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The Conformal Group & Einstein Spaces

The Conformal Group & Einstein Spaces

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The. Con6oJunai. G!toup

&

A thesis presented for the degree of

Doctor of Philosophy

in at Massey University

WILLIAM DEAN HALFORD

January, 1977. THE CONFORMAL

&

EINSTEIN SPACES ABSTRACT

This thesis presents (a) a survey of the use of the from its beginnings to the present time , and (b) a determination of those algebraically special vacuum Einstein -times with an expanding and/or twisting congruence of null , which locally possess a homothetic as well as a Killing symmetry (). Unless the space-time is Petrov type N with twist-free rays , one can restrict attention to one proper homothetic motion plus the assumed Killing mot ion(s). The formalism developed to undertake the systematic search for such vacuum space-times is an extension of the tetrad formalism (1) (2) • used by Debney , Kerr & Sch�ld an d by Kerr & Debney . The spaces which admit one homothetic Killing vector (HKV) plus 2,3 or 4 Killing vectors (KVs ) are completely determined. There are 9 such metrics (12 with 3 degeneracies ) - one admitting 4 KVs , one with 3 KVs , and seven with 2 KVs . Those spaces which admit one HKV plus one KV are not completely determined owing to the equations not being solved in some cases. However , 9 metrics are found, many of which appear to be new.

Petrov type N vacuum spaces with expansion and/or twist which admit a homothety ane poss ible when one KV of special type ts also present , or when the homothety alone is of special type . An extensive bibliography is given .

References: (1) G.C. Debney, R.P. Kerr & A. Schild , J.Math.Phys . 10 _, 1842 (1969) .

(2) R.P. Kerr & G.C. Debney , J.Math.Phys . 1!, 2807 (1970) . iii

ACKNOWLEDGEMENT

It is with the greatest of pleasure that I acknowledge the encouragement and guidance which I have received from my supervisor, Professor Roy Kerr. His deep insights into relativity and differential have been a great source of inspiration . My thanks are due to Roy for the opportunity to undertake this study , which has been an enlightening and enjoyable experience in mathematics .

I want to mention here the debt which I owe my wife Anne and our children Deanne , Sherryn and Leon . Demands made on them were great, and without their ready cooperation this research would not have been possible. Memor gratusque.

Bany thanks are also due to Lyn Stening who typed this thesis. Her patience and skill has been much appreciated. iv

PREFACE

The amount of interest in the use of the conformal group in physics has increased a great deal in the last decade . Most of the current activity appears to be in the microphysical arena , with attention being given to such matters as the breaking of conformal invariance in quantum field theory . But the studj of conformal motions as an external symmetry in the theory of gravitation and cosmology is also developing. This thesis present3 (a) a survey of the use of the conformal group from its beginnings to the present time , and (b) a determination of those algebraically special vacuum Einstein

space-times with an eX?anding and/or twisting congruence of null geodesics, which locally possess a homothetic symmetFJ as well as a Killing symmetYJ .

1 Chapter provides a �rief introduction to the mathe�atical structure of the confor�al group . Besides their group theoretic properties , the place of co�formal motions within the hierarchy of collineations is discussed . Chapter 2 is a survey of the mathematical development of the conformal group and its a??lication to relativity and gravitation, cosmology , and other physical th eories, notably quantum field theory . With the background of the first two chapters , the scene is set 3 in Chapter for the task (b) above. Chapter 4 sets up the formalism which is used throughout the rest of the work . It is an extension of the tetrad formalism used by Kerr and Debney to deterr.�ne vacuum Einstein spaces which possess .

Chapters 5 and 6 contain the bulk of the work involved in determining those spaces which admit one homothetic Killing vector plus 2, 3 or 4 Killing vectors (Chapter 5) or just one Killing vector (Chapter 6). The possibility of Petrov type N vacuum spaces admitting a homothety is considered separately in Chapter 7. There follows a Conclusion , a list of Appendices, and an extensive

Bibliography containing over 460 references . V

CONTENTS

ACKNOWLEDGEMENT iii

PREFACE iv

CHAPTER 1 The Con6oromal G�oup 1 SMvey 2 13 Sc.ena.JU.o 3 59 ForomalMm 4 67 5 Spac.e�.� HKV and 2, o� K-<1.ling V w.U:h 1 3 4 2.c.t.c JtO 101 Spac.e..6 HKV and K�ng Vecto� 6 wUh 1 1 125 P�ov Type Spac.M 7 N 152

CONCLUSION 156

APPENDICES: 1 157

2 160

3 163

4 164 5 165

6 167

7 171

BIBLIOGRAPHY 173 CHAPTER The. 1 Con6oJtma...t Gttoup

1.1 Definitions . Let M be a differentiable n-dimensional manifold and consider the point mapping p - p' ( 1. 1) cp : on M. Suppose there is a geometric object field 0 on M, and pull ' -1 I • back the object O(p ) at p to p by the mapping cp Then we have a geometric object o' (p). Suppose that the mapping (1.1) is a 1-parameter local a transformation of coordinates (x ) on M a 'a .n 1 rn •• x - x (x 3,t· ) , ( a, 2 , ) ,n , ( 1. 2) 't" = r � .....Q = 1 , •.. generated by a vector field X at p. Geometrically , this means that

cp takes the point p a parameter distance t along the integral curves of X, with initial point p, given by the set of differential equations a dx /dt (1. 3) = -fl. The Lie derivative of the object 0 with respect to X is defined at p to be

= lim O' (p) O(p)} . ( 1. 4) �t - t-O It follows (see e.g. [ 1] - [3]) that , if we adopt a coordinate basis {o/o�} {o } at p so that �o , the Lie derivative of a function = � X = � f on M is ( 1. 5)

The Lie derivative of a vector field Y on M is

X,Y] [Y ,X YX ( 1. 6) J. y = [ = - ] - XY - X or , in local coordinates,

( 1. 7)

For any vector fields X and Y, [ , ( 1. 8) ! [X,Y] = £x fyJ' where the bracket is defined in (1.6).

