLVING THE ZERO REST MASS EQUATIONS I 67

whence it is clear thatcp=0.

For the integrand in (8.6) to be well defined on P1, the function fmust be homogeneous of degree -2, or equivalently, a section of the bundleO(-2) discussed in exercise 3g.

Having solved the wave equation we may consider the other z.r.m. equa-tions. These are solved in a similar way by contour integral formulae (Pen-rose 1968a, 1969; Penrose and MacCallum 1973):

17rA' ... 7rB'Pxf(Za)7rc'd7rc (8.8)

27ri

2,7ri J

PxaaA ... 49Bf(Za)1rc,d7rc'(8.9)

8

where in the first f is homogeneous of degree -s - 2 and in the second fis homogeneous of degree s - 2.

To see that these work is a simple application of (8.7). For the first

a 1 'f E'C7xCC,

PA'...B' (x) =27ri

7rA1... 7rB,7rC'PX awC7rE'dhr

which is evidently symmetric on A' ... C' so that

'VCA 'PA'...B' = 0.

The other is similar.Various properties of this general construction may be seen in the first

example. First of all, the field V(x) is singular at points xa in CM whichare null separated from the point ya defined by the dual twistors Aa andB,,,. If we choose L. to lie entirely in PT- (figure 8.1), then W(x) is non-singular for all L., which lie entirely in PT+, i.e. by exercise 7c, for all xin CM+, the future tube. A z.r.m. field with the property of being non-singular on CM+ is a positive frequency field; thus the notion of positivefrequency can be made geometrical. What is needed is that the singularitiesof f (Za) should form two distinct regions on each Lx in PT+ so that acontour always exists for the integral (8.8). (Recall a positive-frequencyfield 0A'...B'(x) with 2n indices is said to have helicity n while a positive-frequency field WA...B(x) with 2n indices has helicity -n.)

Next we remark that there is a great deal of freedom in the function fin (8.6) subject to it giving the same W. In particular, we may change fby adding a function h which has singularities on one side of the contour r


x.

Figure 8.1. For y in the past tube there will be a contour I for each x inthe future tube making cp(x) non-singular.

but is holomorphic on the other, since the contour integral will ignore suchh. With f as in (8.1) for example we could take

f- f=f+ (BaZa)2

where Ba is as in figure 8.1. In general we have the freedom

f ->f+h - h (8.10)

where h, h are holomorphic respectively on opposite sides of F.There is also the freedom in moving the contour. If we combine these

freedoms then the precise relation between cp(x) and the class of objectswhich lead to it via (8.6) becomes rather puzzling! To understand what ishappening we shall introduce sheaf cohomology.

Exercises 8

a) A z.r.m. field cpA'...B is said to be null if it has only one p.n.d., i.e. ifWA'...Bl = for some oA,. Show that a null field arises from thecontour integral (8.8) if f (Za) has a simple pole.

As an extension of this, what are the algebraic types of the linearisedWeyl spinors obtained from fr(Za) = (AaZa)r-5(BpZQ)-r-1 in the casesr=0,1,2?

CHAPTER 8. SOLVING THE ZERO REST MASS EQUATIONS 1 69

b) If the z.r.m. field WA'...B' is obtained from f (Za) show that the fieldsobtained from (Ap Zi) f (Za) and Bp a0 f (Za) respectively are the resultof spin-lowering and spin-raising on OA'...B' (with twistors related to A,,and Ba respectively; see exercise 4d).

c) Solving the Klein-Gordon equation: show that the contour integral

coA'...B'A...C(x) = 21riZA'...ZB'WA...WCPxF(Za,Wa)ZC,dZC WDdWD

wherePxF(Za,Wa) = F(ixAA'ZA,,ZA,,WA', -

satisfies the Klein-Gordon equation

( + m2)VA'...B'A...C=O

if

'AWA)

I`tipWaWZ IyaZ 8Wh F = - Zm2 '.

In this way a cohomological theory of massive fields can be built up.

d) Calculate the field corresponding to the twistor function

RaZa SC'z-PpZp(QtiZ1)2 + (PPZp)2QryZ-y

Show that it is singular on the null cone of the point represented by P!,Qpl .

Show also that, as R,, and Sc, vary, these fields form a five-parameter family(rather than eight, as one might naively think!).

Space-time fields arising from twistor functions of the form

(AaZ')P(BaZa)q(C'QZ,,)r(BaZa)s

are known as elementary states (Penrose and MacCallum 1973; Hughstonand Ward 1979). They are normalisable if C!aDpj is in PT+ or PT-,and are dense in the space of all fields (Eastwood and Pilato 1991). Theyconstitute a useful alternative to plane waves as basis functions.

Chapter 9

Sheaf Cohomology andFree Fields

We saw in the last chapter that in order to make the relationship betweentwistor functions and zero rest mass fields precise we seem to need to studycertain collections of these (holomorphic) twistor functions. These collec-tions are actually well-known to the mathematician: they arise naturallyin the context of sheaf cohomology.

We start by reviewing the essential ideas behind analytic continuationof complex functions. This will quickly lead us to discuss Riemann surfacesand sheaves.

Consider an analytic function f : D --> C, where D C C is a domain(that is, an open connected set). Under what circumstances can we extendf so that it is defined (and still analytic) on some larger set than D? Well,as usual, the analyticity allows us to push out the boundaries of D untilthe `natural boundary' of f (a singularity of some sort) is reached. Itfrequently happens that in doing this the function f gets continued back toa point z E D, but takes a different value there from its original one. Thearchetypical example, of course, is the logarithm. We therefore seem to beobliged to study multivalued functions, unless we can somehow absorb thisbehaviour into the space on which f is defined.

More precisely, we define a function element to be a pair (f, D) wheref is analytic on the domain D. We say that two function elements (fl, D1)and (fn, Dn,) are equivalent (written (fl, Dl) " (fn, D,a)) if there is a se-quence of function elements

(f2, D2), (f3, D3),. - -, (.fn-1, D.-j)

71


such that Di fl Di+i ; 0 and fi = fi+l on Di n Di+i for i= 1,...,n-1.This equivalence relation encapsulates the notion of analytic continuation:the equivalence classes are the complete analytic functions. It is these c.a.functions which are liable to be multivalued, so our aim is given a c.a.f. toconstruct a space R so that the given c.a.f. can be regarded as an ordinaryfunction from R to C. This space R will be the Riemann surface of thec.a.f.; it will be a one-dimensional complex manifold. Broadly speaking,R is constructed as follows. Suppose the c.a.f. is the equivalence classconsisting of the function elements (fa, Da) for a in some labelling set A.If we took the union of all the Da, we would obtain the domain in C onwhich our c.a.f. is liable to be multivalued. So we take instead the disjointunion of all the D. That is, we regard Da and Dp as disjoint even if theyhave some or all of their points in common. So if Da and Dp do have apoint in common, there are now two copies of it, say za and zp. Clearly thisnew space is much too big. We squash it down to the right size and obtainour space R by identifying za with zp wherever there is a neighbourhoodof z on which fa = fp. Actually, for many c.a.f.s (like z2 for example) wecould simply have identified za with zp wherever fa(z) = fp(z). Then Ris the same as the graph of the function f : R = {(z,w) E C2 : z = w2};see figure 9.1. In these cases it is easy to see that R is a one-dimensionalcomplex manifold. Unfortunately, however, this construction would notwork for the function w = (z - i)\. The graph of this function only hasone point (i, 0) above z = i. Its Riemann surface however, has two, one foreach branch of the function. This is why we identify za with zp if there isa neighbourhood of z on which fa = fp, in our construction of R.

So far we have just been considering one complete analytic function andits corresponding Riemann surface. Our aim in the first part of this chapteris to find a way of describing the space of all c.a.f.s. In some sense thereforewe want to study the union of all Riemann surfaces. In order to make thisprecise we need the notion of a germ.

Given a point z and a function element (f, D) such that z E D, thegerm [f, z] of f at z is the set of all function elements (fi, Di) such thatz E Di and there is a neighbourhood of z on which fi = f. Given a germ

[f, z] = {(fi, Di) : i E I}

we can recover the point z (= nD), the value f (z) and the c.a.f. from whichthe germ is taken.

The set of all germs, for all z E C, is denoted 0 and called the sheaf ofgerms of holomorphic functions on C. We see next that 0 has a natural(Hausdorff) topology, and in fact is a one-dimensional complex manifold.

To specify a topology on 0, we have to say which subsets of 0 are open.

CHAPTER 9. SHEAF COHOMOLOGY AND FREE FIELDS 73

z plane

Figure 9.1.

Given a function element (f, D) there is a corresponding subset of 0:

[f,D]={[f,z]:zED}.

We choose these to be the basic open subsets of 0. (It is easily checkedthat they satisfy the axioms for a Hausdorff topology.) An arbitrary openset in 0 is then simply the union of a collection of the basic open sets. Nowthe map it : 0 - C given by 7r([f, z]) = z has some very nice properties. Itmaps the basic open sets [f, D] of 0 homeomorphically onto the open setsD of C. (Again this is easily checked: see exercise 9a.)

In particular, therefore, we can use In [f,D] as a chart and in so doing wecan give the connected components of 0 the structure of a one-dimensionalcomplex manifold. (It is not a connected manifold though.) These con-nected components are the various Riemann surfaces of the c.a.f.s on C.We see this by considering two function elements (f, U) and (g, V). If z E Uand w E V then it can be shown that a path -y in 0 from the germ [f, z]to the germ [g, w] corresponds to analytic continuation from the functionelement (f, U) to (g, V). So -y exists if (f, U) and (g, V) are equivalent.

This is our description of the sheaf 0, except for one last point. Wehave lumped all the function elements together into c.a.f.s, and all the c.a.f.s


together into one (large) space 0. Given 0, how do we recover a functionelement (f, D)? Consider the following diagram:

C

where the map a is given by

a C

a([f, z]) = f(z)

Define a section of 7r over the domain D to be a continuous map s : D --40such that ir(s(z)) = zVz E D.

Theorem 1

For any domain D there is a 1-1 correspondence between function elements(f, D) and sections of 7r over D, such that f (z) = a(s(z)) Vz E D.

Proof

Given a function element (f, D) define

s(z) = If, z] `dz E D.

This is the required section, and clearly

f(z) = a(s(z))

Given a section s of it over D, and a point z in D,

s(z) = [g,z]

where g is holomorphic in a domain U containing z. We define f (z) _a(s(z)), and we have to show that f is holomorphic. The continuity of simplies that the open set [g, U] in 0 is mapped (homeomorphically) to Uby -7r (= s-1). So dz E U f1 D:

f(z) = a(s(z)) = a([g, z}) = g(z)

and hence f is holomorphic on UnD. We repeat this argument for differentsets U until D is covered, and deduce that f = a o s is holomorphic on D.


11

From the point of view of one-variable complex analysis 0 is in itself adesirable goal. For our purposes however it is actually just a very usefulexample. After all, we need to study holomorphic functions on PT+, noton C. So we shall summarise our discussion of 0 by writing the axiomsfor a sheaf, in preparation for looking at some more exotic examples. (Thefirst of these axioms will be no surprise, but we have not made explicit themotivation f o r the others. In fact it is easy to see that the stalk 0.; =1r 1(z)of 0 over z is a commutative ring.)

Let X be a topological space. A sheaf S over X is a topological spacetogether with a mapping 7r : S -+ X which satisfies:

i) 7r is a local homeomorphism;

ii) the stalks Sx = 7r-1(x) are abelian groups;

iii) the group operations are continuous.

The sections of S over an open set U in X form an abelian group S(U).The elements of this group are the function elements we were studyingin the context of the sheaf 0, but in general they need not be analyticfunctions. Indeed if X is a C°° differentiable manifold we could chooseour function elements to be C°° differentiable functions on open domainsin X, or p-forms, or closed p-forms. We obtain in this way the sheavesA.°, AP, and ZP on X. In twistor theory, though, the spaces X on whichwe want to construct sheaves are complex manifolds. Then our functionelements could for example be holomorphic functions, holomorphicp-forms,C°° forms of type p, q, or 8 closed C°° forms of type p, q, and we wouldobtain the sheaves 0, OP, .AM, and Zp,q. (Forms of type p, q and the aoperator are introduced in exercise 9d and are used in chapter 14.)

We can relate all these sheaves to each other using sheaf homomor-phisms. Suppose S and T are sheaves over the same space X. The contin-uous map

cp:S -->T

is a sheaf homomorphism if it preserves stalks

cc: S., -aT VxEX

and is a group homomorphism on each stalk. We shall see an example in amoment. First we define the usual concepts of kernel and image:

ker cp = Is E S : cp(s) = 0 (the zero element in Tx, where s E Sx)}im cp = It E T : t = cp(s) for some 8 E S}.


A short exact sequence of sheaves over a space X is a sequence of sheavesand sheaf homomorphisms

C24

(where 0 is the zero sheaf) which is exact at each stage:

ker cp = 0im cp = ker bim0 = C.

An alternative way of looking at this is to observe that the sequence aboveis exact if Vx E X the sequence of stalks

Ox-Qx-'3x->Cx-Oxis exact (as a sequence of abelian groups and group homomorphisms). Thisemphasises that exactness is a local property, as we can also see clearly inthe following example.

Let X be a complex manifold, Z be the constant sheaf of integers overX (the space of c.a.f.s from X to Z), and 0* be the sheaf of germs of non-vanishing holomorphic functions on X in which the abelian group operationis multiplication. Then

O-4Z O-4e O*-. 0

is a short exact sequence, where i is injection and

e([f, z]) = [exp(21rif), z].

The sequence is obviously exact at Z and 0, but exactness at 0* is moreinteresting. Given a germ [g, z] we need to find another germ [f, z] suchthat

e([f,z]) = [g, z].

For any [g, z] we can find a simply connected neighbourhood N of z anddefine a branch of logg on N (recall that g 0). Then we choose f =zx$ log g.

In contrast, let us now take an open set U in X and consider the sequenceof abelian groups

0-+ Z(U)-*O(U)--+ O*(U) 0

(where S(U) is the group of sections of the sheaf S over the set U, remem-ber). This sequence is also exact at Z(U) and O(U), but not at 0*(U),precisely because the required logarithm may not be available.


This is only one (perhaps the best known) of many examples illustratingthe ability of sheaf theory to express in an elegant way the distinctionbetween local and non-local information. We can go further however. Weshall see presently that sheaf cohomology theory enables us to put a measureon that distinction: we shall be able to decide under what circumstances,and by how much, the sequence above fails to be exact at O*(U).

6ech Cohomology

Consider a sheaf S over a topological space X, and an open cover {Ui}of X. (That is, the Ui are open sets in X such that UiUi = X.) Wewill need to simplify our notation a little by abbreviating Ui fl Uj to U.Our first definition is that of a p-cochain. This is a collection of sectionsfio...ip E S(Uio...ip), one for each non-empty p + 1-fold intersection of thesets Ui, and completely skew-symmetric:

fio...i,, = f[io...i,]

We should refer to this p-cochain as { f20... j,} of course, but we often simplysay `the p-cochain fio...i,,'.

This definition is slightly less intimidating when one learns that in prac-tice p is always between 0 and 3! For example, a 0-cochain is a collection ofsections fi E S(Ui) and a 1-cochain is a collection of sections fij E S(Uij)such that fij = -fji.

The set of all p-cochains has an abelian group structure (inherited fromthe sheaf) and is denoted

Cr({Ui}; S).

Our next task is to define the coboundary map

Sp:C'-*CP+1

and we do this first for p = 0. Given a 0-cochain fi we define a 1-cochainfij as follows:

fij = pjfi - pifj,where pi f j is the restriction of f j to the set Uij . Let us streamline ournotation again and write this as

fij = 2p[j fij .

Then6o({fi}) = {2p[jfi]}

is the coboundary map on CO:

6° : C°({Ui}; S) -+ C'({Ui}; S).


We can see straight away that

kerS0 = S(X).

The coboundary map S, : C" -4CP+1 is defined analogously:

6P({fio...0) = {(p+ 1)P[iP+1.fio...iP]}.

There are special names for the kernels and images of these coboundarymaps:

ZP({Ui}; S) = ker SPBP({Ui}; S) = im SP_1

and the elements of ZP are called p-cocycles, while the elements of BP arecalled p-coboundaries. Again, ZP and BP are abelian groups, and it is easyto see from (9.1) that

6P+106P=0 (9.2)

so that BP is a (normal) subgroup of ZP.The pth Cech cohomology group of the cover {Ui} with coefficients in

the sheaf S is then defined to be

HP({Ut}, S) - ZP({Ui}; S)(9.3)

BP({Ui}; S)

The best way to become familiar with these definitions is to study anexample, of course. We look at a very simple and useful one, namely CP1covered by the two sets

U0

U1

1[-7ro,,7rl,] E CP1 : ?ro, 0 0}

{[lro,,,rl,] E CP1 : irk, 0 0}

and we choose the sheaf S to be O. Then

H°({Ui}; 0)

is the space of pairs of functions fi analytic on Ui (i = 0,1) satisfying

Pi.fo = Po.fi.

In other words fo and fl agree on the overlap U01 and therefore define acomplete analytic function on Uo and U1, which is a global analytic functionon CP1. By Liouville's Theorem this must be constant, so

H°({Ui}; 0) = C. (9.5)


Next we considerHl({U2}; O).

This is the space of functions fol in Uoi modulo coboundaries. (There isno cocycle condition because with this cover there is no triple intersection.)The domain Uoi is an annulus, so these functions fol are precisely thosehaving a Laurent expansion, which we write in the form

0000foi(C) _ bnS-n - E anon

n=1 n=0

(where C = E°; ). Let00

and

9o(C) = E bnC-nn=1

00

gi(C) = ant;".

n=0

Then gi converges on Ug (i = 0,1) and

foi = P19o - po9i

so that foi is a coboundary. Hence

Hi({U1}; O) = 0.

These results are not quite what we will need though. Local sectionsof the sheaf 0 are analytic functions homogeneous of degree zero in thehomogeneous coordinates [iroi, Will for CP1. How do we describe analyticfunctions with nom-zero homogeneity? They cannot really be functions onCP1 of course, and indeed we saw in exercise 3g that they are sections ofcertain line bundles O(n) on CP1.

With this in mind we calculate

H°({Uz}; 0(-1)).

This is quite easy, because the elements here are global sections and in thiscase there cannot be any because a function with negative homogeneitymust have a singularity somewhere. So

fI°({UZ}; 0(-1)) = 0. (9.7)

Indeed, the same argument tells us that

fI°({Ui}; 0(n)) = 0 , n < 0.


Figure 9.2. The splitting formula.

A global function homogeneous of degree zero represents a function on thecomplex plane with no poles and finite at infinity. By Liouville's Theoremsuch a function is necessarily constant so

H°({U$}; 0(0)) = C.

Finally a global function g(7rA,) homogeneous of degree n > 0 has n-thderivative a^° which is homogeneous of degree zero and hence con-ORA/...&'rB,stant. Thus

g=WA'...B'ITA,...7rB,

n

i.e. g is a polynomial and

H°({Uj}; 0(n)) = Cn+1 , n > 0.

Another group we will need is

Hl({Uz}; 0(-1)).

