The Theory of H -space

THE THEORY OF k-SPACE

M. KO, M. LUDVIGSEN and E.T. NEWMAN Department of Physics, University of Pittsburgh, PA, US.A. and

K.P. TOD Department of Mathematics, University of Pittsburgh, PA, U.S.A.

t93t~J.98t

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM

PHYSICS REPORTS (Review Section of Physics Letters) 71, No. 2 (1961) 51—139. North-Holland Publishing Company

THE THEORY OF ~-SPACEt M. KO*. M. LUDVJGSEN~*and E.T. NEWMAN Department of Physics. tJniiersitv of Pittsburgh, PA. LISA.

and KP TOD*** Department of Mathematics. (Jnirersitv of Pittsburgh. PA. U.S.A. Eec’ived June 1 961)

(‘onlents: Introduction I Basic theory 1.2. The spin coefficient farmalisni 3. The good cut equation 2. v-space 2.1. The metric, connection and curvature 2.2. The potentials 2.3 The coordinate systems of Plehanski 2.4. Specific examples of ~-spaccs 3

and ,~.~-spaces I. Asymptotically flat W-spaccs 3.7. ‘~-spaces 3 2. ~W complex Minkowski space ,~

4. The Intrinsic ~-space formalism 4.1. The new formalism 4,2. The ~-conformally invariant operator

57

4.3. The Weyl spinor and its potentials 44. Maxwell fields on k-space

8))

5. Momentum, angular momentum and equations of motion 5.1. Introduction 5.2. Momentum and angular momentum 5.3. The .~-spaceapproach to equations of motion 5.4. The motion of a spinning particle 6. The relation between k-space and twistor theory 6.1. Deformed twistor space

108 108 110 115 117 l2 122

S2 63 84 92

6.2. The twistor space associated with a left-flat space 6.3. Asymptotic projective twistor srace 7. Summary and outlook Appendix 1

126 12S 131 134

~

Appendix 2. Notation and conventions References

136 37

bI (s4 (vi

64 73 7’

9~

98

This work has been supported by grant from the N.S F. (all authors) and S.R C. Research Acsistantship (ML.) and Fellowship (K.P.T.). Present address: Department of Geosciences. Purdue University. ** Present address: Department of Mathematics. University of York. ** Present address: Mathematical Institute, Oxford. Ii K. *

Sin~’Ieorders for this issue PHYSICS REPORTS (Review Section of Physics Letters) 71. No. 2 (1981) 51—139. Copies of this issue max be obtained at the price given betow. All orders should he sent directly to the Publisher. Orders must he accompanied by check, Single issue price Dfl. 41.00. postage included.

100 103

M. Ko eta!., The theory of 7~’-space

53

A#,stract: The theory of X-space, the four-dimensional manifold of those complex null hypersurfaces of an asymptotically flat space-time which are asymptotically shear-free, is reviewed. In addition to a discussion of the origins of the theory, we present two independent formalisms for the derivation of the basic properties of X-space: that it is endowed with a natural holomorphic complex Riemannian metric which satisfies the vacuum Einstein equations and whose Weyl tensor is self-dual. We show the connection of our work on IC-space to that of Plebanski and to the theory of deformed twistor spaces, due to Penrose. Finally, there is a discussion of equations of motion in X-space.

1. Introduction Though the general theory of relativity has been with us for close to sixty years, it has only been in the past fifteen years, beginning with the work of Bondi [1,2], that rigorous results and a deep understanding of the radiation aspects of the theory have been developed. Bondi’s great contribution was the realization (now in retrospect essentially obvious but in 1962 a major breakthrough) that gravitational radiation could best be understood by studying and analyzing the null or characteristic surfaces of a space-time. This was in contrast to previous work which concentrated on space-like surfaces and the associated Cauchy problem. This null surface approach led immediately to an understanding of the radiation degrees of freedom and their geometric meaning as well as to the related characteristic initial value problem. Though one could not, except by perturbation theory, relate the local behavior of gravitational sources (i.e., the local stress tensor) to the asymptotic behavior (along the null surfaces) of the gravitational fields, one could nevertheless gain a great understanding of the radiation process. Asymptotic multipole moments, for example, could be defined and their time evolution studied. In particular a momentum vector associated with each outgoing null hypersurface was defined for the space-time [1]. From the time evolution of this vector, Bondi showed that gravitational waves carry away positive mass from an isolated system. This pioneering work of Bondi’s was extended and generalized by Sachs [3,4] and by Newman and Penrose [5], who systematically introduced into the theory complex combinations of real geometric functions (thereby developing the complex null tetrad formalism [6] and the associated spin coefficient formalism), which had a remarkably simplifying effect on the formal differential equations of the theory. For a review of this material, see [7]. The next major advance was made by Penrose, who formalized the idea of infinity by adding a boundary to space-time [8,9]. This boundary, attached by the process of conformal compactiflcation, consists essentially of the future and past end-points of null geodesics; ~ and respectively. The rather difficult and nebulous idea of “infinity” was thereby made formal and precise. It became possible to speak, instead of relations in their asymptotic limit, of relations existing on ~9. Two basically diverging lines of work and thought (though with a considerable amount of overlap) developed from this general theory. The first was the “practical” or astronomical applications, where one related, usually by perturbation theory, the local source behavior of, for example, black holes or sources for gravitational radiation, to the asymptotic values and behavior of the fields [10].The second line was more formal or theoretical; one studied the geometrical properties of the space in the neighborhood of its boundary ~ in the hope of elucidating fundamental questions about the theory, in order to relate it to other fundamental branches of physics; e.g., particle physics and quantum theory. Some of the questions which arose were: what relationship does the asymptotically flat region of the curved space have with (completely flat) Minkowski space; what, if any, asymptotic symmetries are .~

.~

54

M. Ko et a!., The theory of ~1(-space

there, what is their meaning and how can they be related to the Poincaré group; are there functions analogous to Poincaré generators, i.e., energy-momentum and angular-momentum? It is from this second line of thought and the above type of questions that the developments reported on here, i.e., ~‘-space theory, arose. There is indeed an asymptotic symmetry group, known as the Bondi—Metzner—Sachs (BMS) group [1,3, 6], which first appeared as the set of coordinate transformations preserving some natural appearing coordinate conditions in the neighborhood of It was soon recognized that the group could be understood geometrically as an approximate symmetry (of the space which has no exact symmetries); finally Penrose [11] and from a somewhat different point of view Winicour [12],showed that the BMS group was an exact symmetry of itself. It was early recognized that the BMS group was similar in structure to the Poincaré group [3,30]. The Poincaré group is the semi-direct product of the Lorentz group with the four-parameter (Abelian) group of translations, while the BMS group is the semi-direct product of the Lorentz group with an infinite parameter Abelian sub-group, the supertranslations. There is a unique four-parameter normal subgroup of the BMS group which can be identified with translations and it is this fact that permits the definition of the Bondi momentum [4]. In this formalism, the Bondi momentum is an integral of quantities defined at i~.An analogous definition of angular momentum is hampered by the non-existence of an invariantly defined Lorentz sub-group of the BMS group. One resolution would be to find a space defined at ~ to provide a set of “origins” to which angular momentum can be referred. This can be accomplished in certain cases but not in a general radiating space-time. The search for a set of origins provides one of the motivations for the ~‘-space program. It was also hoped (though, so far in vain) that the BMS group would play an important role in particle physics or in a quantized version of general relativity, either through its representation theory or by its reduction to the Poincaré group [4,56, 57]. One of the distinctions between Minkowski space and a curved physical space is in the behavior of the light cones emanating from points; in Minkowski space these cones have vanishing shear o(essentially a measure of how much a cross-section of a pencil of light-rays becomes distorted [71 as the rays propagate), while for curved space, the shear though zero at the apex of the cone becomes, in general, non-vanishing along the generators of the cone. In the study of asymptotically flat space-times one is, however, usually more interested in null surfaces (asymptotic null cones) in the neighborhood of g~which intersect in a “cut” or smooth two-surface (S2), rather than those which emanate from an interior point. (Ordinary Minkowski space light cones are examples of each type of cone.) These asymptotic null cones almost always (when there is radiation) have a non-vanishing asymptotic shear ~ (One can actually determine the gravitational radiation field from knowledge of the o-°on a one parameter family of these asymptotic null cones.) For Minkowski space, the ordinary light cones have vanishing shear. In fact, the converse statement [13] is true; in Minkowski space every asymptotically shear free null surface is an ordinary light cone and hence determines, from its apex, a space-time point uniquely. The action of the Poincaré group which sends points into points, sends light cones into light cones and hence sends shear free cuts of ~ (of Minkowski space) into shear free cuts. One can in this case actually characterize the Poincaré group [14] as that subgroup of the symmetry group of .9~(BMS group), which maps shear free cuts into shear-free cuts. It appears natural to try to generalize this idea to curved spaces and thereby accomplish the reduction of the BMS group to the Poincaré group. This procedure actually works for those space-times which are asymptotically stationary (i.e., asymptotically flat with a vanishing radiation field), there being then a four (real) parameters worth of asymptotically shear-free null surfaces or cuts of i4 and, further, the subgroup of the BMS group which maps the cuts into each other is the Poincaré group. ~.

.~

,~

M. Ko et a!., The theory of k-space

55

One can thus think of this collection of cuts as a canonically defined Minkowski space (associated with the physical space) on which the Poincaré group acts. The Miiikowski metric can even be obtained by a certain explicitly defined “average distance” between nearby shear-free cuts of This attempt to obtain a Minkowski space (as the set of shear-free or “good cuts” of f~)and the Poincaré group as the subgroup of the BMS group which maps these cuts into each other, fails in~the presence of gravitational radiation. The failure, however, leads to more dramatic and interesting results than perhaps a success could have accomplished. To understand this, it is useful to see why the attempt fails. First of all in the general situation there do not exist any good cuts, let alone a four parameters worth to be identified with a Minkowski space. Though at first sight this appears to end the attempt immediately, there is a way out: if the neighborhood of ~ is an analytic manifold, it can be analytically extended into the complex domain (5~becoming C .~) with some finite complex thickening. Now the search for complex asymptotically shear free null surfaces or good cuts is met with success. One can in fact prove that there exists (for each analytic asymptotically flat physical space-time whose radiation field is not too violent) a four complex parameters worth of good cuts [15], i.e., one obtains a four complex dimensional manifold of good cuts, which is referred to as IC-space [15, 16]. This complex manifold, is, however, in general unrelated to Minkowski space. In fact, there is no subgroup of the BMS group (now the complex BMS group) which maps good cuts into good cuts. We have thus associated with each analytic asymptotically flat physical space-time a four complex dimensional manifold with no obvious associated symmetry. If we define the distance between two nearby points of ~‘ by the same “average distance” between their associated cuts as was done for the asymptotically stationary space-times, we obtain the first of a series of remarkable results: the distance so obtained is a complex (holomorphic) Riemannian metric [16, 17]. One can further show that this metric automatically satisfies the vacuum (complex) Einstein equations and has a Weyl tensor which is self-dual; left flat or right helicity in this case. (Had we worked with the conjugate shear o, we would have produced a space conjugate to ~ namely ~ whose Weyl tensor is anti-self-dual or right flat.) We point out that these solutions of the Einstein equations strongly resemble the self-dual (or anti-self-dual) Yang—Mills fields being used to study instantons [18,19]. For brevity we shall refer to complex manifolds with a (holomorphic) complex metric satisfying the vacuum Einstein equations and possessing a Weyl tensor which is either self-dual or anti-self-dual as a half-flat space. From our original goal of producing, from an asymptotically flat-space, a Minkowski space and its associated Poincaré group, we have instead produced (in what appears to be a natural manner simply asking questions, following the equations and making no ad hoc assumptions) a four complex dimensional space with a metric satisfying the Einstein equations. The original hope of finding a canonically defined Minkowski space associated with the physical space is, however, not completely lost; if the ~‘-space is itself, in some appropriate sense, asymptotically flat in its future and in its past, there is induced on ~ of the physical space two (in general different) canonically defined four complex parameters worth of privileged sets of cuts (not good cuts) which are mapped into each other by the Poincaré subgroup of the BMS group. These two Minkowski spaces arise as the i-spaces obtained from both the r and ..~ of the IC-space. Penrose, beginning from a totally different point of view, has also been led to complex fourmanifolds with half-flat curvature. Via the construction of asymptotic projective twistor space [20,21], which is closely related to the ~‘-space construction, Penrose obtains the general half-flat space by a deformation of the complex structure of flat twistor space [22]. It is this approach which provides the proof that u-space is four-dimensional [15]. The twistor programme has produced a large number of ..~.

—

56

M. Ko et a!., The theory of ~/-space

half-flat solutions, some using sophisticated techniques of algebraic geometry [58—61]. Included in this work are the gravitational instantons which Hawking and others have been studying in connection with a path-integral formulation of quantum gravity [62—64].Essentially the half-flat solutions are expected to dominate the path-integrals analogously with the situation in Yang—Mills theory. A different role for half-flat solutions has been suggested by Penrose [22],as the “one-particle states” in a future theory of quantum gravity. A third approach to the study of half-flat spaces has been that of Plebanski and co-workers [23—27]. In a recent paper [23],Plebanski reduced the equations for half-flat spaces to a single, non-linear second order equation on one function of the four coordinates, obtaining a simple expression for the metric in terms of this function. Large numbers of solutions to this equation have been found. Plebanski’s goal is to find the general solution of the real Einstein equations and to this end he has also considered spaces where one half of the curvature is algebraically general (rather than simply zero) effecting a similar reduction of the field equations. This variety of approaches to half-flat spaces results in a classification into four categories: general half-flat spaces [53], ~‘-spaces, asymptotically flat half-flat spaces and non-linear gravitons. Most generally, any half-flat space can be obtained by the Penrose construction based on the neighborhood of a line in CP3; likewise, any half-flat space will arise from a solution of Plebanski’s equation. It is not known whether all half-flat spaces can be obtained by the ~‘-space construction, thus IC-spaces may be a proper subset of all half-flat spaces. It is known that all asymptotically flat (in an appropriate sense) half-flat spaces are i-spaces and further that the k-space construction applied to such a space gives that space back again, so these provide a smaller sub-class. Conversely, one expects that suitable restrictions on cr°will characterize this class. The most special class defined so far seems to be Penrose’s non-linear graviton [221,defined by him as a deformation of the “top half” of CP3, which gives a geometrical definition of positive frequency together with asymptotic flatness. Again, one might conjecture that the ~‘-space construction based on a further restricted class of shears would generate this class. In the remainder of section 1, we develop the theory of asymptotically flat space-times, introducing the necessary mathematical techniques which culminate in the derivation of the good cut equation characterizing shear free cuts. We remark that the good cut equation will have no real solutions in general, and describe the complexification necessary to give solutions. In section 2, we define LW-space as the space of solutions of the good cut equation and recall the original definition of the metric of ~ in terms of a contour integral. However, to expedite the development we give an alternative approach leading to an explicit expression for the metric, and show that the two methods agree. We calculate the curvature, finding that it is Ricci-flat and self-dual and, proceeding in terms of spinors, we find a chain of potentials for the Weyl spinor and a natural, but gauge dependent, splitting of the metric into a flat part plus a curved part. Two possible choices of coordinates are made on ~‘-space and it is shown how they lead to the recovery of Plebanski’s results. Finally, we present a solution of the good cut equation due to Sparling [28],with the ~‘-space metric arising from it. The Weyl spinor is type N, being closely associated with Penrose’s elementary states of linear theory [29]. In section 3, we consider asymptotically flat, left-flat spaces, first proposing a definition of asymptotic flatness for a complex space. The definition is probably stronger than necessary, but it allows the spin coefficient equations for a left-flat space to be solved in terms of initial data on a null cone. It is then possible to prove, as remarked above, that such an asymptotically flat left-flat space is indeed an ~!‘-space.Further, it follows that if an ~f-space is asymptotically flat, then its own IC-space is isometric

M. Ko et a!., The theory of X-space

57

with it; symbolically, ~ = ~ The section concludes with the proof that, under the same conditions, the X-space of an IC-space is flat; symbolically ~‘~‘ Minkowski space. These theorems justify the assertion that the IC-space construction applied to a physical space-time is a projection onto the self-dual part of the gravitational radiation field, setting to zero the anti-self-dual part (and vice-versa for ~‘). Further, the fact that ~‘~‘ is flat leads to a canonical flat space associated with the original curved space, which was one of the original motivations of the subject. Section 4 presents a different formalism, centering on the key concept of IC-conformal weight (HCW). The formalism is developed in spinors and based on a coordinate system arising from a world line in IC-space, rather than the Bondi coordinate systems of section 2. Again, the metric, connection, curvature and chain of potentials are found with somewhat simpler expressions than in section 2. The formalism of this section provides the foundation for section 5. Section 5 is concerned with an application of the formalism of section 4 to the question of equations of motion. By performing integrals over the good cuts of ~ various spinor fields representing, among other things, momentum and angular momentum, are defined on v-space. With the aid of the Bianchi identities, it is possible to calculate the covariant derivatives of these spinor fields on Y~’-space. In the particular case of an algebraically special solution of the Einstein—Maxwell equations, there is a canonically defined world line in the ~‘-space.Further, this world line follows the Lorentz force law which suggests that such a space-time be viewed as a model for a spinning charged particle, following a trajectory in its own ~‘-space.While the significance of these results is as yet unclear, they are extremely suggestive. One point of view is to regard the theory thus far as a splitting of the (highly non-linear) gravitational field into a transverse part, which determines the geometry of IC-space, and a longitudinal part, which manifests itself in equations of motion in ~‘-space. The connection of ~‘-spacewith Twistor Theory is discussed in section 6. We review the deformed twistor space of Penrose and show how it leads to a left-flat space with the metric in Plebanski’s first form. We then treat the converse problem,showing how Plebanski’s metric form leads to the twistor space. Finally, to connect with the work of section 2, we give the construction of an asymptotic projective twistor space from a solution of the good cut equation. While the physical meaning of the whole theory may be unclear, there are several lines of thought and avenues of investigation which appear promising, and these are discussed in section 7. In appendix 1, we give the results of integrating the spin-coefficient equations for an Einstein— Maxwell space-time in the neighbourhood of In particular this shows the relationship between the asymptotic shear and the asymptotic radiation field. Appendix 2 is devoted to terminology and convections. ~.

1.1. Basic theory We now turn to a review of some important properties of asymptotically flat space-times. We first give the definition of asymptotic flatness, then show how natural coordinate systems and tetrad systems are constructed and then consider the transformation properties of certain weighted functions defined on future null infinity, Next we will briefly review the spin-coefficient formalism and state some properties of asymptotically flat space-times. Since it is assumed that the reader is already somewhat familiar with asymptotic methods used in general relativity, we will not go into great detail but simply describe the results which are relevant to the following sections of this paper. We conclude this section with a derivation of the good cut equation. Under Penrose’s conformal criterion for asymptotic flatness [8],an asymptotically flat solution of the ~.

M. Ko et a!., The theory of ,~C-space

58

Einstein—Maxwell equations (with a spacially bounded source) (M, gab) has the following properties: (1) There exists a non-negative function, ii, on M such that there exists a conformally related (non-physical) manifold (M, 1)2 gab) with a boundary r (future null infinity) defined by ii = O,* where (a) Every future directed null geodesic, which escapes to infinity, intersects ,9~. (b) Vail 0 and VailV~D 0 on ~9±;J~is hence a null surface. (c) .S~has topology R X S2. (d) There exists a diffeomorphism between M and M .9~. (2) (lCabcd and Fat, are well defined and smooth on ..9~,where Cat,cri and Fab are the Weyl and Maxwell tensors on M. As this work is not primarily concerned with null infinity and its properties and in order to avoid an overproliferation of symbols, we will take some notational liberties. In particular though we will be working with the non-physical (conformally related) manifold, most of the symbols used will be appropriate for the physical space-time. When we are forced to use non-physical manifold symbols, they will be given with a carat, e.g. —

Cabcd

=

[l2Cabcd,

~ab =

il~g~,

ta = 1a,

~îa = ‘)2fla,

tha

=

Ilma.

The conformal factor 1) (which vanishes on ~) defines a vector normal to

91+

by

on91~.

‘Ia =~a~

Since ~ is null, ñ’~lies in Y and is therefore tangent to the null generators. On choosing an origin cut (a two-dimensional space-like cross section of .9~),‘Ia defines a parameter, u, on each generator of .Y~’ according to ‘IO~7aU =

Each u

1;

u

=

0 on the origin cut.

const. cut of ,91~defines a null hypersurface in M. Normals to this hypersurface are given by

Ia = ~aU,

where lafl

=1on~.