The Lie derivative of a tensor field S of type (1,2) , say , with respect to X is given by S ( Y , Z) [ X ( Y , Z) ] ( [ X, Y Z ) ( Y , X Z] ) ( 1. 9) (£.X ) = , S - S ] - S [ , or, in local coordinates ,

( 1.10) 2.

For a with symmetric g and affine connection (1.10 ) gives r ( 1. 11) £ 0t:Jy = flol-lg�Y + �yot:lfl + g�}yfl. The local components of the Lie derivative of ! are given by

+ ( 1 . 12) = �Sy

) a where {�Y· , R R are the local components of the metrical connection f-l 1-l f-IY and the curvature tensor respectively , and the semi-colon denotes covariant differentiation with respect to the connection. Two Riemannian manifolds M, M endowed with metrics g, g are said to be conformally related iff 2r/; e g (1.14) g = for some non-zero function r/;. This relationship is calleG homothetic In the degenerate case the if rf; is a non-zero constant. rf; = G, relationship is isometric. In terms of smooth local maps M- M, the 1-parameter � :

infinitesimal transformations (1.2) generated by a vector �ield X M on H are said to form a (local) group of conformal motions on iff

for some positive function � on H; in local coordinates (1.15) lx�v = ·�v · If � is a non-zero constant the conformal motion is homothetic. In case 0 the infinitesimal transformation (1. 15) is a motion W = (isometry ). Using (1.11), we can express (1.15) in the form

. • X X + X = tlr g ( 1. 16) 1-!;V V;!-1 ' -1-lv

· For 1 0 these are the conformal Killing equations, and a vector X which satisfies them is called a conformal Killing vector (C��). When 0 they degenerate to Killing's equations for Killing 1 = [4] vectors (KVs ) X. 3.

The finite transformation equations of the group G1 of 1-parameter transformations (1.2) are obtainc in the usual way by exponentiation. For the infinitesimal transformation generated by a vector field X, the corresponding finite transformation equations are (1.17) t is referred to as the canonical parameter . This result generalizes naturally to a group G of transformations S r �':a. ...a 1 r x = t ( x ; a , ... ,a ) (1.18) in r essential parameters , generated by r linearly independent vector fields X.1., where x. = xYa (i=1, ... ,r) 1. 1. 1-1 in a local coordinate basis {o ) . The commutation relations between 1-1 the group generators are [x.,x.] = c�.x , ( 1.19) 1. J 1.] k k w�ere the structure constants C . . obey the constraints 1.]

k k. c .. = c . ' 1.] ]1.

c ..r c s + c. r c. s + c r . c s. = 0 . ( 1. 20 ) l.J k r J k 1.r k1. J r This last relation , the Jacobi identity , may be expressed in terms of the commutators as [ X. , [X. , X ] J + [ X. , [ X , X.]] + [x_ , [X. ,X.]] = 0. (1.21) 1. J k J k 1. k 1. J

If a Riemannian manifold admits a vector field X which possesses the property (1.16) we say that the manifold has a symmetry (conformal or isometric). In general, a Riemannian manifold will not have any symmetries. However, the manifold may admit r linearly X. independent vector fields ;1. generating a local r-dimensional of symmetries. Then the relations (1.19) and (1.20 ) obtain, and the set of all symmetry vectors X.1. on such a manifold forms a of dimension r, the product of the algebra being the Lie bracket [ , ]. It is well known (see e.g. [5],[6]) that the maximal group of motions of a Riemannian n-space has dimension 21 n(n+1), while for conformal 4.

motions the maximal group dimension is � n+1)(n+2). If these maximal cases occur , the space is flat , conformally flat respectively . The full group of symmetries of a manifold may include some discrete ones , such as space-time reflections , wh ich are not generated by Killing or conformal Killing vectors . In what follows we shall be concerned only with groups which do not include discrete symmetries i.e. the symmetry groups will be continuously connected to the identity.

1.2 Integrability conditions . Substituting (1.15) into (1. 12) we find a. a. £ ( } - s:.O. ' - a. X f3y -?1 osLy + vy� 'f3 gSy,'t'J, , ) (1.22 ) a. a.v , ; o = [ ] where�. a. a.� and�· g V If we put (1.22) into the result 6 �. . ( 1. 23)

we obtain a. a. · R = o .t .!c S�y;� ( 1. 24) � x �Sy 2

where we have written � a. ; � ,a. . Defining the Rice� �e�sor a�d curvature scalar by R a. Sy = Ra. Sy contraction of (1.24) gives - 1 1 � XRf3y ? ;� i.. - n-2) y;S - 2 gpy� ( 1. 25) and £ = - �R- XR (n- 1}W� ( 1. 26) ;� Ma�ing use of (1.24), (1.25) and (1.26) we get a. £ X c� f3y = 0 , ( 1. 27) where

( 1. 28) 5.