This is the space of functions fol analytic on U01 and homogeneous of degree-1 in the coordinates [7ro,, 7r1,], modulo coboundaries. We define (followingSparling : see § 2.4 of Hughston and Ward 1979)

h$(iro ,7r1,) = 1

fr,fol(,\,1)d, i = 0,1 (9.8)

27ri A7r1, - 7ro,

where a a the integrand is homogeneous of degree zero in A°, Al as= a, ;required. The contours To and rl are drawn in figure 9.2 so that ro - Ti


is equivalent to a contour surrounding the point

(Ao, A1) = (7ro', Ir1')

and then by the Cauchy Integral Formula

ho(7fo', 7r1') - hi(7ro', 7r1')1 fo1(A,1)dA

21ri 1 o-P1 A7r1' - 7ro'

fo1(7ro', 7r1').

This exhibits f01 as a coboundary and so

Hl({Ui}; d(-1)) = 0. (9.9)

Of course we could also have shown this using the Laurent series as before,but that would have hidden the fact, which will be of some significance tous, that an integral is involved in representing f01 as a coboundary. (Recallthat the Laurent series coefficients are determined by the Cauchy IntegralFormula.)

Now for Hl({Ui}; 0(n)) with n > 0. Given f0l(7rA') with non-negativehomogeneity, we may divide it by a polynomial of degree (n + 1) to givego1(7rA') of homogeneity -1. This can be split as above. Then multiplyingby the polynomial we reduce f01 to a coboundary. Thus

Hl({Ui}; 0(n)) = 0, n > -1.

For Hl({Ui}; 0(-n)) with n > 1 we have

1f01(ir0',ir1') _ (7r1' )nfoi(',1)

°°

n r

r

a Sr-00-n -1 00I

)n (E ar(r + ar(r + E ar(')

1 000 -n+00 r-n 00 r

)n+r+901(1A')

wheren-1

901(7rA') = Ea-r

1 (7r0')r(7r1')n-r(9.10)


n -4 -3 -2 -1 0 1 2...H°({U;}; 0(n)) 0 0 0 0 C1 C2 C3

Hl({Ui}; 0(n)) C3 C2 Cl 0 0 0 0

Table 9.1.

So fol is cohomologous to g01 (i.e. it differs from g01 by a coboundary) and901 cannot be simplified further. There are n - 1 complex parameters in901 so

Hl({Ui}; 0(-n)) = Cn-1, n > 2.

We may set these results out as in table 9.1.Evidently there is some sort of duality here. In fact, as we shall see,

Hl({Ui}; 0(-n - 2)) = H°({Ui}; O(n))* (9.11)

where the star denotes vector space dual.This a special instance of Serre duality (Wells 1980; Field 1982). Here

it is expressed by another contour integral formula: if

9(7fA') = VA'...B' IrA B' E H°({Ui}; 0(n))

n

and f (lrA') E Hl({Ui}; 0(-n - 2)) is given by a representative like (9.10)then consider

(f,9) =27ri -

y9(7rA')f(7rA')7rC,d7rC' (9.12)

where the integral is round the equator. Evidently this expression is com-plex linear and non-degenerate and depends only on the cohomology classof f. Thus it defines the duality (9.11).

A suggestive way of writing (9.12) is as

(1, 9) = 'A'...B'WA'...B'

where

217rA, ... 7rB, f (7rD, )7rC, d7rC (9.13)V)A'...B' =

1

7ri iwhen it is clear that f determines 1,A1 ...B1, an element of the dual ofH°({Ui}; 0(n)). Now the contour integral formula (8.8) can be recognizedas an instance of Serre duality.


In (8.8) of course, f was a twistor function. We can now see how tointerpret it cohomologically. We cover PT+ by the open sets

U1 = {Za:Z`xA,, #0}U2 = {Za:Z`xBa 0}

and then if f is the particular example (8.1):

f(Zc) = f (Za) =1

12(ZaA,,)(ZaBa)

we have a 1 cochain. In fact, f12 is a cocycle since there are no tripleintersections in the cover {U1, U2}. Further, the freedom (8.10) which wefound in f precisely corresponds to changing f by a coboundary. Thus thetwistor function f is properly thought of as an element of Al ({Ui}; 0(-2)).

Since the resulting space-time field cp was positive-frequency we are ledto conjecture that

H'(PT+; 0(-2)) = {positive frequency-solutions of the wave equation}

or more generally that

} (PT+; 0(n)) = {positive-frequency z.r.m. fields of helicity - n - 2}.(9.14)

We shall see in chapter 10 that this is indeed the case.An unsatisfactory feature of all these calculations has been the apparent

dependence on the cover {Ui} of CP1. It would be very reasonable, forexample, to wonder how to compare two cocycles fib and gab defined withrespect to the different covers {Ui} and {Va} of the same space X. (Weimagine fig and gab to be local sections of the same sheaf S of course.)

To do this, we need the notion of a refinement. The cover {W"} is arefinement of {Ui} if each of the sets Wa is a subset of (at least) one of thesets Ui. In other words, if the indexing sets for {U i } and {W,,,} are I andA then there is a map

r:A -Isuch that

Wa C Ur(a) for all a E A.

This induces a map in the other direction on cochains:

r* : C"({Ui}; S) -+ C1({Wa}; S).

In the case p = 1 for example,

(r* f)ap= fr(a)r(Q) I Wa(j.


In fact a little work (not deep, just fiddly) shows that r* commutes with 5,so that it is actually a map on cohomology classes:

r* : Hl({Ui}; S) - A' (f W.}; S).

Now suppose that {W,,} is simultaneously a refinement of {Ui} and {Va}(with refining maps r* and p* say). Then to compare our ^ocycles f andg we first take their cohomology classes in Hl({Ui};S) and Hl({Va};S),denoted by [A ] and [g], and then compare r* [ f ] and p* [g] in ft' ({W« }; S).If they are equal we regard [f] and [g] as being equivalent. This idea ofequivalence applies to the disjoint union (over all covers {Ui} of X) ofHl({Ui}; S). In other words, given any two elements of Hl, with respectto any two covers, we can ask whether there exists a common refinementof these covers on which the given cohomology classes agree.

Finally, we can defineHl (X; S)

as the set of equivalence classes of this new equivalence relation. (HP(X; S)is defined similarly of course.)

There is no denying that this is messy though, and in practice onealways uses a particular cover. Fortunately for the adherents of the Cechapproach to cohomology (who would otherwise be fewer in number) thereare theorems guaranteeing that under certain conditions on the cover {Ui}and the sheaf S

Hp({Ui}; S) = HP(X; S).

These covers are called Leray covers (see Field 1982), and our cover {U°i U1}of CP1 is one such, so that we have actually showed (in equations (9.5)-(9.9)) that

H°(CP1; O) = C, (9.15)

Hl(CP1; 0) = 0, (9.16)

H°(CP1; 0(-1)) = 0, (9.17)

H1(CP1; 0(-1)) = 0. (9.18)

[There is another way of defining sheaf cohomology which avoids any awk-wardness over covers. A resolution of a sheaf S is a sequence of sheavesand sheaf homomorphisms

0 -> S --0+ ,A° 0° Al 41 AZ ... (9.19)

which is exact at each stage.


By taking the corresponding groups of sections we get the sequence

0 -+ S(X) A°(X) ° A'(X) i (9.20)

This is not necessarily exact, but it is a complex in that im 4n C ker 0m+iWe may therefore take its cohomology:

HP(X;S) _ ker Op(9.21)

im OP-1

To relate cohomology in this sense to the earlier notion of Cech cohomologyone more definition is needed, that of a fine sheaf. A sheaf A is fine if itadmits partitions of unity i.e. if for any locally finite covering {Ui} of Xthere exist homomorphisms hi : A -+ A such that

i) hi(Sx) = 0 for x 0 Wi, where Wi is a closed subset of Ui

ii) Ei hi = 1 (note that this is a finite sum since the cover is locallyfinite).

Typically, sheaves of C°° objects are fine, while sheaves of analytic objectsare not.

Now there is a theorem that if the resolution (9.19) is fine, i.e. all ofthe sheaves Ai are fine, then

HP (X; S) = HP (X; S). (9.22)

Evidently, this gives an effective way of computing cohomology groups aslong as fine resolutions can be found. See exercise 9d and chapter 14.]

We promised before introducing cohomology that it would enable usto measure the extent to which local information about functions on thespace X (expressed in a short exact sequence of sheaves on X) can be madeglobal. We now turn to this question, our last piece of theory before westudy massless free fields in terms of sheaf cohomology on twistor space.

We noted before that although the sheaf sequence

0-4Z-+O-0*-+0is exact, the sequence of sections

0 --> Z(X) -> O(X) -+ O*(X) -+ 0

(9.23)

or equivalently

0->H°(X;Z)--I H°(X;0)-+ H°(X;0*)->0


may not be exact at O* (X ). If however, the set X were simply-connected,then we would always be able to define logarithms and the sequence wouldbe exact at 0*(X). In fact the condition for exactness is precisely that Xshould be simply-connected, or in other words that the de Rham cohomol-ogy group H1(X; Z) should vanish. How do we establish a result like this?

Theorem 2

Given a short exact sequence of sheaves

0_X413-4 C-->0

on a topological space X, there is a long exact sequence in cohomology asfollows:

0 - H°(X; A) H°(X; B) a,* H°(X;C) H'(X; A) H'(X; B) * .. .

Proof

From the short exact sequence we can construct a diagram

1 1 1

0--+ CP(X;A) - CP(X; B) - CP(X; C) --40is is is

0 --+ CP+1(X; A) CP+1(X;13) - CP+1(X; C) --+0

I I I

in which the rows are exact, the squares commute, and the columns are`complexes' (i.e. SP+1 o SP = 0 as in (9.2)). Because the squares commutethe definitions of a* and /3* are clear and because the rows are exact wehave

ker /3* = im a*.

The crucial part of the proof though is the construction of the mapS*, called the connecting map. We start with an element [f] of HP(X;C)with representative cocycle f. (Note that we are suppressing the indices.)Then f E CP(X; C) and 6f = 0. Because /3 is onto, f = /3g for someg E CP(X;B), and /3Sg = S/3g = Sf = 0. So Sg E CP+1(X;13) is in thekernel of /3, which is the same as the image of a, and there must be anelement h E CP+1(X;A) with ah = 8g. We define S*[f] = [h], but weshould check that it is well defined. Suppose we had chosen g' E CP(X; B)different from g but with

f =/3g=/3g'.


Then /3(g - g') = 0 so that g - g' = ak and h - h' = 6(ak) which meansthat [h] = [h']. Similarly, if we had chosen a different representative f for[ f ] we would have obtained the same class [h].

All that remains is to check that

ker S* = im,3* and im 6* = ker a*

which we leave as a pastime for the reader.

Our short exact sequence (9.23) yields

0 -* H° (X; Z) -4 H° (X; 0) - H° (X; O*) - Hl (X; Z) ->Hl (X; O) -* ...

so all we have to do is note that H1(X; Z) is isomorphic to the de Rhamcohomology group H1(X; Z) (Bott and Tu 1982) to get the result claimedearlier.

In addition though, there is some more interesting information furtheralong this same long exact sequence:

1 '(X; 0) Hl(X; 0*) 112(X; Z) H2(X; 0) __+... (9.24)

Now in the particular case when X is CP1, we have seen that

Hl (X; 0) = 0 = H2(X; 0)

so that (9.24) reduces to

0--+ FI1(X;0*)LH2(X;Z),0

where exactness implies that S* is 1-1 and onto, that is, an isomorphism:

H1 (X; 0*) = 112(X; Z). (9.25)

The significance of this isomorphism is the following: a consideration ofholomorphic line-bundles on CP1 (exercise 3g) shows that they are definedby 1-cocycles C1 (X; 0*). Further two line-bundles may be thought of asequivalent if the cocycles differ by a coboundary, since this just representsa coordinate transformation. Thus Al (X; 0*) represents the equivalenceclasses of line-bundles. This will be true for any X. For the case consideredhere, one also knows that H2 (X; Z) = Z (which is a particular instance ofthe general result that H'(X; Z) = Z for any n-dimensional real manifold


M). So (9.25) states that the group of equivalence classes of line-bundlesover CP1 is isomorphic to the integers. This observation can be extendedto say that every line-bundle over CP1 is equivalent to 0(n) for some n.

In the next chapter, we return to the problem of solving the zero restmass equations and find the theorem conjectured in (9.14).

Exercises 9

a) Show that the map ir : 0 -- C is a homeomorphism from [f, D] to D.

b) Is there a sheaf 13 over R whose sections over the set U C R are boundedfunctions on U?

c) The procedure used in the definition of tech cohomology defines a map

Hq({Ui}; S) -+ Hq(X; S).

Show that in the case q = 1 this is injective. (In other words, by taking therefinement one obtains a larger group.)

d) One particular fine resolution of the sheaf 0 on a complex manifoldX which is of great importance is that leading to Dolbeault cohomology(Morrow and Kodaira 1971; Field 1982). To introduce this we need somedefinitions. A C°° 1-form a on, say, twistor space is said to be of type (0, 1)if it has the form a = AO (Z, Z)dZp. Likewise a 2-form is a type (0, 2) formif it has the form A00 (Z, Z)dZc, A dZp. In general a (p, q) form has p dZ'sand q dZ's. We write £p'q for the sheaf of germs of (p, q) forms. Now define

8 = dZa A aZ : £p,q £p,q+l

which is the anti-holomorphic version of the exterior derivative d. Showthat

d=8+a; 82=0=a =ae+ae.Thus there is a complex

... £P,q - £p,q+1 . .

The fact that this is exact is the Dolbeault lemma (Field 1982).Further these sheaves are fine since any cover {Ui} of X has a partition

of unity consisting of C°° functions {pi} subordinate to it. Deduce that

e,...


is a fine resolution of O.The cohomology groups

HP = ker a : £°,P _, £0,p+1im8:£°'p-1 _, V&

obtained in this way are known as Dolbeault cohomology groups. In theseterms a p-cohomology element is represented by a globally defined C°° aclosed (0, p)-form rather than a set of holomorphic functions defined onp + 1-fold intersections. As a further exercise, can you see how these tworepresentations are related for H1(CP1; 0(-2))?

e) Fill in the gaps in the proof of Theorem 2.

f) If V and W are open sets in a complex manifold X, consider the sequence

0 - CP(V u W; S) - CP(V; S) ® CP(W; S) - CP(V n W; S) -- 0

where

o-(f) = (Pvf, pw.f)T(9,h) = pw9-pvh.

Show that this sequence is exact. The resulting long exact sequence is theMayer-Vietoris sequence:

0- H°(V U W; S) H°(V; S) ®H°(W; S) H°(V n w; S) b

Note that 6* : H°(V n w; S) - H1(V U W; S) is an instance of the Cechmap:

C: H°(U°nUl n...nUp;S) -+HP(U°UU1 U...UUp;S)

which interprets a function on an intersection as a representative cocycle.

Chapter 10

Solving the Zero RestMass Equations II

With the machinery of sheaf cohomology we now set out to demonstratethe generality of the contour integral formulae of chapter 8. We wish toprove two theorems:

1.1 (PT+; O(-n - 2)) = {zero rest mass fields (pA'...B' (x) of helicityn holomorphic on CM+}. (10.1)

2.Al (PT+; O(n - 2)) = {zero rest mass fields WA...B(x) of helicity-n holomorphic on CM+}. (10.2)

These correspond to (8.8) and (8.9).The method of proof is to find a suitable short exact sequence of sheaves

and deduce the theorems from the corresponding long exact sequence ofcohomology. The right hand sides in (10.1) and (10.2) are globally definedobjects, that is they have the character of H°is on CM+, while the left handsides are H1's on PT+. Accordingly we consider sheaves on an intermediateobject, the primed spin bundle, which is traditionally referred to as F inthis context.

We have the double fibration (Eastwood et al. 1981; Wells 1981; seeexercise 7f) in figure 10.1. Here v is the usual projection; the inverse imageof a point in PT under µ is the whole a-plane in F. In particular, afunction f (xa, 7rA') on F pushes down to a function on PT if it is constanton a-planes or equivalently if

7rA VAA'f = 0. (10.3)

91


F

PT CM

v (x', 7rA') X'It (x',7rA') -> (ix AA 7rA', TA'),

Figure 10.1.

We may restrict (10.1) to the future tube CM+ and corresponding regionsF+ and PT+. We define Zn(m) to be the sheaf of germs of symmetric n-index primed spinor fields WA'...B'(x,7r) holomorphic on F+, homogeneousof degree m in 7rA' and satisfying the zero rest mass equation:

DAA'<PA,...B' =0- (10.4)

Consider the map

7rA' :.in+l(m - 1) -+ Z, ('m,)

VA'B'...C' --> 7rA' cPA'B'...C' (10.5)

It is a straightforward exercise to see that this is onto, so we may considerthe short exact sequence

0 (10.6)

where T is just the kernel of the map 70'. If 0A1...B' E T then this meansthat 0A'... B17A = 0 so that 1&A'...B' = IrA' ...7rB' f (x, 7r). Further the zerorest mass equation (10.4) on 0A'...B' requires that f(x,7r) satisfy (10.3).From the homogeneities chosen in (10.6), f is therefore a twistor functionhomogeneous of degree -n - 2. We may write (10.6) as

0 -, T(-n - 2) '`'"-+ a' in+1(-1)A'-> in (0) 0.

The corresponding long exact sequence of cohomology includes the sec-tion

...-+ H°(F+; Zn+l(-1)) -' H°(F+; Zn(0)) - H1(F+;T(-n - 2))II'(F+; Zn+1(-1)) - ..

CHAPTER 10. SOLVING THE ZERO REST MASS EQUATIONS II 93

The first term is global sections of Zn+1(-1) over F+. For fixed xa, such asection would give a global section homogeneous of degree -1 on P1, andthere are no such sections. Thus H°(F+; Zn+1(-1)) = 0.

A similar argument, slightly more technical, shows that

Hl(F+; Zn+1(-1)) = 0

so that we are left with an isomorphism

H°(F+; Z, (0)) a- H1(F+;T(n - 2)) (10.7)

The term on the left represents solutions of the zero rest mass equations onF+, homogeneous of degree zero in 7rAI. They must therefore be indepen-dent of 7rs', and defined on CM+. The term on the right is a cohomologygroup on F+ but the coefficients are twistor functions, so that they don'tnotice that they aren't defined on PT+, i.e.

Hl(F+;T(-n - 2)) = H1(PT+; O(-n - 2)). (10.8)

With these observations, (10.7) is the required isomorphism (10.1).(Evidently there are some non-trivial details, which we have glossed over

here, to do with relating cohomology on different spaces and to do with thetreatment of infinity in CM+. These and similar details omitted belowmay be found in Eastwood et al. (1981).)

The map S*-1 which, given an element of H1(PT+; O(-n - 2)), pro-duces a z.r.m. field, may be described in terms of the following diagram ofcochain groups:

I I I0 C°(T(-n - 2)) - C°(Zn+1(-1)) - C°(Zn(0)) -4 0

I I I0 - C1(T(-n - 2)) - C'(Zn+1(-1)) --> C'(Zn(0)) --+ 0

I I I

Suppose fij defines an element of H1(PT+;O(-n - 2)) with respect tosome chosen cover. Then

7rAi ...7rB'7rC' fij E C'(Zn+1(-1))

and

S(7rAI ...7rC' fij) = 7rA' ...7rC'Sfij = 0.