A parameter, P, along the generators of these null hypersurfaces is now defined by 1aT1;

If

~‘

il=~=0on~~.

is the stereographic coordinate labelling the generators of xa=(u,1,~,~)

*

Strictly $~need only be part of the boundary; for a careful treatment see [30].

~,

it can be seen that

M. Ko et a!., The theory of X-space

59

is a good coordinate system in the neighborhood of r, with I~given by ~=0. (The physical space r — —P”.) Such a system has been referred to as a Newman—Unti (NU) coordinate system in the neighborhood of 7 [31].In terms of such a system, the line element can be seen to have the form di2= _(2du dP+%~~F).

(1-1)

The vectors iI’~and fa on 7 can be completed to form a null tetrad by the addition of two complex null vectors th’~and nr (tha is the complex conjugate of tha) such that _th*~iñ 4= 1 and IOtha = ñath0 = 0. In terms of the NU system, such a pair of vectors is given, on 7, by th°= —2Pô~

~

=

—2P8~.

It should be noted that ~

tha*=e~tha;

(1-2)

with A an arbitary function, also satisfy the null tetrad orthogonality relations. We call an NU null tetrad in the neighborhood of 7. In terms of this tetrad we define the following quantities on 7 [5]: 1

,~ —

1

,~

,~ ta”bIc’d ‘~~‘abcdt

m ~m

4A

~a)

,

fa”bfc”d

n

m

(1-3)

cabcdmfltm,

*2 =

(f~,iIa, ,1Z*Z,

1Am

‘~a’bfc”d ~ fl

P3~’-”abcd

‘~a”b~c”d

~‘

5tP4~

7j~abcdm

n m n

,

çb~EabI**thl*, =

Fat,(lfl

+

(1-4)

,~2thl~),

cb2=Fat,mn, =

(15)

thathb~1f,,

where Cabcd is the Weyl tensor and

Fat,

the Maxwell tensor on ~f. Under the condition of asymptotic

M. Ko et a!.. The theory of

60

~if -space

flatness, each of these quantities can be shown to be finite and smooth on 7 [81.0.0 can be shown to be a measure of the asymptotic shear of the null hypersurfaces in M corresponding to the u = const. cuts of 7 From their definition it can be immediately seen that 4~and 0.0 transform according to ~,

=

e~12~i~

~o*

=

e’’2~~~

0.0* = e2iAo~0

under transformation (1-2). In general we say that a quantity

i~, on

,1~,which transforms according to

(1-6) under (1-2), has spin weight (SW) s. Strictly, a spin weighted function should be regarded as a section of a certain line bundle over the sphere [65]. Associated with the concept of SW there are two differential operators ~ and ~ (edth) defined by [341 —

=

th~~afl SXI1, —

(1-7)

+ S,~’fl,

=

where x = tht, and ,~has SW s. It is easily checked that ~ has SW s + 1, and l5,~has SW s In the special case of a NU tetrad system, equations (1-7) become =

2Pt’ a(ps~r,)It9~

=

2P1~3(p_s)/~9~

—

1.

(1-8)

From now on we assume that all quantities are defined with respect to a NU tetrad system. Using equations (1-8) it is easily seen that the commutation relation between ~ and ~ is given by

(~t5 5~q= 2sk~,

(1-9)

—

where k = t~TóIn P is the Gaussian curvature of the u-const. cut on which ~ is defined. Consider now the transformation to a new u coordinate on 7 given by u’~=G(u,~,fl.

(1-10)

If this is accompanied by a transformation to a new ~ coordinate, plus a change of conformal factor, given by (1-11)

=

=

(1-12)

Oil,

we obtain a new NU system

Xa =

(u*, F,

~,

~) near 7, in terms of which the line element is

M. Ko et a!., The theory of s-space

d~*2= _(2du* dP* +

61

on

(1-13)

where 1” = In terms of this system, the new NU tetrad can be easily computed and is given by =

= Ota

+ ~G tha + ~G flia + 0° ~G ~G ‘Ia,

(1-14)

th~Otha+~G’Ia.

If u and u~have the same origin cut, i.e. u = u* = 0 on the origin cut equations (1-14) become =

Opt,

I~= Ola,

th

=

Otha

(115)

at the origin cut. Using equations (1-12) and (1-15) one can see that to ,o*_f’—3,o — ‘-‘ ‘/n,

‘Pa

~o*_/’—2j,o ~~‘h

—

‘-J

~

o*_/’—i fT

—

‘-1

and

l/I~, çb~

cr°transform

according

o CT

at the origin cut. In general we say that a quantity ~ which transforms according to P1*

=

at the origin cut, has conformal weight (CW), w [34]. Using the transformations given by (1-10), (1-11) and (1-12) it is always possible to transform to a special NU system in which P =P

0~(1+C~).

(1-16)

Such a system is called a Bondi system. Throughout this paper, Bondi systems will always be (UB, PB, C,.C) and the corresponding edth operators by 2P~’ö(P~-q)/c9~, ~ = 2P~’8(P~~)/DC.

represented by x~= =

~o and ~o

where (1-17)

With the choice (1-16), the metric of each cut of 7 is that of a sphere. A regular spin weight s function ~ (i’, ~) is_a sectiop of a line bundle over this sphere and should properly be regarded as a pair of functions ‘q(~,~), ~ C), C = each non-singular at finite values of its argument and related by ~

(1-18) 1.2. The spin coefficientformalism The starting point of the spin coefficient formalism [5]is the introduction, into physical space-time, of

l’4. Ko et a!.. The theory of ~i’-space

62

a null tetrad A ~ = (l’~,~a m’~,~ 21(aflb)

gab

—

with the metric given by

2m(arnb).

‘

The spin coefficients are complex linear combinations of the Ricci rotation coefficients (AIlk) which are given by Aqk =

(VbAai)A~A~.

There is however an economy of notation in using the spin coefficients since the twenty-four real rotation coefficients are combined into twelve complex spin coefficients and it becomes feasible to introduce a separate letter for each one. A similar economy is achieved with the Riemann tensor (in vacuo) or the Weyl tensor whose ten real components are combined into five complex tetrad components.

For future use, we recall the definitions of some of the spin coefficients: p

=m’~Fn”Vt,la; p ,ñ~~mbVl; mam~~Vbla; o~= m”rn”Vt,l

(119)

0. =

When Ia is a tangent field to a geodesic congruence, these are the optical scalars of the congruence and have simple geometric significance [7];in particular o-, the shear, provides a measure of the distortion of the congruence and p is essentially the complex combination of the divergence and twist of the congruence. In the special case when the congruence forms light-cones in Minkowski space, we have p=

15=—lJr,

(1-20)

cr~=0,

where r is the affine length measured from the apex of the cone. WeyI tensor components are written *0,. *~,*0,. . *4 with a typical component being .

~a

—

c~

—

‘‘-‘abcdi

bic

d.

7’

—

.

.

ja m ~m

1—~abcd~ I~’ —bjc —d

m ~m ‘P0 Using the four differential operators (the tetrad components of the covariant derivative operator) the spin coefficient version of the Einstein or Einstein—Maxwell equations, can be written out explicitly, that is, without use of the summation convention [5, 71. (They form three sets of first order equations; the definition of y,~from A ~, the Ricci identities and the Bianchi identities.) We point out and emphasize, that though the spin-coefficient formalism was invented to study the real Einstein equations, it is particularly well suited for studying the complex Einstein equations (i.e. equations which are formally identical to the real Einstein equations but where the (complex) metric components are holomorphic functions offour complex coordinates). The only change in the spin-coefficient equations is that all barred quantities are no longer complex conjugates of their unbarred counterparts but _are independent holomorphic variables. This is indicated by replacing bars by tildes (e.g., ~ a ~ ~a ~ *o) in all the equations. Though we will not write out the spin coefficient equations here, we will use them extensively. In section 3, they are used to study the complex half-flat Einstein equations, i.e. ‘P0

—

—~

Rat,

=

0;

C~b~d =

M. Ko et a!., The theory of W’-space

63

or çfr~=O;

cfra0.

In appendix 1 we summarise the results of integrating the spin-coefficient equations in the neighborhood of 5~for asymptotically flat space-times. The solution is given in an arbitrary NU coordinate system and is determined by a knowledge of the asymptotic shear CT°(u,~ C) together with a number of functions of two variables. In particular, the radiation parts of the Weyl spinor are given in terms of the shear alone. 1.3. The good cut equation

In order to seek good cuts, we must first know the transformation of the asymptotic shear under a change of Bondi coordinates. To find this we consider the special class of transformations (1-10) of the form U~~— uB—a(C,~).

(1-21)

These preserve the Bondi coordinate conditions and are the supertranslations [34].The unphysical tetrad is transformed according to (1-14) to PP

a= Ia

=

th~

—

l5oatha

—

~joatha

+ ~oa~oaña

(1-22)

thatljana.

According to (1-5) the shear is =

th*ath*~Tat~.

Substituting from (1-22) and using the definitions of the other spin coefficients and their known values from appendix 1 leads, after a straight-forward but lengthy calculation, to C~~) = O~t(UB, ~ C)—

~a.

Now demanding that the cut u~=0 have vanishing shear, we are lead to 15~a= o~(a(C,~), C~C).

(1-23)

It is worth remarking that the translation sub group of the BMS group is the subgroup of supertrans-

64

M. Ko et a!., The theory of ~(-space

lations (1-21) with

15~aO.

(1-24)

We shall see in section 2.4 that this has a four-parameter family of solutions. From (1.23) we may observe that the shear is unchanged by translation. The question then arises as to whether a(C, C) can be chosen such that the new shear o~ vanishes; in other words, do there exist asymptotically shear free cuts of 5”? If UB = Z(C, C) were such a cut then ~Z(C, ~) = o~(Z,

~,

(1-25)

~).

However, this will in general have no solutions, since o-°~ is a complex function, corresponding to two real functions, while Z is just one real function. The resolution of this difficulty is to complexify the original 5~,by allowing u8 to be complex and treating ~ and ~ as independent complex quantities thus freeing ~ from being the complex conjugate of C and denoting it by ~. All barred quantities now become tilded quantities independent of their former complex conjugates. Provided o(uB, C, C) is analytic in uB and in the real and imaginary parts of it will be holomorphic as a function of u, ~ and ~ at least in a thickened neighborhood of 5~which we shall call C 5~.The amount of this thickening will depend on the singularities of the analytic extensions of geometric quantities such as o. The equation for complex asymptotically shear-free cuts is now ~,

~Z(C,

1)

o~(Z,~,~)

(1-26)

which will be referred to as the good cut equation. It is the study of the space of solutions of the good cut equation which makes up the theory of ~‘-space.

2. ~‘-space In this section, we give the definition of IC-space as the space of solutions of the good cut equation derived in section 1. The metric on ~‘-space is defined in terms of a contour integral and we show that it is Riemannian. Earlier calculations of the connection and curvature of ~‘-space [15] relied on manipulations and repeated differentiations of the contour integral but here we present a different development, obtaining simpler, explicit expressions for the metric, connection and curvature. Use is made of certain spinor fields on ~C-space,further simplifying calculation but tensor and vector equivalents are given where appropriate. A sequence of potentials for the Riemann tensor is exhibited in subsection 2.2, and in subsection 2.3 we use our expressions for the ~‘-space metric to obtain the particular coordinate-systems and forms of the metric given by Plebanski [23]. 2.1. The metric, connection and curvature

In terms of a Bondi-coordinate system

(UB,

C, ~) on

an asymptotically shear free cut, or good cut,

M. Ko et a!., The theory of ~‘-space

65

is given by UB

=

Z(C,

h

satisfying (2-1)

where cr°~(u, ~ ~) is the asymptotic shear of the Bondi system, and all quantities are suitably cornplexified as remarked in subsection 1.3. The good cut equation (2-1) has a four complex parameter solution provided o-°~ is not too large. This fact, long suspected, has been proved by Penrose on the basis of Kodaira’s deformation theorems on complex manifolds [15]. Writing the solution of the good cut equation as Z(z”, C~C)~the four parameters Z’~are coordinates on ~‘-space, the space of good cuts. The complexification of 5~should only be regarded as a complex thickening of the real 5~since the functions o and Z will inevitably encounter singularities if arbitrary values of their complex arguments are allowed. Similarly, there will be restrictions on the allowed ranges of the coordinates Za. We shall write the thickened 5~as C5~and think of it as large enough to permit a solution of (2-1) for Za in some open set of C4.

To define the metric on ~°-space,we first see how a vector at a point P in ~°-spaceis to be interpreted. If two infinitesimally separated points P, Q have coordinates Za, Z’~+ EVa where v°are the components of a vector then the corresponding good cuts are given by

where V(C, ~)

=

VaZ,a.

From (2-1) we obtain =

(22)

ft°BZ.a.

so that a vector is a linear combination of the four-independent solutions to (2-2), the linearised good cut equation. The metric length of the vector V~zis now defined to be 1 df? -‘ gabvv

=

~

(V2Z.a)2)

where

do

Oi(1+Cj)2

and the contour of integration is a sphere, 52•

(2-3)

66

M. Ko et a!,, The theory of ~C-space

The question of existence of contours is a little tricky [15]but for small enough ó~and V’~ near enough to “time-like” there is no problem. The definition of the metric can then be extended to all vectors by analytic continuation. To see that this definition leads to a quadratic metric, rather than Finslerian as might at first appear more likely, we find an expression for the general solution to (2-2). Provided d~is not too large, the deformation theory of complex structures ensures a solution V of (2-2) which is regular on the real sphere C = C and has no zeroes there. By a constant rescaling of V we can ensure that 1 £d110,.1

24

8irTV~

(

so that V corresponds to a unit vector at P, and then find a unique s.w. —1 function W such that 2=1+~W.

(2-5)

1/V

Writing the general solution to (2-2) as (2.6)

Za = Vla*

for some quadruple of s.w. 0 functions l~(C,~), we find =

0.

Thus

V2~o

0l~ = ma

(27)

where ma is a quadruple of regular s.w. + 1 functions satisfying ~joma

=

0.

(2-8)

From the properties of ~o [34], (2-8) has only three linearly independent solutions. Further, we can find a quadruple la of regular s.w. 0 functions unique up to addition of constants with ~0la

=

(2-9)

ma.

(2-7) now becomes ~j0la~=~-

ma

=

ma +~Wma

=

~t(la + Wma)

where use has been made of (2-5) and (2-9). Thus I~= la

+

Wma

(210)

M. Ko et a!., The theory of J~-space

67

where the possible additive constant has been absorbed in Ia. Finally (211)

Za = V(la + Wma).

A convenient choice of Ia

~5

\/~

(242)

~1_C~).

ia_~(i,~C~),

1+~~ 1+~’C 1+~

Any other choice l’a is related to this one by a constant non-singular linear transformation l’~= ~

(2-13)

Since we are working at a single point P of ~‘-space,a different choice (2-13) simply corresponds to a coordinate transformation on IC-space. For later use, we define two further covectors

(244)

fla = la + ~O~jOla.

tha = ~O1a,

By direct calculation from (2-12) and (2-9) we then find 21(aflb) —

2m(athb) =

Thus the tetrad (la, abi

Pta,

at,

—

flab =

diag(1, —1, —1, —1).

(2-15)

ma, tha) is a null tetrad with respect to —

flab,

that is

—

fi ianb mamb and all other scalar products vanish. To evaluate the integral (2-3) we —

—~

—

C-coordinate =

C+ W

1-WC

Then 1

I

a

a

a

\

~1~* .L.

I~’~I +Wm =—li

~

—~

‘

=

dQ~I(U*)2

where dfl*_2’~C”’C. 0 — i (1+r~)2

I

‘~b

I

1+~~’(1+Cj)

and dfl0/U2

_~(7* —

u~— aI* ‘V

a~

‘

—

r*

—

i+~’j

first

make a transformation of the

M. Ko et a!., The theory of ~‘{-space

68

Thus /1

dfl*

1

\~~l

(2-3)= ~~8ir~(U*)2) and this integral can be evaluated by substituting C* = e~”cot(O/2),

~‘=

e~cot(O/2) and integrating over

the polar coordinates 0 and q5. The result is 1’=

gat,v’~v”= ‘qahv’~v

(v°)2 (v’)2 (v2)2 (v3)2 —

—

—

(2-16)

and the metric is explicitly seen to be quadratic. This particular form for the metric implies that the choice (2-12) for 1, corresponds to a choice of coordinates in ~‘-spacewhich are orthogonal at P. It is

now possible to obtain the connection and curvature tensor by repeated differentiation and manipulation of (2-3) [15].However there is an alternative, more direct approach which is based on the four quantities Za;

~Za;

~oZa;

(2-17)

~

which, at fixed (but arbitrary) C and ~ are four covector fields on W-space [66]. For notational convenience, we shall omit the commas in (2-17). If we can find all the scalar products of the covectors (2-17) we can explicitly write out the metric in terms of them. From (2-15) and (2-16), since l~and m 0 are null and orthogonal at fixed C and ~ in the metric ~ we have gZaZh

0.

(2-18)

Applying ~ to this we successively obtain

=0

(2-19a)

gab~~fj~,~t,

gab~)Z~Zb =

0;

(2-19b)

with the aid of (2-2), and applying 15o we obtain gabZa

ti0Zt,

=

(2-19c)

0

gab

~~Za ~30150Zt, = 0

gab

~Z0 t50Zt, + g’~Za~i~Zt,

(2~~l9d) =

0.

(2-19e)

Substituting from (2-11) and using (2-14) and (2-15), we find gab

~50Za150Zi,

=

~g”Za ~~~0Zt,=

1.

At this stage we have seven of the possible ten scalar products. To proceed further we define

(219f)

M. Ko et a!.. The theory of p-space gab

~Za

~OZb = _2 ~F(z’,

C, ~)

69

(2-20a)

which implies gab~jZa~‘6

(220b)

0f50Zt, =

gab

150 j0Za lSoóoZb

=

—2 + 2~&~15~.

(2-20c)

—

Given a knowledge of ~ we can therefore construct a null tetrad e~’= {La, Na, Ma, J~a}where tetrad index (js

=

.t

is the

0, 1,2, 3) by

LaZa

(2-21a)

Ma =

(2-21b)

15oZa

Ma = ~OZa + ~15o~Za

Na = Za

—

(2-21c)

~15oZa

+ 15of5oZa + ~

—

(221d)

S~&°~)Za — ~15o~150Za

in terms of which the metric is gab =

2L(aNby’2M(a!11b)

=

(2-22)

rj~~e~’et,~

where

/0 1 0 0 110 0 0 0 0—1 \o 0 —1 0 In terms of the holonomic basis (2-17) this can be written gab = flab

+ 2Hab,

(223)

where flab

= 2Z(aZb) + 2Z(a 15o~JZb)—

Hab =

(~15~&°p,~F)ZaZt, —

—

(2-24)

215rJZ(a 1504),

15OS~Z(a15oZb) +

F150Za150Zt,.

(225)

which we find in subsection 2.3 to have We observe that the metric appears as the sum of a part the appearance of the Minkowski metric, with a part Hab which we shall find is a potential for the curvature. However, we have not yet determined S~,which leaves a circularity in the above definition. From (2-20b) we have tlab,

M. Ko et al., The theory of ~{-space

70

=

(2-26)

~

so we expand the vector ~Za in our basis as oZa

A

1Za +A215~Za+A315t)Za +A415~)Za.

The A1 can be determined algebraically and we find A4 = 2~. In terms of the solution (2-11) to (2-2), the function ~ is found to be 2). = V~(~W + W

(2-27)

Directly from (2-27) or from (2-20a) one can show that .~Fsatisfies the equation =

—215,~&~, = 2~.

(2-28)

This equation will be used in section 4.

For covariant differentiation, we see that a knowledge of the covariant derivative of Za suffices to determine the covariant derivatives of the entire basis by using l5~and 15,,. Denoting the covariant derivative of Za by Vt,Za = Zat,, (2-3) gives 152oZat, = &°

(229)

8Zat,+ tT°BZaZh.