When written out in full, (1.27) reads (see [5], p. 285, or [7])

( 1. 29)

Equations (1.27) or (1.29) are the integrability conditions of + x - g � + ) = o ( 1. 30 ) a.;�y \/\ex.� %< a.� Y ga.y�S - g�y�a. which follow from (1.15) and the Ricci identities Xa.; �y - Xa.; yS = X Ifa. Sy ( 1. 31) � L 1 Defining Sy = 2(n-1) gSyR - RSy ( 1. 32) and C a.Sy --1-- (LSy ;a. - L ) a.y;S ' (1.33) = n-2 further integrability conditions can be obtained [6]. Thus we have the following C:Theorem 1.1� In that a Riemannian n-space admit a group of infinitesimal conformal transformations generated by the vector field X, it is necessary and sufficient that the equations i..X ga.S = � ga.S' ( 1. 15) 0 � = ' ( 1. 27) £ ia.Sy (C � £. v ) = X a� ;

( 1. 34)

1 � c = c �·\11 ' � X a.Sy - 2 a.Py � (1.35) p p __ L ( 1 c 1 Ill c 3 �X ca.�y v ) = - _( 4_t L. vp ) a.S - a.�y v � ca. ; n- 2 x y 2 'p ; 2 \1 �y 1 c c c - � vSy�a. + a.vy�S + a.svWy) p 1 ( C c c + 2 � � PSy + �S a.py + � a.SP), ( 1. 36)

and further� higher order conditions (C £ X a. ) = • ' �y \1 \1 ' ' ' ; 1 2 ( £..X Ca.S y; v? . • • • V 1 . . ) = , 6.

be algebraically consistent with respect to �' �a. , � and

X " If there exist among equations (1.27), (1.34), (1.35), a.;1-'Q (1.36), .•. exactly s equations which are linearly independent among themselves and linearly independent of (1.15), then the Riemannian n-space admits a group of conformal transformat ions of dimension %

Theorem 1.1 reduces to the corresponding one for motions when * = 0 (see [8], p.62). Collinson & french [9] have expressed the conformal Killing equations (1. 15) and the integrability conditions given by the equations of Theorem 1.1 explicitly using a null tetrad and the Newman-Penrose formalism [10].

1.3 Group structure. O(p,q) is the (real) non-compact which leaves invariant the

E(p,q) is the corresponding pseudo- i.e. O(p,q) plus translations . The 0(3,1) of rotations and "boosts" has M six generators �� which form a Lie algebra with the Lie bracket [ , ] of generators as the product. If we add to these the four translation generators P we obtain the Poincare , or inhomogeneous 1-l Lorentz, group E(3,1). The conformal group � is obtained by adding D five generators K and to the Poincare group . The 4-vector K� 1-l generates the special conformal transformations (SCT), and D is the generator of dilations. The conformal group � is a non-compact 15-parameter . a. If (x ) are local coordinates on a Riemannian 4-space (space-time) , the action of the 15 generators of the conformal group (�, � is as follows � = 1,2,3,4): 7.

x 1-l' l-l p Jl- = )l + b ' 1-l ' V MI-L V Jl- x 1-l = a�-L V X a�-L V = -aw ' ' ' > o, D Jl- x1-l = )..)l ).. a scalar 2 (1.37) K ,f- x'l-l = (}l- Jlx)/t,, 1-l c2 l-lx 2 2 2 ,fx, ll = 1 - + C X ' X = 1-l 1-1 2 2 l-l c = c e . 1-1 The SCT generators and translation generators are related by N P N = K , 1-l 1-l where N is the inversion operator

The SCTs depend upon the four parameters � and form an Abelian which is continuously connected to the identity (c=O). However, inversion is a discrete operation on points of the manifold, so N is not an element of the connected group . Writing the generators in the form

P 1-1 = oil'

- X 0 ' V 1-i (1.38)

where { g = diag - , + , + , + ) , the Lie algebra is given by [M ] = M M - �-tv'\p �PM V ).. + g V)..I-L P - �).. vp �p�)..'

= P [\Iv•P)..] �A I-1 �)..Pv' [ K K K \Iv' )..] = �A I-1 �).. .v' ( 1. 39) [PI-1,�] = �VD M1-t v'

[PI-l,D] = p ' 1-1

[K ,D] = - K 1-1 1-1' all other commutators vanish. 8.

(Note: It is conventional for most physicists to take � and o IJ. as conjugate variables in the sense that o is replaced by id , IJ. IJ. introducing a factor i on the right hand side of (1.39), where i2 = -1. ) O(p,q) groups are defined as the real linear transformation groups on a (p+q)-dimensional real space. Unitary U(p,q) groups are defined as the complex linear transformation groups on a (p+q)-dimensional complex space; with indefinite metric they arise naturally in the study of the geometry of hermitian spaces (see e.g. [ 11]) . It has long been known [ 12] that SU(2,2) is the group of Dirac spinor space, a four-dimensional complex vector space equipped with a metric of signature 0 . The conformal group � is realized linearly in six dimensions as 0(4,2) and in four dimensions by 4X4 complex matrices as SU(2,2). Its non-linear realization in four dimensions is given by the real transformation equations ( 1. 3 7) above. As mentioned earlier we restrict our attention to the group component which is continuously connected to the identity , so it is the special orthogonal and unitary groups ....-:-:.ich are involved. Thus we have the following local isomorphisms:

� � S0( 4,2) � SU(2,2). See also [ 13]. Among all locally isomorphic groups there is one which is simply-connected; it is the universal covering group . for the conformal group on M 4 , the universal covering group is SU(2,2). According to Klotz [14], SU( 2,2) is the connected component of the symmetry group of twistor space , and �istors can be thought of as the spinors appropriate to compactified Minkowski space M4 . The groups SU(2,2) and S0(4,2) are respectively 4:1 and c 4 2:1 homomorphic to the conformal group on M c ([14],[15]). The conformal group is the smallest Lie group containing a subgroup isomorphic to the Poincare group. All connected analytic of the conformal group are not yet known. However,

Belinfante & Winternitz [16] in 1971 classified the 1-parameter subgroups of U(p ,q ) and SU(p,q) by using canonical forms for Hermitian linear operators on finite-dimensional vector spaces with indefinite metric. In an outstanding sequence of papers Patera, Winternitz, 9.