But fI1(Zn+1(-1)) = 0 so that

IrA' ...7rB'7VC'fij = P[iVj]A'...B'C' (10.9)


for 0jA'...B'C, E C°(i,+1(-1)). Define

WjA'...B' =0jA'...B'C'7rCI E H°(Z, (0)) (10.10)

then by (10.8)

whence

P[%cPj]A'...B' = 0

OjA'...B' = PjcPA'...B' 10.11)

and cPA'...B' is the required z.r.m. field. To see that it satisfies the z.r.m.field equations we have from (10.10)

P[$QAC',j]A'...BIC' = 0

so that VAC' 0jA'...BIC' is globally defined, homogeneous of degree -1, andso vanishes. Thus by (10.10) and (10.11)

AA VA'...B' =0-

The contour integration of chapter 8 is done in the splitting implicit in(10.9) (recall equation 9.8).

So much for (10.1). The negative helicity result (10.2) is different in thatit involves potentials. We start with the short exact sequence of sheaves onF+:

0 -> T(n) --> K(n) D,, QA(n + 1) -+ 0 (10.12)

where

K(n) is the sheaf of germs of holomorphic functions on F+ homogeneousof degree n in IrA,;

DA is irA'VAA' and T(n) is the kernel of this map, i.e. T(n) representshomogeneity n twistor functions;

QA(n + 1) is the sheaf of germs of spinor-valued functions OA (X, 7f) homo-geneous of degree (n + 1) in IrA' and satisfying DADA = 0.

The only part of exactness which isn't obvious is that DA is onto QA(n+1),and this is quite straightforward.

The corresponding long exact sequence of cohomology includes the sec-tion

0 - H°(F+;T(n)) -> H°(F+;K(n)) -* H°(F+; QA(n+ 1))--> Hl (F+; T(n)) - Hi (F+; )C(n)) --> ... (10.13)

and we must identify these various groups:


H°(F+;T(n)) is the space of global twistor functions on PT+ homoge-neous of degree n. These must be of the form of polynomials µ =µA'...B'7r

7.B'satisfying

C' A' B' C'7r VCC,µ = ... Oc(c'µA'...B') = 0. (10.14)

H°(F+;JC(n)) is the space1of polynomials like it but without (10.14), sayA = AA'...B'7rA ...7rB

H° (F+; QA(n + 1)) is the space of spinor fields OA = ' AA'...c,irA ' ... 7rc'with (n + 1) primed indices and satisfying

DA OA = 7rD'7r`4' ...7rC'V(D'AOA,...C')A = 0 (10.15)

We shall write these spaces respectively as Tn, An and Tn. Next, bysimilar arguments to those leading to (10.8), we have H1(F+; T(n)) =H1(PT+; 0(n)). Finally, and this we must ask the reader to take on trust,H1(F+;1C(n)) = 0.

(10.13) becomes

4n-'-4H +;O(n))-*0 (10.16)

where i is inclusion, v is the map

AA'...B' - VA(A,AB'...C'), (10.17)

and 5* is the connecting homomorphism (which we shall describe explicitlyin a moment).

Now how does this help us to identify H1(PT+; 0(n))? The answeris in terms of `potentials modulo gauge transformations'. Given a fieldOAA'...B'C' satisfying (10.15):

DA(D,II,A'...C')A = 0

we defineWAB... D = V (B...V D''r,bA)B'... D' . (10.18)

Then this is a helicity -. (n+2) z.r.m. field, i.e. it has (n+2) indices and

DAA'PAB...D = 0- (10.19)

Deducing (10.19) from (10.18) with the aid of (10.15) is a straightforwardcalculation (and a good exercise in spinor calculus!). The transformation

OAA'...BIC' - 7GAA'...B'C' +VA(A'AB'...C') (10.20)


is a gauge transformation, taking solutions of (10.15) to solutions of (10.15)but leaving unchanged WA...D as defined by (10.18). Finally, if AB'...C'is a solution of the twistor equation (10.14), then (10.20) actually leaves'+'AA'...C' invariant.

Thus there is an exact sequence

0 -> T,, 24 An _04 T,, %+2 0 (10.21)

where 4n+2 is the group of helicity -a(n+2) z.r.m. fields on CM+ and vis the map (10.18). The only part of exactness not checked is that v is onto.This is readily done, and then a comparison of (10.21) with(10.11) yieldsthe claimed isomorphism (10.2). (For Maxwell fields, the sequence of fieldsintroduced here is more familiar. The z.r.m. field is 'PAB, the potential is'AA' satisfying

VA (A"PB')A = 0

and the gauge transformation is just

0AA' ' OAA' + VAA'A

Finally To = C.)The map 6* can be described quite explicitly in this case and provides

a partial answer to the problem of defining an `inverse twistor function';i.e. given the field, how do you find the corresponding Hl? Suppose thefield is WA...D and that 0AA1 ...C' is a potential for it. For a given twistorZ f' E PT+ define a neighbourhood Ui C PT+ as the union of all the linesthrough Za which lie entirely in PT+.

Now given two such neighbourhoods Ui and Uj we define fi j (Zn) by anintegral: if Z' E Ui fl Uj then the a-plane defined by Za meets the a-planesdefined by Za and Z,' at Pi and P. in CM+ (see figure 10.2). Define

fij(Za) = I OAA'B'...C'1rB' ...7rC'dxAA' (10.22)

where Za = (wA, WAS) and the integral is along a path F lying in the a-plane Za and joining its intersections Pi, Pj with the a-planes Z' and ZJ.Because of (10.15), the integral is independent of the path chosen so that(10.22) is a function of Z. By the same token, this defines a cocycle in that

fij+fjk+fki=0since this represents an integral round a closed contour in the a-plane.

To see that this is the right cocycle, we return to the question of ex-tracting the field from a given cocycle.


Figure 10.2. The inverse twistor function.


Given a cocycle fij E H(PT+; 0(n)), first restrict it to a line in PT+to get px fij E H(P'; O(n)). This group is zero so

Pxfij = P[iojl, Y'j E C°(P1; O(n)) (10.23)

also irA'VAA'Pxfij = 0 So irA'VAA'' i = 7rA'DAA'oj defines a polynomialof degree (n+1) in irA', 1,bAA'...B' (x)irA' ....B', which produces the potentialOAA'...B' (x)

With the cocycle (10.22), if we restrict to a particular point P, thenwe are considering all a-planes through P as in figure 10.2. The splitting(10.23) is now accomplished by writing the integral (10.22) as the differenceof an integral from Pi to P and an integral from Pj to P. The rest followsimmediately.

Chapter 11

The Twisted Photon andYang-Mills Constructions

Up to now we have encountered twistor functions (or more accurately, co-homology classes) in a purely passive role: a z.r.m. field results from doingsomething to a twistor function. In the next two chapters, the twistorfunctions have a more active role in that they are used to change twistorspace.

The first example is the twisted photon (Hughston and Ward 1979) wherea function fo(Z') homogeneous of degree zero is used to define a line bundlewhich encodes the information of a Maxwell field.

We recall that the space PT is the space of a-planes in CM. Given onesuch a-plane, the extra information in T represents the choice of a constant7rA,-spinor associated to the a-plane. Equivalently, the fibre of T thoughtof as a line bundle over PT consists of the solutions 7rA, of the equation

XbVb7rA, = 0

where Xb denotes the tangent vectors to the a-plane,

Xb = AB7rB', for arbitrary AB.

Writing this equation as7rB VBB,7rA, = 0

we consider ways of modifying it so as to construct different bundles overPT. One possibility is to consider

7r B'DBB,7rA, = 7rB, (V BB, - ieABB, )7rA, = 0 (11.2)

99


for constant e and some vector ABB' . Solutions will exist if the relevantcurvature vanishes, i.e. we must require

EAB7r A' DAA'7rB' DBB' 7rC' = 0

which is just7rA'

7rB'VAA,AB,A = 0. (11.3)

This is true for all ir A' if

VA(A,AB,)A = 0. (11.4)

Thus given AAA' satisfying (11.4), (11.2) is integrable on all a-planes andwe have a holomorphic complex line bundle T over PT (or at least overthe part of PT corresponding to the part of CM on which AAA' is defined)whose fibre is the one-dimensional vector space of solutions to (11.2). Infact, given (11.4) we can solve

7rA DAA'SeB' = 0 (11.5)

on each a-plane, i.e. we don't have to consider only solutions proportionalto 7rA'. If we focus attention on one point p of CM and consider all the a-planes through it, then we can compare solutions of (11.5) on the differenta-planes by comparing them at p. Thus the bundle T restricted to the lineLp in PT must be trivial.

Now from the discussion in chapter 10, if we define

VAB = DA'(AAB)A' (11.6)

then cOAB satisfies Maxwell's equations:

V A,AWAB = 0 (11.7)

as a consequence of (11.4). Conversely, any a.s.d. Maxwell field cOAB has apotential AAA' satisfying (11.4) so we may enunciate a

Theorem

There is a one-to-one correspondence between anti-self-dual Maxwell fieldson CM+ and complex line bundles on PT+ which are trivial on each lineL.Proof

The correspondence one way is shown above. We consider below the ques-tion of going the other way. First, a construction analogous to that of

CHAPTER 11. THE TWISTED PHOTON 101

Figure 11.1. Constructing T.

figure 10.1 provides an explicit coordinatisation: suppose A', Ba play theroles of Z' and Zj0' there (figure 11.1). Then for the solutions at p of theequation (11.5) on the a-plane Za we may start with 7ro ' at a to obtain

/7r

A' = 7ro' exp(ieJ Abdxb).a

P

or start with iro ' at b to obtain

ftA' _ 7roexp(ie f

PAbdxb)

b

The two different answers are related by

A' irA exp(ieja

Abdxb)

7rA' exp(iefo(Z")). (11.8)

The twisted photon construction may therefore be described as follows:given a line bundle L over PT+ trivial on lines, trivialise L with respectto some cover {Ui} of PT+. Then L is coordinatised with coordinates Zapatched according to

Za = Zj' exp(ie.fij (Z')) (11.9)


where fij(Za) defines a cocycle in Hl(PT+; 0(0)). To extract a field, wereverse the process leading to (11.8): given (11.9) we restrict to a line Lxand `split' f

Pxfij = P[i''j]

then

7r A'VAA,''i = 7fA'DAA10j = AAA'(x)7rA'

which yields the potential AAA' (x) and hence the field.

There is an interesting example of charge integrality associated with thetwisted photon. We consider the complement of a world line in M. Thishas topology R x S2 so that the corresponding region, say X, in PT istopologically R x S2 X S2 where the second S2 is the celestial sphere. OnX consider the familiar short exact sequence of sheaves

0->Z-+0 ->Q*->0

and the portion of the corresponding long exact sequence of cohomologygroups

H'(X; Z) -+ H'(X; 0) -* H1(X; Q*) H2(X; Z).

A topological calculation shows that H'(X; Z) = 0 while H2(X; Z) = Z®Zso we are left with

0-+HI(X;0) -4 H1(X;0*) z®z.

The first group H'(X; 0) represents Maxwell fields which can be obtainedby the contour integral formula (8.9), while the second group H'(X; 0*)represents Maxwell fields which can be obtained as line bundles by thetwisted photon construction. We claim that the map b* goes to zero inone of the Z's but goes to the value of the charge of the Maxwell fieldin the other. (This requires choosing representatives carefully and is dis-cussed in Hughston and Ward (1979).) Consequently, Maxwell fields con-structed as twisted photons have integral charge, while Maxwell fields con-stucted as contour integrals have zero charge (since the sequence is exactat H1 (X; 0*)).

The twisted photon construction can be generalised in a natural wayto produce solutions to the anti-self-dual Yang-Mills equation. This israther more remarkable since these equations are non-linear. A Yang-Millsfield is essentially a connection on a trivial GL(n, C) bundle B over M.

CHAPTER 11. THE TWISTED PHOTON 103

We may represent the connection by a matrix of potentials AAA'ij wherei, j = 1,. .. , n. The covariant derivative is

DAA'i2 = c5 VAA' - ieAAA'ij (11.10)

and curvature is obtained by commuting derivatives:

Fabij = DaAbij - VbAaij - ie(AaikAbkj - AbikAakj). (11.11)

We may write this in a matrix notation omitting the Yang-Mills indices as

Fab = 2O[aAb] - ie[Aa, Ab] (11.12)

where the large brackets denote a matrix commutator. From (11.12) oneobtains a Bianchi identity

V[aFbc] + ie(F[abAc] - A[aFbc]) = 0. (11.13)

Finally, the Yang-Mills field equations are

0[a*Fbc] + ie(*F[abAc] - A[a*Fbc]) = 0. (11.14)

In general, (11.14) represents a second-order equation on the potentialmatrix Aa. However, if the original choice of Aa produced a field Fabwhich was self-dual or anti-self-dual then the field equations (11.14) are aconsequence of the Bianchi identities (11.13). In this case the equation tobe solved is first order in Aa, namely

*Fab = fiFab.

To find a twistorial approach to this problem we consider the analogue of(11.5). That is, on a particular a-plane specified by i.A' we consider theequations

7r A'DAA'ij 0j - 7CA'(VAA'Si3 - 0- (11.15)

This equation will have n linearly independent solutions on the a-plane if

EABiA'iB'DAA'iiDBB'jk = 0. (11.16)

This is the counterpart of (11.3) and implies

VA(A,AB,) Aii - ieAA(AlikAB,)Akj = 0 (11.17)

or equivalently*Fab = -iFab.


Given this condition of anti-self-duality on the field, one may constructa bundle £ over some part of PT whose fibre above a particular a-plane isthe vector space of solutions to (11.15). By the same argument as before,£ is trivial when restricted to a line Lp in PT so that we may enunciateanother

Theorem

There is a one-to-one correspondence between GL(n, C) Yang-Mills fieldson CM+ and rank n holomorphic vector bundles on PT+ which are trivialon each line, L.

Proof

The procedure for extracting the field from the bundle is precisely analo-gous to the previous method. If the transition matrix is G(ZO) then whenrestricted to a line it must split but now as a product:

P.G(Z') = 9-1(x, ir)9(x, ir)

or

gG = g.

Now 9.A' V AA, , G = 7rA' DAA'9 since irA' DAA, G = 0. Thus

irA,

DAA'9 9-1 7rA DAA,g 9-1ieAAA'7rAt (11.18)

where AAA' is the matrix of potentials. Applying 7rB'VAB, to (11.18)immediately gives (11.17).

For subgroups of GL(n, C), notably SU(n), extra structures are re-quired on the bundles £ but we shall not consider this further. Instead werefer to the original papers of Ward (1977) and Atiyah and Ward (1977)and the book of Ward and Wells (1990) where also specific examples aregiven. In conclusion, we remark that what allows the constructon to workis that an anti-self-dual connection is flat on a-planes. In the next chapterwe shall see another instance of this.

Chapter 12

The Non-Linear Graviton

Twistor theory as discussed so far has been concerned solely with flat space,or at most with conformally flat space. There are various ways of attempt-ing to generalise different parts of twistor theory in the presence of con-formal curvature, and these meet with varying degrees of success (Penrose1976; Penrose and Ward 1980; Ward and Wells 1990; Bailey and Baston1990). In this chapter we wish to consider the problem of finding a-planesin a space-time M with conformal curvature. Naturally we shall need tocomplexify M since an a-plane is a complex surface, so we need to supposeM is analytic.

Two tangents Xa, Ya to an a-plane necessarily have the form

Xa = XA7rA' Ya = ,UA .A'(12.1)

for some 7rA'. To be surface-forming, the commutator of Xa and Y' must

be a linear combination of Xa and Ya:

XbvbYa - YbvbXa = aXa + pya. (12.2)

The RHS of (12.2) has the form vA7rA for some VA so that (12.2) is equiv-alent to

7rA' (X bvbYa - YbvbXa) = 0.

Substituting from (12.1) we are led to

A' B'7r 7r V BB' 71 A' = 0 (12.3)

which we recognise as the geodesic shear-free condition of chapter 6. Ap-plying 7rC' V C' B to (12.3) we are led to

TA/B'C'D/7rA17rB'7rC'7rD' = 0.

105


F

PT M

Figure 12.1.

(This could have been deduced from the Sachs equations (6.13,6.14.)Thus if at a point p in M we _want an a-plane for each primed spinor

7rA', then the primed Weyl spinor YA'B'C'D' must vanish. For a real space-time this necessarily entails that T ABCD vanish and M be conformally flat.However, if we consider instead complex space-times M then we lose theconjugation of spinors, and the primed and unprimed Weyl spinors becomeindependent curvature quantities. Now we can set one part equal to zero.The Weyl tensor is required to be a.s.d. in that

*Cabcd = 12EabpqCpgcd = -ZCabcd (12.4)

(In a four-dimensional manifold of some other signature, either (++++) or(+ + --), the duality operation has real eigenvalues and one can have realsolutions with a.s.d. Weyl tensor. The present discussion will still coverthe complexification of these solutions.)

Thus given (12.4) we can define a three-dimensional complex manifoldPT, the space of a-planes in M. The equation

7rA 0

has solutions on the a-plane defined by 7rAi, so we can also define T. Theextra information in T is the choice of scale for the spinor 7rA1 on eacha-plane.

Another way to define T and PT is in terms of the primed spin bundleF of M, mimicking the double fibration of chapter 10 (see figure 12.1).Consider the distribution defined on F by the two vector fields 7rA' DAA,This distribution is integrable if (12.4) holds. The space of leaves is PT,with T as above. The vector field T = 7rA1 8 on F projects to thehomogeneity operator on T, and PT is the space of integrable curves of T.

A point p of M is represented in PT by the set Lp of a-planes through p.As in the flat case, this is a projective line P1, and so PT has a special four-parameter family of compact holomorphic curves in it. Furthermore, null

CHAPTER 12. THE NON-LINEAR GRAVITON 107

separation in M is mirrored by intersection of the corresponding curvesin PT. The fact that the conformal structure in M is quadratic mustbe reflected in PT by some restriction on these curves; specifically, thecondition for two nearby curves to intersect must be a quadratic condition.To make this condition precise, we must define the normal bundle N of asub-manifold Y in a manifold X (Hirzebruch 1966; Ward and Wells 1990).At a point p E Y the fibre Np of N is

Np = TpX/TpY

i.e. Np consists of vectors modulo vectors tangent to Y.If Y is a P 1 corresponding to a line Lp, then N is a rank two vector

bundle over P1 and so, by a theorem of Grothendieck (1957), is a sum ofline bundles:

N = 0(m) ® 0(n). (12.5)

(The vector bundle 0(m) ® 0(n) has transition matrices diag(f, g) wheref, g are the transition functions for 0(m), 0(n) respectively.)