A particular solution of this is

(230)

aZaZb + 2I~Zta15oZb) + y15oZa 150Zt,,

=

where f3

= —~15oy,

a

=

—

and y has to satisfy —

~15o(&~15oy)~y15~á~ + ~y(ó~)2= —

=

If the particular solution (2-30)_actually is the covariant derivative of contracting both sides with ts0z” 150Z” leading to y

=

(2.31)

_Vi~.

rFa,

Za,

we may determine y by (2.32)

which is seen, after some calculation, to satisfy (2-3 1). Taking this expression for y, the general solution of (2-29) is Zab

=

Z~’t,+ ycbZ

(2-33)

M. Ko et a!., The theory of ~‘C-space

where the yCab are independent of C and ~ The the metric to vanish:

7”ab

71

are found by requiring the covariant derivative of

~0

Vcgab

and substituting from (2-23) and (2-33). This leads to C

7

ab

—

—

so that the covariant derivative of

Za

is simply the particular solution given by (2-30) with y given by

(2.32).

At this point, we remark that Penrose’s bracket {Za, Zb}

[15],

2Z

(234)

1a150Zt,i

which by (2-3) satisfies 15o{Za,Zt,}0

is by (2-30) covariantly constant, Vc{Za, Zb}

0.

(235)

From (2-34): {Za, Zb} is a simple anti-self-dual bivector and corresponds (by convention) to an unprimed spinor. Thus (2-35) states that an ~‘-spaceadmits covariantly constant unprimed spinors and, varying C~ (2-34) gives all unprimed spinors. The curvature of 7C-space then necessarily has a particularly simple form, being Ricci-flat with self-dual Weyl tensor. In terms of a spinor decomposition Rabcd =

*ABCDEA’B.EC’D’

+

~~ABC’D’A’B’ECD

+ A (A~QDEA’C.B’D’

—

+

‘~CDA’B’EABEC’D’

+ IIIA’B’C’D’EABECD

EADEBCA.D’EB’C’)

1IA’B’C’D’.

theWhile only non-zero termtensor is ~ may be found directly from second derivatives of Za via (2-30) and the Riemann V[bZc]a =

~Rbca”Zd,

we continue the development in terms of spinors, giving tensor equivalents where appropriate. Since we have a null tetrad e~,spinors may be introduced with the aid of the usual flat space Infeld-van der Waerden connection symbols 0’2AA’. Alternatively, we can adopt the abstract index notation (or “Battelle conventions” [67]of Penrose) and write directly La

°A°A’

Ma

(2-36) Ma = TA °A’

Na =

M. Ko et a!., The theory of ~1t’-space

72

with the normalisations OAI’~= OA,IA’ = 1.

(2-37a)

The bracket (2-34) becomes 2Z~0150Zt,1= OAOBEA.B’

so that by (2-35), °A is covariantly constant. Similarly ‘A is covariantly constant. Since the bracket has SW 1, the correct assignment of spin-weights is ~ to 0,, and ‘A’ and to °A and ‘A~ The linearised good cut equation (2-2) now requires —~

~)0A=0;

15Ad1A’OBOA.

~0A=IA;

(2-37b)

Collecting results from (2-21) to (2-30) we find 1A”1500A —~15~OA’ + ~1A’

=

F

=

Hat,

=

OAOB((~15&0B~)0A,0B,)~O(A,IB,)+~IA,IB,) 2f3O(AIBY~71A’IB’).

Zab

OAOB(aOAOB+

The derivatives of the primed spinor basis are found from (2-30) to be VAA’OB’ VAA’IB’

=

=

OA(aOAOB’ + 2f30(A’IB’) + ~YIA’IB~),

OA((t5a + 2~f3)OA’O8’— 2a0(A.IB.) — 131A’IB’),

r7A’c~ ,,,r7A’T V A L’A’ — V A 1A’

(2.38a) (2.38b)

— —

and LIOA.=LIIA =0,

where

E

—

~7AA’r1 V

The Weyl spinor may be obtained directly from the Ricci identity VAA’V’~’0C.=

ii,, ,B,(,,E,OE

(2.39a)

M. Ko et a!., The

theory of X-space

73

and 1C’ — — ‘PA’B’C’E” TI VIAl’ ~‘ VAA’V B’

as —

_-f’~

*A’B’C’D’

TI

VIA,

y

~,.‘D’VAA’V B”C’

—

~7

~Aç~

ID’VAA’V B”-.’C’.

In performing this calculation we make use of the identity ~ -~a~— I’7aar \ OOL~ ‘7’a — 3OO~~‘pa),

which follows from the definition of ~, (2-20a), and its corollaries, =

=

—Z’2aa,

=

z” (~a+ 2ó~B)a

and the identity (for any OAVAA,fl

(239b)

i~)

= (15~b~)0

—

(Z”fl,b)IA’.

Finally *A’B’C’D’

=

zaYa’A”B’IC”D’ + 4Zf3IJIO + 6ZaaaI(A.IB,Oc,OD,) 4Z” (15 0a + 2&0B/3) aI(A’OB’OC’OD’) + ~a (15~a+ 2(fr~f3)a0A’0B’0C’0D’.

—

(2-39c)

2.2. The potentials

As mentioned in the Introduction, it is possible to find a chain of potentials for the Weyl spinor (or equivalently, for the Riemann tensor). These potentials are constructed from the functions Z, ~ and y introduced earlier and thus depend explicitly on C and C (unlike the Weyl spinor which is, of course, independent of C and C~depending only on position in ~‘-space).The potentials are all defined for one fixed choice of ~ and C~then a change to another choice corresponds to a gauge transformation. As a result of their definitions, each potential automatically satisfies a field equation_which has the appearance of a gauge condition although the only gauge freedom is in changing ~ and C as referred to above. The first potential FAA’B’C’ is essentially the spinor connection in the dyad (OA; I,,.) for a fixed choice of C and C: r’A I

,‘~

VIA

r

j’

VIA

ç~ 1C’

A’B’C’ = — ~JB’V A’IC’IB’V A”0A((15~a+ 2&~f3)0A.0B.0c. —

3a0(A.0Q’Ic’) 3130(A’IwIc’) —

—

YIAIB’Ic’)

74

M. Ko et al., The theory of ~?‘-space

so that

FAA’B’C’ = FA(A’B’C’). ABC’ =

VAD’1

=

Then directly from (2-40) we see

OB,VAD,VAA,IC, — IB’VAD’VA’OC’

OB,,/,A,c,D,E,IB

IB,l/IA,C,D,E,OE’ = ~frA’B’C’D’,

—

(241)

where use has been made of (2-39a) and the observation that ~VI

j’

\IVIA

f’I

~,V AA’(~~’B’)~,VC”D’

—

Again, directly from (2-40) we find VIBC’rA V I ABC’

— —

VIBC’I) VIA r V ~...‘B’V A’1C’

also with use of (2-38) Thus 1AA’B’c’ is a potential for

The second potential for H~A’B’ =

—

VIBC’r VIA ç~ — rA rB C’D’ V LB’V A’~—’C’— 1 A’C’D’1 B’

-

-

and satisfies the gauge condition, eq. (2-42). or equivalently the potential for TAA’fi’C’, is from (2-25),

*A’B’c’D’

*A’B’C’D’,

0A0~~((~15~eJ_ ó°BS~)0A’0B’

—

‘ó0.~i~0(A’I~~’)+ .?JIA’IB’)

(2-43)

which is the “non-flat” part of the metric in the splitting (2-23). To see this, direct calculation from (2-43), using the identities of (2-39b), leads to VI

y.j’AB _rA V BC”1 A’B’ — 1 ABC’

-

as required and also to the gauge condition on T7C’A’ ~jAB V

—

C’ rC’A’

l’JAB

A’

A’B’11

II

HABA’B’,

I

-

BC’.

For the third potential we define ~ABC

=

~000(15O15

Z0A’

—

15~ZIA’).

(2-46)

Using the identity 15~oZa=

~a

Z” 15o~Za 15o~

(2-47)

which is obtained from (2-20), it can be shown that Ti — ~jAB VCB’Y ABCA’11 A’B’, VYA’D v

whence

ABC

7

YABCA’

—

A’

—

~ ~jAB ~2.U1

-

yj’CDA’B’ AR’T1

is indeed a potential for

—

HABA’B’

and also satisfies a gauge condition, (2-49). For the

M. Ko et a!., The theory of k-space

75

fourth and last potential, we remark that it is possible to find a function 9(Z’2, C~~) such that =

~,152 0Z;

~bØt,

=

~15O15~Z,

(2-50)

or equivalently 28A’ø

2OAVAA,ø

=

~3~OZ0A’

—

~OZIA’

(2-51)

where —

0A’~-”

VAA’

since the integrability condition for (2-50) is just (2-47). Further it is possible to show that, by manipulations of (2-51), 0 automatically satisfies the equation E149

=

“ÔA’ t5B’OSSO.

(2-52)

The fourth potential is now defined to be ØABCD = ØQAQBQCQD

(2-53)

so that (2-51) and (2-46) imply ri ~ABCD VA’D~J

—

7 ABC A’

-

and from (2-52) El ØABCD = _H~~A,E,H~’R’.

(2-55)

Collecting these results on potentials together we have Ti £~ABCD VA’DrJ

—

7 ABC A’,

—

VI ABC — LrAB VB’CY A’~1 A’B’,

-

VI rj’AB VBC’.fI A’B’

-

— —

rA 1 A’B’C’,

VAD’FA’B’C’ = ~A’B’C’D’

(2-41)

together with gauge conditions on the potentials,

o

ØABCD =

— !tj’AB 7 ABCA’2’1

T7A’B

v

_HABA,B,H~~’ YJCDA’B’ A’B’El

(2-55) -

M. Ko et a!.. The theory of ~/!‘-space

76 V7CA’LTAB A’B’

11

~‘

rIBC’rA V 1 ABC’

~jAB

— —

1~

— —.

~

rA

A’

C’rCA’ I BC’,

—

1 rB /3’ CD’ .

-

A’(”D’

We may introduce tensor equivalents of the potentials by @ahcd = @ABCD~A’B’~C’D’,

(256)

Yahc =

(2-57)

Hat,

YABCA’f BC’,

(2~58)

= HABA’h’,

“abc = 1AA’B’C’~BC,

(259)

Rat,cä = ~IA’B’c’D’~AB~c’D,

(2-60)

and write out the tensor equivalents of the spinor potential equations. However the first potential has a more direct interpretation in terms of the null tetrad eBa, namely I-’

cab

Mv

—

a

chM

L~VcN~, + NaVeLj,

—

~

—

MaVcMt,.

(261)

The usual definition of Ricci rotation coefficients is e’/3e/3VCe~ so that

is a tensor whose components in the null tetrad are just the Ricci rotation coefficients:

1ahc

a at,ce

,

1e ~e,. is skew on the last pair of indices and further, since the anti-self-dual bivectors constructed from —

Tcab

the tetrad are covariantly constant, is actually self-dual on this pair. The Ricci identity becomes V[dFc]aj, + IfcIeaIFdlCb

=

‘~Rabcd

(2-62)

which then splits into self-dual and anti-self-dual parts as m ~V[d1

r

In

\±

dab)

(V[dlc]ab)

—

“21’abcd

= 1[d~eaI1c]b.

(2-63)

The tensor 1’Cab has a further interpretation as the difference between two connections: Given a null tetrad, one can define a flat connection which preserves the metric (and so necessarily has

M. Ko et a!., The theory of ~‘C-space

77

torsion) by VaeTMb

0.

=

(2-64)

From the definition of 1ab~, we also have

M

Ti Vae

_rc b1

M b ae

so that for any covector

Va

=

vMea,

F

VaVi,

Thus

“ba

VaVb

=

—

(265)

Fb’aVc.

is the difference between the flat connection and the metric connection.

The next potential equation can be written Vit,Hcia

—

He~Jciae

~Fat,c

(266)

which also splits as ITI

LI’

tr

\±

!,,V[bIlc]a)

‘‘21abc

(V[bHcla)

= HcEJ~]ae.

(2-67)

The last two potential equations seem to have no simple interpretation. They are —

V

~-“abcd —

—u

Tic ‘

Ycab

Yabc

ab

with corresponding gauge conditions E 0abcd

=

(VICydlbC)

HacHbd + HadHbc =

‘~(HacHbd — HadHbc).

2.3. The coordinate systems of Plebanski

In a recent paper [23],Plebanski reduced the complex Einstein equations for self-dual curvature to a single second order non-linear equation on one function of four complex variables. He does this in two different ways by making specific choices of coordinate system adapted to the left-flat geometry and obtains the metric in two canonical forms each depending on one function. While, as remarked in the Introduction, it is not clear which left-flat spaces are IC-spaces, it should be possible to express the IC-space metric (2-23) in Plebanski’s two forms. We do this by choosing coordinates appropriately and thereby obtain the expressions for the metric, curvature and potentials in particularly simple forms.

M. Ko et a!.. The theory of ~C-space

78

(Plebanski’s remaining equations are for us identities.) This yields some new insight into Plebanski’s 0

function. Likewise in section 6 new insight into his fl-function will be gained. For Plebanski’s second form we choose as coordinates, for fixed C and C p=Z;

q=—15

0Z°

x=Z+15ot5oZ;

y=~Z,

(2-77)

then from (2-36) et seq., 4V,,,, = —0,, alay IA’ a/ax.

(2-78)

—

0’ The chain of potentials become 0ABCD = OA°B°C°D0,

OAOBOC(O.yOA’ +

YABCA’ =

HABA’B’ = °A °B

OJA’),

(O,yy°A’OB’+ 20,xyO(A”B’) + O,xxIA’IB’),

(2-79) TAA’B’C’ =

1/IA’B’C’D’ =

0A(O,yyyOA’°B’OC’ + 30.XYYO(AOBIC)+30,xxyO(A’IB’IC’ + O.xcxlA’IB’IC’),

0 .yyvv°A’°B’°C’°D’ + 40.xy%’v°(A’°B’°C”D’) + 60,xxyy O(A’OB’IC’ID’)+

40,xxxy°(A”B”C”D’)

+

0xxxx’A”B”C”D’~

In particular, this gives the metric in the form ds2 = 2 dp dx +2 dq dy + 2(0,,,,~,dp2

—

~

dp dq

+

fr~.

dq2).

where the first two terms correspond in the splitting of (2-23) to the assertion that flab is a flat metric. The equation on 0, (2-52) becomes 0xx0yy

—

~

)2

—

—

0qy =

flab

(2-80) and the rest to 2Hab. This justifies

0

which is Plebanski’s second Heavenly equation, but for us is an identity. A different choice of coordinates on ~1C-space leads to Plebanski’s first form of the metric and is useful in the twistor picture. Define p = Z(C 0, Co), q = —151~Z(C~,, Co) as before, but now define the two other coordinates as r=

Z(C0, i,);

s

=

—150Z(C0, ii),

that is, as Z and —150Z at the same fixed Co but a different fixed ~,. A convenient notation for the coordinates is to reserve Z and 150Z for quantities at (Ce, ~,) and write Z, 150Z for quantities at ~ ii).

M. Ko et a!., The theory of X-space

79

Then p=Z;

q=—~Z

r=Z;

s=—150Z.

(281)

From (2-18) and (2-19) all four coordinates are null and, further, p and q are orthogonal, as are r and s. The metric takes the form t, = 2(g~.dp dr + g~ gab dza dz 5dp ds + gqr dq dr + gqs dq ds), where -

~a

150Z

-

_Za2a

Z”15O2a

,

gq,—

150Z”Za

g~—

,

gpr—

gqs—

1 = {Z”, Z~H~a, Zb} (282) 4 = (g,,~g~5qpsgqr) so by (2-35), 4 is independent of position. This has Plebanski’s first form if we can express the remaining metric components as partial derivatives of a single potential, 11. To this end, we express the vector 15oZa in our basis as —

SZa = IT 1Za

—

1T2’óijZa + 1T3.~a

—

1T415t2a

(283)

or d(15Z) =

i~ dp

+ IT2 dq + 1T3 dr + i~ ds.

Contracting (2-83) successively with za and ~ 1r3=gqr;

and using (2-18) and (2-19) we finally obtain

1T4=gqs

so that gqr = (~oZ).r;

gas = (~o0Z).5.

In a similar, we consider 150~Z,15~2,15o~to obtain gqr =

(~0Z).~ = (15o~2),q,

gas =

(~0Z).5= (~~2),a,

gpr =

(15o~xjZ),r=

gps =

(150~Z).5= (~o2).p’

~

(284)

M. Ko ci a!.. The theory

80

of ~‘(-space

and these are just the integrability conditions for the existence of a single function 12(p, q, r, s) with g,,~=

gp.s

2,pr,

‘

=

12,,,., (2-85)

gq~=

gqr = 12.qr’

Further, by (2-82) 12.pr.12,qx

—

12,ps 12,ar =

-

is constant. We may now choose C~relative to so as to make 4 be unity. This then satisfies the last integrability condition for making the identification ~

Z + I50tSOZ

12,,,

~5oZ = 12,q.

(286)

In conclusion, the metric is gab

dza dz” = 2(ul,~,dp dp + fl.,,~dp ds + flat dq dr + 12,qr dq dr),

(2-87)

with ul,p011,qs

—

11,ps12,qr

1.

2.4. Specific examples of ~1(-spaces As a conclusion to this section, we present a solution of the good cut equation (2-1) for a particular

shear. This solution, first found by Sparling, was the first non-trivial explicit example of an ~‘-space. We begin by reviewing how the ~‘-spaceconstruction leads to flat space in the case of vanishing shear, o~ 0. The good cut equation for this case is 15,~Z=0.

(2-88)

Substituting for 15 from (1-8) in (2-88), we fInd =

~(~) + ~

(2-89)

I + CC

where p and q are arbitrary functions of ~i If Z as given by (2-89) is to be defined as a regular spin weight zero function over the whole of the Riemann sphere of C when C = C, then p and q are necessarily linear in ~ p=u+Y~

q=X+v~

which gives the four complex parameters (u, X, V. v) on which the good cut function depends.

M. Ko eta!., The theory of Xspace

81

Thus Z

=

Z~!a(C,~),

(290)

(u, X, Y, v),

(2-91)

where =

(a

~l+Cj(l~C~C,C~).

(292)

We may proceed in various ways to calculate the metric. The simplest is to observe from (2-90) that =0 so that, from (2-26) ‘=0. From (2-23) this gives the metric gab =

2ZaZb + 2Z(a 15

(293)

015O.Zb) — 2150Z(a 150.4)

which is independent of position since, by (2-90), Z is linear in the coordinates. In fact, (2-93) yields gab

dzz dz”=2dudv—2dXdY,

which is the Minkowski metric in null coordinates. Sparling considered the following form for o~[28] o

Af(C)

—

—

- 2 -

a2

Y2m u~(C,

h

u~(1+CC) where f(() is a quartic polynominal in ~ and the

-9 2Y2m are spin weight 2 spherical harmonics. While the

solution can be found for arbitrary f, it is simplest when 1(C) is a fourth power and without loss of generality we may then take f to be unity. The good cut equation becomes (2-95)

and the solution to (2-95) may be written 2 + As2, =

z

(2-96)

M. Ko eta!., The theory of ~‘t’-space

82

where

s=s~(C,~), (u, X, Y, v),

=

~a

=1(Y,v00)~

8= uv—XY

j)

and (a (C~ is as in (2-92). Sparling has calculated the metric by twistorial techniques from (2-96) and finds ds2=2dudv_2dXdY_~(Ydv_vdY)2.

(2-97)

We remark that this is in Kerr—Schild form [45]and thus the curvature must be algebraically special. The simplest way to find the curvature is to remark that with the correspondence u=x;

X=—y

v=p;

Y=q

the metric (2-97) is in Plebanski’s second form [23]

ds2 = 2 dp dx

+ 2 dq dy + 2(0~~ dp2 — ~

dp dq + 0,,,,. dq2)

where 0=

—A 2(,px+qy)

—A

28

It then follows rapidly from Plebanski’s work [23],that the curvature is type N. Further, the space is non-singular everywhere except at 8 = 0, which is the light cone of the origin. Regarded as a linearized field on Minkowski space, the metric (2-97) is one of Penrose’s “elementary states” [29].

3.

~a2

and ~

spaces

Three natural questions about ~‘-spacesthat can be asked are (1) Given an IC-space (arising from a physical space-time M) denoted by ~C(M), it is possible to find ~t°of ~‘(M)or ~‘2(M)?(2)_What is the relationship of X2(M) to ~‘(M)?(3) What is ~‘~M)? We remind the reader that ~‘ and ~‘-spacesare obtained respectively by asking for cuts of CI~such that o-°= 0 and ê°= 0.