Zassenhaus & Sharp [17] have more recently (i) constructed explicitly all (q+1) maximal solvable subalgebras > of the algebra of SU(p,q) for p � q 0 over the field of real numbers ; (ii) determined all conjugacy classes of the maximal solvable $ 6, subgroups of SO(p,q) and O(p,q) for p � q � 0, p+q with some more general results ; (iii) presented a general method for reducing the problem of finding all continuous subgroups of any Lie group G, with a non-trivial continuous invariant subgroup N, to that of classifying the subgroups of N and the subgroups of the factor group G/N. They applied the method to find all classes of continuous subgroups of the Poincare and Weyl groups with respect to conjugation under the groups themselves i.e. with respect to their inner automorphism groups ; (iv) listed all isomorphisms between different subgroups of the Poincare group , and for each isomorphism class of the corresponding subalgebras found all invariants ; (v) given the invariants of all real Lie algebras of dimension $ 5 6; and of all real nilpotent algebras of dimension (vi) classified all subgroups of the de Sitter group 0(4,1), which is a subgroup of the conformal group , into equivalence classes with respect to inner autornorphisms of the group and found all invariants of the corresponding subalgebras. This systematic development has paved the way for a complete analysis of the continuous subgroup structure of the full conformal group. The subgroups of the Lorentz group have been known for some time [18]. Unitary analytic representations of the conformal group and its subgroups have been considered, for example , in [19].

1.4 Collineations. We can also view the group of conformal motions in the perspective of a classification of Riemannian space-times by means of collineations. A M Riemannian space with curvature tensor � Q y is a..... said [20] to admit a symmetry called a curvature collineation (CC) if ( 1. 40) 10. for some vector field X on M. A subclass of the CCs is the set of special curvature collineations (SCCs ) on M defined by

o, <£ e J > ;� = ( 1. 41) x �Y where denotes covariant differentiation with respect to the ; �y) metrical connection { on M. [6] The affine collineation ( AC) on M is defined by £X{�} = 0 . (1.42) [21] ACs are special cases of pr ojective collineations (PCs ) defined by ( 1. 43) where + = � ; � �; S . (n 1) � A special pr ojective collineation (SPC) is a PC for whic� �;�y = 0 .

The Heyl projective collineation ( WPC) on M is defined "::;j' ( 1. 44) where

.u � . \-. apy 6�'"' Ra y )

�n (1.15), A conformal motion (Conf M) has bee(1.15)n defin ed y ; S 0 and a special conformal motion (S Conf M) is wit�- a = . These form subclasses of the conformal collineations (Con:= C) given by = a a 2 o + o � ;y ; S - g £ x{�Y} s � y � Sy lji ;o.) which is just (1.22), and the special conformal collineations (1.22). (S Conf C) for which r ;a,.,Q = 0 in All of these are contained within the Weyl conformal collineations (W Conf C) defined by which is just (1,27). Motions (M) and homothetic motions (HM) as d�ined earlier form subclasses of the ACs and S Conf Ms . 11.

CCs belong to a more general class of Ricci collineations (RC) g1 ven by 1. ( 45) The hierarchy of collineations is displayed in the [20]: following diagram due to Katzin, Levine & Davis

- 0 W Conf C Ra. �-

Conf M I

I S Conf M

M

The arrows indicate direction of increasing generality, linking subclass to class. The broken arrows indicate a relationship only when R 0. al3 = The significance of these collineations is that (a) they represent an invariant classification of Riemannian space­ times on the basis of symmetry groups, which includes and extends the Petrov classification [8] ; they serve as a source of new field conservation laws in general (b) [22] relativity e.g. the Komar identity appears as a necessary condition

for the existence of a CC - see also tne paper by Trautman [23]; 12.

(c) geometrically , the existence of a sec implies the existence of a cubic first integral of a geodesic particle trajectory in space­ time; (d) every space-time with an expansion-free, shear-free, rotation­ free, geodesic congruence admits groups of CCs , in particular pp-waves do. Further details are given in [20]. 13.

CHAPTER SuJtve.y 2

There is a vast literature on the conformal group , nearly all of it in the form of original research papers , although the basic facts are contained within the books of Eisenhart [sj, Yano [6], Petro� [8] and Schout en [21] in particular . I have attempted to give here a fairly extensive bibliography , but it is not claimed to be exhaustive m- to library resources available to me being limited. I have chosen to present this account in several sections : ( 1) the conformal group in general; (2) the us e of tne conformal group in physics ; (3) its application to relativity an d gravitation in particular ; ( 4) cosmological applications. liaturally there is a great deal of overlap among the sections since many authors have in a single �ork dealt with matters pertaining to more than one area. Further , this survey is "local" in nature i.e. the global properties of the conformal group are mentioned o�ly

b�iefly; for a recent account of a global nature, see e.g. Yano �24].

2. 1 General.