An `infinitesimally neighbouring' line Lq to Lp is more accurately asection of N, the normal bundle to Lp. By (12.5), L. is defined by a pairof sections, one of 0(m) and one of 0(n). There is supposed to be afour-parameter family of nearby lines and by table 9.1

dimes°(P1; 0(m)) M+1 m > -10 m<-1

so that m and n in (12.5) are severely constrained! Clearly

m+n=2 ifm,n>-1n=3 ifm<-1

so that the only possibilities for (m, n), if we order them with m < n,are (-1, 3), (0, 2) and (1, 1). In the first case, all sections are of the form

(0,VA'B'C'lrA'lrB'lrc') where 7rA, is the coordinate on P1 and so all have3 zeroes. In the second case, all sections are of the form (C, coA'B'7A'TB' )and so have either no zeroes (if C $ 0) or 2 zeroes. In the third case, thesections are of the form Such a section has a zero if thezeroes of the two parts coincide, i.e. if aA',3A, = 0. This is the requiredquadratic condition and so the restriction that we are looking for is thatthe four-parameter family of compact holomorphic curves in PT must havenormal bundle N = 0(1) ® 0(1).

A twistor space PT with these properties corresponds to a space-timeM with a.s.d. Weyl tensor. In fact PT corresponds to the whole class of


space-times conformal to M, since the construction so far is conformallyinvariant.

We next suppose that M satisfies the vacuum field equations, i.e. that

4'ABA'B'=0=A (12.6)

as well as 41A'B'C'D'. The extra structure that this determines on PT arisesfrom the fact that the equation

DAA'7rB' = 0

now has solutions. That is, we have a global parallelism of primed spinors;we can compare primed spinors not just all over each a-plane, but from a-plane to a-plane, and we can determine when two separate a-planes havethe same 7rA'. This gives PT the structure of a fibration with projection 7rover the projective primed spin-space PS, and T the same over the non-projective primed spin-space, S. Also, the base-space has a preferred choiceof EA'B' inherited from the space-time M. Equivalently, there is a two-form

A'B'T = C d7ry' A d7rB' on T.The final extra piece of structure, which encodes the other piece CAB of

the space-time metric, is a two-form it on each fibre of the projection it. Todefine µ in space-time terms, suppose we are given a particular a-plane zwith a particular 7rA', and a pair of vectors X, Y tangent to the fibre of Tcontaining z. X and Y determine connecting vectors Xa and Ya as fieldson the a-plane z, connecting it to two `nearby' a-planes with the same 7rA'.Now form

µ(X, Y) = EABX AA' YBB' 7rA' 7rB' . (12.7)

We claim that this is independent of the point on the a-plane z at which itis evaluated and independent of the particular choices of connecting vectorsXa and Ya. This is easily verified, so that (12.7) does define a two-formon the fibres.

(Alternatively, µ may be defined on the primed spin-bundle F as

1µ= 1EAB7rA'7rB'dxAA AdxBB

and projected down to T.)In summary, we have constructed from a solution M of the vacuum

equations with a.s.d. Weyl curvature a four-dimensional complex manifoldT with

i) a projection it to primed spin-space;

ii) a homogeneity operator T;


S

Figure 12.2. The arrangement for defining the metric.

iii) forms r on the base and µ on each fibre with LYT = 2T; LTIL = 2µ;

iv) a four-parameter family of holomorphic curves which in PT are com-pact and have normal bundle 0(1) ® 0(1).

We must show that conversely such a T produces an M, and then howto set about constructing such manifolds.

For the first part we need a direct construction of the metric and con-nection of M from T and PT. The conformal metric is, we know, definedby intersection, and condition (iv) ensures that it is quadratic. To fix theconformal factor of the metric at each point p it is sufficient to define theinner product of two null vectors, say Ua and Va, at p. In PT we sup-pose U' and Va correspond to lines L,,, L infinitesimally neighbouring toLp. On each fibre of the projection it, L, and L define vectors U andV. Further, since Ua is a null vector, U vanishes at one value of 7rA', sayIrA' = aA', and similarly V vanishes at another, say IrA, = /3A" (figure 12.2).At a general value of IrA, we may form p(U, V). This is a function on theRiemann sphere Lp homogeneous of degree 2 and with precisely two zeroes,one each at aA' and 3B,. We consider the expression

g(U,V) =(aA'gA,)µ(U,V) (12.8)(aB"7rB")(#C'7rC")


Figure 12.3. The connection.

The RHS is homogeneous of degree zero in WA' and has no zeroes. Itis therefore constant, i.e. independent of IrA', and we take this as thedefinition of gabUaVb. (Note that in order to construct (12.8) we have usedboth the forms T and µ of condition (iii).)

A point in PT defines a totally null two-plane (that is, an a-plane inM) by all the lines Lp through it. A one-parameter family of lines througha point z in PT therefore defines a curve lying in an a-plane. If we definea geometrical procedure for parallelly propagating a null vector along sucha curve this is sufficient to define a connection on M.

Suppose we have the curve y through a point p and a null vector Va atp. In PT, two fibres are picked out, one F1 corresponding to V a and oneF2 corresponding to y (figure 12.3).

To define vectors along y corresponding to the parallel propagation ofVa, we take the lines joining the point of intersection of L with F2 to thepoints of intersection of Lp,, Lp27 ... with F1. This is evidently a linearoperation on vectors at p. Further, if y describes a closed curve in an a-plane then this connection is integrable. It is clear that this connectionpreserves the conformal metric and it is straightforward to check that itactually preserves the full metric. Thus if the connection is torsion-free, it isthe metric connection, is flat on a-planes and provides a global parallelism


of primed spinors, whence the Ricci tensor vanishes and the Weyl tensoris anti-self-dual. This completes the proof that curved twistor spaces T(satisfying (i)-(iv)) lead to half-flat space-times M, if we can establish thatthe connection defined by figure 12.3 is torsion-free. This is a small technicalpoint and we remove it to an exercise.

The final question we face is that of constructing curved twistor spaces.Any method that generates manifolds satisfying (i)-(iv) will do (for exampleHitchin 1979) but we shall consider the problem of starting with flat twistorspace T and arriving at T by a deformation of the complex structure ofT (more accurately, of part of T). As usual, we shall be informal in ourtreatment of holomorphic deformations. A fuller account adapted to theproblems of twistor theory may be found in Burns (1979).

The simplest way to define a deformation of a complex manifold X is interms of a coordinate atlas {Ui} with coordinate functions Zi and transitionrelations

Zi=fij(Zj)onUzflUj.

The manifold Xt is a holomorphic deformation of X if Xt has the sameunderlying real manifold and atlas but the transition relations are holo-morphic also in t:

Zi = fij(Zj,t), ffj(Zj,0) = f%j(Z3). (12.9)

Given (12.9) the deformation is defined to first order by the partial deriva-tive of the transition relations with respect to t. In fact this infinitesimaldeformation is correctly described by a vector field cocycle. If X is n,dimensional and a = 1,. .. , n is the coordinate index then (12.9) standsfor

Zai = f0ij(Z0j, t) on Ui n Uj.

Define

00 =9Za I a af; a

(12.10)at t=o aza

-at aZa

when there is an implied summation on the Greek index but not on theLatin index. (Also, for notational convenience, evaluation at t = 0 will beunderstood.)

Oij is evidently a 1-cochain of vector fields; in fact it is a 1-cocycle. Tosee this, note that on a triple overlap we have

Zj- _ .fk(Zk,t)Za = fij (Zf, t) = f (f k(Z", t), t)

112

so that

since

Ojk =

eik

AN INTRODUCTION TO TWISTOR THEORY

a.f,k 8atazj

af; a afi; a.f k aat aza + azp at azaeij + ejk

af; a - a

aza aza azf

whence eij + ejk + Oki = 0. Now if each Za is subjected to a t-dependentcoordinate transformation on Ui so that

then

so that

eij

where

Za = h% (h. 1 ry(Zj , t), t), t)

aza aat aza

aha a aha aza a aha azf ahj aat aza + aza at aza - azf az; at aza

i+eij-Sj

aha abi = at aza

Here a defines a 0-cochain of vector fields. Thus the coordinate-independentdescription of an infinitesimal deformation is as a 1-cochain modulo co-boundaries, i.e. as an element of Al (X; O) where O is the sheaf of germsof holomorphic vector fields. (See figure 12.4.)

As an example, we consider the case X = P1. Since X is one-dimen-sional, O is a line bundle (see exercise 3g) and we have only to decidewhich one. A vector field Y can be written in homogeneous coordinates asY = aA' (lrB,) a

e where aA' is a pair of functions homogeneous of degree1. Furthermore the vector field iA' e

a on C2 represents the zero vector


Figure 12.4. The vector field O j defined on Ua fl UU.

field on P1 so we only care about the part of aA not proportional to WA',i.e. we must consider

CY[A'7rB/] =1fA,B'OC17rCi

= 2fA,B,V(7r).

Thus Y is determined by the single function V which is homogeneous ofdegree 2 in 7rA, and so O = 0(2). By the argument above, the infinites-imal deformations of P1 are in one-to-one correspondence with the groupH1(P1; O) = H'(P1; 0(2)). However, by the results of chapter 9, we knowthat this group is zero, so there are no infinitesimal deformations of P1.

In fact this result holds generally for Pn (Morrow and Kodaira 1971),so we must consider deformations of only part of twistor space, typicallythe neighbourhood of one particular line, Lp.

The deformations which we seek must preserve conditions (i)-(iv). Topreserve the first, we must restrict to vector fields tangent to the fibres of7r:

y = fA(w,7r)A. (12.11)

For (ii) we demand that Y and T commute:

[T,T]_(TfA_fA)aaA=0

which means that fA must be homogeneous of degree 1 in (WA, 7rAl) -

If we think of Y as defined on the overlap between two sets U, U withcoordinates WA, 2,A, then the infinitesimal deformation is given by

wA = WA + tf A(WB,7rB,) + O(t2) (12.12)


so that, at fixed 7rA',

AdwA = dwA + t,gfB dWB + O(t2).

To satisfy (iii) we must have

li=EABdWAAdWB=EABdWAAdWB=µ.

(12.13)

Substituting from (12.13) we find this reduces to

B a fACA

aWB=

which implies fA = CAB for some f (WC , xrc') homogeneous of degree 2.Thus we satisfy conditions (i)-(iii) of five pages ago by taking a 1-cocycle

f E H'(X; 0(2)) and forming the vector field cocycle

AB Of a=CaWAaWB.

From chapter 8, we recall that such an f is precisely the `twistor function'for a linearised gravitational field. Here we are seeing this cocycle in anactive role, determining an infinitesimal deformation. To define a finitedeformation, we consider the case when the cocycle is defined on a two-setcover of the neighbourhood of a line. If the sets are U, U with coordinatesWA, wA we first seek the integral curves of Y, i.e. we solve the equations

dwA _ _EAB of(12 14)d

WA(o) =aWB

WA

.

and then setWA = WA(t). (12.15)

Here t plays the role of the deformation parameter. (We should emphasizethat this is one way of getting a finite deformation from an infinitesimalone, there are others, and further this method depends on the choice ofrepresentative of the cohomology class of f. Although the infinitesimaldeformation is a cohomological object, the finite deformation isn't.)

So far we have neglected condition (iv). However, there are some the-orems of Kodaira (Morrow and Kodaira 1971; Hansen et al. 1978) whichassure us that we are justified in doing so, that in fact (iv) will be true auto-matically for small enough deformations (i.e. for the deformation parametert in a neighbourhood of zero). The relevant theorems are as follows:


1. Suppose Xt is a holomorphic family of complex manifolds for t in aneighbourhood U1 of zero in C, Y is a complex submanifold of X0and No is the normal bundle of Y in X0. Then there exists a neigh-bourhood U2 of zero in C and a holomorphic family of complex sub-manifolds Yt C Xt fort E U2 with Yo = Y provided H1(Y; No) = 0.

2. A sufficient condition for Yt to lie in an n-dimensional family of sub-manifolds in Xt is H'(Yt; Nt) = 0 and then n = dimes°(Yt; Nt).

We want to use these when Yo is a particular P1, say Lp, in PT andXo is a neighbourhood of Yo. We know already that No = 0(1) ® 0(1) sothat H1(Yo; No) = 0. Thus by Theorem 1 above, there is an analogue Yt ofYo in Xt. If we knew that Nt = No then, by Theorem 2, Yt would lie in ann-dimensional family where

n = dimes°(Yt; Nt) = dimes°(Yo; No) = 4.

Here we have used the two facts that Yt and Yo are the same as complexmanifolds, namely P1, since there are no infinitesimal deformations of P1,and that 4 is the answer in PT.

In this case, we would know not only that there was a four-parameterfamily, but also that the conformal structure was preserved.

Since there are no deformations of P1, to see what happens to Nt wemust consider the question of deforming vector bundles over P1. If thetransition matrices of a vector bundle E in some trivialisation are allowedto depend on a parameter t say, so that

9«p = 9«0 (Z,t)

neglecting the cochain indices) then the infinitesimal deformation is definedby

aetgap

t=o

which is a 1-cocycle of endomorphisms of E. By a similar argument tothe previous one, what is actually involved is a cohomology class inH1(P1; End(E)) where End(E) is the sheaf of germs of endomorphismsof E. When E = 0(1) ® 0(1), an endomorphism is locally a 2 x 2 matrixof homogeneity zero functions so that

Hl (P1; End(E)) = 04 copies'l(P1; O) = 0.

Thus in this case there are no infinitesimal deformations of No and so, for tin a neighbourhood of zero, Nt = No = 0(1) ® 0(1). (This neighbourhoodof zero may be strictly smaller than the one supplied by Theorem 1.)


All we needed for Theorem 2 was

H1(P1; Nt) = 0; dimes°(P1; Nt) = 4. (12.16)

If Nt = 0(1) ® 0(1), these are certainly satisfied and furthermore thequadratic conformal structure is still defined. However, it could happenthat for some value of t the analogue Yt of Y° persisted but Nt became0(2) ® 0 or even 0(3) ® 0(-1). Now (12.16) is still satisfied so that Ytstill lies in a four-dimensional family but the quadratic conformal structureis not defined. This would be reflected in the space-time M correspondingto Xt by a singularity in the conformal structure. These questions areconsidered further in Tod (1982).

Armed with these theorems of Kodaira, then, we may construct de-formed twistor spaces by forming the transition relations (12.15) and thenseeking the four-parameter family of holomorphic curves whose existence,at least for small t, is guaranteed.

We conclude this section with a simple example of this process dueoriginally to G.A.J. Sparling. Spading considers the 1-cocycle

f4700711

defined with respect to the two-set cover of PT+ (Ward 1978):

U = {Z`ti E PT+ - I : 7ro, 0}

U = {Za E PT+ - I : 1rl, 0}.

The line I, where 7rA, = 0, is removed. (12.14) becomes

dw0 Cowl (w01)3= 0' ' - - l

dt dt 7r0i7rl/

whence

CJ0 = W0; W1 = W1 + t(w0)3

7rot7r1,

(12.17)

These are the transition relations. Note that w° defines a global function,homogeneous of degree 1. On any holomorphic curve Lp, this function mustbe a linear polynomial in 7rA':

w° _ 7ro, + u7r1', u, E C.

Substituting (12.18) in (12.17) and rearranging:

(12.18)

)a3

(2 t2 2 l3wl 33 2 1912

0C u .u (7r1 )u7ro,7rl, + ) = w ++i (e . )(

(w0)4


This defines a global function homogeneous of degree 1, in other wordsanother linear polynomial in 7rA', on Lp so

,27rwl = -tu3 ( 1) + vir°, + C7r1', v, E C (12.20)ira

and (12.18) with (12.20) give the holomorphic curves in terms of the fourparameters xa = (u, v) e, ).

If two nearby curves C1, C2 are given by x' and xa +Sxa then the vectorva = Sxa is null if the equations

iro'S + 7rl'Su = 02

ir°'Sv+ici'6 -3t(f1,) u2Su=07ro'

have a simultaneous solution. The condition for this is

Su5v - 3tu2(6 )2 = 0 (12.21)

so that this is the conformal metric, and it is quadratic as anticipated.For the full metric, we consider two null vectors Va and Ua satisfying

(12.21). For simplicity, we may take Va = (Su, 0, 0, 0); Ua = (0, AV, 0, 0).Correspondingly SwA and OwA are given by

6w° = 7r1'Su2( ')S 1 = 3t 1 2Sw - u u

pio,Owo = 0

Owl iro' Av.

By (12.8), the contraction gabUaVb is given by

EAB6U AAWBUaVb =gab

,7r0/1r1'

= SuLv= goi5uw

so that go, = 1 and finally the metric may be written

gabdXadXb = 2du dv - 2d d - 6tu2d 2. (12.22)

It may be checked (if desired!) that this rather simple metric does indeedsatisfy Einstein's vacuum equations. The Weyl tensor is anti-self-dual andin fact Type N.


We have called this chapter the Non-Linear Graviton since this is thename which has become attached to this construction. Penrose's motivationfor this name is his suggestion (Penrose 1976) that these spaces should formthe one-particle states of a future quantum theory of gravity. Each stateis genuinely a non-linear object, in the sense that it has some curvature,rather than a solution of a linear field equation in Minkowski space.

As he also emphasised at the time, however, he would wish to preservethe name non-linear graviton for those deformed twistor spaces which wereactually positive-frequency in an appropriate sense. The appropriate sensewould be related to the space T being a deformation of the whole top halfT+ of twistor space. Each fibre would have to have a boundary correspond-ing to a deformed version of the line I and these would be related from fibreto fibre.

The implications of this extra restriction have yet to be fully workedout.

Exercises 12

a) Suppose E is a holomorphic vector bundle over a complex manifold Xand s is a holomorphic section of E which vanishes to first order on asubmanifold Y of X (i.e. s = 0, ds j4 0 on Y). Show that the normalbundle of Y in X is canonically E ly. (Now if E is the bundle 0(1) ® 0(1)over CP3, a section vanishes on a line, so that the normal bundle of a linein CP3 is 0(1) ® 0(1). We are grateful to Dr. M.G. Eastwood for thisexercise.)

b) For a two-set cover of the neighbourhood N of a line in PT, the linearisedversion of (12.14) is

wA=WA - AeABofawB

with f E ft' (N; 0(2)).Find the linearised version of the four-parameter family of cross-sections

and the metric and show that the linearised curvature is proportional towhat would be obtained by treating f as a twistor function in the contourintegrals of chapter 10.

c) Show that the connection defined geometrically in this section is torsion-free.

Chapter 13

Penrose's Quasi-LocalMomentum and AngularMomentum

In general relativity there is a long-standing problem of defining momentumand angular momentum in a general curved space-time. In this chapter, wedescribe an application of twistor theory which aims to provide a solutionto this problem. As we shall see, this aim is not entirely achieved and theprogramme has both successes and failures.

We recall that in general relativity, all the local matter content is de-scribed by the stress-energy tensor Tab. Gravitational energy, whetherin gravitational waves or in the form of gravitational potential energy, isnotoriously non-local and one cannot expect to characterise it by a localdensity. Instead, Penrose (1982) has suggested that one should seek a non-local invariant associated to any two-surface S and representing the totalmomentum-angular momentum flux through that surface. This non-localinvariant will be constructed by twistorial techniques tailored to give theright answer for linearised general relativity where there is a clear rightanswer.