M. Ko eta!.,

The theory of i-space

83

The answer to the first question is in general “no” since 7C-spaces do not in general have a CI~. However for those special IC-spaces which are in some appropriate sense asymptotically flat, the answer is yes. The difficulty is in defining the appropriate sense. In subsection 3.1 this question will be

discussed. In subsections 3.2 and 3.3 we will show that, when the 7’-space is asymptotically flat, ~‘2(M)= and ~‘X(M)= complex Minkowski space. 3.1.

Asymptotically flat IC-spaces

The purpose of this subsection is to define those ~C-spaces which can be called asymptotically flat. This will be accomplished, in a slightly roundabout fashion, by first defining asymptotically flat left-flat spaces and then showing that each of them is an X-space. (Note that, as remarked in the introduction, though all i-spaces are left-flat, not all left-flat spaces are necessarily YC-spaces.) An asymptotically flat left-flat space is a left-flat space that has a real slice R (by a real slice is meant a real four dimensional submanifold which in some holomorphic coordinate system (Z~z)is given by Z’~

real) such that (1) R is difleomorphic to (real) Minkowski space. (2) The metric gab is holomorphic on R and in a thickened region, YR, around R; R which is not unique, can be moved around in this region. (3) There is a conformal compactification in the sense of Penrose (see section 1) of R with .r(R) as the boundary and CI~(R)as the thickened boundary. (4) Within the light-cone of every point p of YR, there is a sphere worth of preferred geodesics which intersects CI~(R);further there is a neighboring point of p such that the two sets of preferred geodesics do not intersect. The later condition introduces the idea of a “time-like” world line in R. Basically what we have attempted to do with these conditions is to mimic for the left-flat spaces the conditions needed in physical space to obtain the I~and Cit The difficulty is that there is no conformal compactification for the entire complex manifold and we are forced to formalize the idea of a region for compactification. Though the conditions we have given here are sufficient for our needs, it seems almost certain that they are far stronger than is really needed. Furthermore the ones that are needed probably can be reformulated in a more economical manner. We, however, have not been able to see exactly how this should be done [68]. 3.2. v-spaces

We will now study the properties of asymptotically flat left-flat spaces (in the YR region) by means of the spin-coefficient equations [53]. From conditions (3) and (4) and the meaning of conformal compactification (section 1), we can introduce an NU coordinate and tetrad system based on a “time-like” world line L in R, r being the affine length on the generators of the cone emanating from L (r =0 at L), u labeling the null cones or cuts of CI~(R)and C and C labeling both the generators of the null cones and the generators of C5~(R).The coordinate ranges are: the real part of u and r go from to +QO with a small thickening around zero for the imaginary parts, C extends over the completed complex plane with C being close to (or thickened around) The tetrad vectors are the same as in (1.19) (fig. 1). We now consider the spin-coefficient version of the complexified Einstein equations in which all barred variables are replaced by the independent tilded variables (e.g. o —* ê) and all variables are —~

~.

M. Ko et al., The theory of ~-space

84

/

/ // /

/

/

~/ 1/

\ \~-~ ~u=O

~

and

\~/

Fig. 1. A good

\\ ~

II ‘\\\

/1

“

affine length along generator

(o’°= 0) nuH cone and its associated good cut in p-space.

considered as holomorphic functions of u, r, C and C~The condition of left-flatness (C~b~d = ~Cabcd)is equivalent to the vanishing of IIIABCD or ~ ‘J’l, çl’~,~ and çfr4. Though one could write out all the spin-coefficient equations (the number of equations is doubled because the tilded variables satisfy independent equations) this is a tedious and unnecessary task since these equations have appeared frequently in the literature [5,7]. When needed in our argument a few specific spin-coefficient equations will be explicitly given. Since the vector l’~is the tangent vector to the generators of the cones from L, the NU tetrad (by an argument almost identical to that used in the real case [5])can be chosen such that the spin-coefficients satisfy 5 ‘r=a+f3; ~=&+$ p=j (3.1) K = K = S =

= IT = IT =

0.

We will now concentrate on a particular cone, C (labeled u = 0) coming from L. By giving appropriate data (a single holomorphic function on C) all the spin-coefficient equations on C will be integrated. A final equation then yields the propagation of this data from null cone to null cone along L and thus a solution will be determined by the original data. From the spin-coefficient equations (on u = 0) [5], Dp=p2+u6,

Dc~2pr,

M. Ko eta!., The theory of 7t’-space

with the fact that at r =0, p p~5——1Ir;

85

—hr and r —~0 (see (1-20)) we obtain immediately (3.2)

o’~0.

It is this simple but very important result that permits the exact integration of all the equations onC. ê, which does not vanish, except at r =0, satisfies Dê2pê+ifro

(3.3)

so that e=c+4Jr12~odr1

(3.4)

and hence we have =

J

r’2

(3.5)

t~odr’.

We have the first example (of many) where, when an equation is solved, the “constant” of integration (ê°in this case) is uniquely determined by some knowledge of the variable at r =0. (This effect does not occur for the real case because r =0 is not in general a well-defined point, either it is not in the allowed coordinate range or the fields are singular there.) The integration of the remaining spin-coefficient equations is straightforward and the results are: spin-coefficients p=i5=—1/r, 0~=0,

(3.6)

13

=~-~lnP+~—15lnP [!Jr12

ciodr’_J r’ iiodr’]—~—Jrbodr’+~JrI315c&odT1,

M. Ko eta!., The theory of ,W.space

86 r

a

r

i/odr’]~

—~-~6lnP+~-15lnP [Jr’ ~odrI_.~Jrl2

=

I)

a

=

—

2r

15 ln P, -

r

1I~

~

r

r

1 0

r

r

P

1J152~

1

T

J2

r

—~--~

152~Iodr1+~JrI3152lIIodr1,

1)

0

r

r

A

lnP [Jr’ =i{(l5 lnP)2—~2lnP—15~

&dT1_~Jrl2

~odr’]}~

0

-

=0, r =

—~15~ lnP+~_Jr’

rJr12152&dr1+~Jr13 152~OdrI

152~odTI

o

o

r

~=5~(~)+~lnP

r

[~J15cIodT’]+15~lnP[—~fT’15&dr’+~-sJr’315lfrodr’}~ 0

r

‘F’

1

r

~

1

0

J

J

1)

0

r

1r’ 15~ç&odr’—~-~T~215~çfiodr’+~ 2rJ 0

153~iodr’;

T~3

tetrad variables (with tetrad as in appendix 1) =

(—2P,

0)/r,

[J‘/~ r

=

(0,

2P)~(‘2P,

0)

r’

I

dr’

Jr12 i~odT’]~

—~

0

(a)

r

r

—

~ —Jr1

15çfrodr’

—~

Jr12

15~

0dr’+~ Jr’3

0

r

XA =(_2P,0)[_~ ~

(37)

15*odr’,

1)

r

2r 0

U=

~[L~f

152~drI]

r

15~lnP

+~J r’

152~

dr

1

r ~JrI2

152~ 0

dr’ +~Jr’3 0

152çj~~TI;

M.

Ko eta!., The theory of St-space

87

Weyl tensor =

~o(r,C~ = O(r’5), data on u =0,

j)

=!~4?_~J r13 15~ 0dr’,

2 15~dr’,

I~1

*2

=!~T_5fr’

(3-8)

with (from the condition that the i/’s be regular at r =0) i)?=—615~2Y,

~=~Jr~&dr;

(3-9)

i~=_2~2~~,

~~Jr2’~odr;

(3.10)

i~=-~15~, ~=Jr~odr;

~=—~t5~

(3-11)

c~=—Jiiodr;

(3-12)

where ~, ~, ~ and ~ are functions of C and ~ as well as of the apex of the cone, and are the same as ~ and ‘~of subsection 4.3. Finally, if we include a left-flat (4~ = 0) Maxwell field (the coupling to the metric is only one-way since the stress tensor of a half-flat field vanishes) we have =

~i

t~o(r,~, j)

0(r3), data on u =0,

=~—~rJ r~~odrI,

~?=—15.~,

.~t Jr’*odr’, 0

~

2ç2_~J~1d~,

(3-13)

~=_Jccodr’;

~

of

with .s~iand ~ being the Maxwell potentials subsection 4.4. We see from (3.6)—(3.12) that all the information of the gravitational field on the cone u =0 is

M. Ko et al., The theory of 5-space

88

determined by knowledge of çfro(0, r, C~~), P(0, 1) and I~(0,~, ~). We will now show that P and P are determined modulothe choice of worldline through the apex, i.e. thereis a fourparameter choice of P corresponding to the tangent to the world line, and a four parameter choice of P corresponding to the ~‘,

acceleration of the world line. First note that P plays essentially the role of the conformal metric on Ci~(R) d~2= —(2du d~+ d~dJ)

(3-14)

where P = —l/T, and that it can be written P= VP 0=~V(1+Cj).

(3-15)

If the metric on C, which is an explicitly known function of r, is expanded in powers of r around the apex, or r =0, we obtain T

2[_15~lnP

=

g’

g~’ 2P [~15~

r(~_~ 152Q)+

~

0(r2)],

+ 0(r3)1, (3-16)

r2g”

=

=

4p2

8P2~+ 0(r2).

The line element on the surface u = 0, r = constant (small) (which is a sphere in the limit r —~ 0) is thus ds2 =

—

r2d~dj

r2

~d(d~

(3-17)

If we seek a transformation of the coordinates in (3-17) of the form =

q(C, ~)

=

W(C, ~) (3-18)

-

ñ

=

such that ds2=_T~J~1,

we obtain the equations

1’,, ~(1+3p~),

M. Ko eta!., The theory of 7-space

V2 = 1+ 150W,

89

(3-19) ~3-20

1+150W

—

which defines V from ~(or ~ (Equations (3-19) and (3-20) could simply have been hypothesized but in the absence of our derivation an existence proof for solutions would have been difficult.) We now prove that the general V derived from (3-19) and (3-20) has the form VZaVa

(3-21)

and further that there exists a function S(C, ~) such that (3-22)

15~Za= SZa.

We know from the existence of solutions to (3.20) and (3.19) and from condition (4) that there is a W0 which yields a regular V0. It is easily checked that

w~’~’°~

323

mW0+l

is also a solution of (3-20) with 1!a(C,C)v12,

(3-24)

m

=

15~l,

(3-25)

in

=

~ol,

(3-26)

1+ ~

(3-27)

n

=

where 1,, are four independent solutions of 15 Ua =0 and v°are four complex constants. This freedom to choose the va is essentially the freedom inherent in the original choice of ~ and C to label the generators of C. From (3-23) and (3-14) with VO~= 1 + 150W0 it follows immediately that V

V0(l+ Wom)

Vo(ta + W0150!a)V”

(3-28)

or V = ZaVa

where Za = Volja +

W01501a).

(3-29)

M. Ko eta!., The theory of 5-space

90

By defining S= it is easily seen that 15 2oZa

=

(3-30)

SZa

which concludes our proof. Finally, it is possible to show that V and hence P can be determined (by differentiating (3-19) and (3-20)) in terms of the acceleration of the line L i.e. in terms of t3”. We thus have the result that all quantities on C are determined by ~I,0,t~ and v” To move off C the last of the Einstein equations yields a propagation equation for 4”o, i.e., an equation of the form, =

(3-31)

~

for which the right hand side is an extremely complicated functional of çt’~whose details are irrelevant for us. This completes the proof that an asymptotically flat left-flat space is completely determined by a single function 9~fo(r,C~C) and the choice of a world line. Note in addition that the ~, ~, 3~and ~ functions will be defined at the apex of every cone and therefore are functions of Z’1, the coordinates of the apex, and C and C. These functions have the strange dual role of being both direction dependent local functions and giving as well, by eqs. (3—9).-(3-12), the asymptotic values of the Weyl tensor in all null directions from the point z°.We will now show another interesting feature of these functions. If we define, analogously to (3-9)-(3-12) 9~(ro)=_~J(ro_r)3~

0dr,

TI)

=

~(ro) =

—~

J

To

—

J

(r0

(r

0 r) —

=

2 ~ dr,

—

r)

& dr,

(332)

—J ~

dr,

= “~,&o(T~, C~j),

so that ~(0) =

~,

~‘(0)= ~ etc., we have with ô/9r0 =

~

3/sf’ (and evaluated at r0 = 0)

M. Ko eta!., The theory of 7-space

=

21~a1”,

~

91

=

(3-33)

~al,

~

we

To conclude this subsection define an asymptotic shear o~associated with a Bondi slicing of Ci~(R)and show that the metric of the u-space derived from that o~is the same as the metric o! the asymptotically flat left-flat space we are dealing with. Since the Bondi o~could have been used as the data in a physical space-time M to yield the asymptotically flat left-flat space we will have proved that (a) an asymptotically flat left-flat space is an X-space; (b) X~=X. If we construct an asymptotically flat left-flat space by the methods described earlier in this section using an NU coordinate system based on a “time-like” line L through YR we obtain a one parameter slicing of C7(R) labeled by the proper-time u along L and a uniquely defined V(u, C~C) and hence conformal metric on C7(R) d12= _(2du dP+~,~). It is always possible to introduce a Bondi slicing of C7(R), UB =

G(u, C~~)

(UB =

constant) by

(3-34)

where G satisfies G = ~9G/êu= V(u, C~~).

(3-35)

This condition is the complex version of the usual method [54],of obtaining the conformal factors of a new slicing (see section 4). (The solution to (3-35) can be made unique by requiring 0 = G(0, ~, j). Doing this is sometimes useful but by no means important. The general solution is obtained from the unique one by supertranslations.) Since the slices u = constant are (by construction) shear free, (3-34) can be thought of as a one parameters worth of solutions to the good cut equation (3-36) where the o~B(uB,~, ~) is calculated by applying ~ to (3-34) and afterwards replacing the u by u

=

G’(uB, C~C), i.e., if OG = g(u, C~C) then (T~(UB, C~~) = g(G”(uB, ~, ~), ~, ~).

(3-37)

By our method of construction the points of the asymptotically flat left-flat space correspond to the good cuts of (3-36). Furthermore, it is easy to see that the S of (3-30) is o~(G(u,C~C), C C). By using (3-30), (3-19), and (3-20), it is trivially seen that the ~ constructed from the solutions of (3-36) and that given by —J~rtfr~dr are the same. From this and (3-33) it follows that the Weyl tensors and hence the metrics are the same. This concludes the proof. Notice that we have proven, in addition to our theorem, a rather attractive reciprocity theorem,

M. Ko eta!., The theory of 5-space

92

namely that the functions ~, ~, ~ are the local potentials defining the local metric and curvature properties of ~‘-spaceand also, essentially, the asymptotic curvature components. An almost identical result is true for left-flat asymptotically flat Maxwell fields on an asymptotically flat ~‘-space.From (3-13) and ~ have a dual role as local direction dependent functions and as essentially the asymptotic values of the Maxwell field. By an argument identical to that leading to (3-33) we have .~‘

(3-38) 0, C~C) = ~,a1.

=

If we define the vector potential =

15

0PA1a

—

(339)

~

it is easily shown that ia~ib_1i

!7

vEbaala

Ut

—

—(po(~T—

and that hence Fab = ~(z’~,

V[baa].

C~~) is thus (via

(3-39)) the vector potential of the Maxwell field with the choice of C and ~ being

the gauge freedom. One can, in addition, show that 11[a150’bJ = S~iOAOBSA’B’ °ab

= .9

is the Hertz potential for the field. 3.3. ~W = complex Minkowski space The proof that the ~‘-spaceconstructed from an asymptotically flat ~‘-spaceis Minkowski space is quite simple. Using a Bondi coordinate system in the neighborhood of Ci°(R)and remembering that the

i~çvanish

=

because of the left-flatness, one has immediately from (Al—13) that

0.

From this it follows that

o~=o~(C,j) and that by a supertranslation one can always find a Bondi system such that

o~=o.

(3-40)

M. Ko eta!.,

The theory of 5.space

93

The good cut equation for IC-spaces ~2

0Z =

0

then yields, from subsection 2.4, Minkowski space. Combining these two results, we see that the IC-space of an ~C-spacewhich arises from a physical space M and is asymptotically flat, is itself completely flat. Identifying the future null infinity C7(~’) of the ~‘-spacewith the physical future null infinity C7(M), this gives a Minkowski space of preferred cuts (but not good cuts) of C 7(M). In this way, the space M has a canonical Minkowski space associated with it, which was one of the original goals of this study. This point is considered further in section 7.

4. The intrinsic IC-space formalism

For the most part, the results of this section are the same as in section 2. The point of view here is somewhat different, however, in that emphasis is placed on the behavior of quantities under conformal rescalings of complexified null infinity and under coordinate transformations of k-space. This point of view leads to the introduction of quantities and operators more intrinsic to the geometry of v-space and, in many cases, better adapted to computation [33].Furthermore, this new formalism lends itself more easily to many applications, including equations of motion. 4.1.

The new formalism

In section 2, extensive use was made in the physical space of Bondi coordinates on C7 to discuss ~‘-space.These, however, are clearly irrelevant to the geometry of 7C-space; more intrinsic coordinates on C7 would be helpful. In this section we shall introduce coordinates based on families of curves in IC-space, which lead to considerable simplification in subsequent calculations. A world line Z~z(u) in ~‘-spacedetermines a one complex-parameter family of good cuts, given by UB =

1).

Z(z’~(u) C~~) = F(u, C~

(4-1)

In a neighborhood of C7, we introduce coordinates u and r by setting UBF(U,C,~),

TB—

VI~ 2, so that the line element on C7 becomes

where V = E; and rescale the metric by the factor V di2 = V2 d~=

—

(2 du dP + d~dC)

(4-2)

where P = VP 0.

(4-3)

94

M. Ko eta!.. The theory of 5-space

Since, by construction, the u = const. cuts are shear free, the coordinate system (u, C~~) on C7 will be referred to as a shear-free NU (SFNU) system. We have du = V’ duB; thus the quantity La in the new system corresponding to Za ~5 (4-4)

La = V’Za.

That is, if u = u0 is the cut corresponding to U

Z” (uo),

the cut corresponding to

Z’~(uo) + dz’~is

given by

= Uo+La dz”.

In view of (4-1) we have V=F=Z0f

where

V’~=

1a

(45)

ZaVa

and dot means u-derivative. From (4-4) we obtain

LaVa = V_lZaVa =

1.

(4-6)

Since Za satisfies =

(4-7)

U°BZa

we have for V,

or t3~V/V=o-°B.

(4-8)

Substituting (4-8) and (4-4) into (4-7) we get lS~(VLa)

(15~V)La,

or, equivalently, from eqs. (1-8) and (4-3), 2La 0.

(4-9)

15

Equation (4-9)is equivalent to eq. (4-7) in terms of our new formalism. The quantity &~has been coded into the edth operator. Defining dA2 = da 2, we see that the u-space metric, defined by (2-3), can be written as 0/4irV a b II dfl \—1 gabu U = ~j 2(LaU1~)2) (440) It is convenient at this point to impose the condition line.

VaVa =

2 on the parameterization of the world

95

M. Ko eta!., The theory of 7-space

Thus dfl \—‘ = If 2 = gabv a V b = II 2(LaV~~)2) ~j dfl\’ ~)

(~J

(4-11)

,

or J di2=1.

(4-12)

Differentiating (4-11) with respect to v b, we obtain Va

2

J La dfl.

(443)

This gives a relation between Va and quantities defined on C7.

-

In addition to the quantities V, La and dD, we also have the quantities ifr~,~ ~, ~t and ê° defined on each u = constant cut of C7, or, in other words, on the cuts of C7 defined by the world line. In order to develop the intrinsic formalism further it is necessary to define these quantities not only on the cuts associated with one single world-line but on any good cut of C7 (point of a”). We achieve this by using a space filling congruence of world-lines rather than one single world line. A point z° of ~ determines a good cut of C7, and the tangent vector (VIZ) to the world line of the congruence which passes through Z’~ determines a unique scaling on the cut and hence a NU tetrad system on the cut. Thus, given a point Z” together with a vector V”, the above quantities are defined uniquely the cut of C 7 associated with Z”. They may therefore be written in the functional form (z a~C1 ~ V”), where z a corresponds to the cut of C7 on which i is evaluated, and v” is tangent to the world line which passes through Z’~. Let us now investigate how these quantities transform under a change of congruence equivalently, a change in the v” field. Consider a change in the coordinates on C7 associated with a change from t~’(Ut) where z”(O) = z*~(0).On C7, the transthe world between line Z” =the Z”(U) to ut the coordinates world line will Z’~=have Z formation u and the form

on

,~

or,

u*=G(u,C,j) where G(0, C~~)=0. We have du duldu*

=

La

= La

dz”; thus, at the origin cut, we have

dxaldu* = LaV~~* = O_l.