In this section I s�all comment briefly on the �a?ers in c�ronological order . The Di�liograpnical reference is [2c]. Special importance is at tache� to the work of Brinkmann . As long ago as 18U7 Liouville was using the idea of conformal mapping of spaces. It is now well known that e·:ery analytic function of a single complex variable is a confor�al map?ing in some domain of the plane. However , these mappings do not form a Lie group , because no finite-dimensional algebra can be associated with them (see Guggenheimer [25]). In higher dimensions the situation is different. If one extends Euclidean 3-space by the addition of the point at infinity , one can give this extended space the coordinate 3 structure of the 3- s in Euclidean 4-space. Then inversion in a 2- sphere in Euclidean 3-space reduces to orthogonal geometry on 3 s in Euclidean 4-space ( Guggenheimer). Liouville proved that the 3 conformal mappings in S form a Lie transformation group generated by simili tudes (Euclidean motions + dilations) and inversions . 14.

Although Klein (1872) was aware of the local isomorphism between the group of collineations in flat 5-dimensional and the conformal group of point transformations in a 4-dimensional space imbedded 1n the projective space, it was not until the work of Lie & Engel at the end of the 19th century that the study of the conformal group as a continuous group of transformations began systematically. Most of the work has been carried out in the last 40 years. Bianchi (1902) announced that "the most general of a Euclidean n-space upon itself for n> 2 is obtained as the product of inversions with respect to a hypersphere , motions and transformations of similitude". Fubini (1903) showed that two infinitesimal conformal transformations of a Riemannian n-space (n> 2) cannot have the same G. paths. Another of his theorems is: If a group r of conformal transformations of a Riemannian n-space admits minimum invariant G varieties of order m, the r may be reduced by means of a transform­ ation of variables to a group on m variables with only m linearly independent transformations . Kasner (1913) began the development of the of sets of curves. Weyl (1918,1921) characterized conformally flat spaces of dimension> 3 by the vanishing of the tensor (1. 28) which bears his name, and published his unified field theory which made use of a subgroup of the conformal group . Kasner (1921) proved a special case of a theorem of Brinkmann (Theorem 2.1 below). Schouten (1921) obtained a necessary and sufficient condition for a given Riemann space to be conformally flat (see

Section 2. 3, p.37 ), and Schouten & Struik (1921) proved that if an Einstein n-space (n> 3) is conformally flat it must be of constant curvature. Brinkmann (1923, 1924 ) proved that a conformally flat Riemannian n-space can always be imbedded in a Euclidean (n+ 2)-space, and considered the question: "Hhen can a Riemann space be mapped conformally on some Einstein space?" He determined (1925) all 15.

Einstein spaces which can be conformally mapped non-trivially (i.e. non-homothetically) on Einstein spaces. His main results appear in the following three theorems :

Theorem 2.1: The only Einstein 4-spaces which can be conformally but non-homothetically mapped on (possibly different ) Einstein spaces are spaces of constant curvature .

Theorem 2. 2 : V An Einstein space n can be conformally but non­ homothetically mapped on another Einstein space iff its metric takes the form 2 -1 2 + 2 ds = f dcp f

K = R/n(n-1), R being the curvature scalar such that R = a.S ( R/n)ga.S. ·'·

Theorem 2. 3: The most general form of the metric of an Einstein v4 which admits a conformal but non-homothetic map on some (possibly different ) Einstein 4-space is 2 + + f(x,cp) dx + 2 ds = 2 dx dy 2 dcp dS 2 dcp g(y,cp) dy dcp. If f and identically , the most general Einstein xcp 1_0 �cp � 0 4-space v4 conformal to v4 is given by - 2 - 2 = 2 ds ( arnY + b ) ds , a, b constants. If f idenL..:ally but ;; neH coordinates can be xcp i 0 �cp 0, 16.

chosen so that ds2 takes the form 2 ds = 2 dx dy + 2 d� d8 + 2 f(x,�)dx �. and the most general Einstein 4-space V4 conformal to V4 1s given by

-2 = -2 2 ds ( ax + bcp + c) ds , a,b,c constants. If f 0 and the Einstein 4-space is flat. � -

The homothetic case was dismissed by Brinkmann as being "without interest". This is, in fact , far from being true , and we shall take up this point later. A result of Thomas (1925) is: At corresponding points of two Riemannian n-spaces whose metrics are conformally related by g = e 2!/J g the tensor a� a�' a 11 a v . .a = { } ( } -a(v Q.l.l v } S - + J - g g { ) K�Y y TI'o� vy oy v� �Y � has the same values. Knebelman (1938) showed that if a Riemannian n-space V n admits an intransitive gro�p of motions Gr , there ex.i sts (n-r ) functionally independe�� S?aces conformal to V n admitting the same (1951). group Gr of motions . T�is result was generalized by Yano Modesitt (1935) considered some singular properties of conformal transformatio�s between Riemann spaces. Fialkow (1939-1945) investigated the properties of curves

in a Riemannian space \. which are invariant under conformal n transformations of the S?ace, and studied a Vn imbedded in a . conformally flat V n+ 1 Coburn (1941) found a sufficient condition for two Riemann spaces to be conformally related in terms of the components of their affine connections and principal directions in the spaces. He also gave a theorem for two unitary spaces to be conformally related. Following on Kasner' s work, Coleman (1942) further developed the conformal geometry of 1-parameter families of curves. A leading me��er of the Roumanian school of differential geometers , Vranceanu (1943) studied the geometry of spaces defined by a conformal connection. 17.