We begin by reviewing the definition of momentum and angular mo-mentum in special relativity and in linearised general relativity.

A material system in special relativity is defined by its stress-energytensor Tab which we may suppose for definiteness to have support withina world tube W in Minkowski space, M. (The case of say electromagneticfields spread throughout M is a simple generalisation.)

Conservation of energy is expressed by the condition that Tab has zero

119


divergence:

VaTab = 0. (13.1)

Given any Killing vector kb the current Ja = Tabkb is then conserved:

VaJa=0

and we may define a conserved charge by integration over an arbitrary3-surface E:

Q[k] = fTabkLdOa. (13.2)

Here we are thinking of Q as a linear functional on the space of all Killingvectors. Q has ten real components corresponding to four componentsof momentum, when kb is a translation, and six components of angularmomentum, when kb is a rotation.

In fact all the information of total momentum and angular momentumis contained in (13.2) when kb is a self-dual (s.d.) Killing vector, that is,one for which the derivative Vakb is a self-dual two form. These comprisethe four translations and the s.d. rotations, which are necessarily complex.

In linearised general relativity, the source Tab produces a linearised Weyltensor, which can be thought of as a spin-2 field on M related to Tab bythe linearised Bianchi identity

VA/AOABCD = V(BB''bCD)A'B' (13.3)

where 4'ab = 2 (Tab - ygabTcc). Now one anticipates being able to measurethe momentum and angular momentum of the source with a two-surfaceintegral involving the gravitational field in an analogous way to the formulain Newtonian theory for the total mass:

M= f pdV

47rGV p.ds

The two expressions, the 2-surface integral and the 3-surface integral, arerelated by the divergence theorem.

Here the desired formula is

Q[k] = 4irGRabcdfcddaab = f Tabkbdva (13.4)

S E

where Rabcd is the linearised Riemann tensor, f ,b is a bivector potentialfor the Killing vector kb, in a sense to be explained, and dash is the volumeform for S.

CHAPTER 13. QUASI-LOCAL MOMENTUM 121

We saw in exercise 4e that if wAB is a solution of the twistor equationVA,(AWBC) = 0 (13.5)

thenVA'AWBC = -ZEA(BkC)A' (13.6)

where kb turns out to be a s.d. Killing vector.In coordinates the solution of(13.5) is

WAB(x) icPA'B'xAA'xBB' +iTA'(AXB)A' +QAB 13.7)

and from (13.6)kAA' = TAA' - 2icPA'BI xB'A. (13.8)

Thus to make (13.4) work, one takesfab = WABEA'B' (13.9)

where wAB is related to kb by (13.6). It is then a simple exercise to verify(13.4). Another way of looking at (13.4) is in terms of spin-lowering (ex-ercise 4d). Outside the world tube W, YbABCD is a z.r.m. field. For anysymmetric twistor TO corresponding to a field wCD, it follows that

VAB = '0ABCDWCD

is a solution of Maxwell's equations. We may therefore calculate its chargeby integration over any two-surface S. This gives a charge for each symmet-ric twistor T°p and so defines a twistor A. Then (13.4) relates A, p, themomentum-angular momentum twistor or kinematic twistor of the source,to explicit integrals over the source.

Not every twistor A,p can arise as a kinematic twistor (Penrose andMacCallum 1973). Note that if the chosen element wAB of T ®s T is actu-ally a constant spinor field on M, then the corresponding charge vanishes.Further if wAB is such that ka is a real translation, the corresponding chargeis real.

To see what this means for Amp we substitute (13.7) and (13.8) into(13.4). If the matrix of Aap is

AAB PAB'` ap L PBA' -2it A'B'

then

Q[k] = AapT'AABQAB + PAA'TAA' + 2ip 'B'<oA'B'

= fTab(Tb + 2iVB'C'XBC,)&,a


so that AAB = 0, Pa = Pa is the momentum and µA'B' is the s.d. angularmomentum. These restrictions on Aap can be written concisely with theaid of the infinity twistor Iap as

AapIp'y = A76Iaa. (13.10)

It is these conditions which reduce the ten complex components of AP tothe four real components of momentum and three complex components ofangular momentum. Note that one also has the formula

AapAap = -2PaPa = -2m2 (13.11)

for the total mass threading through S. The formula (13.11) for the massrequires for its definition the norm on twistor space, while the formula(13.10), which reduces the number of components to ten real ones anddistinguishes the components of Aap which are the momentum, requires inaddition the infinity twistor. These points will be significant later on.

In a general curved space, there are no Killing vectors and a formulalike (13.2) is of no use. However, one might be able to use the first equalityin (13.4) if one had a way of recognising or defining fat solely at S. Weconsider this problem next.

Any solution of (13.5) is a linear combination of products of solutionsto the valence one twistor equation

DA'(AWB) = 0; DA'AWB = -iSAB7rAi. (13.12)

An arbitrary 2-surface S defines a pair of spinors (oA, tA) tangent respec-tively to the out- and in-going null directions orthogonal to S. If we nor-malise the dyad by oAtA = 1 then there remains the freedom

(DA, tA) -> (AOA,A-1tA) A E C*. (13.13)

The complex null vector ma =oAGA'

is tangent to S and we may considerthe parts of (13.12) which involve derivatives only tangential to S. Theseare

oBSWB = 0 ; LBSWB = 0; (13.14)

oBSWB = -iOA 7rA, ; tBSWB = iLA 7rA1 (13.15)

where S = ma DaHere (13.14) is to be viewed as a pair of equations on the two components

of WA, and (13.15) as the definition of 7rA', in terms of w A. We shall referto (13.14), (13.15) as the two-surface twistor equations and to a solution(WA,7rA,) as a two-surface twistor. Note that (13.14) defines an elliptic


system for wA. (This is apparent when the equations are written out incoordinates. (Essentially it is because S is closely related to the 0-operatorof the complex structure of S, which is elliptic; see exercise 9d and Wells(1980).)

Before embarking on a more detailed disussion of (13.14) and (13.15) wedescribe a formalism introduced by Geroch, Held and Penrose (1973: theGHP formalism) precisely adapted to the situation considered here, namelywhere a spinor dyad is specified up to the freedom (13.13).

We suppose that the dyad is chosen in a four-dimensional neighbourhoodof S. As a first step all spinorial and vectorial quantities are replaced by theset of scalars which are their components in the chosen dyad. These scalarsare weighted, in that they change under transformation (13.13) of the dyadand, specifically, a scalar 77 is said to have weights (p, q) if it transforms as

77 aagrl. (13.16)

Next, separate letters are introduced for the components of the spinor con-nection which are weighted scalars. These are

OADOA;

OASOA;

OASOA;

OAAOA;

/CI = -tAOtAP' = -tA&AO'I = -tAStArl = -tADtA

(13.17)

where D = laVa, A = naV0, S = maVa. It can be checked that these areweighted scalars, whose weights are left as an exercise. (Some of the spincoefficients (13.17) have already been encountered in chapter 6.)

The prime introduced in (13.17) indicates a symmetry generated by thetransformation

OA 2tA; LA --+ ioA; OA -ZGA ; GA __+ -2OA

under which e.g.P -'P; P ' -'P

(13.18)

and so on. Note in particular that S' = 6. This symmetry is systematicallyexploited in the formalism and effectively halves the number of equationswhich need to be written out. The remaining parts of the spin connectionare

Q = LASOA; 0' = -oA&AE = LADOA; E/ = -OAALA. (13.19)

These do not transform as weighted scalars but can be combined with theoperators D, A, 6 and 3 to produce weighted operators p and J pronounced,


`thorn' and `eth' respectively, according to the scheme

prl = (D - pE - gE)rl; P'rl = (A +pe' + qe')r/(S-p,3+q;')?1; X77 = (13.20)

when acting on a quantity 77 of weight (p, q).Again it is left as an exercise to show that these operators have well-

defined weights and to calculate what they are. (To say that e.g. p hasweight (1, 1) is to say that p? has weight (p+l, q+1) if 77 has weight (p, q).)

To write (13.14) and (13.15) in the GHP formalism we introduce thescalars w° and w1 of weight (-1, 0) and (1, 0) respectively which are thecomponents of wA, and the scalars lro, and irl, of weights (0, 1) and (0, -1)which are the components of IrAl:

WA = w°oA + w111 A; ,rA, = 7r1,5A' - Iro,tA'. (13.21)

Then (13.14) and (13.15), the two-surface twistor equations, become

C wl - v0° = 0; Cfi'w° - v'wl = 0 (13.22)

a1w1 - Pw° = -i7ro ; c w° - p wl = -i7rl'. (13.23)

In each case, the second equation is the prime of the first and of courseevery term in any one equation must have the same weight. These twoobservations provide useful checks to calculation!

The remainder of the GHP formalism consists of three sets of equations,which we relegate to an appendix. Firstly, there are six equations andtheir primes defining the curvature in terms of the spin coefficients. Next,and related to these, there are the commutators of the weighted operators.Finally there are the Bianchi identities.

As an exercise in the use of the first two sets of equations, we invite thereader to show that the commutator of 4 and a' applied successively to w°and wl in equations (13.22),(13.23) leads to the pair of new equations:

Jiro, + P7rl' = 402 - (Pu - A)wl + i(Ol - wol)woJ'7r1, + P'Iro' = i(03 - W21)w1 + i(02 - wll - A)w°.

(13.24)

These will be useful later. Returning to (13.22), we have claimed thatthis system represents an elliptic system from pairs of functions of weight(-1, 0) and (1, 0) to (by inspection) pairs of functions of weight (-2, 1) and(2, -1). We regard this as a map

T:E -+ F


of vector bundles over S (see exercise 13b). Picking a Hermitian innerproduct on F, we may define the adjoint

Tt:F ->Eand the index

ind T = dimkerT - dimkerTt.By the Atiyah-Singer Index Theorem (see e.g. Palais 1965), this number

is a topological invariant of the surface S and the bundles E, F; i.e. it isindependent of the metric, connection and curvature of the space-time. Infact it turns out to be 4(1 - g) where g is the genus of S (Baston 1984).We shall generally be concerned only with topologically spherical surfaces,where the index is four. This implies that the solutions of (13.22) form acomplex vector space T(S) with dimension at least four. With more care,it can be shown that when S is `near' to a metric sphere in flat space,dimT(S) is precisely four (Eastwood 1983) though examples are knownwhere the dimension is higher (Jeffryes 1986). Thus we think of T(S) as atwistor space defined at S by the two-surface twistor equations.

Returning to (13.4), this means that we may recognise the differentchoices of fab solely by solving (13.22) at S. The remaining componentsof the twistor equation (13.12) are then propagation equations for takingthe two-surface twistors off the surface S. In a general curved space therewill be curvature obstructions to doing this (since a space-time must beconformally flat to admit four linearly independent solutions to (13.12)).However, Penrose's proposal is to use (13.4) to define a kinematic twistorA at S even in curved space, by using for fab spinor fields constructed fromT(S). This defines A as a special element of the dual of T(S) ®s T(S).Writing out (13.4) in terms of curvature spinors in the GHP formalism wefind:

AaaZ0'Z0 =Z

f{(woi -'i/&I)w°w° + (vii +A - 12)(w°wi +w1w°)1 2 41rG 1 2 1 2 1 2

+ ((p21 -')3)W121}dS (13.25)

where wA, WA are the spinor fields corresponding to the elements Za, Za of1 2 1 2

T(S). Here we are assuming that T(S) is four-dimensional and introducingan index a for its elements. With the aid of (13.24) we find a simplerexpression for (13.25):

AapZ`YZR =2 f{-°(o' + p7'i) - w1(a'w1, + p'wo,)}dS

1 2 47rG 1 2 2 1 2 2

2

47rGf(o#ii'+2°,11')dS (13.26)


where the second equality follows on integrating by parts.At this stage, we have no norm on T(S) and no infinity twistor so that we

cannot ask whether (13.10) is satisfied, nor can we use (13.11) to calculatethe total flux of energy through S. Put another way, the `symmetry group'of T(S) is still GL(4, C) since we lack the norm to reduce it to SU(2, 2)and the infinity twistor to reduce it further to the Poincare group.

In an attempt to rectify this we consider the norm

E = WA7FA + WA 7rA1 (13.27)

appropriate to surfaces S in flat space.The point is that, for such surfaces, E is a constant although constructed

from fields wA and 7rA' which satisfy (13.22) and (13.23) and which thereforevary from point to point on S. One cannot prove the constancy of E solelyfrom (13.22) and (13.23) however and this is a sticking point for surfacesin curved space.

We may proceed from here in two different ways: we may attempt tomodify (13.27) so as to find a definition which is constant on an arbitrary Sin curved space, or we may seek surfaces on which (13.27) is constant andcontent ourselves with a study of them. Following a suggestion of Penrose(1984), we call a surface S on which (13.27) is constant for every choiceof two-surface twistor non-contorted so that if E varies, S is contorted.As we shall see below, a contorted surface is one which `knows' that it isin the presence of conformal curvature. Our strategy will be first to saywhat is known in the non-contorted case, before a brief consideration of thecontorted case.

Suppose then that the two-surface twistor space T(S) has dimensionfour and that E defined by (13.27) is constant on S and so defines a normon T(S). What is special about such an S? First we observe that anypoint p of S defines a two-dimensional subspace Lp of T(S) consisting ofthose fields wA which vanish at p. Since the space of two-planes in a four-dimensional complex vector space (G(2, 4) in the terminology of exercise3e) is known to be complexified compactified Minkowski space, CM', thisprovides a map of S into CM'. This much is true regardless of E, but withthe aid of E we distinguish N(S) in T(S), where E is zero, and hence realcompactified Minkowski space.

Further the map found above maps S into this real Minkowski space.Since the construction is conformally invariant this suggests that if E isconstant on S then S can be embedded in conformally flat space with thesame first and second fundamental forms (since these are what enter into thetwo-surface twistor equation). The converse of this is evidently true, since ifS can be so embedded, the two-surface twistors on S are the restrictions of


twistors in the ambient space. We are thus lead to the proposition that E isconstant on S if S can be embedded in conformally flat space, and indeedthis can be proved (Jeffryes 1984; Tod 1986). (Essentially, the constancy ofE provides the integrability conditions for the conformal factor Q and itsderivatives pQ and p'S2 at S.)

This then is what is special about non-contorted surfaces S, and for suchsurfaces we have a good definition of norm which we can use to calculate atotal mass using (13.11). To verify (13.10) we need further to locate infinityin the real compactified Minkowski space associated to T(S). To see thatthis is a separate problem over and above that connected with the normconsider a surface S in de Sitter space (exercises 5e and 7h). Here T(S) isobtained by restricting twistors in the ambient space. The natural infinitytwistor might be expected to be the non-simple one associated to de Sitterspace, but the surface S cannot be expected to `know' that it is not inMinkowski space!

We shall therefore begin with calculating the total mass for examples ofnon-contorted surfaces S.

The obvious first case to try is a sphere of symmetry in a sphericallysymmetric space-time (Tod 1983a). On such an S, we may choose thescaling of the dyad so that all GHP quantities are spherically symmetric,i.e. are constant. It follows from exercise 13b that all the ones with p - qnon-zero must vanish, leaving only p, p' and N = X011 + A - 02 as possiblynon-zero real constants. Further these are dependent in that K = 2(N-pp')is the Gauss curvature of S and so is related to the area A by the Gauss-Bonnet theorem, KO = 4ir. There is therefore essentially only one numberdistinguishing spheres of symmetry of the same area in different sphericallysymmetric space-times. Any such sphere is therefore equivalent to one ofthe same area in a de Sitter space with an appropriately chosen cosmologicalconstant. This defines an embedding of S in a conformally flat space-timeand so the norm is constant. To proceed further, we introduce a tetrad andcoordinate system at S and write out the two-surface twistor equationsexplicitly.

The general spherically symmetric metric can be written

ds2 = A2(t,r)dt2 -B2(t,r)dr2 -r2(d92 +sine dc'2)

where r is an area distance. We define a null tetrad by

laOn = 1 [A-1 - B-1

n°Va = 1 [A-1 +B-1--];

(13.28)


a9

Va =m r/ as

where = tan 2 Then la and na are orthogonal to a sphere S of constantr and t, and m' is tangent to S. In this tetrad we find, as anticipated, thatv and a' vanish while

2r /For (13.22) we obtain

490

49 1a ((1 +CC)2W°) = 0

whence1 a+bS ° c+dCW _

(1 +CC)1

W =(1 + CC)2

(13.29)

and (a, b, c, d) are the four coordinates on T(S), the two-surface twistorspace of S.

Substituting (13.29) into (13.27) we find

E=WATT+WA/7f,q,= -2 (cb-bc+ad-ad). (13.30)

As anticipated, the angular dependence has dropped out leaving a constantwhich is the expression for the norm in the coordinates (a, b, c, d).

Finally, we substitute in (13.20) to find the kinematic twistor:

A,#ZaZp =27

i(`9ll + A - 02) w°w1dS

-(,ii + A - &2)(ac + bd)r2. (13.31)

Although it will have no Lorentz invariant significance, we may relabel thecoordinates on T(S) as

a=0°; b=521; c=irVP1,; d=-irv"2-P°,

so that (13.30) becomes

E = 52°15° + 521151 + S2°'P°, + S2"P1,


which is the norm in its usual form. Now (13.31) appears as

A,,,pZ"Za = 2PAA'SlAPA,

where

so

1, r3 N o, o, i'Po =-G7=-Pl ; Po =P1 =0

Poo, = Pu, = r3NGvf2-

Therefore the quasi-local mass defined by (13.11) is

3

MP = --2AapAaa = Papa = (rG )2,

3

Mp=rG (13.32)

We have been led to a simple invariant expression (13.32) for the massinside a surface S of area A in terms of A and the one remaining curvaturecomponent at S.

For the Schwarzschild space-time,

N=-b2= GMsR3

where Ms is the Schwarzschild mass parameter. In this case (13.32) be-comes

MP = Msso that the Penrose mass at any sphere of symmetry in the Schwarzschildsolution is just Ms, the Schwarzschild mass parameter. For the Reissner-Nordstrom space-time one calculates

Ms e2N=w11- 2=G( r3 2r4

where e is the charge of the solution, and so

e2MP=Ms-.2r

Here the mass has a contribution, due to the electromagnetic field, whichdepends on r. For a Reissner-NOrdstrom black hole, the outer horizon isat ir=r+=Ms+(MS-e2)2.


At this radiusMp = 2 r+

which is the irreducible mass, i.e. the mass of a Schwarzschild black holewith the same area.

Finally, we consider the Friedmann-Robertson-Walker (FRW) cosmolo-gies. These are spatially homogeneous and isotropic cosmological modelswith perfect fluid matter tensor. They have spatial cross-sections of con-stant curvature labelled k = 1 for the 3-sphere, k = 0 for flat space andk = -1 for hyperbolic space.

Now for Mp we find the product of the matter density with a volume,the volume of a sphere in flat space having the same surface area as thesphere S. For k = 0 this is straightforwardly density times volume withinS but for the others it is different. In particular, one sees that in the k = 1case the mass within a sphere of radius r increases from zero, attaining amaximum when the sphere is equatorial in the 3-sphere, and then decreasesto zero again.