From the discussion of conformal weight in subsection 1.1 it can immediately be seen that

P*=OVP t Since we are now in the complex domain ~1s~ and ~ etc. are no longer complex conjugates of each other except on the real It

96

M. Ko eta!., The theory of 5-space

di’l* =~2d11 o

*

— —

°1r3, 0

at the origin cut, where L~=

V=LaV~~*=

G’. Since du*

=

O du, we also have

~1La.

The simple behavior displayed by these quantities under a change of vector field v a suggests the introduction of a general definition: We say that quantity 7~(z”,C~C~Va) which transforms according to ~*

(zaCjv~~)=~mn(za,C,~va)

~

has ~‘-conformalweight (HCW)w. Obviously any quantity with CW w also has HCW w. However, since our transformations are now restricted by the condition 1521V = 152(LaVa *) = 0, the converse, in general, is not true. By virtue of the fact that 152 ‘V =0, one can prove the following important theorem: Theorem 1. If ,~has SW s and HCW o.,, where w s, then 15~0_~1~has HCW s 1 and, of course, SW —

w+1.

Similarly, if a quantity e, with SW w + 1 and HCW s 1, can be written in the form —

= 15w—s+1~

then ~ has SW s and HCW w. The proof of Theorem 1 is straight-forward and will be omitted here. The demonstration uses an induction argument analogous to that employed in the proof of a similar theorem in [6]. Quantities with well defined HCW will have a central role in what forpreferable, such quantities depend 1. Itfollows, would be of course, to in as simple a way as possible on our choice of the vector field V’ circumvent entirely the need for a preferred vector field, since the vector field is unconnected with the geometrical quantities of interest; unfortunately, this does not appear to be possible within the framework of our formalism. To use quantities having well defined HCW seems the best possible alternative. We conclude this subsection with a discussion of the spinor decomposition of La. Since Za can be written Za

=

OAOA~

we may write La

(4-14)

= °A°A’

where =

V”20A

and

0A’ =

V1120A’.

(4-15)

97

M. Ko eta!., The theory of 7-space

From this it can be seen that

°A

has SW ~and HCW ~, and

°A’

has SW

—~

and HCW ~. In terms of our

formalism equations (2-3m) become (4-16)

and (4-17)

Note that Theorem 1 guarantees that equations (4-16) and (4-17) are ~°-conformallyinvariant. Two other spinors

LA

and

LA’

are defined by

LA—150A

(4-18)

LA’~~’6OA’.

(4-19)

and

It is easily checked that neither LA nor LA’ has well defined HCW. The combinations OALA and OA,LA, however, have zero HCW and are therefore invariant. Since OAIA = OA,IA~= 1, it is easily checked that OAt = OA’L = 1. Thus =

@-20)

2O[A’LB9,

=

(4-21)

20[A’LB’],

where gab

=

(422)

6ABSA’B’.

Another spinor quantity which will be found useful in subsequent sections is ~OA’.This quantity has SW ~and, after a short calculation, can be shown to have HCW The results of this subsection are conveniently expressed in the following table: —

—

sw

HCW

2—n n—2

—3 —3

q5~

1—n n-i

—2 -2

dfl

0

Quantity

2

0 OA 0,4

—

1

1

1

2

2

1 2

1 2 1 2

32

98

M. Ko eta!., The theory of 7-space

4.2.

The ~-confonnallyinVariant operator, J) a

In this subsection we investigate the action of the covariant derivative operator Va on the quantities defined in subsection 4.1. Since Va does not preserve HCW (that is, if ~ has a well-defined HCW, Va?? in general does not) we introduce the differential operator J,,,, defined by J’a’q = (Va + W PaIP)

(4-23)

~

where 0 is the HCW of ~ and Pa

=

V~P.Under a change of the vector field

V”,

we have

and thus p*/p* where

Va

=

‘VaIV+PaIP

=

VaV.

(]‘afl)* =

We therefore have

{Va +w(~+~)}V’°i~ = r~)an.

Thus, 1’~’ihas HCW w, and so the operator J’a preserves HCW. The remainder of this section is devoted to calculating the action of J’ a on the quantities defined in subsection 4.1 in terms of known quantities on C7. Since dO has HCW 2, we have ~adVadi2+29dQ~0

(4-24)

dD0/(4V2ir) and P = VP0. We next consider the action of ~ on ~. Consider the transformation

where we have used dIi

where

=

dz’1. We have 1P(u, C p*(u* C~~)= G 1 ~)=(1—á)P(u, C~j). t~=

La

Thus, on the cut defined by u’ = 0 or, equivalently,

U =

p*(~~, 1) (1 —á)P(—a, C~1) =

P(0, C1 ~) —

aI~(0,~, 1)— áP(0, C~~).

—a, we have

M. Ko eta!., The theory of 5-space

99

Hence

5pp*(OCj).p(OC~) P~0,~,C)

—

P ap a

425 (-

at u * =0. Using equations (1-14), (1-3) we see, from the definition of ~, that =

Thus, on the cut defined by u * =0, we have ~,?*(O, C~1) ~fr~(—a, C1 ~)+ 315 a~fr~(—a, ~, h 3thifr?(—a, C~1). —

Hence =

~?*(0,C~~‘~) ~?(0,C~1) —

=

—~/‘?a+315a~—3áçb?

=

—ifr?a

We therefore have —3 ~

=

315a~ a —

(~ —3 ~

and, since a = L~dx”, this equation is equivalent to I’a~I’?= 315Lac1’~+La

(~~— 3~~).

Finally, on introducing the definitiont V’1J~a,

this becomes I’ail’? = 315La~+ La]~çb?.

Proceeding on a similar way, one can obtain the corresponding formulae for ~, t J~is essentially the same as the J,’ operator defined in [35].

~

etc:

)

M. Ko et a!.. The theory of ~(-space

100

ja1/1~n=

(n

1’ac1~=

(n ~4)~La~±i

kaçO°n

(n2)15Laco~+i+LaJ~ç~

I2aç~°n =

(n

4)15Lacti~±i +L4~//,~ +La~c~

(4-26) 2)~’6La~±i +La~(,~.

We now consider the action of Ja Ofl La. The simplest way of obtaining this is to translate eq. (2-30) into the intrinsic formalism. This is achieved by substituting equations (4-3), (4-4) and (4-8) into (2-30) and using the definitions of the differential operators ], a and 15. After a short calculation this yields ~aLb

=

~~LaLb ~15~L(a15Lb)+

(4-27)

~15La15Lb

where

c~=7iv.

(4-28)

Note that, by construction, the HCW from 1 andequations SW 2. The action of 1aandon(4-27): the spinor 0A’ of L~ canfunction similarly‘~behas obtained (2-38a), (4-15) components °A and

4.3.

0

(4-29)

IAA’OB

=

]‘AA’OB’

= °A

°A’°B’—

~15 ~O(A’LB’) + ~tA’LA.}.

(4-30)

The Weyl spinor and its potentials

In section 2, expressions for the Weyl spinor of ~C and its potentials were obtained in the non-intrinsic formalism (c.f. equations (2-39c), (2-40), (2-43), (2-46) and (2-53)). In this subsection we translate these expressions into the intrinsic formalism. As might be expected, the intrinsic form of the Weyl spinor and its potentials will be seen to be much simpler than their non-intrinsic form. We start with the Weyl spinor given by (2-39c). This is translated into the intrinsic formalism by means of equations (4-3), (4-8) and the relations (4-15). After a lengthy but straightforward calculation,

we find that IPA.B’C’D. =

~4j’

+

where

~

O(A’OB’OC’LD)

OA’OB’°C’°D’ —

O~’OB’&C’LD’)

—

O~’LB’Lc.’tc.)+

~LA’LB’LC’LD.,

(431)

101

M. Ko eta!., The theory of 7-space =

L” ~

(4-32)

and ~ is given by (4-28). In a similar way the potentials for IIIA’B’C’D’, given by equations (2-40), (2-43), (2-46) and (2-53), can be translated into the intrinsic formalism. They are given by =

~

=

~

0A0B

~LA’tB’l~C’}

°A’°B’

7AA~C =

oAo8o915~oA, —

ØABCD

= 0A0B0C0DØy

O~’L~’~ + SLA,LB,)

—

(4-33) (4-34) (4-35)

~‘LA’}

(4-36)

where g = V~Z/2 =

(4-37)

V~Ø,

(4-38)

and ~ is the s-function of section 2. Equations (4-33) and (4-27) yield the following interesting relationship between 1)AA’OB’ and A A’OB’

J~

FA~B~C~:

...rA C’ — I A’B’C’° .

—

This equation will find an application in section 5. We have thus shown that, to each of the weighted functions ~, ~‘,~, ~ and ~‘, there corresponds a potential. This fact is expressed in the following table: Function

7

SW HCW

Potential

-2 —2

-2

8~

—i

L

U

4

1

.y,4APC AB ~AB A

—2

2

I*ABC

By translating equations (2-32) and (2-50) into the intrinsic formalism we obtain the following relationships between these functions: WLaJ~a~F

(4-40) (4-41)

~S=L~,a2~.

(4-42)

M. Ko et a!., The theory of 5-space

102

In section 2, two equations, (2-28) and (2-31), relating ~ and (and hence ~) to the asymptotic components ifr~and fr~of the physical space Weyl spinor, where derived. When translated into the intrinsic formalism, these equations reduce to the following important relationships

~/6

=

_154

=

—15~~/2.

(4-43) (4-44)

An immediate consequence of these two equations is that they imply that the ~ and ~ functions of this section are the same as the corresponding functions of section 3, if the IC-space is asymptotically flat. To see this we first note that, for an asymptotically flat i-space, the results of section 3 imply that (4-45) =

(4-46)

~,

where çi~and ç~are asymptotic components of the ~‘C-space Weyl spinor, and then compare equations (4-43) and (4-44) with equations (3-11) and (3-12). Similarly, by using equations (4-40)—(4-42) (which are equivalent to equations (4-33)) together with the condition of asymptotic flatness, one can show that the 1~’and ~ functions of this section are the same as the g and ~ functions of section 3. Another important consequence of eq. (4-44) is that it enables us to find a relationship between ~(z”, C1 C) and the Gaussian curvature K = 1515 ln P of the good cut associated with the point Za of ~. By eq. (A1-12) we have =

(the

~JO

15K

(4-47)

term vanishes since ç~is evaluated on a good cut). Thus, equations (4-44) and (4-47) imply

15K =

_153

.~/2

(4-48)

or, equivalently,

2~/2 K=C(z’1)—15 where C is a function independent of C and ~ If we now make use of the fact that

J dfl

=

1

(see eq. (4-12))

together with the Gauss—Bonnet theorem, it is easily seen that C = 1. Thus K=1—152~/2.

This equation will find a number of applications in the following sections of this paper.

(4-49)

103

M. Ko eta!., The theory of 7-space

4.4. Maxwell fields on f-space In subsection 4.3, we saw the i~ essentially determines a potential TA’B’C’ for the Weyl spinor ~IIA’B’c’D’of X-space. In this subsection we show that the analogous situation holds true in the Maxwell case; that is, ~ induces a left-flat potential, ~ for a Maxwell field cA’B’ on ~°-space. By analogy with equation (4-43), we define a function ~ by -

(4-50)

=

Since ~ has SW 1 and HCW—2, Theorem 1 implies that ~ has SW —

we define ~ a~.=

—

1 and HCW0. Using the function

by (4-51)

oA (15~oA.— PIdLA’).

It is easily checked that a ~ has zero SW and HCW, and is therefore independent of the

V”

field. We

wish to show that ~ is a left flat potential for a Maxwell field, ~PA’B’ on ~. That is -

A

_T7

~PA~B’’ VAB’aA

and

-

-

(L...~7 A’ vBA’aA.

-

Note that eq. (4-53) automatically implies that T7A’

-

çA’R’

satisfies Maxwell’s equation [32]:

—

VA cAB’

—

We start with the last equation of (4-26) for n

=

2:

J’AA’Q2 = OAOA’~2.

(4-55)

Using eqs. (4-55) and (4-50) it can be shown that =

oAoA’152P~&

(4-56)

Writing VAA’P~as =

X1LALA’

+ X2OAOA’ + XIOALA’ + X4LAOA’

(4-57)

and substituting into (4-56) we obtain after a short calculation 152X

4=0

2X4+15X1=0

(4-58) (4-59)

M. Ko eta!.. The theory of 5-space

104

—2K15X

4—

—

2X 15KX4 + 15 2 =

(4-60)

KX1 + 2X2 + 15X1 =0.

(4-61)

We now introduce a new function, the v-function, defined by 4 V,~.PJ3= X~. =

(4-62)

O”O’

Using equations (4-57) and (4-59) it can immediately be seen that O’4VAA’9/~=

OA’

—

(4-63)

~

Using eq. (4-51) one can easily show that cOA’B

=

VB.Aa~’ =

15(oA VB’API~)oA (oAVB,AP~)tA,.

(4-64)

—

On substituting OA VAA~~from (4-63) into (4-64) we obtain -

cAB’

152(~

=

‘~j 0A’OB’

jT O(A’LB’) +

Since, by equations (4:58) and (4-59) independent of C and C. The dependence of aAA on C~C

(4-65)

~‘LA’LB’.

153~

vanishes, we have

15c°A’B’ =

0. Thus, unlike

aAA,

cPAB

is

1 is essentially a reflection of the gauge freedom in the choice of ~

each value of (~C) corresponds to a different gauge. It now only remains to show that aAA~ satisfies (4-53). After a short calculation, we have VBBa N

=

OA{(VBB,~)LB 15(VBB,~)oB’}.

(4-66)

—

From eq. (3-88) we have (VBB.PA)L’~= X2oB + X4LB

(4-67)

and 15(VBB,PJ~)oB’=

—(15X1 + X4)t2

—

(15X3 + X2

—

(4-68)

KXI)oB.

On substituting (4-67) and (4-68) into (4-66) and using (4-59) and (4-6 1) we have VBB.a~’=

OA{OB(2X7 + 15X3 — KX1)+ LB(2X4 + 15X1)}

=

0.

Thus a~.is a left-flat potential, and ‘PA’B’ satisfies Maxwell’s equations.

(4-69)

M. Ko eta!.,

The theory of 7-space

105

It is easily seen that eq. (4-53) implies the existence of Hertz type potential, =

which satisfies (4-70)

VBA,OAB.

By using a method similar to that used to determine ~ in terms of terms of ~.

0ISB,

~2,

0AB

may be determined in

Defining a function 4 byt (4-71)

(note that, by Theorem 1, 4 has SW —1 and HCW —1) it can be shown that OAIS

(4-72)

S/IOAOB

=

satisfies eq. (3-101) and that

The charge, e, of the physical space Maxwell field can be shown to be e

=

J

c? di2.

(4-73)

Using equations (Al-li) and (4-26), we have I’AA’ci

(4-74)

15(OA0A’c2).

=

Thus VAA’

e=

J

(1,AA’co~)dQ

=

—f

15(oAoA.co~)df~ =0,

and hence, as one might expect, e is a constant. If e is non-zero, equations (4-73) and (4-74) enable us to obtain a new chain of potentials for cAB’ which are in a canonically defined gauge. Let (4-75)

and ç~JAB =

~j.

c? gAB

tIn general ~?has the form

dll.

eV2

—

fid rather than —fi.~.However, .~ can still be defined by ~?— eV~2= ~

(4-76)

M. Ko eta!., The theory of 5-space

106

Note that, since the total weight of each of these integrals is zero, both 4~and

P/JAB

are well defined

quantities on Using eq. (4-52) w~have =~JQB.Ac?)a~. dQ +jc?VB~Aa~dIl

VB.A4AA,

~J(1BAc?)a~’

=

Since both I’B’Aci, and ~7 ,jA_ B’A’2~A’

V

aAA

dfl + cAB.

contain a factor of

0A,

(1B.A~?)a~’vanishes.

cAB’.

—

Thus, -

Similarly r

(J1IAB

—

VA’BJa

‘~~A’~

—

Therefore, 4~.and pJJAB form a chain of potentials for ‘,bA’B’. A useful identity connecting 4~ and a may be obtained as follows: ~.

VBB4AA’

~ J{(VBBaAA.)c? + aAA’kBB’cc’?} dO

=

=

1J{(VBB.aAA’)c?

=

~

f {(Va~’)~

BLA’) 15(oBoB’co~)}dLI

—

OA(ISB

—

oAoBoA’oB’c~co~} df?,

OA

—

(4-79)

where we have used equations (4-74), (4-51) and (4-50) and integrated by parts. Contracting (4-79) with VBB’, we also obtain S4AA’ =

~

J

(aAA’(pi

—

oAoA’ç02c2) dO.

(4-80)

Similarly we have VA’CPJJAB

=

~

J

{(VA’cOAB)c? oAoBoC~A’coic2)dO —

(4-81)

and P/JAB

=

~

J

{é.2.Bc?

—

oAoB(plc2} d(l.

(4-82)

M. Ko eta!., The theory of 7-space

107

Equations (4-80) and (4-82) will find an application when we come to the section on equations of motion. For some applications, it is useful to have cAB’ expressed in the form of an integral. This can be achieved as follows. Since f dill = 1, we have

f dill J~ J(~ —,

=

cA’B’

=

O~’O~’

=

°A’OC’

dill — 15S~’O(A’LB’)+~&A’LB’}

15(oA’oB’)

+ ~ 152(oA.oB.)} dill

=~

J

152~OO

dill,

(4-83)

where we have integrated by parts. Using equations (4-57), (4-58), (4-59) and (4-60) one can show that can be expressed in the form (4-84) where X

3 ‘~‘I5F

X~+T.

2’~from (4-84) into (4-83) we obtain

Substituting for 15 =

=

=

J J J

=2

(~2~

+ 153x)OA’oB’ dill

J

1529/J~~

dill

~

dill

(4-85)

dill.*

(4-86)

J

~LA’tB’

—

X153(OA’OB’) dill

A quantity which will prove useful when we come to consider equations of motion is Contracting (4-86) with v we obtain

c~A’B’V~’.

~‘

‘PA’B’VA

=

2J

~LA’LB’VA

dill.

Using the expression for the metric given by eq. (3-58), plus the fact that *

(4-87) LaVa =

1 we have

For flat space-time (4-85) is the usual Kirchofi integral [14].In general, there is a correction term dependent on the curvature of 7-space.

108

M. Ko eta!.. The theory of 5-space

b _OAOA~+LALA’

VagabV

(4-88)

2OALA’.

Thus, LB’VA

(4-89)

= ~~°A•

Substituting (4-89) and (4-87) we finally obtain ~OA’B’VA =

2

J

PA0ALA’ dill

=

2J P~15Ladi?.

(490)

5. Momentum, angular momentum and equations of motion 5.1.

Introduction

In general relativity there have been a number of approaches to the problem of defining momentum and angular momentum of an isolated gravitational system in terms of ~ [1, 3, 5, 12, 14, 32, 41]. In these approaches, which lead to more or less equivalent results, momentum and angular momentum are defined by means of certain surface integrals over arbitrary cuts of rather similar to the way in ..~,

which charge is obtained in classical electrodynamics by means of a Gaussian integral. However, even though these definitions possess many physically reasonable properties, they are defective in two

important respects: the momentum and angular momentum tensors do not “live” in any canonically defined space, and the angular momentum tensor transforms in an un-physical way under supertranslations [32].These difficulties are essentially related to the fact that the homogeneous Lorentz group is not an invariant subgroup of the BMS group. In this section we overcome these difficulties by using a new definition which is very much in the spirit of ~‘-space. We define momentum and angular momentum by means of integrals of certain ~‘-conformallyinvariant expressions over good cuts (points of ~‘) of C~ rather than over arbitrary cuts. Furthermore, the integrals are designed such that the momentum and angular momentum tensors lie in the tangent space of Before discussing the full General Relativistic situation, it is instructive to consider first a special ~‘.

relativistic system with Minkowski coordinates {Za} on which is defined a stress-energy tensor Tab, satisfying V”Tab = 0. If the system is non-radiating (i.e. Tab = O(r~”),n > 2), the total momentum Pa and total angular momentum (with respect ot the origin) Mat. of the system are given by pa

=

J T” d.~b

(5-1)

J M~~c~

(5-2)

Mab =

where ~

=

2z ~“ T” ]c and the integrals are performed over any non-timelike hypersurface £

M. Ko et a!., The theory of 7-space

109

If, however, the system is radiating (T~= O(r2)), M°”and F” depend critically on the choice of hypersurface £. To take this effect into account, we use a special class of hypersurfaces, £~which are asymptotically tangent to the future null cone emanating from a point P. The momentum with respect to any of these hypersurfaces is given by

J T” d.~b.