Groups of conformal point transformations on even­ dimensional spaces with a non-singular skew-symmetric metric tensor received the attention of Lee (1945). De Cicc (1946) showed that the Kasner measure of horn angle (in the Kasner plane) is conformally invariant, and

Kasner & De Cicco (1947) proved that the only groups in the set of harmonic transformations of the real cartesian plane are the conformal group , the affine group , and the subgroups of these two . Rozenfel'd (1948) studied the conformal of m- and m-planes in metric n-spaces. Su (1949) generalized the concept of Lie derivation to attack the problem of whether a space with a normal conformal connection admits infinitesimal transformations . A concircular transformation of a Riemannian n-space is a conformal transformation of the space which maps geodesic circles onto geodesic circles . The Japanese school has done considerable work in this area. See e.g. Yano & Adati (1949). Kuiper (1950) treated the conformal grou� as a subgroup of t�e general (n+1)-dimensional projective group leaving invariant a quadric hypersurface of n c�mensions in the projective (n+1)-space. Homothetic correspondences between Riemannian spaces V n were investigated by Shanks (1950 ). Amongst his results �as : A V n is flat iff it admits a group of homothetic transforoations of dimension %n

Yano (1951) showed how , with the use of the Lie derivative , the results of Shanks (1950) may be obtained more easily . Yano also proved that an Einstein n-space given by R = (R/n )� with R i 0 �v v cannot admit a proper (i.e. non-isometric) homothetic motion , thus strengthening Brinkmann's (1925) theorem. He also generalized a theorem of Knebelman (1930 ), obtaining: If a Riemannian space V n admits an r-parameter group G r of homothetic transformations (r

The invariant construction of the geometry of c hypersurface of a conformal space by means of the subgroups of the con=ormal group associated with the hypersurface was the subject of a paper by Akivis (1952). Dzavadov (1952) generalized to pseudo-Euclide�� spaces a

result due to Study (� 1909), namely , that the conformal transformations of a Euclidean space of 3 or 4 dimensions can be represented by linear fractional transformations of quaternions. Kurita (1953,1955) did some. work on projective and conformal correspondences between Riemann spaces. In particular , he re-derived certain results on conformally flat spaces of imbedding class one due to Schouten (1921), Matsumoto (1951), and Verbickii (1952). Kobayashi (1954) proved that in certain important cases the group of transformations which leaves invariant an infini'esimal connection in a differentiable fibre bundle is a Lie group. These cases include the affine , projective and conformal groups . 19·

Takano & Imai (1954) contributed a theorem on the conr�rmal invariance of coefficients of the connections in subspaces of a Riemannian space. The canonical forms of the metrics of Riemannian spaces which admit r independent infinitesimal conformal transformations whose paths constitute normal or geodesic congruences were determined by de Vries (1354). It is well known (see e.g. Eisenhart [5], p.223) that the coordinates of a Riemannian n-space V n can be chosen so that the contravariant components of the generators of 1-parameter � infinitesimal transformations are 6 . It seems that de Vries � = 1 was the first to use this result to give the following general result : A canonical form of the metric of a Vn which admits an infinitesimal conformal transformation generated by (i) the non-null vector X, a (ii) the null vector X, with components � in local coordinates (x ) is 2� a 13 e hal3 dx dx ,

= i = = h h (x ) (a ,P 1, ... ,n;i 2, ... ,n), a13 aS , and (i) h11 = ±1, = 0 = (ii) h11 , h12 ±1 , 6� respectively , and � = 1 in both (i) and ( ii). His results include those of Brinkmann (1925) and Shanks (1950 ).

Ishihara & Obata (1955) showed that if the group of all conformal transformations of a Riemann space is transitive and if there exists a point p which is left invariant by all homothetic transformations of the group , then the space is conformally flat. Kano (1956) did some groundwork on the conformal geometry of a special Kawaguchi space with a specified metric. Blum (1956) considered the equations which the satisfies as a result of the Bianchi identities of any Riemannian n-space.

Brickell & Clark (1962) constructed conformal connections using methods due to E. Cartan.

Pirani & Schild (1966) set up a geometry of conformal space which parallels the usual Riemannian treatment . For example, the procedures which in yield the are modified so that in conformal geometry they yield the Weyl tensor. 20.

Grgin (1968) used a projective geometric formalism for the description of the conformal compactification of Minkowski space and of its invariance groups. The classification of a Riemannian v4 according to groups of proj ective motions was undertaken by Aminova (1971), who showed inter alia that the complete group of projective motions is determined by the group of homothetic motions and one (non-homothetic) proj ective motion (cf. Kuiper (1950 ) and Shanks (1950 )).

De Cicco & Anderson (1971) defined conformally equivalent Riemannian n- spaces , conformal , conformal covariant derivative , and the Weyl tensor in terms of the relative conformal metrlc. tensor g* = g -1/ng g = det(g ), and obtained some results, aS aS ' aS many of which do not appear to be new. Lovkov (1971) determined the generators of a group of homo­ thetic transformations acting simply transitively on a Riemannian v4 with metric of the form ds 2 = -dx 2 + e -2zdy 2 + 2e -zdy dt - x 2 dz 2 . In addition , he deter�ined all Riemannian 4-spaces adm�tting this homothetic group. Egorov (1971) and Egorova (1971) studied how.othetic motions in Riemannian spaces with 2 ( i) ds = 2dx1dx2 + and ( ii) 2 s 2 2 +co h x1 (dxm+1 + ... + dxn ) and proved that these groups have dimension (i) n + �(m-1) + %< n-m-1)(n-m-2) + 1, 1 1 ( ii) n + ¥m- 1)(m-2) + i'

Suguri & Ueno (1972) gave an incomplete summary of known results. However, included was the result , first proved by Yano (1951), that we shall use later: the commutator of two infinitesimal homothetic Killing vectors is a Killing vector. Another significant result proved in this paper is: 21.