One may continue in this fashion and calculate Mp for various non-contorted two-surfaces in various space-times (Tod 1983a, 1990a). Theresults are broadly in agreement with physical intuition in a wide varietyof circumstances. In particular, the original quasi-local mass definitionsuccessfully `detects' the following aspects of mass and energy:

1. total mass-energy: in the sense of both the ADM and Bondi masses`at infinity',

2. gravitational-wave-energy at infinity: as measured by the Bondi mass,

3. rest-mass energy: in, for example, the k = 0 FRW cosmology,

4. matter kinetic-energy: in the tilted Bianchi-type-V cosmology,

5. gravitational potential energy: in, for example, the time-symmetricinitial-value-problem,

6. electro-static field energy: in the Reissner-Nordstrom solution,

7. gravitational-wave-energy 'quasi-locally': in cylindrical gravitationalwaves, and

8. the conserved quantity identified in the 'post-Newtonian' approxima-tions to general relativity as energy.

(See Tod (1990a) for a review of the work which leads to this list, andfor a bibliography of Penrose's quasi-local mass up to that date.) This


is a remarkable list for a single, uniform definition to encompass. It isparticularly striking that the definition, which was designed to work inthat limit of general relativity which is linear theory (G -- 0), should alsobe correct in the other limit which is post-Newtonian theory (c -> oo).

This completes the discussion of the non-contorted case and we turn to abrief discussion of contorted two-surfaces. To provide an example of these,we review the 'small-sphere' calculation (Kelly, Tod and Woodhouse 1986;Woodhouse 1987). The idea here is to calculate the quasi-local mass as apower series in the size of a sequence of small spheres, or small ellipsoids,as the surfaces shrink to a point.

The simplest way to construct such a sequence is as follows: first choosea point p in space-time M, and a unit time-like vector ta at p; let thegeodesics ruling the null cone N of p be generated by null vectors la whichare normalised at the vertex p by

gabtall = 1.

If r is now an affine parameter determined by the parallelly-propagatedgenerators la according to

laVar=1,r=0atpthen the surfaces of constant t are `small spheres'. We write S(r) for thesphere of constant r. It is a straightforward, if lengthy, exercise to calculatethe metric, connection and curvature of the space-time M on the null coneN as a power series in r. One may then solve (13.22) and (13.23) for thetwo-surface twistors, and calculate the kinematic twistor Aa/j from (13.25)or (13.26) to any desired order in r.

In the presence of matter, the leading non-zero term in the kinematictwistor Aaa is O(r3). The norm defined by (13.27) is constant to O(1), andthe norm of Aa/j to this order is the Minkowski-metric norm of the vector

P. = 37rr3Tabtb.

In other words, the quasi-local mass at this order is the norm of a momen-tum vector defined as the volume of the small sphere times the momentum-density obtained from the stress-energy tensor. This is an entirely satisfac-tory result, but is probably just a reflection of the fact that the quasi-localmass definition is correct for linear theory.

In a vacuum space-time, the leading non-zero term in Aa/j is two ordershigher, at 0(r5). (Note therefore that if one divides by the volume of S(r)the result vanishes at r -> 0; the quasi-local mass is not the integral ofa density, which would be a local mass.) This time however the result of


the calculation is quite unsatisfactory. The norm of the kinematic twistorcan be negative even for small-spheres in the Schwarzschild solution; otherinvariants of A,p, such as its determinant, can be imaginary e.g. for small-spheres in the Kerr solution. At the same time, the norm defined by (13.27),although constant on S(r) at O(1) and O(r), is varying at O(r2); the small-spheres are contorted at this order.

Before proceeding further, we make another observation about the con-torted/non-contorted dichotomy. Given four linearly independent solutionsto the two-surface equations, (WAi, 7rA') for i = 1, 2, 3, 4, we may define adeterminant

W°1 W11 7ro,1 7r1'1

W0 4 W14 7ro,4 7r1,4

(13.33)

For a two-surface S in flat or conformally flat space, and therefore also fora non-contorted S, this determinant 77 is constant. However, just as forthe norm E, 77 cannot be shown to be constant just from the two-surfacetwistor equations (13.22, 13.23) and it will generally vary on a contortedS. Note that 77 has something of the character of a Jacobian. Penrose(1984; see also Penrose and Rindler 1984) has given arguments why theoriginal definition of quasi-local quantities (13.4) should be modified by theinclusion of a multiple of 77 in the integrand.

Returning to the small-sphere calculation, we find that 77 from (13.33)is constant on S(r) at O(1) and O(r) but varies at O(r2). Remarkably,if we modify the integral by including 7j as a factor in the integrand, assuggested by Penrose, then the integral vanishes at O(r5); in other words,this modification removes this pathology. The difficulty that remains isthat there is still a leading non-zero term but it is now O(r6), and it hasso far defied calculation.

We may summarise the situation as follows:

There is a complete definition of quasi-local mass for non-contortedtwo-surfaces; it gives intuitively satisfying results in a wide variety ofcases; however there are at present no general theorems about, forexample, positivity.

For contorted surfaces, there are indications that the definition isin need of modification, and suggestions for what the modificationsshould be; however there is not at this point a clearly satisfactory wayforward.

Given the need of the original definition for modification, and given theattractiveness of the idea of a quasi-local mass definition, it is natural that


other definitions have been formulated and investigated (see e.g. Ludvigsenand Vickers 1983; Bartnik 1989). To end this chapter, we shall considerone of these definitions which is quite close to the spirit of the original.This is the definition of Dougan and Mason (Dougan 1991; Dougan andMason 1991) which is related to the definition of quasi-local charges forYang-Mills theory (Tod 1983b) which was in turn inspired by Penrose'soriginal definition of quasi-local quantities (Penrose 1982).

Given a space-like, topologically-spherical two-surface S and a spinorfield AA on S, one considers the integral over S of the Witten-Nester inte-grand:

Q= 4±G f AAA VB)AAA - XAVA(A4AgI)Cla`4'B'. (13.34)

This integrand is the 2-form which enters into the Witten proof of positive-energy (Witten 1981) and its subsequent extensions (e.g. Nester 1981;Horowitz and Perry 1982; Ludvigsen and Vickers 1982; Reula and Tod1984; see also Penrose and Rindler 1984). As an exercise, the reader mightshow that the integral is real (integration by parts turns it into its complexconjugate).

For an asymptotically-constant spinor field AA' and a two-surface S `atinfinity', (13.34) yields a component of the Bondi or ADM momentum. Ifone formally replaces AA by WA and AA' by IrA' where these are the com-ponents of a two-surface twistor then (13.34) becomes (13.26), though ofcourse one has now given up the reality of (13.34). The idea of Douganand Mason is to use (13.34) with spinor fields AA' which are either `holo-morphic' or `anti-holomorphic' on S in the sense that one or other of theequations

TAA' = 0 (13.35)

6AA' = 0 (13.36)

which in the GHP formalism are

;W, - p/al' = 0; Al'd'- vA°' = 0;W' - AI, = 0; aA1' - pA°' = 0

holds on S. Each of these equations has a 2-dimensional complex vectorspace of solutions on S. Call these vector spaces V, V respectively. We maylabel a basis of either by

A' A'A A'_(AA'.,

,A.')


and then the quantityAt B'10,1' = eA'B'Ao, Al, (13.37)

is easily seen to be constant on S (exercise). Provided it is non-zero, thistherefore defines an 1A'B on each of V and V.

In more sophisticated language, equations (13.35, 13.36) define two dis-tinct holomorphic structures on the spin-bundle SA' of the space-time at S;since SA' will generally be trivial as a holomorphic vector bundle in eithersense, both of equations (13.35, 13.36) will generically have two linearly in-dependent solutions, i.e. solutions for which (13.37) is non-zero. However,for exceptional two-surfaces, SA' will fail to be holomorphically trivial forone or other definition; solutions of the corresponding one of (13.35, 13.36)will be proportional as spinor fields and it will not be possible to define(non-zero) 1A'B, by (13.37). For such an exceptional S, one of the con-structions will fail. From (13.35), SA' will be exceptional if p' is zero onS, since then A°' will be zero and all solutions will be proportional to LA';likewise from (13.36), SA' will be exceptional for the other definition if pis zero. These conditions for exceptionality are sufficient, but they are notnecessary.

While exceptional two-surfaces present a problem, this problem cor-responds to the existence of `extra solutions' to the two-surface twistorequation in the original Penrose construction. There is no equivalent inthe Dougan-Mason construction to the contorted/non-contorted dichotomyand the difficulties which it brings to the Penrose construction.

The integral (13.34) defines a momentum vector PA'A according to

PA A 4IfGf {A A,VB')AXA -

B,(13.38)

S

and we use the e-spinor together with its complex conjugate to calculatethe norm of the momentum, i.e. the mass.

This then is the Dougan-Mason definition of quasi-local momentum andmass. Strictly speaking, it is a pair of definitions depending on whether oneuses V or V, (13.35) or (13.36). What properties do these definitions have?

First of all, in flat space, solutions of either of equations (13.35, 13.36)are necessarily constant on S, so that the momentum defined by(13.38) necessarily vanishes for any 2-surface S in flat space.

By extending the spinor fields AA' to a 3-surface E spanning S ina suitable way, it is possible to show that (13.38) gives the correctdefinition of momentum in linear theory.

At spatial infinity one recovers the ADM momentum using either ofequations (13.35, 13.36), but if one takes a two-surface S tending to


null infinity then there is a difference between the two choices. Specif-ically, going to future-null-infinity one must use the definition basedon (13.35), and one then recovers the Bondi momentum; the integralusing (13.36) diverges towards future-null-infinity; the converse holdson the approach to past-null-infinity.

For small spheres in the presence of matter, one again has `volumetimes momentum density' for the momentum vector. In vacuum oneobtains at O(r5) a future-pointing time-like vector constructed fromthe Bach tensor. At O(rs) one begins to see a difference between theuse of V and V.

The most striking feature of this definition is that it can be used toprove a positivity result:

If the Dominant Energy Condition (see e.g. Penrose and Rindler 1984) holdson a 3-surface E spanning the 2-surface S and if at S the spin-coefficientp' satisfies p' > 0, p' # 0 somewhere on S (respectively p < 0, p 0 some-where) then the momentum defined by (13.38) using (13.35) (repectively(13.36)) is time-like and future-pointing.

The condition on the spin-coefficients is a condition of convexity on thetwo-surface S; if p' > 0 then the area is non-decreasing at all points inthe past-pointing, out-going null direction; if p < 0 then the area is non-decreasing at all points in the future-pointing, out-going null direction.Provided one of these two conditions holds, one of the two definitions ofmomenta is time-like and future-pointing. The proof involves a modificationof the Witten argument (Witten 1981) using the spinors AA' at S to provideboundary-values for the Witten equation on E (Dougan 1991; Dougan andMason 1991).

This theorem makes the Dougan-Mason definition a particularly inter-esting one.

Exercises 13

a) Calculate the weights of the spin coefficients (13.17) and show that theoperators (13.20) have well-defined weights.

b) Show that quantities of weight (p, q) on the sphere S are sections of aline bundle which may be identified with O(q -p) (see exercise 3g). Deducethat weighted functions can only be constant on S if p = q.

Chapter 14

Functionals on Zero RestMass Fields

Given the powerful constructions of chapters 8-10, which show how thePenrose transform describes zero rest mass fields and neatly geometrisesthe concept of positive frequency, it is natural to seek a twistor descriptionof interactions. One might expect such a description to arise out of a twistordynamical principle, replacing the usual space-time Lagrangian, and indeedthe early work (Penrose and MacCallum 1972) proceeded in just this way,the resulting twistor integrals being referred to as twistor diagrams. Twistordiagrams share with Feynmann diagrams a simple combinatorial structure,but are not supposed to share their divergences! The analogy betweentwistor diagrams and Feynmann diagrams has been extensively explored,for which see Hodges (1990) and references therein. In this work twistordiagrams are regarded as determining integrals of holomorphic functionsover compact contours, and it is only fairly recently that the diagrams havebeen reinterpreted as functionals on the cohomology classes representingthe fields. This latter question, which is reviewed in Huggett (1990), is oursubject here, and in order to discuss it adequately we will need to draw onsome more cohomology theory. We start, however, by making a relativelysimple observation on the geometry of products of twistor spaces.

In the product space {(Za,Wa) E PT x PT* : ZaWQ 0} the fol-lowing holomorphic differential forms arise quite naturally. Let DZ =ea06. ZadZQdZadZ'r, where eaa67 is as in (7.7), define DW similarly andlet DZW be their wedge product. Then for each homogeneity we have theform

137


DZW(14 1).

(ZaWa )nThese forms can be used as kernels for integrating products of homogeneousfunctions f (Z) and g(W) which are holomorphic on some parts of PT andPT*. Indeed, given such an f and g, both homogeneous of degree -2, wecan evaluate

f (Z)g(W)DZW(14.2)

(Z ,,)2

Remarkably, this integral has the following simple interpretation inspace-time. Regard the functions f and g as representative cocycles forelements of first cohomology groups, and let 0 and X be the scalar fieldscorresponding via (10.1,10.2) to these cohomology elements. Then (14.2) isequal to the usual (Gross 1964) Hermitian symmetric positive definite con-formally invariant norm, which we refer to as the scalar product betweenthese fields.

One may imagine generalising this to interactions involving more thantwo fields and more than two twistor spaces (or their duals) in the product.The kernels are still combinations of forms like (14.1) (perhaps includingIap), but they quickly become cumbersome to write out explicitly. It waspartly for this reason, but more to emulate the Feynman diagrams of quan-tum field theory, that the twistor diagram notation was devised to representthese kernels. We are not going to describe twistor diagrams here; we sim-ply observe that their theory aims to provide a (finite) twistor description ofinteractions between fields (z.r.m.s in the first instance). If a given such in-teraction involves a number of fields, then there would be a twistor diagramrepresenting a kernel against which to integrate the product of representa-tive cocycles for these fields. The diagram should also prescribe in someway a (compact) contour over which to perform this integration, althoughthis is often quite problematic. Indeed a good part of the work in estab-lishing that (14.2) is the scalar product comes from the identification ofthis contour. Even when such a contour has been identified, there remainsthe question of whether it defines a functional on the cohomology classesand not just the cocycles. (It also remains to demonstrate the Hermitiansymmetry and positive definiteness in the twistor picture. See Baston andEastwood (1989).)

In this chapter we will address the problem of finding all functionals ona given collection of z.r.m.s, these functionals all corresponding to a givenkernel (which arises from a specified twistor diagram, say). Before we cando this though we will need some more cohomological theory. For a slightlymore leisurely account, see Huggett and Singer (1991).

The reader will probably already be aware that there are various co-homology theories; de Rham cohomology is well known, and in chapter 9

CHAPTER 14. FUNCTIONALS ON ZERO REST MASS FIELDS 139

we introduced Cech cohomology. Here we introduce Dolbeault cohomologyand discuss its relationship with the de Rham and Cech theories. Beforethat, however, let us briefly review the de Rham cohomology.

On a (paracompact, Hausdorff, C°°) manifold M of dimension n wedefine SlP(M) to be the group of C°° differential forms of degree p, and welet d denote the usual exterior derivative. Then there is a complex (i.e. asequence of groups and group homorphisms such that `d2, = 0):

1l°(M) - c1(M) ... d, QP(M) ... -+ Q'(M)

and the de Rham cohomology groups are defined by

(14.3)

HP (M) _ {wEQP(M):dw=0}dQP-I(M)

(= closed p-forms modulo exact p-forms).

In fact it can be proved fairly readily that these cohomology groups arethe same as the Cech groups in which the sheaf is the constant sheaf R:

H* (M) = ft* (M; R).

If our manifold M happens to be a complex manifold (of complex di-mension n) the groups QP(M) have extra structure. This is most familiarwhen p = 1. Then

111(M) = c1'°(M) ®Q°'1(M)

where in local complex coordinates zi the group Q1'°(M) consists of formslike

fidzi

while cl°"(M) has elements

For a general p we have

QP = ®f °QP-i,im

where r8(M) consists of forms `of type (r, s)':

fjl...j,.kl...k,dzjl n ... A dzj,. n dzkl n ... A dzk,ji...jrk,...k,


The derivative operator d also decomposes:

d=8+8which (again) is most familiar when applied to functions in our local coor-dinates zi:

Here d : S2°(M) -+ 12'(M) has split into

8 : 1 °(M) -4 QI'°(M)8 : 00(M) --> S2°,1(M).

The induced decomposition of the action of d on (r, s) forms is obvious (butmessy to write out explicitly!). It is clear, though, that we get

e : Qr's(M) -* Sir+183(M)

e : SZr's(M) Qr,s+1(M).

Furthermore,

d2=(8+8)2=82+88+88+82=0

and each of the terms 82, 88 + 88 and 82 maps into a different directsummand, so in particular

We can therefore define, for any r, the Dolbeault complex

Qr,°(M) e, Qr'1(M) --+ ... e, Qr,s(M) e, ... S2*'n(M) (14.4)

and the Dolbeault cohomology groups are defined by

Hr'$(M) _ {w E Qr,s(M) : 8w = 0}e OQr,s-1(M)

(The alert reader will recall these definitions from exercise 9d.) It can quiteeasily be shown that

He's (M) = IIs(M; SZr).

which gives us a very useful relationship between Dolbeault and Cech co-homology.

Let us return to the de Rham cohomology groups for a moment andrecall the pairing between HP(M) and HP(M) given, simply, by integration.


Poincare duality guarantees that this pairing is non-degenerate, and in thefinite-dimensional case we have in fact

HP(M) = HP(M)*.

There is another class of linear functionals on HP(M), though. We canconsider the group QP(M) of p-forms with compact support, and define

H' (M)

in the obvious way (noting that d : QP -> QPc+1). If w E 1P(M) andr/ E Q'-P(M) then

JMwA77

is well defined and only depends on the cohomology classes of w and 77. So

Hn-P(M)

is a space of linear functionals on

HP(M).

Indeed, it can be shown that

Hn-P(M) = H,(M).

We can do just the same for the Dolbeault cohomology groups in thecase when M is a complex manifold (of complex dimension n). Given

[a] E H"8(M)[fi] E

Hc -r'n-s(M)

we have a pairing given by integration

fMaA fi.

We can see how this generalises our usual integration by putting r = n.Then we have

H"'s(M) -' H"+s(M)(9 0

H°'"-s(" ) Hc (11'I) - H"+s(M)1 pairing j usualC with a C integration

compactDolbeaultform

(14.5)


In fact we will need our Dolbeault cohomology to have coefficients insheaves of germs of holomorphic functions of various sorts. Let such a sheafbe denoted by S. Then we tensor each group in the complex (14.4) andtake cohomology (with respect to 8) to obtain

H'''9 (M; S)

and in the compactly supported case

Hg's (M; S).

We now have a pairing between

H'''s(M; S) and S*)

where S* is the dual sheaf to S. (For example, 0(-n) is dual to 0(n).)We need one last refinement of all this theory before we can actually

use it, namely relative cohomology. Relative de Rham cohomology comesfrom the complex whose pth term is

1P (M, U) = QP(M) ED QP-1(U)

(where U is an open subset of M) with the differential

d(a, /3) = (da, o I u - d f3)

mapping 1 P(M, U) to V '(M, U) (and satisfying d2 = 0). We have theshort exact sequence

0--+ I1P-'(U) -> S2P(M, U) --> QP(M) 0

Q -' (0, 3)

and the corresponding long exact sequence

...-4HP-1(U)-+ HP(M,U)->HP(M)-+ HP(U)-+...

called the relative exact sequence, in which the connecting map is restrictionto U.