P”(P) =

(5-3)

Similarly, on taking the origin of the coordinate system to be P, the momentum with respect to P is given by

J M”’~dXc.

=

(5-4)

It is easily seen that P~(P)and M”(P) depend only on the point P, and are therefore well defined tensor fields on the space-time. For a non-radiating system, P~~(zc~) is independent of Z”, and M”(f) satisfies (5-5)

M”(Z’~~

We therefore have VaP~’= 0

(5-6)

and =

—2g~”P”1.

(5.7)

In terms of spinors, equations (5-6) and (5-7) are equivalent to vi VAA’.U

-

—

and VCC’WAB

—

C(AP$)C’ =0

(5-9)

where Mab =

EABWA’B’

+ A’B’WAB

and = WAil,

WAB = (al(AB).

(510)

M. Ko ci al.. The theory of 5-space

110

For a radiating system a certain amount of momentum and angular momentum is radiated away between the cuts of ~ corresponding to the points Z” and Z” + dz”. Thus the right hand sides of equations (5-8) and (5-9) will now be non-vanishing. We therefore have VBB’PAA’ =

(5-il)

—NABA’B’

and Vcc.wA~

—

C(APB)C’

=

JABCC’

(5-12)

where NABA’B’ dZBB’ is the amount of momentum and J ABCC’ dz’~’is the amount of (self dual) angular momentum which has escaped between the cuts corresponding to the points Z” and Z” + dza. For a bounded system which emits electromagnetic radiation, it is possible to obtain explicit expressions for NABA’B’ and JABCC’ using the electromagnetic energy momentum tensor Tab = cABcA’B’.

It turns out that they are given by NABA.B’(z)

JABCC’(Z) =

_f

J

co2c~2oAoBoA’oB’dill

c ico 2oAoBococ’ dill

where the integral is performed over the cut of ~

(5-13)

(5-14) corresponding to the point z”. (For the definitions

of quantities in the integral, see subsection 5.2.) If all relevant quantities are analytic in f’ then P”, WA’B’ and WAil may be extended into the complex by allowing the coordinates Z” to take on complex values. At any point1’~AB• off Using the real Pa will in theslice, transformation general be complex and w A’B’ will no longer be the complex conjugate of law of WAil’ under infinitesimal translations, it is possible to show that there always exists a unique complex world line along which WAil’ vanishes. We call this world line the complex centre of mass world line of the system.

5.2. Momentum and angular momentum We now return to the full General Relativistic case. We use the Tamburino—Winicour linkages [12] to obtain definitions for momentum and (self dual) angular momentum of an asymptotically flat solution of the Einstein—Maxwell equations. The T—W linkage expressions have the the form

=J F(~)

L~

dill

where ~a is a descriptor (generator) of the BMS group, F(~)is some functional of ~ (the exact form of

111

M. Ko eta!., The theory of 5-space

which need not concern us here) and the integral is performed over some UB = const. cut of I~(or C.~).In [12]the components of the momentum are defined as the four linkage expressions corresponding to the four infinitesimal BMS translations at the UB = const. cut; and this turns out to be the Bondi

momentum. Similarly, the linkage expressions corresponding to the infinitesimal BMS Lorentz transformations on the UB = const. cut are defined to be the components of the angular momentum. These definitions are, however, not quite satisfactory. The momentum and angular momentum vectors live in a space of cuts defined by the BMS translations and, without invoking some further restrictions, such a space cannot be canonically defined. Furthermore, the angular momentum expression transforms in an unphysical way under supertranslations. We overcome the difficulties by using a different definition for momentum and angular momentum which is more in the spirit of ~‘-space.We consider the linkage expressions only on good cuts of C~, and define momentum as the four linkage expressions corresponding to the four infinitesimal supertranslations which take the good cut into another good cut (i.e. the supertranslations corresponding to an increment dza in i-space). According to this definition, the momentum vector is given by 4JiIi~Za dill

Pa(Z”)

0,

where the integral is performed over the good cut of C~corresponding to the point In terms of the intrinsic formalism, this expression has the form Pa(Z”) = ~

J

i/J°~La di?.

Z”

in k-space.

(515)

Note that the weights (HCW and SW) of the integral in (5-15) are both zero. Thus Pa (z”) is a well

defined vector field on ~C-space. A complex Lorentz transformation of the form * — ITB °A’-’A°B,

* — °A’°A’

IrTB ~

I’)

JA~)J-4’~,

at a point of u-space, induces, via the simple dependence of and leaves C fixed:

°A

on C1 a bilinear transformation on

C*=C.

cC+d Thus, an infintesimal Lorentz transformation of this type determines a complex Lorentz descriptor of the complexified BMS group. We use these descriptors to define the self-dual angular momentum WAB. It turns out that (SlAB is given by ~)AB

=

~

J

fr? OA OB

dill.

(5-16)

Note that, since the weights of the integral of (5-16) are zero, (SlAB is a well defined spinor field on ~‘-space.

112

M. Ko eta!.. The theory of 5-space

Due to the very complicated dependence of °A’ on ~ and ~, it is difficult to find a natural definition for the anti-self-dual angular momentum WA’B’ by the same method. It appears likely that a natural definition for WAR’ can be given only in terms of YC-space. In order to obtain expressions for the covariant derivatives of pc~and WAB, we use the Bianchi identities and Maxwell’s equations on C5~.In terms of a shear free NU system, these are =

—‘~+2c~

j~’?=—~~±4c?~ °o~fr?+6co°~ =

~

(5-17) (518) (519)

—~+ê°~+2~c~

(5-20)

—~+2ê°~+4~?c~

(5-21)

1,~=—?+3~°~+6~c~

(5-22) (5-23)

1’c~= —~c?

(5-24) (5-25)

+

= ~

-0~O

(5-26)

where

~=‘oK+t51ê°

(5-28)

=

~O

(5-27)

~2~ji/p)~J,2~0

(5-29) (5-30)

= ~2(~/p)

and =

~2e0

(5-31)

These equations combined with equations (4-26) enable us to obtain expressions for I’AA’~I’~ etc., in terms of known quantities on C~. We are now in a position to calculate VaPb and VaWAB. Using equation (5-15) we have =

~

J

{(J~bcb~)La + tIi~La} di?.

M. Ko eta!.,

113

The theory of 7-space

Equations (3-24) and (5-18) imply that 2~La~+ La(~iib~ + 2c~).

J’açl’~=

Thus

J (lf1~J~

~

VbPa

i,La +

J

2LaLb~°2c°2) dill + ~ (2~LaLbtIJ~ + LaLb ~ ifi~)dill.

After a short calculation it can easily be seen that

~

can be written in the form (5-33)

~x.

=

(532)

On substituting (5-33) into (5-32) and integrating by parts we obtain

J

VbJ~a= ~

(c~I’iLa+ 2LaLb~°2c°2) dill.

(5-34)

In spinor notation, (5-34) becomes VBB’PAA’

=

—~

J~

1A’B’C’O

Since written1~~‘O~’ in the= form viB nA V ~‘r A’

— —

+ 2oAoRoA’oB’c2°~)di?. ,

vi(B DA) — V (BE A’) —

where

F~’B’c’ is

(5-35)

one of the v-space potentials (see eq. (4-39)), (5-35) may be

~ j’.rBA BA’

-

-

where

J

N~

3.= ~ ~

Using a similar method, VCC’WAB

=

c(APB)c’

dIll.

+~ VCC’WAB

—

(5-37)

may also be calculated. It turns out that this is given by (5-38)

JABCC’

where JABCC’

=

J

0H Aflc,B,OB~Oc + 2~p?c2°oAoBo&c.}dill. ~ (~b 2

(5-39)

Equation (5-38) may also be expressed as V~’WAB =

3PAA’

.

(5-40)

M. Ko eta!.. The theory of 5-space

114

and IC,

—

V (CWAB) J ABC. If gravitational effects are neglected (e.g. by setting the gravitational constant to zero) equations (5-37) and (5-39) become — —

NABA’B’

J

=

-

co2c2oAoBoA’oB’

di?

and

JABCC’

J

=

cic2oAoBoCoc’

di?

which are identical to equations (5-13) and (5-14). This shows that, in the flat space limit, our equations

give the correct result. The pattern of N~BB’and ~

in equations (5-37) and (5-39) suggests the introduction of a new quantity,

X’~’~g1iven by XABCD

I J{~frOyABCoAoD+2~6~c2oo~~oBoCoD} di?.

=

(5-42)

Using equations (4-26) and the potential equations: A’B’C’D’ = rA I A’B’C’

TJAB

EIA’B’

v,

~yAB AR’

VBC.ITI

—

ABC

~

VCB’YA’

it is comparatively easy to show that 1ARC

sTAB JVA’B’

— 1~7 —

VDA’X

_~

ABCD

1ABC VCB’JA’

and DD’,vi ~PA’B’C’D’r

A

—

5JAB VBC’IV NB’S

Thus, XABCD, JAABC and N~.act as potentials for ~‘A’B’C’D’PA. Note that equations (5-45) and (5-36) immediately imply V’VcPa

=

EPa

=0.

(5-46)

M. Ko eta!., The theory of 7-space

115

As was shown in subsection 5.1, N~J and J~BC essentially determine the flux of momentum and angular momentum escaping through C~. However, as yet, we have been unable to attach any physical interpretation to XABCD. 5.3. The 7C-space approach to equations of motion

General relativity is probably the only field theory to predict (or more accurately to have the re~l hope of predicting) the ponderomotive force law for massive bodies and thereby their equations of motion. Though the study of motion in general relativity has been the subject of intense activity for

years, it is still beset with considerable difficulties. Basically there are two approaches; the first being a local one, where one, from the stress tensor, defines the center of mass and studies its motion in the background space-time. The second approach, which is more relevant for us here, is to define physical quantities (such as the momentum, angular momentum, higher multiple moments, etc.) by surface integrals of the Weyl tensor (appropriately weighted on cuts of I~)and then to see how these quantities change on some canonically chosen one parameter family of cuts. Though this method does lead to exact equations of motion, they are complicated and suffer the serious drawback of not having a canonically defined space in which to consider the motion occurring. However with the advent of IC-space theory this situation has improved considerably. The basic idea remains unchanged; the definitions of the physical variables are slightly altered (by changing the weighting factors in the integrals), not all cuts are permitted (only “good cuts”) and a “center of mass” is defined with the result that simple and natural appearing equations of motion result, the motion being in X-space itself. Though the results are very attractive and suggestive, their physical meaning is quite obscure. It is the purpose of this subsection to outline this approach to motion. Before considering the full general relativistic approach, it is instructive to consider first a nonradiating special relativistic system (described in subsection 5.1) for which the fields Mab, (DAB, WA’B’ and Pa were defined. The centre of mass world-line of such a system is usually defined as that world line on which M”Pa vanishes. However, within the present context, the concept of the complex centre of mass world line (CCMWL) is better suited to our purposes [38].This mathematically more elegant, but physically more obscure entity, is defined as that complex world line on which WAR’ vanishes. Using equation (5-5) it is easily seen that, on this world line, Pa and (DAB satisfy the following equations of motion Pa = Pa

0

and WAR =

0,

where m is the total mass of the system, Va is tangent to the CCMWL (and normalized such that VaV” =2) and a dot indicates a covariant derivative along the CCMWL. Furthermore, the Pauli— Lubanski spin vector, S’~,is given by iS”

=wAilp~’

116

M. Ko

ci

a!.. The theory of ~)C-space

(with ~AB and P~’evaluated on the CCMWL) thus showing that ~AB provides a measure of the intrinsic spin of the system. In order to carry out a similar procedure in the full general relativistic situation it is necessary to introduce an appropriate definition for (DAB’. As we saw in subsection 5.2, such a definition does not arise naturally from the T-W linkages. We can, nevertheless, introduce a tentative* definition for w NB’:

=

4

J

~lJi0,~’Owdill.

Since the total weight of the above integral is zero,

furthermore, in the case of stationary space-times,

is a well defined spinor field on ~W’-space~ can be shown to give the correct result for the

~A’B’

~A’B’

self-dual part of the angular momentum. Even though we do not have a rigorous proof, it appears almost certain that there exists a world line in ~‘-space on which w NB’ vanishes. We define this world line to be the CCMWL of the system. After a lengthy calculation it is possible to show that, on this world line, Pa and ~0AB satisfy equations of the form ~V’~mva + Xa,

Pa Pa

=

Na,

XaVa

=

0

(LIAR = JAR.

These equations constitute the equations of motion for the CCMWL for a general system. Since this section is intended merely as an outline of the JC-space approach to equations of motion, the explicit form of the functions Xa, Na and JAB will not be given here. The interested reader is referred to [541 where these functions are given explicitly. Since the above equations of motion describe the motion of an arbitrary spatially bounded source, they are, as one might expect, relatively complicated and difficult to work with. A much more interesting situation, and one for which the elegance and simplicity of the YC-space approach becomes more apparent, is when we consider the above equations of motion when the original physical space-time is that of a charged algebraically special solution. It has been noted in a number of recent papers [37,55] that such a solution bears a strong resemblance to a classical charged spinning particle.

For instance, when the gravitational effects of such a solution are neglected (e.g. by setting the gravitational constant to zero), the remaining Maxwell field is precisely that of a charged spinning particle. Furthermore such a solution, when complexified, possesses a privileged one (complex) parameter family of good null cones (privileged in the sense that not only 0.0 but 0. vanishes) which, loosely speaking, emanate from the singularity; a situation closely resembling that of the null cone emanating from the world line of a special relativistic point particle. For these reasons we interpret such a solution as a general relativistic analogue of a classical charged spinning particle. Furthermore, we

interpret the world line in the ~‘-spaceof the solution described by the privileged family of good null cones as, in a certain sense, the world line of the particle. In terms of a NU tetrad system adapted to the family of good cones i/o, ~ and çb~can be shown to vanish identically [37]. In particular, the vanishing of ~ implies that ‘-°A’B’ is zero on the world line of

the particle. This world line is therefore also the CCMWL of the system. *

We emphasize that there are several possible definitions for

SOAB

only studied one of them. The basic idea of defining the CCMWL by

each having the Correct correspondence limit to flat-space and that we have

WA’B’

=

0 would be unchanged.

M. Ko eta!., The theory of 7-space

117

As we shall see in the next subsection, the equations of motion associated with a charged algebraically special solution are extremely elegant and physically suggestive. However, since they are set in IC-space (which is an essentially complex space) rather than in the original physical space-time, it is difficult to place any precise meaning on them or to be certain that they have any relevance at all to

the real physical world. At the moment they are an enigma, but an attractive one. 5.4. The motion of a spinning particle* In this subsection we use a diverging algebraically special solution of the Einstein—Maxwell equations [36,40]as a general relativistic model of a charged spinning particle, and use the results of subsection 5.2 determine its equations of motion. When complexified, a diverging algebraically special solution can be shown [37]to possess a one (complex) parameter family of diverging null hypersurfaces on which ci, *o, 1//i and ~ can all be made to vanish. These null hypersurfaces intersect CI~ in a one parameter family of good cuts and thus describe a world-line in the ~‘-spaceof the solution. We interpret this world-line as the complex centre of mass world-line of the particle [38,39]. In terms of a shear-free NU system on C5~based on this world-line, ~“i, ifr0 and 4o vanish and equations (5-18), (5-23) and (5-24) become

to

0= —~~+4qi~

(5-47) (5-48)

—2

~

=

~

(5-49)

Equation (5-48) implies that ~? is independent of C. Also, since it has SW zero and is assumed regular, it must be independent of C. Thus ç~hasthe form ~ ?(u). Since f dill = 1 and the charge ofthe solutionis given

by e=Jq~?dill=c?Jdi?,

we must have e

=

c?.

(5-50)

Equations (5-47) and (5-49) now become 2eP/P~5ço~

(5-51)

and *

This work is an extension to 7-space of the method pioneered in [35,40,41,42] thus avoiding the difficulties encountered in the earlier work.

M, Ko eta!., The theory of X-space

118

i~=4e=—4et52~, where ~

=

—~

(5-52)

(see section 4).

Equation (5-52) implies that ~ has the form (553)

where the coefficient of m has been chosen such that, in the special case of a Schwarzschild solution, m is the mass. Equation (5-53) implies that

—2V’~m=f~di?.

(5-54)

Differentiating (5-54) with respect to u, and using equations (5-17), (5-53) and (5-51), —

2\/~in

=

J ,4~di?

+~

we get

di~ dill

=J{(~_3~)4~}di?

~

—2\/~-m—4e

_2f{~2~+~6~ ~p~}dill

Thus in

=

(5-55)

0.

Since Pa

~Jç1I

di?

20La

(5-15)

and Va2JLa

di?,

(413)

119

M. Ko eta!., use theory of 5-space

eq. (5-53) yields Pa

=—~J(—2V2m—4e~)Ladi?

(556)

=~mva+2eJ~eLadi?.

Recalling that the vector potential, A~=1J~?aadi? where, in our case, Aa

=

J(~8La

Aa,

is given by [eq. (4-75)]

J~?(~P~La~’óLa)dill,

e, we have

*~

~La)dill

+~La)dill 2Jl5~La

di?.

(557)

Thus, on substituting (5-57) into (5-56), we obtain mV~+ eAa.

Pa =

(5-58)

It must be remembered that Va is normalized such that VaV” = 2. If Va is replaced by the four-velocity, Ua, of the centre of mass world line, where UaUa = 1, eq. (5-55) assumes the very familiar form Pa =

mu~+ eAa.

In order to obtain the time development of Pa we use eq. (5-34) contracted with =

~

J

(cb°~La + 2La~çp~) dill.

(5-59) c”:

(5~)

It is easily checked that

~Ladi? = 0. Thus, on substituting for ifr~from eq. (5-53) into (5-60), we obtain Pa2eJ~I~Ladill_J~c~Lad11.

(5-61)

M. Ko eta!., The theory of 5-space

120

Since e e

=

~?,

Jcia

eq. (4-80) gives

di? eAa —

=

J~

La

di?.

(5-62)

After a short calculation it can be shown that á.~

LaLa+1~LaPJJ’6(1~La).

Thus J~adi?2J~Ladi?+2J~1’La di?;

and, on substituting this into (5-62), we obtain 2e

J ~Pi~Ladill +2e

J)

La dill

eAa

J~°co

2°La di?.

(5-63)

Equations (5-61) and (5-63) now yield Pa

=_2eJ~Pi~La di?+eA~.

Finally, on using eq. (4-90), we have PAA’ =

(5-64)

+ eAAA.

—eçoA’B’VA

0A’B’ and AAA’ on YC. This gives the time development pa in terms fields ~ Toequation obtain the equation of motion for theofcomplex centreofofthe mass world line of the particle, we simply substitute for pa from eq. (5-58) into eq. (5-64). This yields, remarkably, the familiar equation

mvAA’

=

—V’~ecoA’B’VA.

(5-65)

To obtain the time development of (DAB, we use eq. (5-38) contracted with v~”: (DAB

—

V~PB)C’=

—~

J

di?

S/I~5~FOAOB

—

J

~1

OAOB

di?.

(5-66)

Substitution for ~ from eq. (5-51) into (5-66) yields (DAB

Since ~?

—

=

V~PB)C’=

e

J~

e, eq. (4-82) gives

~3.9FOA0j3

di?

—

J

c~?co2°OAOBdi?.

(5-67)

M. Ko eta!., The theory of 7-space

e

J

OAB

Also, since OAB =

dill

—

eBAR

S/JOAOB

0AB =

(5-68)

OAOB(,~?cO~ dill.

=

and

)~OA=

0, we have (5-69)

}~S~0AOB.

Using eq. (5-25),

121

one can easily show that (5-70)

= ~

Thus OAB

=

(5-71)

~J~~OAOB

and

J dO From eq. (5-57) we have =2 J ~ OAB

—J~OAOB

=

AA,(AVAB

dill = 2

J

~O(A~B)

dill.

(5-72)

OA’O(AVB; dill;

(5-73)

also, using eq. (4-88), we have A’

OA’VB

Thus, eqs. (5-73), (5-74) and (5-72) yield

J ~ dill J ~ J dill J ~

AA’(Av~=2

=

O(A&B)

—

(lAB

ö~OAOBdill

—

(5-74)

~OAOB.

Substitution of f (lAB dill from eq. (5-74) into eq. (5-68) —e.BAB

J

+ eAA.(Av~+ ~

dill

~OAOB

=

J

now gives

oAoBço 1c02

di?.