If g and g are conformally related metrics on a Riemanr.ia.• syace M, and M is not conforrnally flat , then the complete isone��; group on (M,g) is the complete conformal group on (M,g). This result , apparently first obtained by Defrise (1969) (see under �e�rise-Carter (1975) below), may be restated as : the conformal gro·..:-;, on (1-:,g) is conformally isometric. Yano (1951) had previously �r��ed the theorem for the case when the conformal group is homothetic. Suguri & Ueno also proved some theorems regarding conformally flat c�.ifolds.

Sigal ( 1973) showed that for Petrov type �; ·iacuurr. spaces the Lie derivative in the principal null direction c� ��e Wey� tensor and its dual is equal to a conformal transformation a:-.: a duality rotat ion of the original tensor. Mavry�ev (1973,1974 ) showed that the rela���� �et�een analytic functions and conformal transformations o� t:-.<:: ::o;::,:lez plane does not generalize to a complex matrix representat�s:-. �� s;;a-:::e-time (cf. Guggenheimer [2:] , py.223-224) , and appliec cc::-�:::-=a� 7ra�sform­ ations to the ecuations of relativistic mechanics.

A topological descri?tion of Killing ve:�:::- ��e::� and conformal Killing vector fields , aimed at a unifie: ��a:��a���'=

. (Poincare - Bendixson) theory of stationary and co:- .:"::-=-.2.�:�.­ stationary timelil

Hori, Sakoooto & Sato (1974) presente� a �=�-::.�e� realization of , which is line� �:.a:- �he ?ci.ncare transformations , using a method based on the techn.:.�·..:a :.e·:elo;eG. by

Coleman , Wess & Zumino [59] in connect ion with chira2. s:.--==.e7r:.; (see also Borisov & Ogi evetsl

Loewner & Nirenberg (1974) made a useful c���i.�uti.on to mathematical physics with a paper on partial differe��:.a: e� a�ions invariant under conformal or projective transformatio�s.

Tilgner (1974) gave a description of the f� collineation group and the conformal group of arbitrary finite-dic��si.onal pseudo­

orthogonal vector spaces over the real field. He arg�e:._that sets of

(finite) rational transformations like special collinea�i ns a�d SCTs

form a groupoid, rather than a group , since there are �2�-zero

denominators . With every such groupoid there is ass�-.:.ated a unique Lie group. 22.

Using the modern theory of bundles Schmidt (1974) showed that a conformal structure on a manifold M defines in a natural way two principal bundles over M and a parallelisation on one of these , which then can be used to define a boundary of M in analogy to the b-boundary wh ich is intrinsically defined by the Lorentz metric

(Schmidt 1971, Penrose 1969, and Hawking & Ellis [1]). A similar construction is possible for a projective structure on M. The general mathematical background is the theory of prolongations of G-structures (see e.g. Kobayashi 1972) but this theory is not used by Schmidt. Fedi�cenko (1975) considered special conformal mappings of Riemannian spaces . No details are available to me . Lord (1975) applied generalized quaternion methods in a paper on conformal geometry . Mayer (1975) considered the compactified Minkowski space 4 M c as a closed subspace of a 5-dimensional projective space, and investigated the transformation properties of vector and tensor fields 4 or M c under the conformal group . See also Go, Kastrup & Hayer ( 19 7 4) . A Riemannian n-space ( n > 3) is conformally symmetric iff its Weyl tensor ob eys c�Sy;v = 0 , and admits a special quadratic first , integral (SQFI ) defined by the symmetric tensor af..lV iff a f..lV ;II. = 0 . Every confor!!lally flat n-space (n > 3) is conformally symmetric, but the converse is not true. Roter (1975) proved some theorems about conform­ ally symmetric spaces with indefinite metric tensors which admit SQFis and applied his results to conformally flat spaces , extending the work of Levine & Katzin [92]. The r independent generators of a conformal group C X.1 r acting on a Riemannian space (M,g) with metric g will also generate ...... 2r/; a conformal group C acting on (M, g = e g). Thus , 1n certain cases, r it will be possible, with an appropriate choice of �. to find a space (H,g) for which the conformal group C r is reduced to a group of either (i) isometries, or (ii) homotheties. In these cases C r is said to be (i) conformally isometric , or (ii) conformally homothetic, respectively . (X.f..l) Yano [6] proved that C is conformally isometric when rank = r. r . 1 This implies that C r is simply transitive and hence excludes all conformal groups with isotropy subgroups. Defrise-Carter (1975) showed that C r is conformally isometric under much more general conditions . 23.

If a scalar concomitant F of the metric g is such that 2P� f = e r, where p is a constant , then F is said to be a conformal scalar of weight p (du Plessis, 1969). F is proper if p 1 0 ; a conformal invariant if p = 0 . A vector concomitant Y is said to 1-! ,..., = a conformal vector of weight p iff Y Y + 2p� , and is proper 1-! 1-! 1-! i 0 = when p . y is a conformal gradient iff Y , Y \1 V,IJ Defrise-Carter 1-!(1 975) proved that a conformal 1-!grou p C acting on a r space (M,g) which admits (i) a proper conformal scalar , (ii) a proper conformal gradient , is respectively (i) conformally isometric, (ii) conformally homothetic. These results include the one obtained independently by Suguri & Ueno (1972) and the special case proved by Yano (1951). Defrise-Carter went on to prove the converse of these results for spaces with positive definite metrics , and conjectured that the converses hold in general . In particular , she proved that a Lorentzian 4-space with no proper conformal scalar is conformally equivalent to the pp-waves , or else is conformally flat . The pp-waves admit a proper conformal gradient and the conformal group is reduced to a group of homothetic motions of dimension 6

(= 5 isometries + 1 homoThetic trans formation), or 7 (= 6 isometries + 1 homothety ) if the pp-wcve metric takes a special form .