Relative Dolbeault cohomology comes from the complex whose sth termis

Qr,s (M, U; S) = cr,s (M, S) ® nr,s-1(U S)

with the differential

8(c,' M _ Oct, 8,8 + (-1)saI U),


and we have the analogous short exact sequence of sheaves and long exactsequence of relative cohomology groups.

For relative Dolbeault cohomology with compact support, we define

QA's (M, U; S)

from the exact sequence

0 -> (U, S) -+ c2's (M, S) --+ Qr'3 (M, U; S) -+ 0.0'

C

extend

by zero

Then the qth cohomology group of the complex

cr'*(M, U; S), 8

is Hr 9(M, U; S). Again there is a relative long exact sequence, and thistime the connecting map is induced by 8.

Now, at last, consider an interaction between m zero rest mass fields.Let Uj (j = 1,.. . , m) be any open subset of Pj (which is PT or PT*)swept out by projective lines corresponding to points in a convex set Vj inMinkowski space. Then the theorems of chapter 10 (which were given forthe case U = PT+, V = CM+) apply:

H1(Uj; O(-n - 2)) = {massless free fields of helicity n2

holomorphic on Vj I.

We will also adopt the notation Fj = Pj - Uj, Lj for a projective linecontained in A=L1x...xLm.

We will be studying the space of functionals on the tensor product

H1(Ul;O(ri))®...®H'(Um;O(rm.)) (14.6)

with the object of picking out those determined by a given twistor diagram.This tensor product is much more tractable if we use relative H2s to rep-resent our fields, instead of H's. We can do this, because of the followingrelative long exact sequence:

H1(Pj;O(rj)) -, H'(Uj;O(rj)) - H2(Pj,Uj;O(rj)) -, HI(Pa;O(r3))=0 =0

So

H1(Uj; 0 (r3)) Uj; O(rs))dj.


Theorem

The tensor product of cohomology groups (14.6) is isomorphic to

H2m(lI, II - F; 0(r))where O(r) = 0(rl) x ... x

Proof

We sketch the proof in the case m = 2: the scalar product. The prooffor general m is only notationally more fiddly. We have, by the KiinnethTheorem (Spanier 1989) for relative cohomology,

H4(II, II - F; O(r)) = ® H'(Pi, Ui; O(rl)) ®Hq(P2i U2; 0(r2))-Mp+q=4

But we can easily see from the relative long exact sequence and the geo-metric properties of Uj that

Hp(Pj, Uj; O(rb)) = 0 for p < 2.H4(11'11 - F; O(r)) = H2(Pi, Ui; O(ri)) ®H2(P2, U2; 0(r2))

= H'(Ui; 0(rl)) ®H1(U2; 0(r2))as required.

Our next task is to represent, in cohomological terms, the differentialform specified by the interior of the diagram. In the scalar product case wehave the element (WaZa)-r'DZW of

Hs'°(II - E; O(-r))where E = {WaZ' = 0} and r = (r, r). In general we will allow ourselvesmore freedom than this, and think of the diagram as determining a kernel

h E H3m'q(II - E; O(-r)).

The space of all functionals on H2`(1I,1I - F; O(r)) can be deducedfrom the following commutative diagram (for r = 0, s = 2m and n = 3m),in which the complex dimension of M is 3m:

--> H'"'3 (M; S) -> HT's (U; S) - H'''s (M, U; S) -4

HH -''"-3(M; S*) _ HH -'''"-3(U; S*) E- H,'-r,'-'(M, U; S*) <--1 1 1

C C C


We also use the fact that the first two pairings (which are those we saw ear-lier) have the property that any continuous linear functional on Hr>9 (U; S)(say) can be obtained by pairing it with an element of Hn-r,n-8(U; S*).

So our space of functionals is

HH',m(II, II - F; O(-r))

and we need to see how our kernel h picks out one (or a small finite number)of these. We first use the open inclusion

i:II - E -- II

to cut out E:

i* : HHm'm(II - E, II - E U F; O(-r)) -+ HH',m(H, II - F; O(-r)).

and then we note that cupping with our kernel h defines a map

Uh: H°'m_ (II-E,II -EUF)-HHm'm(II-E,II -EUF;O(-r)).

So our functionals are determined by elements

a E HO,--q(H - E,II- EUF).

But these elements are more than just contours. They can be thought ofas 'function-contours', i.e. contours with germs of holomorphic functionsdefined on them, as was described very neatly by Jozsa (in Hughston andWard 1979). In order to insist that our functional arises from a straight-forward contour, we look back at (14.5) and see that we require a to be inthe image of the map

H5ir,.+q(II-E,H-EUF)- HO,--q (II-E,H-EU F),

which embeds the ordinary contours into the space of function-contours.In the scalar product case q = 0 and m = 2 so we are looking for

elements ofH10(II-E,H-EUF) (14.7)

and if our fields are elementary states (see exercise 8d) we have F = A.More generally, suppose only that (II - E, H - E U F) is homotopic to(H - E, II - E U A). Then (14.7) becomes

H10(lI-E,II -EUA)

which is isomorphic toH2(A - E) (14.8)


by the Thom isomorphism theorem (Bott and Tu (1982)). This lattergroup has one generator, so the scalar product diagram determines onefunctional, and this functional can be shown to coincide with the contourintegral (14.2), as we would hope!

In general our functionals would be given by elements of

Hm+q(A - E) (14.9)

for some q and E, or if the Thom isomorphism theorem were unavailable,

H5m+q(lI-E,l-EUF).

Finally we make a few comments on how to tell if a contour is cohomo-logical. The question is this: a twistor diagram with m elementary stateshas at least 2m poles in the integrand. A practical way of proceeding withthe evaluation of such a diagram is to take residues at each of these poles,and then to tackle the remainder of the integral. Does this correspond to acohomological evaluation, and are all cohomological evaluations capturedin this way?

Having taken all these residues, one would be left seeking a contour inHm+q(A - E), so it seems from (14.9) that both answers are yes. Well,they are, but one has to be a little more careful: taking 2m residues isdual to applying 2m cobord maps. (A cobord map, applied to a contourlying in a real codimension 2 submanifold S, replaces each point x of thecontour by an Sl lying in the normal space to S at x.) On the otherhand, the Thom isomorphism we used in (14.8) involves the applicationof one `big' cobord map. (Here the submanifold is of real codimension8 and the Thom isomorphism replaces each point x in the contour by anormal 8-dimensional disc. This, followed by a boundary map, yields anS7 above each point x in the original contour, which is our `big' cobordmap.) An interesting technical lemma (Huggett and Singer 1990) allowsus to show that these procedures coincide for contours without boundary,so that any such cohomological evaluation can be implemented by takingthese 2m residues first. The remaining question concerns contours withboundary.

Chapter 15

Further Developmentsand Conclusions

In this book, our aim has been to provide a brief introduction to twistortheory, sketching the mathematical background and indicating where itmakes contact with the physics of space-time. It is possible to carry onfrom here in directions which we may characterise as inner or outer, andmathematical or physical.

By inner we mean developments which make rigorous and treat morethoroughly what has been treated informally here. For spinor theory themain reference is the two volume Spinors and space-time of Penrose andRindler (1984, 1986). This is a mighty compendium of results for spinorsin flat and curved space-time, and has more detail, in volume 2, on thequasi-local mass construction described in chapter 13. For more on sheafcohomology and the twistor theory of massless fields and the active, curved-twistor-space constructions of chapter 11 see Twistor Geometry and FieldTheory (Ward and Wells 1990).

By outer we mean developments that go beyond what has been coveredhere. Other parts of the theory of z.r.m. fields have been given a twistordescription, and the formalism of chapters 8 to 10 has been much extended.Hyperfunction z.r.m. fields are discussed in Bailey et al. (1982) and fieldswith sources, which can be described by relative cohomology, by Bailey(1985). This material is reviewed in the article of Bailey and Singer inBailey and Baston (1990).

There is a large body of theory concerned with twistor diagrams whichare the counterpart of Feynman diagrams, in that twistor diagrams definecontour integrals which represent scattering amplitudes for various scatter-

147


ing processes involving z.r.m. fields. In chapter 14 we saw how contourintegrals of this type may be described in terms of sheaf cohomology. Forthe theory of the twistor diagrams themselves, see Penrose and MacCal-lum (1973); Sparling (1975); Hodges (1982, 1983a, 1983b, 1985a, 1985b)and the articles by Hodges and by Huggett in Bailey and Baston (1990).This theory at present lacks a twistor theoretic generating principle (suchas Feynman diagrams have in space-time Lagrangians). An interestingproposal, though, is the twistor conformal field theory for four space-timedimensions (Hodges et al. 1989; Singer 1990; Huggett 1992). The Riemannsurfaces and holomorphic functions of the more usual two-dimensional con-formal field theory are here replaced by (a class of) complex three-manifoldsand holomorphic sheaf cohomology, with the aim of generating interactionamplitudes between the space-time fields represented by the cohomologyelements.

Massive fields have a twistor contour-integral formulation (Penrose andMacCallum 1973) with an analogous cohomological apparatus (Hughstonand Hurd 1981; Eastwood 1981). A geometrical view of the spin-and-statis-tics theorem can be given from this point of view.

In chapter 7 we saw that a null geodesic is represented by a null twistor.A time-like geodesic can be represented by a combination of twistors, whichleads to the representation of a massive spinning particle twistorially (Pen-rose and MacCallum 1973; Tod and Perjes 1976; Tod 1977). This com-bination has an internal symmetry group which is typically a semi-directproduct involving SU(n). From representations of these twistor internalsymmetry groups it is possible to build up a twistorial classification of el-ementary particles which is similar to, but differs in important respectsfrom, the standard classifications (Hughston 1979).

Many examples have been given of the constructions of chapter 11 solv-ing the a.s.d. Yang-Mills equations (Atiyah and Ward 1977; Atiyah et al.1978; Corrigan et al. 1978; Woodhouse 1983; see also Ward and Wells 1989)and of the construction of chapter 12 solving the a.s.d. vacuum equations(Curtis et al. 1978; Hitchin 1979; Ward 1978; Tod and Ward 1979). Theconstruction of chapter 11 has been generalised to give monopoles (Ward1981; Hitchin 1982, 1983) while the construction of chapter 12 has beengeneralised to give a.s.d. Einstein (but non-vacuum) spaces (Ward 1980;LeBrun 1982) and scalar-flat, Kahler manifolds (LeBrun 1991).

A development from the study of a.s.d. Yang-Mills fields has been thestudy, in twistor terms, of integrable equations (see the article of Wardin Bailey and Baston 1990). Many of the known, completely-integrablep.d.e.s turn out to be reductions of the a.s.d. Yang-Mills equations, andthe Lax-pair formulations associated with integrable equations often turnout to be interpretable as curvature conditions on a connection like (11.16).

CHAPTER 15. FURTHER DEVELOPMENTS 149

(By a reduction we mean that the Yang-Mills potential is assumed to beinvariant under a subgroup of the conformal group of Minkowski space,possibly with some extra assumptions on the form of the potential.) Thereare a few known integrable equations which have not been obtained froma.s.d. Yang-Mills, and it has been suggested by Mason (recorded in Tod1992) that these will arise from the a.s.d. Einstein equations. Accordingto this conjecture, the non-linear graviton is a universal integrable systemfrom which all others are obtained by reduction.

The real, Lorentzian-signature Einstein vacuum equations for station-ary, axisymmetric space-times are equivalent to a certain reduction of thea.s.d. Yang-Mills equations. This fact has been exploited by Woodhouseand Mason (1988) and by Fletcher and Woodhouse (in Bailey and Baston1990) to give a a twistor construction of such space-times in terms of holo-morphic bundles over non-Hausdorff Riemann surfaces. Their formalismprovides a geometrical way of understanding and unifying the considerableliterature on this class of solutions of Einstein's equations.

One type of reduction of the a.s.d. Einstein equations is the Einstein-Weyl equations. These are a kind of 3-dimensional conformally invariantgeneralisation of the Einstein equations which therefore provide another setof non-linear p.d.e.s with a geometrical interpretation which is solvable bya twistor construction (Jones and Tod 1985; Tod 1990b).

Returning to a.s.d. Einstein spaces, since these can be real for Riemannsignature, the Riemannian case has been of particular interest to mathe-maticians. There is a large literature on positive-definite twistor theoryfollowing the original paper of Atiyah et al. (1978) (see e.g. Besse 1987). Arelated development of great mathematical significance has been the studyof the moduli-space of a.s.d. Yang-Mills fields on a given 4-manifold as ameans of studying the topology of 4-manifolds (Donaldson and Kronheimer1990).

A twistor characterisation of the full, as opposed to s.d. or a.s.d., Yang-Mills equations has been given (Isenberg et al. 1978; Witten 1978). Theconstruction uses bundles over ambi-twistor space, a space of pairs of twist-ors which can be identified with the space of null geodesics in Minkowskispace. A similar thing has been done for the Yang-Mills-Dirac system(Henkin and Manin 1980; Manin 1988).

Super-symmetric extensions of twistor theory, of the non-linear gravi-ton and the Einstein-Weyl equations have been given by Merkulov (1991,1992a,b).

Other ways of describing twistors in curved spaces are discussed in Pen-rose and MacCallum (1973); Penrose (1975); Penrose and Ward (1980). Inparticular, there is a close connection with Newman's H space (Newman1976; Hansen et al. 1978). The holy grail of general relativity is a con-


struction of the general, real, Lorentzian-signature solution of the vacuumequations. A twistor characterisation of these has been given in terms ofdeformations of the complex structure of ambi-twistor space (Baston andMason 1987; LeBrun in Bailey and Baston 1990); this is a characterisationrather than a solution since the deformations are not given in terms of freedata. A more recent attempt to find the general solution of the vacuumequations has been to try to define twistors as charges of spin-! fields, sincethe vacuum equations are the integrability conditions for such fields to exist(in the sense of `potentials modulo gauge' (Penrose 1992)).

There is an almost separate tradition in twistor theory which invlovesthe approach to complex manifolds outlined in exercises 3h and 9d. Holo-morphic objects are thought of as constructed from real differentiable ob-jects satisfying Cauchy-Riemann equations (Woodhouse 1985). In thistradition, cohomology classes for example are globally defined differentialforms rather than functions defined on overlaps. Likewise the bundles andfibrations of chapters 11 and 12 are defined in terms of modified Cauchy-Riemann equations. In particular, Newman's good cut equation (Newman1976; Tod 1982) can be interpreted in this way.

Other material not previously mentioned can be found in the book ofcommissioned reviews (Bailey and Baston 1990), in the collected reprintsfrom Twistor Newsletter (Mason and Hughston 1990), in the book of Bastonand Eastwood (1989), and in the proceedings of the 1993 Twistor Theoryconference in Devon (Huggett 1994).

In this book, one of our aims has been to convey some sense of the varietyof twistor theory. Initially the theory is one of space-time geometry withan emphasis on conformal invariance and the group isomorphisms whichintroduce the complex numbers:

SL(2, C) '--4' O+(133)

SU(2,2) O(2, 4) ' C(1, 3).

The discussion of z.r.m. fields leads us into the complex calculationsof sheaf cohomology and gives a first instance of the disappearance of fieldequations into complex analyticity.

The active constructions of chapters 11 and 12 involve more elaboratecomplex manifold theory and, as recompense, solve non-linear field equa-tions by complex analyticity.

The further developments which we have just sketched here lead on intoa massless quantum field theory, classification schemes for elementary par-ticles, geometrisations of other branches of physics and the possible solutionof a completely classical and long-standing problem in general relativity.

CHAPTER 15. FURTHER DEVELOPMENTS 151

Through this wide-ranging activity, the unifying thread is the centralimportance of holomorphic and geometrical ideas in space-time physics.

Chapter 16

Hints, Solutions andNotes to the Exercises

Chapter 2

a) For Rabed in terms of Sabed, consider Sa[bc]d

b) For eabedFabFcd = 0 Fab simple, first look at components to provecde = 1 fcdethat Ca 49abe FfcFde

d) You need GXeabed = 0; write it out and use the result of (c).

e) We have chosen to define the Lie derivative from the covariant derivative.An exercise like this then requires a calculation in tensor calculus. In adifferent, more abstract approach, the fact proved here would be elementarybut then other facts would be more obscure.

Chapter 3

a) Show first that t E SL(2, C) has a unique polar decomposition t = RUwhere R is a positive definite Hermitian matrix and U is in SU(2).

d) The formula for Rabed follows from repeated applications of the techniqueleading to (3.7).

e) See Field (1982) or Griffiths and Harris (1978).

g) For the line bundle O(-1) over CP1 the trivialisations are

t0([lrA'J, AIrA') = ([lA'1, Alr0')

153


tl(L7fA'],)t7rA') _ ([IVA'1, X7ri').

So yo = A7ro, and It, = A7r1!, and hence the transition function fol is Wnias required. Local sections si satisfying the transition relation

so = foisi

have the property that

Define

Si so

7r1' 701

s1where 7r1, ; 0

otherwise.7ro'

F(7rA') _7r1'

so

Then F is homogeneous of degree -1.

h) For the last part, if V° is a type (1,0) vector-field then V(LVb) definesa (contravariant) conformal structure; conversely given a conformal struc-ture and a real vector field Ua, one has a unique vector field which is `90°anti-clockwise' from Ua and the same length; this is JabUa.

Chapter 4

a) The problem is to find an example of the geometric situation illustratedin figure 4.1. One begins by imagining the rotation through 27r of an or-thonormal frame at the point corresponding to the north pole in CP 1 = S2and the origin in C. One then seeks a sequence of paths on the S2 begin-ning and ending at the north pole, with the first and last in the sequencebeing the zero path and the others gradually covering the S2. Finally oneconsiders the effect on the chosen orthonormal frame of being carried alongthese paths. For more details see Geroch (1968).

e) Differentiate again and use the Ricci identities (4.4, 4.5).

Chapter 5

b) See Penrose and Rindler (1984) section 6.8.

c) Writing xa(s) as xa, and xa(0) as pa, (5.6) becomes

xa _ pa - sBap21 - 2sBbpb + s2B2p2

Now calculate dxa (for s and Ba constant). There will be a dpa term, apbdpb term, and a Bbdpb term. Finally, calculate dxadxa. By a miracle,

CHAPTER 16. HINTS, SOLUTIONS AND NOTES 155

only the dpadp' term survives, and with the correct factor!

d) GX E

= 9rAXbVbWA + C.C._ -iXBBB VBTB' + C.C. from the twistor equation= 0 because Xb is real.

GX(WAWA+WA7rA')IrALXWA + W AGX FA + complex conjugate9rAXbVbWA - 9AdiABWB - ZIrAWA4+W AXbVbwA +WAc5ABiB + 4WAIrA + C.C.

e) After the coordinate change the metric becomes

cosh2 t{sech2t dt2 - [d'p2 + sine 1(d92 + sine 9dcb2 )] }

which is conformal to

dr2 - [d 2 + sine 1'(d02 + sine 9dO2 )] .