(575)

Finally, on using eqs. (5-75) and (5-67) we have — B)C’ —

C’~ WAR

—

V(A.U

A

—

A’~

~

erlA’(AvB) ~ e~~ 48.

-

122

M. Ko eta!., The theory of 5-space

This equation gives the time development of WAR in terms of the fields AAA’ and BAB. Equation (5-76) may be simplified further by substituting for P” from eq. (5-58). This gives 0AB

~

=

(5-77)

eBAB.

Equations (5-55), (5-58), (5-64), (5-65) and (5-76) constitute our equations of motion for a charged spinning particle. Note that, if the charge e vanishes Pa =

m Va,

t0AB

_

0,

and the complex centre of mass world line is geodesic in ~‘-space [43]. Since these equations of motion are set in an essentially complex space-time (namely JC-space), it is difficult to attach a precise physical interpretation to them. However, considering their simplicity, elegance and physical suggestiveness, it is difficult to believe that they are no more than a mathematical curiosity. We conclude this section by mentioning that, when gravitational and spin effects are neglected, equations (5-66) becomes

\r2m~Y~ _1e2{üa

~

where the right hand side can be recognized as the Abraham—Dirac—Lorentz radiation reaction term [44].

6. The relation between

~‘-space and twistor theory

In this section we explore the relations between the theory of IC-spaces developed in earlier sections and the Twistor Theory of Roger Penrose and coworkers [22,29, 47—5 1]. We begin with an account of Penrose’s geometrical construction of the general left-flat space in terms of a deformation of part of “flat” twistor space [22].By introducing suitable coordinates via twistor

theory on the left-flat space, we show how the ~‘-spacemetric, defined twistorially, emerges naturally in Plebanski’s first form, yielding a twistor interpretation of Plebanski’s i?-function [231. We then obtain a system of differential equations for the solution of the converse problem, to

reconstruct a deformed twistor space given the i?-function of a left-flat space [511. To show the connection of the methods of section 1 to Twistor Theory, we construct the asymptotic projective twistor space, P p3” [21,291,associated with a solution of the good cut equation. The metric defined from P ~ by Penrose’s method is then found to agree with the i-space metric obtained from

the good cut equation. 6.1. Deformed twistor space

The deformed twistor spacet P ,‘T’~of Penrose is a bundle over C2 —0, to be considered as unprimed spin space, with fibres a part of C2 [221. t In fact, we are here dealing with dual twistor space to accord with the notation of earlier sections.

M. Ko eta!., The theory of 5-space

Using the spinor

ITA = (IT

0, IT1)

Uo={IrAIIro0};

as a coordinate on the base space, two neighborhoods are defined

Ul={ITAI’irlO}.

The fibre coordinates over U0 are taken as a spinor ~A determined by the transition functions =

0)A

123

+ fA~(B

1TB),

ITB

=

(w°’,0)1’) and over U1 as 6jA’ and the bundle is

(6-1)

E U0 fl U1.

Thus a twistor Wa consists of a pair (ITA, 0A) Additional structures are required to exist on the bundle, restricting the function f”~’.Firstly the

global existence of the homogeneity operator y = 0)A

f9/8(DA

+

ITA DIaITA

is demanded, requiring 0)A’

alawA’ =

ójA’ 8/&iA’

so that fA’ must be homogeneous of degree one in the twistor Wa fA’(AITB

Ao.~’)=

AfA’(ITB,

= (ITA, 0)A)

i.e. (6-2)

B’).

This allows the definition of homogeneous functions. Secondly, the transition, (6-1), is required to be symplectic in that a two form ~a dWA’AdwA,d~’A do3A’ is required on the fibres i.e. at fixed 2~—+-~-~

3(,/~

~

—0 .

ITA.

This results in an equation on 63 -

The bundle, P ~‘, is now a deformation of part of CP 3, flat projective twistor space. Further f’s’ is to be chosen so that P ,97* includes a deformation of a neighborhood of one of the linear subspaces, CP 1, in CP3. By a theorem of Kodaira, provided the deformation is not too large, there will exist a four parameter family of compact holomorphic surfaces in P 9~which correspond to the linear subspaces of flat twistor space in the limit when the deformation vanishes [52],so that one of these surfaces is fixed by specifying two twistors which lie on it. It is this four parameter family of surfaces (referred to as curves, for brevity) which Penrose takes for the points of a left-flat space. Before proceeding to Penrose’s definition of the metric, we introduce coordinates on the space of curves, that is, on the left-flat space, as follows: fixing two spinors XA, YA in the base space once for all, curve is parametrised by its intersections ~ yA’ with the fibres above XA, YA respectively. A curve is then given as a pair of functions

a

124

M. Ko et a!., The theory of 5-space =

wA(xB

y8’

ITB),

ITB

EU

=

~A’(xB’

yB

ITB),

ITB

E U1,

homogeneous of degrees one in 8MB’ 8w i9(U 8(0.

0, (6-4)

and zero in X’~’and 8WB’ 8w 8w 8(0

ITA

~

yB

Using (6-3) it is possible to show that 65

—

8YA’8Y8YA’8Y

(- )

so that these are two globally defined functions homogeneous of degree two in quadratic polynomials in ITA.

ITA

and thus are

We now make an ansatz for the curves, (6-4), in one neighborhood, say (Jo: A

~

A

~XA’+~ (VBX)

A’_(X ITA)yA~+(Y

(X8Y~~)

where Xa = (XA’, X’~’),Va

=

A

B

ITAXY ITB)GA~X V (X’~Y~) ‘,, a,7TB

66 -

(YA’, yA)

The first two terms are a linear interpolation between XA’ and

yA’,

exact for vanishing deformation.

The deformation hence is GA’. Using this ansatz in (6-5) we find G

GA’

GB’

~

=

0, (6-7)

,,8GA’

/

A

\8GB’8G_

L~A7~t.Y ITA)~~-O. UI (JIA’UI

We may now give Penrose’s definition of the metric on the space of curves. First, for the conformal metric, two points of the left-flat space are null separated if the corresponding curves meet in P ~“.

From the definition of the coordinates on the space of curves, this means, in particular, that two points which differ only in their XA’ coordinates but have the same y-~’ coordinates will be null separated (since the corresponding curves will meet above VA). Thus the X°’and X1’ coordinates are null and orthogonal, and likewise the Y°’and V’. The full metric must therefore take the form ds2 =

2HA’B’ dXA’ dYB’.

(6-8)

Now for the full metric we take three curves P, Q and R such that Q differs from P only in the XA’ coordinates and R differs from P only in the ~~A’ coordinates. Taking these differences to be small, suppose P, 0, R, have coordinates (XA’, yA’) (XA’ + dXA’, yA’) (XA’, y~’ + d Y’~’)with corresponding curves WA’, (DA + do/”, 10A + öwA’ in Pr. Then Penrose defines the contraction of the two null vectors PQ and PR by HA’B’

dXA’ dVB’ = (Xi1TB)(Y~~irc) ~A’B’ dWA’ p~B’

(6-9)

M. Ko eta!., The theory of 7-space

125

Following Penrose [50],we observe that -the right hand side of (6-9) is homogeneous of degree zero in ITA and that the zeroes in do/” and ôw~’ at YA and XA. respectively are cancelled by the zeroes of the denominator. Thus it is independent of ITA, and a function the space of curves. Obtaining ~~A’ and ~A’ (6-6) and substituting (6-9) leads to an expression for HA’B’ which apparently depends ITA. Substituting XA and YA respectively for ITA in this gives

from

on

into

on

1 8GA’ HA’B’—7eA’B’+-~ Ui ,FA’~XA _1

8G~ IFAYA

UI~

where 4

XA YA. Thus

=

8GA’

8GB’

—

UI

,FAXA

—

wA

U~

16-lla)

,

1FA~’YA

and similarly, from (6-7), A’

A’

but (6-11)

=-~v~ ~‘ IFAXA ,

= 1FA~YA

OI~

are precisely the integrability conditions for the existence of a function ill(Xa,

aill

Ya) such

that

‘.‘B’~

_~

8XA’ —

(6-’llb)

~ ‘-‘A’ IFA=XA,

E~4’J3’J

(6-12)

—1

8(1 =

B’

EA’B’X

—

GA] ~TA

YA

and in terms of which the metric takes the particularly simple form 82(1 HA’B’

=

ÔXA’ 8YB’~

(6-13)

Further, with the aid of (6-7), we find det HA’B’ = HA,B,HA’B~=

~.

(6-14)

(6-13) with (6-14) is essentially Plebanski’s first form [23] of the right flat metric as may be seen by introducing coordinates (pq)

yA’;

(r,s)=X”~’.

126

M. Ko et a!., The theory of 5-space

Comparing (6-12) with (6-6), we find a geometrical interpretation of [l(Xa, Va) pointed out by Penrose. If, for fixed Va, we consider the congruence of curves through Ya~then the tangent to the congruence at a point Xa is 1(43 8i?/8X,

3,

where Ia$ is the infinity twistor [29]. Plebanski’s coordinates defining 0(XA’,other /~B,XA, VA) by may be introduced by eliminating 820

—

8/.LA’8/.LB’

—

AC’

4A’ = yA’

in favor of

8i?/8XA’

and

/-

82i?

B’D’

This is essentially 0 of sections 2 and 3. The integrability conditions for such a 0 are automatically satisfied, and the metric becomes .i

U5

2

x.’A’ LUA ‘,

.j

~

___________ .jUAA’UAB’. .j ~.‘

U/.LA’L~

(I/LA’

U/LB

Just as the coordinates (XA’, yA’) are thought of as determining a curve by two points on it, so the coordinates (XA’, I-IA’) determine a line by an initial point, XA’, and initial direction /LA’.

The fact that the metric defined in this way on the four-dimensional space of curves gives rise to a left-flat curvature tensor was proved geometrically by Penrose, while in our formalism it follows (by the work of Plebanski [231)from (6-13) and (6-14). The general left-flat space may thus be obtained by twistor methods. Penrose imposes global restrictions, notably positive frequency and asymptotic flatness in an appropriate sense, on the left-flat space by restricting the original transition relations (6-1). This gives a second class of left-flat space, which he calls the non-linear gravitons [22].The question which

~‘-spacesare non-linear gravitons and vice versa is still under investigation. In particular, one would like to know exactly what conditions on o-~in the good cut equation give rise to an SW-space which is a non-linear graviton. 6.2. The twistor space associated with a left-flat space We now consider the converse problem to subsection 6.1, namely given a metric in Plebanski’s first form, to obtain the associated deformed twistor space. In a left-flat space, all the integrability conditions for the equation

VAA’ITB =0, where

VAA’

is the spinor covariant derivative, are automatically satisfied. This means that covariantly

constant unprimed spinors fields can be found. Given such a field Hab

=

(6-15)

‘WA,

the two form,

ITAITBA’B’,

will be closed and so will give rise to a family of two surfaces. Further any tangent vector to such a two

M. Ko eta!.,

surface

The theory of 7-space

127

wifi have the form =

ITAAA

for some A A, so that any tangent vector is null and any pair are orthogonal. Thus a left-flat space is ifiled with families of completely null two surfaces, one family for each unprimed spinor. These are the twistor surfaces of Penrose, and constitute the points of p7*, the deformed twistor space of subsection 6.1. To see this, we fix a point *~ of P 9~*and consider all the curves, in the sense of subsection 6.1, which pass through Wa. All these curves correspond to null separated points of the left-flat space. A simple counting argument ‘ives the dimensionality of the surface, and the associated unprimed spinor labels the fibre in which Wa lies.

The problem of finding the twistor space P Y” of a given left-flat space is therefore the problem of finding these completely null two surfaces. From (6-4), the problem described above is that of finding two functions 0)0’, W~’so that the two surfaces are the set of (XA’, yA’) satisfying wA’(xB’,

yB

~B

=

(6-16)

~‘

since this gives all the curves through

Wa =

(I~A,~A’)

In Plebanski’s coordinates, we take the general

left-flat metric to be gab dz°dz” = 2(ill .pr dp dr + ill,,,~dp ds + ‘II,qr dq dr + (ll.qs dq ds)

(6-17)

-

with l2,pjll,qs — ill.psill,qr =

1.

(6-18)

The surfaces of constant p and q and the surfaces of constant r and s are two families of completely null two-surfaces and our object is to “fill in” the others between them. Writing ir~= (1, i’), since (6-16) is homogeneous of degree one in ‘WA, we are seeking two functions P(z°;i’), Q(z°;C) such that the surfaces of constant P and 0 are completely null for each ~ and with

initial conditions P(p,q, r, s;O)p;

Q(,p,q,r,s;0)=q.

(6-19)

We fix P and 0 by requiring dP

A dO

=

dp A dq +t(il,prdp

which is essentially the two form

A dr+ill,~,dp

j.~of subsection

Ads +ill,q. dq

A dT+i1,qs

dq A ds)+ j2dr A ds,

6.1.

These requirements lead to fl,qsP,r — ill.qrP.s = -

ilpsPr

—

(1.,,.P,5

and the same for 0.

=

CP.~

(6-20)

128

M. Ko eta!., The theory of 5-space

While we could solve these equations as a power series in we show instead that this is equivalent to solving a Hamiltonian system which is essentially the same as finding the integral curves of the vector field described at the end of subsection 6.1. The problem is that, given only i?(p, q, r, s) satisfying (6-18), we do not know its dependence on XA in the notation of subsection 6.1. The procedure is to find the solution of (6-20) and (6-19) to first order in C; then to introduce this P and 0 as coordinates in place of p and q. If the metric is to retain the form (6-17) this will yield a new if differing from the old (1 by terms of order ~, thus discovering the dependence on XA, and the process can be iterated with the new ~‘,

coordinates. The first order solution to (6-19) and (6-20) is

P=p—~i?,q;

Q=q+~i?,~,

(6-21)

where there is freedom of choice P*P+~K,q;

Q-*Q-~K,,

for arbitrary K(p, q) but since i? is arbitrary up to addition of a function of p and q (and a function of r and s) we disregard this.

Writing i?’(P, 0, r, s)= i?(p, q, r, s)+ ~w(p,

q, r, s),

we find W,r =

i?,qi?.pr

—

ul,pQ .qr

=

i?,qi?ps

—

i?,pi?.qs + r

—

5,

(6-22) with similar freedom, which we again disregard. Thus the twistor surfaces unfold as solutions of a Hamiltonian system (6-21), where (1 acts as

Hamiltonian and itself changes according to (6-22).

-

The initial conditions (6-19) mean that the twistor surfaces are unfolding from ~ = 0 towards ~ = ~. We may start instead by seeking functions R (p, q, r, s; i’), S(p, q, r, s; C) again satisfying (6-20) but with R(p,q,r,s;~x)=r;

S(,p,q,r,s;cc)=s

and obtain a power series solution in ~‘. Then, in_the notation of subsection 6.1, P and 0 play the role of 0)A’ and R and S play the role of ~jA’ With ~ standing for ITA, the transition functions (6-1) are found, in principle, by obtaining R and S as functions of P, 0 and ~. 6.3. Asymptotic projective twistor space Associated with any asymptotically flat solution of Einstein’s equations, it is possible to define an asymptotic projective twistor space [21,291, which is a deformation of flat twistor space, as in subsection

129

M Ko eta!., The theory of 7-space

6.1. Further, the left-flat space obtained from this deformed twistor space by Penrose’s construction is just the IC-space given by the construction of section 2. To define the asymptotic projective twistor space, P ~,

we recall the picture of C ~

as a cube (fig.

2). The good cut equation (2-1), for fixed ~, is a second-order ordinary differential equation, so that a solution is a curve in a plane of fixed C, determined by two initial values. It is the space of these curves, or twistor lines, which forms P ~“. Thus P ~ is a bundle over the part of C2). This is in agreement with subsection 6.1, if we use the C-sphere withoffibre C2 (orthere homogeneity everything to write 17~A= (1, C). The good cut equation is -

=

o~(Z,C,

1)

(6-23)

so that a twistor line is given by uB=Z(C);

1=11.

We define p(C)=Z—Ct52, q(C)=

(6-24)

~Z+~ooZ,

and as coordinates on P r we take ~i and p and q at an arbitrary fixed C~say Ci. An asymptotic projective twistor is then given as a triple ~p,,q 1 C~)where -

Pi

=

p(Ci);

(6-25)

q1 = q(Ci).

~2boundary of non-singular region of Cf

I I

agoodcut

K I UB

~—‘~~“

atwistline

~

Fig. 2. C5~showing a good cut ruled by twistor lines. -

+

M. Ko eta!., The theory of 5-space

130

Taking a different fixed p2=p(C2);

q2 =

~,

say

C2,

would give coordinates (p2, q2 ~ where

q(C2)

and the transition functions of the bundle would be determined by solving for P2 and q2 in terms of Ps, q1 and Ci using (6-24). The four parameter family of curves, in the sense of subsection 6.1, which are to be the points of the left-flat syace, is given by the solution of (6-23) with the C-dependence determined by regularity, Z(z’~,C~C). In (6-25), this gives the curves as Pi

Ci, 1),

=pi(z’; ~)p(za;

(6-26)

qi= qi(z” ; C) = q(z” ; Ci, C) and as ~ varies, (6-26) will eventually encounter singularities when ~ gets far from ~, i.e. far from the real slice of C5~.Thus for a particular ~ there is only a neighborhood, U1, of Ci in the C-sphere where

p, and q, are good coordinates on P ~. This is the statement that the bundle is non-trivial. Changing to C2 gives another neighborhood U2 of C2 and in this way we obtain a covering of the base-space. While we cannot in general obtain the transition functions explicitly, we can find an “infinitesimal” transition function if C~and C2 differ infinitesimally. If C2

Ci + e,

=

Pi =p(Ci);

p2’_p(Ci+e)

q,

=

q(Ci),

q2=q(Ci+e),

then 2),

Pi e 8H/8q1 + 0(e q 2), 2 = q1 + e 8H/8p1 + 0(e where P2 =

—

H(,p

1, q,; C~~)= h(Z(C1,

1), C~~),

and

1). Thus the infinitesimal transition functions appear as a canonical transformation with the time-integral of

131

M. Ko eta!., The theory of 7-space

the shear,

o~, as

Hamiltonian. This incidentally points out the existence of the two form

js,

~ =dp,Adq,=dp2Adq2 of subsection 6.1, on the fibres. Finally, we wish to verify that the metrics defined by the Penrose construction in terms of curves and by the methods of section 2 agree. For the conformal metric, if two curves meet in P r, representing points null separated in ~‘-spacein the Penrose metric, then the corresponding good cuts intersect along a whole twistor line. This makes the integral of (2-2) diverge so that the corresponding points of X-space are null separated in the metric of (2-2), and the metrics agree conformally. For the full metrics, it is necessary to consider three infinitesimally separated points of IC-space, P, 0 and R, with the vectors P0 and PR null as in subsection 6.1. It is a straightforward matter to check that the Penrose construction for the contraction of the vectors PQ and PR does indeed agree with the contraction in the metric of (2-8), (see [17]for details). 7. Summary and outlook In this paper we have attempted to survey both previously known and more recently discovered results about ~‘-spaces.A topic not yet discussed is why should one be interested in ~‘-spacesor more precisely what relationship does ~‘-spacetheory have with physics? There is, unfortunately, no simple answer to this — it may well be that from a physicist’s point of view the subject is empty though perhaps filled with beautiful mathematics. The evidence in favor of a relationship is certainly far from being conclusive and what there is rests more firmly on aesthetic and philosophical grounds than on its ability at clarification of physical ideas or its predictive value. In defence of the subject, however, we point out that it in no sense should be considered a theory yet. It is simply a rigorous and logical extrapolation of

the Einstein theory of gravitation (extrapolated perhaps beyond its natural domain) with no ad hoc elements or assumptions (except the assumed analyticity of o-~, and even this, it is suspected, is not needed). A point of view for studying ~‘-spacescould thus be — push this extrapolation to its extreme; find what structures arise naturally (oras special cases) and then study their relationships. The questionthen could be restated — are these structures and theirrelationships interesting? Do they seem to be related to physicalideas or concepts? Is it possible that they are underlying and offering clues to a more basic theory? The remainder of this section is devoted to reviewing these structures and describing some conjectures concerning their possible physical relevance. We began with an asymptotically flat physical space-time and, using only the data for the self-dual of the radiation field (namely o~), we found, in the analytically extended space-time, complex asymptotically shear free null surfaces or good cuts. This led immediately to the X-space construction. (If the anti-self-dual part of the radiation field, namely J-°,had been used ~‘-spacewould have resulted.) The point here is that it is solely the radiation field (the self-dual or anti-self-dual part) the physical space which determines the ~ or f-space and that the longitudinal parts of the field play no role in the intrinsic geometric properties the ~‘ or ~‘-space.The longitudinal parts do play a most important role in the theory of equations of motion on the ~‘-space.(It appears likely that o’~or ã~are not irreducible data and that the set of all shears having the same second time derivative determines the same ~‘ or ~‘-space.Though this seems reasonable and works for one example, we see no way as yet, to prove or disprove it.)

part

of

of

132

M. Ko et al., The theory of 5-space

One of the most important questions to be asked of the o~,is what must its properties be so that the derived ~~‘-spaceis asymptotically flat? It seems almost certain that if o~ (in some special Bondi frame) vanishes sufficiently rapidly as u +~ (probably as u ~3) then the ~t( will be asymptotically flat with a “future” ..~~(R); likewise if it vanishes as u it will have a “past” 9~(R).(The two Bondi frames —~

—~ —~

needed for this would, in general, be different.) From the discussion in subsection 4.3, we see that any asymptotically flat physical space-time, whose W is also asymptotically flat, possesses one or two canonically defined Minkowski spaces depending on whether the o~vanishes sufficiently rapidly in one or both directions. Each of these Minkowski spaces manifests itself as a four (real only if the special

Bondi frame is a real one) parameter set of privileged cuts of the physical ~ all of which are connected by Bondi translations. If the physical space is such that its o~vanishes in the past and future in the same special Bondi frame, then the two Minkowski spaces coalesce and become one. (These Minkowski spaces are apparently identical to the ones proposed many years ago with a different motivation and from a different point of view [34].)Though there is a satisfactory feeling about having a well defined Minkowski space available for gravitating systems (it has been sought for years) it is not

clear how it is to be used. There is a poorly understood connection between these asymptotically flat ~‘-spaces and positive frequency decomposition of certain fields. For example, in the integral expressions in section 4 for the potentials, e.g.