Now T = sinht and t = log(tan211), so

T = -cotr

and so 0 < r < -7r as required for the Einstein Static Universe.

Chapter 6

a) Necessity is easy. Sufficiency is a case of Frobenius' Theorem (Spivak1975); the problem is to show that for any two vectors orthogonal to Va,their Lie bracket is also orthogonal to Va.

b) la is h.s.o. e=> l!aVbla! = 0 from part (a)q *{V albs}la = 0 where * means the dual

. la{VAB'lBA' - VBA'lAB'} = 0OAOA V ABA OB OAS = OAOA VBAS OAOB'

oADOA = 0 and p = p

c) Solve f = 0 for OA in terms of xAA'. You get

01 x + iy

00 z± x2+y2+z2

Now show that la = OA0A', which depends on xAA', is coplanar with tAA'AAand x


Chapter 7

a) Show that the eigenvalues of E are +1 (twice) and -1 (twice).

b) DAA'aBI3B' +VBB'aAI3A' = aBVAA'13B' +PB'DAA'aB+ aAV BB'I3A' + 13A'VBB'aA

_ -aBOAB'I3A' -;aB'VBA'aA+ aAOBB'QA' + I3A'V BB'aA

from (7.1)= (aAV BB' - aBOAB' )13A'

+ (QA'V BB' - /B'V BA' )aAEach term is now skew in both AB and A'B'.

c) Suppose (wA, 7rA') is an arbitrary twistor on the line corresponding toxa - iya. Then

WA = i(xAA _ iyAA )7rA,

Show that this is in PT+ yAA'TA7rA, > 0, so that the condition on yais that

yap. > 0

for all null future-pointing pa.

d) X[aPQ'r1 = 0 * X°QQ7PQ7 + (XIR7 - X7Rp)Qp7Sa+ (X7Sp - XQS7)Qp7Ra = 0

where Q,67 is any line skew to Pp7 (so Qp7Pp7 0), and P167 = RLOS71.Thus Xa is a linear combination of Ra and S.

e) First show that a eap76P76 is skew and simple, so that

1 E076P76 = A[cBp1

Next show that Aa and Bp are (distinct) planes through the line L definedby P. If Ra and SO are any two (distinct) points on L then

Pap = R[cSp],

so L is real R[aSp] = A[, Bp] . Now it is easy to show that if L is real thenRI must be null. Conversely, if Ra and SO (and any linear combination ofthem) are null then it is easy to show that RaSa = 0 and hence that anyplane through L is a linear combination of Ra and Sa.

f) This double fibration is a special case of a correspondence between twocomplex manifolds Z and X given by a third complex manifold Y and two(surjective, maximal rank) mappings 77 and r:


Y

The pair (77, -r) must embed Y as a submanifold of Z x X. The Penrosetransform (described in chapter 10) can also be studied in this more generalsetting. See Baston and Eastwood (1989), for example.

g) Through any two points in PT there is a unique line. Now consider apoint xAA' on the a-plane

A A = ixAA' AA,

and on the ,(3-plane

It is immediate that AaBa = 0. The converse is much easier in the twistorpicture: if AaBa = 0 there is a line L through Aa lying in Ba, so thereis a point common to the a- and /-planes. In fact there is a 1-complex-parameter pencil of lines of this type, all meeting at A. Hence the result.

h) We refer back to (5.6) and (5.7), in which we considered the 0(2,4) nullcone N. Instead of taking the space of generators, which yielded compact-ified Minkowski space, we now simply put V = -1, obtaining the de Sitterspace described in exercise 5e. This hyperplane intersects all the generatorsof N except those for which V = 0. In RP5, the V = 0 hyperplane is nottangent to the quadric Q (being the polar plane of the point I' not on

Z X

BA'_ -iXAA'BA.

Q).

Chapter 8

a) Put

where

f(Za) = ZA)

g is holomorphic and non-zero on ZaAa = 0. Then

0A'...B'(x) =1

IrA' ... IrB' px ( )Irc, d?rC27ri ry

ZAa1 1

P. (Z Aa ) A, whereoA'

(x) =AA'

+ iXAA'AA.


Now for a given x choose y to surround this pole in 7rA' with a simple loop,and calculate

OA QsA'...B'(x)

In general, the cases r = 0, 1, 2 correspond to the types N, 3, D.

b) , 7rA' ... 7rB' Px (Za Aa f )7rc, d7rc'f 7rA' ... 7rB'

aD'lrD, Px (f )lrc, dire , where CA' =

AA'+ ixAA' AA

'aD 0A'...BID'

and similarly for spin-raising.

c) See Penrose and MacCallum (1973) and Hughston and Hurd (1981).

d) LetpA' = pA'

+ iXAA'PA etc., as in (8.2). Then the integral is

E'1 f pA'IrA,rB'R.B' +gC'7rc,sD'7rD'd7r

27ri (p F' lF, qG' 1rG,)2

Choosing a local coordinate z = 2B and calculating the residue at z = 0qlr BI(for which the `epsilon identity' EA[BECD[ = 0 is needed), one obtains thefield

pA'rA, + sB'gB,(pC'gc, )2

This is singular on the null cone of the point represented by P[,,Qp] , asrequired, and it is invariant under the transformations

r H r+ap+/3qs H s-/3p+yq

where a, , 3 and y are complex parameters.

Chapter 9

b) No, because the groups of sections of a sheaf over open sets have thefollowing property: if f E 13(U) and g E 13(V) are such that

f l unV= 9 I unv

then 2h E 13(U U V) such that

h lu= f and h Iv= g.

This is one of the properties of a `complete presheaf'; see Field (1982)chapter 6, especially Proposition 6.1.4.


c) Consider the mapping

r* : Hl({Ui}; S) -> Hl({W-}; S)

where {W,,,} is a refinement of {Ui} (with refining map r). Observe thatthe mapping

r* : C°({Ui}; S) - 0°({W.}; S)

is onto, and use the commutativity of r* and 6.

d) For the second part of the exercise, first note that there is an isomorphismbetween O(-2) and 521, the sheaf of germs of holomorphic 1-forms. Thenuse the short exact sequence

0 - 521 -+ E1'° - Z1'1 -4 0,

where Z1,1 is the sheaf of germs of 8-closed (1,1) forms, to construct thefollowing commutative diagram:

0 - C°({Ui}; st') -+ c°({Ui}; £1'°) -+ c°({Ui}; Z1'1) - 016 16 16

0- 01({Ui}; Q1) 01({Ui}) £1'°) - cl({Ui}; Z1'1) -o

Now the isomorphism

H1({Ui}; 521) -+ H1(P1; Q1)

is effected by mapping the Cech class [w] to the Dolbeault class [8(6-1(iw))].The map 6-1 is multiplication by a partition of unity. If this cover {Ui} isas in (9.4), then the appropriate partition of unity is

I7ro, I2 I1rl_ l2{170I2 + 171I2

,170I2 + 171I2

e) See Spanier (1989).

f) It is easiest to use a fine resolution of S. Suppose S = 0, for example,and use the resolution in exercise 9d. Then CP = £°,P.

Chapter 12

b) This is a difficult exercise, and getting signs and factors right is a trickybusiness. If A = 0, then a typical line L,, is given by

cvA = WA - ixAA' lrA,. (16.1)


Suppose the `linearised deformed line' is given by adding terms of order Ato this. The main idea is to restrict the linearised transition relation to L.,and split it. Explicitly:

Px aw = 9A (X, 70 - 9A (x, 7r) (16.2)

for some gA, gA so that, from the linearised transition relation given in thequestion,

cvA+.kA(x,7r) = WA +)tgA(x,7r).

This gives us the O(X)-correction to (16.1):

WA = -ix AA 'TAI - AgA(x, 7)

(16.3)

(16.4)

and a similar expression for hatted quantities. To find the metric, first takean infinitesimally neighbouring deformed line with SxAA' = VAA':

SWA = -iVAA'irA - )VCC'VCC,gA(x,7r). (16.5)

The conformal metric arises from the condition imposed on VAA' if SWA isto vanish at say 7rA' = aA'. To investigate this condition, we use the styleof argument which led to (10.23): from (16.2)

7rA'DAA'9B = 7TA'VAA,9B

= hAA'BB'(x)7rA'7rB' (16.6)

since the first equality defines a global, homogeneity-2 function on Lx, whichmust therefore be a quadratic polynomial in 7r. If VAA' in (16.5) is to be anull vector whose primed spinor part is aA' then

VAA' = QAA'BB'QBaB' (16.7)

in terms of the `connection symbols' o,AA' BB'- (These appear because allindices in this question are concrete and not abstract.) We have

oAA'BB' = SBASB,A' +O(A)

so substituting (16.7) into (16.5) and requiring this expression to vanishwhen aA 7CA'=

0 we find

aAA'BB' = SBA5B,A' - iAhBB,AA'.

Finally the linearised metric satisfies

9abcOCC'C bDD, = ECDEC'D'


from which it follows that

gab = 71mb + 2iAhab (16.8)

where hab is determined by the original f(w,7r) by (16.2) and (16.6). Itremains to be checked that this hab is a potential for the linearised Weylspinor of the linearised metric (16.8).

c) One approach is to introduce a simple coordinate and tetrad systemas follows: choose any two fibres in T and coordinatise a holomorphicsection by the values XA, yA of wA in these two fibres. Observe that thesecoordinates are all null and that the metric can be written 2HABdXAdYB.Now consider the holonomic tetrad

e0 =a

e1 = e2 =a

e3 = aax0, 8X1' aY°' aYl

From the geometric definition of the connection show that

Veoe° = Veoel = Deie° = Deiel = 0

and similarly for (2,3), and that

Veoe2 = ve2e°

and so on. Now it is possible to deduce that the torsion vanishes.

Appendix

The GHP Equations

The spin coefficients and GHP operators have been defined in (13.17),(13.19) and (13.20). The components of curvature in the chosen spinordyad are labelled:

weight'rb0 = OABCDOAOBOCOD (4,0)01 = OABCDOAOBOCLD (2,0)02= OABCDOAOBLCLD (0,0)03= OABCDOALBLCLD (-2,0)04 = '+GABCDLALBLCLD (-4,0)

-A'O-B'000 = GABA'B'OAOBO

' '(2,2)

LB-001 = -OABA'B'OAOBOA

' '(2,0)

TB002 = OABA'B'OAOBTA

' '(2, -2)

OB-010 = OABA'B'OALBOA' '

(0,2)GB011 = OABA'B'OALBOA

' '(0,0)

LB012 = QOABA'B'OALBLA

' '(0, -2)

OB020 = 4ABA'B'LALBOA (-2,2)-A'B'021 = OABA'B'L

ALBOL (-2,0)

-A'L-B'022 = OABA'B'LALBL (-2, -2)A = 24R (0, 0).

The GHP equations form three groups. First the equations relating thecurvature to derivatives of the spin coefficients:

ap - c'v = (P-A)T+(P -P,)r, -01+¢01 (A.1)

,pP - XYv = p2+Q5-9T-TIC+q500 (A.2)

163


pa - arc = a(P + P) - K.(T + T) + (A.3)

pT - p'!c = p(T - T') -}- a(T - T') + 001 (A.4)aT - p'a = -p a - v'p +T2 + rc,c + 002 (A.5)

p'P - c'T = p7 p'+ aa' - TT' - !r.' - 02 - 2A. (A.6)

Next the commutators of the GHP operators acting on a quantity 77 ofweight (p, q):

TT(pp' - P'P)rl = (T - (T -

(pj - J1p)71

(ai' - J'c)7l

-p(kf -TT'+02+Oil -A)i7- q(ICr + ';2 + Oil - A)77pan+ac'r1-T p77 -r-ji'r1

-P(Pr- -Ta+'+/1)11-q(v'7-Fm'+Ool)fl(P -P)P7l+(P-P)p'ii+p(PP'-ai +'02-Oil-A)71-q('-'+2 - Oil - A)1).

(A.7)

(A.8)

(A.9)

Finally the Bianchi identities become:

p4'l - X00 - pool + c,¢oo = -T'5o + 4p &1 - 342 + T'000- 2p¢ol - 2aq io + 2noll

/ /

+9002 (A.10)

PV'2 - '"4'1 - C'0ol + p'qS00 + 2pA = a'*0 - 2T'01 + 3p b2 - tic &3+ P Ooo - 2TOoi - 2T0lo

+ 2p01, + 70o2 (A.11)

P03 - X02 - P421 + x020 - 2C'A = 2o,'01 - 3-r'02 + 2pb3 - n04- 2p'qS1o + 2T'oll + r'020

" "- 27021 +9022 (A.12)

P'4 - X03 - X021 + 7'4520 = 3o,'02 - 4T'03 + p04 - 2n'o10+2a'o11+;020-2Tq521+ F022 (A.13)

APPENDIX: THE GHP EQUATIONS 165

and the contracted Bianchi identities are

poll + P'coo - 410 - X001 + 3pA

P412 + P'¢01- aO11 - x'002 + 3aA

(p' + P )c5oo + 2(p + p)q5u

-(T +2T)¢iol-(2T+T)Olo-'9012 - X021 + Q-020

+ U7002 (A.14)

(p' + 2P )O`ol + (2p +;5)012

- (T + T)c¢02 - 2(T + T )O11

- j'O00 - r-022 + 0'021

+ Q'&10 (A.15)

A great economy of notation is achieved since many of these equationsare actually four equations; the original, the complex conjugate and theprimes of these. Thus for example the commutator (via - a p) tacitlyincludes the complex conjugate (po"' - J"p) and the primes (pi's' - a'p')and (p's - chi').

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Index

a-plane 55, 105fl-plane 55

abstract indices 5

algebraically general Weyl tensor

30

algebraically special Weyl tensor30

almost complex structure 24ambi-twistor space 149anti-self-dual bivector 17anti-self-dual Yang-Mills

equation 102astigmatic focussing 48Atiyah-Singer Index Theorem

125

Bianchi identity 103bivector 6boost 8

causal curve 42causal infinity 42Cech Cohomology 77Cech cohomology group 78Cech map 89celestial sphere 19, 57charge integrality 102charge of the Maxwell field 102cobord map 146coboundary map 77compactified Minkowski space 36compactly supported p-forms 141

complete analytic functions 72complex 85, 139complex manifold 18complex projective space 18complex shear 47complex structure 24complexified compactified

Minkowski space 58conformal curvature 105conformal group 33conformal invariance 41conformal Killing vector 9, 33conformal rescaling 41connecting map 86contorted surface 126contour integral formulae for

z.r.m. fields 67correspondence 156

CPn 22cut of the future null cone 20

de Rham cohomology 139de Rham cohomology group 86de Sitter space 43, 63deformation of a complex

manifold 111density of elementary states 69dilation 34Dirac equation 31Dolbeault cohomology 139Dolbeault cohomology groups 89Dolbeault complex 140Dolbeault form, of type (0, 1) 88

175

176

Dolbeault lemma 88double fibration 63, 91, 106Dougan-Mason construction 134dual of a bivector 7dual sheaf 142dual twistor 55, 61dyad 15

eigen-bivectors 7Einstein field equations 8, 29Einstein Static Universe 38Einstein-Weyl equations 149elementary states 69eth 124Euler operator 60

fine resolution 88fine sheaf 85first Bianchi identity 8flag-pole 17F riedmann-Robertson-Walker

cosmologies 130function element 71future tube 63future/past null infinity 39future/past time-like infinity 39

geodesic deviation 46geodesic shear-free condition 105geodesic shear-free congruence

49germ 72GHP equations 163GHP formalism 123Goldberg-Sachs theorem 49Grassmann manifold 22Gravitational energy 119

helicity 31, 67hermitian spinor 14holomorphic deformation 111homogeneity operator 106

INDEX

hypersurface-orthogonal vectorfield 50

infinitesimal generators ofSU(2,2) 59

infinity twistor 57, 64inner product on twistor space

54integrable almost complex

structure 24integrable equations 148inverse twistor function 96

Jacobi field 45

Kerr Theorem 50, 60kinematic twistor 121Klein representation 58Klein-Gordon equation 69

Leray covers 84Lie derivative 9Lie derivative of a spinor 33, 39line bundles over PT 99line bundles over CP 1 23linearised general relativity 120Lorentz group 7

massive fields 69massless neutrino equation 31Mayer-Vietoris sequence 89momentum and angular

momentum 120momentum-angular momentum

twistor 121Moving Sphere 21

negative frequency 31Newlander-Nirenberg Theorem

24Newman's H space 149non-contorted surface 126normal bundle 107

INDEX

normalisability of elementarystates 69

normalised dyad 15null bivector 22null cone 6null flag 17null geodesic congruence 45null geodesics in twistor terms 56null Maxwell field 31null separation of points in

twistor terms 57null tetrad 15null twistor 55

orientability 25orthonormal frame bundle 25

p-coboundaries 78p-cochain 77p-cocycles 78past of a set 42Petrov-Pirani-Penrose

classification 29Poincare duality 141Poincare group 34polar decomposition 153positive frequency 31, 118positive-frequency field 67potentials for z.r.m. fields 95primed spin bundle 91principal null directions 16, 29projective null cone 19projective spinors 18projective twistor space 55, 63proper orthochronous Lorentz

group 7PT, PT+, PT- 56

refinement 83Reissner-Nordstrom space-time

129relative cohomology 142, 147

177

Relative Dolbeault cohomology142

relative exact sequence 142resolution 84Ricci identity 8Ricci scalar 8Ricci tensor 8Riemann surface 72Robinson congruence 61RP" 22

Sachs equations 48, 106scalar product of fields 138scalar-flat, Kahler manifolds 148Schwarzschild space-time 129screen-space 46second Bianchi identity 9section of a bundle 23section of a sheaf 74self-dual bivector 17, 32self-dual Killing vector 120Serre duality 82sheaf 75sheaf homomorphism 75sheaf of germs of holomorphic

functions 72shear 47, 49shear-free congruences 49short exact sequence 76simple bivector 9SL(2, C) 11small-sphere 131source-free Maxwell equations 30space-like infinity 39special conformal

transformations 34spherical lamp-shade 13spin coefficients 123spin-and-statistics theorem 148spin-lowering 32, 121spinor 13

spin-raising 32

178

spin-space 13splitting 81, 102, 104splitting formulae 80stalk 75stereographic projection 19SU(2,2) 54symmetric spinors 16symmetrisation 5

T, T+, T- 56thorn 124time-orientability 25, 26topological obstruction to

building spin bundle 27torsion-free connection 111, 118totally null 2-plane 55transition function 19transition matrix 104twisted photon 99twistor 53, 60twistor conformal field theory

148twistor diagrams 137, 147twistor equation 31twistor internal symmetry

groups 148twistor space 53, 54two-surface twistor 122two-surface twistor equations 122

weighted scalars 123Weyl spinor 29Weyl tensor 29Witten-Nester integrand 133

Yang-Mills field 102Yang-Mills field equations 103

INDEX

zero rest mass free fieldequations 31

s. a. huggett and k. p. tod- an introduction to twistor theory

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