~=Jn~odr the integral is taken (for fixed ~ and ~) over a contour in the complex r plane from 0 to ~ The contour was intended to be in a thickened region around the positive real axis. Alternatively, if the integral were taken along the negative real axis (if such a contour exists) a second set of potentials, defined now from the asymptotic curvature properties of Y~(R),would be obtained the local curvature properties at r =0 being unchanged and the 5~and r of an asymptotically flat IC-space would thus, in general, be different. If, however, the ~ had no poles in the upper half r-plane or equivalently had a positive “frequency” Fourier decomposition, the integrals could be deformed into one another and the two ~‘s could be identified. Though in linear theory this would translate into positive frequency decomposition and proper vanishing in both limits in a fixed Bondi frame for o~, we have no hint of whether this is true in general. Aside from their possible connection with Minkowski space, asymptotically flat IC-spaces appear to be special cases of a more general situation—with even the beginnings of an interpretation. The basic idea or question is that if one has a real solution of some field equations, the Maxwell equations, the Einstein equations or even the Yang—Mills equations how could the decomposition of the field into its longitudinal parts and radiation parts be accomplished? In Maxwell theory starting with a real solution one could, because of the linearity of the theory, take the radiation part as being half the real retarded field minus half the advanced field; one could then finally decompose this into its self-dual and anti-self-dual parts. How, a priori, to do this for the gravitational field would not be obvious we have, however, seen that the ~‘ and ~‘ construction does just this. Our conjecture is that this idea extends to Yang—Mills fields as well; namely, for appropriate real Yang—Mills fields on either —

—

—

Minkowski space or asymptotically flat physical space there is, from the asymptotic data, a means of constructing complex self-dual (or anti-self-dual) fields on the Minkowski space of ~Y(or ~‘) space that

M. Ko eta!., The theory of 7-space

133

represent the pure radiation parts of the Yang—Mills fields. It seems likely that this separation must play an important role in a quantum theory of these fields. We mention in the context of quantum theory that Penrose’s non-linear gravitons are a special subset of X-spaces (probably those arising from positive frequency o~) and that further it is possible to introduce a scalar product between gravitons and from that produce a ~ravitonFock space. Since, as was shown in section 3, there are, to each IC-space, a curvature field 1/JA’B’C’D’, as well as a chain of potentials, 0ABCD, YABCA’, HABA’B’, FAA’B’C’ one can try to duplicate the classical Pauli—Fierz.scalar product [46]between two graviton states ~ and 7C 2.

if

=

1,bA’B’C’D’ VA

(1)

(2)

dS~’

where the integral is taken over a real three dimensional space-like surface in Minkowski space. Unfortunately, in our case, though the potentials are well defined, the complex conjugation operation is not well defined on the complex manifold; if, however, the ~‘-spaceis asymptotically flat so that its CI~(R)can be identified withobtains the physical ne can then take the integral over the real slice of 4(M).One after CI~(M) a lengthyocalculation the C5~,i.e. on ~

(~iI7C

2)iJ(i)o~â~du dfl0. (2)

Though the connection between ~‘-space theory and quantization appears promising, it is the

connection between ~‘-spacetheory and classical equations of motion that is both astonishing and enigmatic. As was seen in section 5, integrals, over the good cuts, of the longitudinal parts of the real fields as well as the use of the physical space Einstein—Maxwell equations, produced momentum and angular momentum fields satisfying classical looking equations of motion on the f-space. For example, beginning with either the uncharged or charged algebraically special Einstein—Maxwell fields, we obtained, with no approximations and no ad hoc assumptions that the center of mass motion in k-space was either geodesic motion or motion induced by the Lorentz force law with the Maxwell field being the pure self-dual radiation field just mentioned. One obtains similar results for the evolution of the angular momentum. These results are beautifully simple but mystifying; how does one physically interpret a complex

world-line in a complex manifold; what is moving and where is it moving? (It was pointed out in a different but related context that intrinsic spin angular momentum could sometimes be interpreted as a displacement of a particle into the complex extension of Minkowski space. This is perhaps relevant here [38].)Clearly, much work and a bright idea are needed here. However, even in the absence of an interpretation of the equations of motion (and, of course, of the entire IC-space idea) it seems likely that by coupling the Einstein equations to the Yang—Mills fields and constructing the X-spaces associated with the asymptotically flat solutions, it will be possible to derive the Yang—Mills equivalent of the Lorentz force law. In conclusion, we recall the strong and intimate connections- between ~‘-space theory and the Penrose Twistor Theory with the most natural generalizations of flat-space twistor theory by the deformations of the complex structure leading immediately to the i-space ideas and with a slightly altered procedure to the self-dual Yang—Mills theory [19].If twistor theory and self-dual Yang—Mills theory are of physical importance, as they seem to be, this is clearly true also for IC-space theory. It

134

M. Ko eta!.. The theory of 5-space

would be a cruel and unaesthetic God who would lay such a scheme before us and not have it mean something.

Acknowledgments The ideas and influence of Roger Penrose are evident throughout the theory of LW-space and it gives us great pleasure to acknowledge his seminal role in the continuing development of the subject. We wish to thank Dr. W. Halliday for permission to quote his, as yet unpublished results on the asymptotic solution of the Einstein—Maxwell equations in appendix 1, and Dr. G.A.J. Sparling for permission to present the solution of the good cut equation in section 2.

Appendix 1 In this appendix we summarize the results of integrating the spin-coefficient equations in the

neighborhood of 5~for asymptotically flat space-times. If the tetrad and coordinate system assigned to J~and its neighborhood by the choice of conformal factor and origin cut of section 1 are used in the spin coefficient equations (actually, it is more convenient to use physical space variables e.g. r = 1’ + O(P2), and the vectors la, fla, ma, lña) one obtains for the asymptotic solution of the Einstein—Maxwell equations* in a general NU coordinate system Xa = (u, r, C~C) the following: the tetrad vectors have the form I

—

—

ôxa

—

—

~9r’

(Al-i)

n=n~=j_+Ui_+X~”-, A =2,3

with

(C, ~)‘

=

=

w

=

~

+ O(r2);

~OA

=

(—2P,

o°/r+ 0(r2),

0), (A1-2)

=O(r3), U = —(y°+ ~°)r + U°+ O(r~);

XA

We are indebted to Dr. W. Hallidy for permitting us to use some of his, as yet unpublished, results on the asymptotic solution of the

Einstein—Maxwell equations.

M. Ko eta!.,

The theory of 7-space

135

the spin coefficients are p=

hr + O(r3);

—

r = O(r3);

A

a

=

~

=

~jO/,.2+

O(r4),

a°/r+ O(r2),

=

fJ°/r+ O(r2),

y

=

‘y°/r+ O(r2),

=

A°/r+ O(r2),

~

=

~°/r + O(r2),

(A1-3)

v = v°+ 0(r’),

with ,,°~~‘2P/P• ~ê/3u a°= —f3°=

—~

~ ln F,

~(P/P)

~o=

(A1-4) 0)0 =

U°=—~lnP.

~

The Weyl tensor and Maxwell field become çfro =

ç11~/r5+ O(r6),

=

çb~/r~ + 0(r5),

=

~/r~

=

1/~/r2+

1//3

~if4 =

+ 0(r4),

(A1-5)

0(r3),

~r~/r+ 0(r2),

and =

q~’i= =

çb~/r3+ 0(r4), ~?/r2

+ 0(r3),

4~/r+ 0(r2),

(A1-6)

136

M. Ko eta!.. The theory of 5-space

with the relations = ~2~o

—

—

+ ~°A°— o°A°,

~2~o

(A1-7) (Al-8)

2(j~/p)— A°+ 2A°P/P

=

(Ai-9)

and final equations —

3~frO

p/p

=

~

—

3~P/P

=

—~1/i~ + 2ff°1/~ + 4qS?4~,

31/is P/P

=

—~44?+ 3u°1/~ + 6~qS~,

çb?—2çb?P/P

=

P/P

=

—

+

tT01/I~+

2~°~° (Al-b)

(Al-li) _?+o04~.

There are also the barred versions of these equations, e.g. =

ö~/r2+ 0(r4),

1fr~= ~

ln P + ~

=

(Al-12)

2(j~/p)~o~

pjp

2ço

etc. If one works with the special conformal factor which yields a Bondi coordinate system, i.e. P = Po = ~ (1 + CC), then as a special case of eqs. (1-26) and (1-27) we have —0.

I0_

‘P4

—

08,

0B, ~0.

,0.,w ‘P3

—

g0_

‘P4

..O

—

(~3 — ,0—~

UOB. •0

-

00

In the purely gravitational case, with 4, = 0, the solution is determined by a knowledge of o~(u, C~C), a function of three variables, together with ifr~,~‘? and Re 1/’~at one particular value of u. To see this, observe that (A1-13) gives 1/’~and ~ from 0°B. Then (Al-b) can be integrated, given initial values, to determine ~, 1/’? and 1/’~. Since the gravitational radiation field is essentially ~ and 1/’~there will always be a non-zero o-°~ in the presence of radiation. Appendix 2. Notation and conventions 1” — flat space null vector. fa (C~C) flat space tangent vectors to a null cone parametrized by the stereographic coordinates C and C. null vector field in flat or curved space. —

—

M. Ko eta!.,

137

The theory of 7-space

~)— null vector field on ~‘-spacefor fixed C and ~ normalization induced by Bondi scaling in physical space. La (Z’~,C~C) = ZJ V null vector field on IC-space for fixed C and C; intrinsic 7C-space scaling. v = VaZa scale factor; V’~is normalized IC-space vector field, V”Va = 2. °A, ‘A, 0,4’, ‘A’ spinor fields on ~‘-space;Bondi scaling. °A, 1,4, °N, 1,4’ spinor fields on ~‘ space; intrinsic scaling. Za °A°A’ Batelle convention; omitting OAA’~ Za(Z”, C,

-

-

—

—

—

}

L0

—

—

—

, Ca~,cci— conformally rescaled physical space-time. K — gaussian curvature of good cut with intrinsic conformal factor. ~, ~‘,~, ~ — potential functions for gravitational field. PA — potential functions for Maxwell field. u — retarded time coordinate. UB — Bondi retarded time. 0.0 — asymptotic shear of a one parameter family of null surfaces. asymptotic shear of a Bondi family of null surfaces. edth operator with Bondi scaling. edth operator with intrinsic scaling. F,, = ~ e0123 >0—definition of dual. = iFaj, if Fat, is self-dual. = iFOb if Fab is anti-self-dual. Fa~b= ~(Fab j*1~ab) projection onto the self-dual part. Fb = ~(Fab+ j*Fab) projection onto the anti-self-dual part. “Left-flat” means self-dual Riemann tensor. “Right-flat” means anti-self-dual Riemann tensor. ‘a, ‘Ia.

..

—

—

—

—

Conventions for quantities associated with ~Wand * spaces * ~2z.~o Rabcd R abcd == 1//A’B’C’D’EABSCD lRabcd — left-flat

0 =0 integrable helicity 2h if positive frequency 1TA’~(1,C) ~ — projective dual twistor space Wa dual twistors ~//ABCD

=

VAA’A

—

References [1] H. Bondi, Proc. Roy. Soc. (London) A269 (1962) 21. [2] H. Bondi, Nature 186 (1960) 535. [31R.K. Sachs, Proc. Roy. Soc. (London) 270 (1962) 103.

[4]R.K. Sachs, Phys. Rev. 128 (1962) 2851.

Rabcd R abcd ==

1/JABCDE A’B’8 C’D’

lRabcd — right-flat

0 =0 integrable helicity —2h if positive frequency

1/JA’B’C’D’ =

VAA,AB

1TA.~(1,C)

P ~ — projective twistor space Z~— twistors

138

M. Ko eta!., The theory of 5-space

[5] E.T. Newman and R. Penrose, J. Math. Phys. 3 (1962) 566. [6] R.K. Sachs, in: Relativity, Groups and Topology; the 1963 Les Houches Lectures (New York, Gordon and Breach, 1964). [7] F.A.E. Pirani, in: Brandeis Lectures on General Relativity, eds. S. Deser and K.W. Ford (N.Y., Prentice-Hall, 1965). [8] R. Penrose, Proc. Roy. Soc. A284 (1965)159. [9] R. Penrose, in: Relativity, Groups and Topology; the 1963 Les Houches Lectures (N.Y., Gordon and Breach, 1964). [10] C. Misner, K. Thorne and J.A. Wheeler, Gravitation (San Francisco, W.H. Freeman, 1973). [11] R. Penrose, in: Group Theory in Non-Linear Problems, ed. A.O. Barut (Boston, D. Reidel, 1974). [12] J. Winicour, J. Math. Phys. 9 (1968)861. [13] That u=Oct’o’°=O,is implicit in E.T. Newman and L. Tamburino, J. Math. Phys. 3(1962)902. [14] E.T. Newman and R. Penrose, Proc. Roy. Soc. A305 (1968) 175; See also ref. [11]. [15] R. Hansen, E.T. Newman, R. Penrose and K.P. Tod, Proc. Roy. Soc. (Lond.) A363 (1978) 445. [16] E.T. Newman, in: General Relativity and Gravitation, eds. G. Sharir and J. Rosen (John Wiley & Sons, N.Y., 1975). [17] M. Ko, E.T. Newman and K.P. Tod, in: Asymptotic Structure of Space-Time, eds. P. Esposito and L. Witten (N.Y., Plenum Press, 1976). [18] A.A. Belavin et al., Phys. Lett. 59B (1975) 85. [19] R. Ward, Phys. Left. 61A (1977) 81. [20]R. Penrose, in: Quantum Gravity: an Oxford Symposium, eds. C. Isham, R. Penrose and D. Seiama (Oxford, Oxford University Press, 1975). [21] E. Flaherty, Hermitian and Kahlenan Geometry in Relativity (Springer-Verlag, Berlin, 1976). [221R. Penrose, G.R.G. 7 (1976) 31. [23]J. Plebanski, J. Math. Phys. 16 (1975) 2395. [241J. Plebanski and S. Hacyan, 1. Math. Phys. 16 (1975) 2403. [25] J. Finley and J. Plebanski, J. Math. Phys. 17 (1976) 585. [26]J. Plebanaki and I. Robinson, Phys. Rev. Lett. 37 (1976) 493. [27]C.P. Boyer and J. Plebanski, J. Math. Phys. 18 (1977) 1022. [28]G. Sparling, private communication. [29] M. MacCallum and R. Penrose, Phys. Reports 6C (1973) 242. [30] R. Geroch, in: Asymptotic Structure of Space-Time, eds. P. Esposito and L. Witten (N.Y., Plenum Press, 1976). [31] E.T. Newman and T. Unti, J. Math. Phys. 3(1962)891. [32]B. Bramson, in: Asymptotic Structure of Space-Time, eds. P. Esposito and L. Witten (N.Y., Plenum Press, 1976). [33]R. Hansen and M. Ludvigsen, ORG., to be published. [34] E.T. Newman and R. Penrose, J. Math. Phys. 14 (1973) 874. [35] R. Geroch, A. Held and R. Penrose, J. Math. Phys. 14 (1973) 874. [36] I. Robinson and A. Trautman, Proc. Roy. Soc. A265 (1962) 463. [37] R. Lind and E.T. Newman, J. Math. Phys. 15(1974)1103. [38]E.T. Newman and J. Winicour, J. Math. Phys. 15(1974)1113. [39] Z. Perjes and K.P. Tod, G.R.G. 7 (1976) 903. [40] E.T. Newman and R. Posadas, Phys. Rev. 187 (1969) 1784. [41] R. Lind, J. Messmer and E.T. Newman, J. Math. Phys. 13 (1972) 1884. [42] E.T. Newman and R. Young, J. Math. Phys. 11(1970)3154. [43] M. Ludvigsen, ORG. 8(1977) 357. [44] F. Rohrlich, Classical Charged Particles; Foundation of their Theory (Reading, Mass.. Addision-Wesley, 1965). [45] G. Debney, R. Kerr and A. Schild, J. Math. Phys. 10 (1969) 1842. [46] M. Fierz, Helv. Phys. Acta 13 (1940)45. [47] R. Penrose, J. Theor. Phys. 1(1968)61. [48] R. Penrose, J. Math. Phys. 8 (1967) 345. [49] See also all issues of the Twistor Newsletter some of which appear in: Advances in Twistor Theory, eds. L.P. Hughston and R.S. Ward (Pitman Advanced Publishing Programme, 1979). [50] M. Ko, E.T. Newman and R. Penrose, J. Math. Phys. 18 (1977) 58. [51]J. Porter, El. Newman and K.P. Tod, ORG. 9 (1978) 1129. [52] K. Kodaira, Amer. J. Math. 88 (1963) 79. [53] A. Janis, W. Fette and E.T. Newman, J. Math. Phys. 17 (1976) 660; G.R.G. 8 (1977) 29. These references contain all algebraically special half-flat spaces. For examples of non-algebraically special half-flat spaces, see the work of Plebanski and coworkers. [54] M. Ludvigsen, Ph.D. Thesis, University of Newcastle-upon-Tyne (1976). [55]E.T. Newman, J. Math. Phys. 15 44 (1974). [56] E.T. Newman, Nature 206 (1965) 811. [57] P.J. McCarthy, Proc. Roy. Soc. (Lond.) A330 (1972) 517; A333 (1973) 317, [58] G.A.J. Sparling, Twistor Newsletter 1. [59] R.S. Ward, Proc. Roy. Soc. (Lond.) A363 (1978) 289.

M. Ko eta!., The theory of 7-space

139

[60] R.S. Ward and K.P. Tod, Proc. Roy. Soc. (Lond.) to appear. [61] N. Hitchin, Polygons and Gravitons, preprint, Mathematical Institute, Oxford. [62]T. Eguchi and P. Freund, Phys. Rev. Left. 37 (1976)1251. [63] S.W. Hawking, Phys. Left. 60A (1977)81. [64] S.W. Hawking, Phys. Rev. D18 (1978)1747. [65] W.D. Curtis and D.E. Lemer, J. Math. Phys. 19 (1978) 874. [66] M. Ko, E.T. Newman and K.P. Tod, in: Asymptotic Structure of Space-time, eds. P. Esposito and L. Witten (New York, Plenum Press, 1976). [67] R. Penrose, in: Battelle Rencontres, eds. de Witt and Wheeler (Benjamin, 1968). [68] K.P. Tod, in: Complex Manifold Techniques in Theoretical Physics, eds. D.E. Lerner and PD. Sommers (Pitman Advanced Publishing Programme, 1979).