Boyer & Kalnins (1976) have given a detailed discussion of the infinitesimal symmetries of the Hamilton-Jacobi equation. The group of point transformations locally isomorphic to the conformal group 0 (3,2) is studied. They show that the separation of variables of the corresponding Hamilton-Jacobi equation in the form of a sum lS related to orbits in the Schrodinger subalgebra of o( 3,2). Recently Berger (1976) announced results relating to the Cauchy development of homothetic symmetries in space-like hypersurfaces of empty and non-empty space-times . Constraints which must be satisfied by the spatial metric and extrinsic curvature in the presence of a nomothetic Killing vector are obtained by proj ecting Killing 's equations into a space-like hypersurface. 24 .

2. 2 Use of the Conformal Group in Physics . Chronology will not be preserved in this and the remaining sect ions , and we revert to separate references in the bibliography . The conformal group has appeared in physics both as an Cexternal ) symmetry group of space-time coordinate transformations and as an (internal ) dynamical group of transformations of' rest-frame states of a quantum system having internal degrees of freedom. 0 Bateman & Cunningh am (191 ) [27] appear to have been the first to explicitly link the conformal group with physics . They showed that Maxwell's equations are invariant under the 4-parameter Abelian subgroup of special conformal transformat ions (SCTs ), and therefore under the full 15-parameter conformal group ; it was already known that the Maxwell equations are invariant under dilations and the Foincare group (translations and rotations ). Mayer (1975) [28] has recently given an account of this covariance working on compactified 4 Minkowski space M instead of the usual Minkowski space 4 c M

In their 1962 paper [29] , as well as giving a brief history

o� conformal transformat ions in physics and a discussior. of conformal

ir.variauce in quantum mechanics , Fulton , Rohrlich & Witten distinguished between three transformation groups , all bearing the name "conformal

group" . These are : (1) the group c 0 of conformal transformat ions in flat Minkowski space ; (2) the group C of conformal transformations g o� the metric tensor �v of a (non-flat) Riemannian space; and (3) the group C, which consists of conformal transformations of the metric plus equations wh ich characterize the tensor nature of �v · called C the "extended conformal group" ; it has c , C and the They 0 g group of all coordinate transformations as proper subgroups , and so includes Einstein's theory of within its domain

of applicability. The group c0 is defined as the set of those transformations in C that transforms flat space into flat space;

either c 0 : Minkowski - Minkowski, or c0 : Minkowski - conformally Minkowski. 0 Haantjes [3 ] showed in 1937 that every element of c0 can be composed of motions and inversions only . Fulton et al. [29] showed that Maxwell 1s equations and the conservation of charge and of energy-momentum are invariant under

C. This appears to have been noticed first by Schouten & Haantjes [31] in 1934. The C 0-invariance of Maxwell' s equations , discovered by

Bateman & Cunningham , is a special case of this result . 25.

The conformal group is the lowest-dimensional Lie group containing the Poincare (inhomogeneous Lorent z) group as a subgroup . The Poincare group is important as the symmetry group underlying the kinematics of any (special) relativistic theory . It is natural that many attempts have been made (see e.g. [32]) to generalize physical symmetries in terms of the conformal group . One might then adopt the definition that a physical theory is conformally invariant if physical laws do not change in form under coordinate transformations of the conformal group . This is not the only possible definition of conformal invariance in physics (see e.g. Sigal [79]). Weyl [33] published his unified field theory in 1918, the underlying group of that theo�J being the 11-parameter group of

Poincare transformations plus dilations , to whicJ-. his na<.,e is given. but His t:-,eory was not fully conformally-invariant , of course, re::-resented an attempt to incorporate the conforr..ally invariant t1ax-v: e.:..l equations into general relativity . The attem�� itself had las�i�6 impact , even though the theory was found to be p�ysically uns =.�isfactory .

The conformal grJ�� is the largest gro�? whic� ?reserves nul2. ]._ine elements [ 32] , so there arises the possbilit)· t:-,at this grou� is an exact symmetry grou? for massless particles (?hotons , neu-cc-i:1os ) in space-time . This conformal symrnetry is bro:-:en, however , oarut [35] in t�e presence of massive parti cles [34] and , as � Sornzin of conformal ha\·e remarked , "limits the us e the gro'J.p to the high energ:: domain" where the rest-nasses of the particles are negligible in co�parison. This situation is in kee?ing with the general observation that in physics S)��etry breaking often reduces the symmetry group to one of its subgroups (in this case , the 11-parameter [ 36] ). Barut & Haugen have argued that by re;:>lacing the usual rest-mass by a scale factor times a new , conformally-invariant [ ] mass (first considered by Schouten & Haantjes 37 in 1936), and by interpreting conformal transformations as a space- and tine-dependent change of scale , conformally invariant equations of motion can be �T itten for massive particles as well . It is knoHn that relativistic wave equations become conformally invariant when the mass term is removed [32],[3�]. In fact , Dirac [32]

in 1S36 g�ve such wave equations for the electrocagnetic field and for 2&. spin-� fields, thereby placing the conformal group firmly in the quantum arena. Pauli [38] considered the invariance of the Dirac equation under the conformal group C defined by fulton et al. (see above ). The next success in this area appears to be that of Hoffmann [39] who in 1948 provided a conformal treatment of the equations of a meson field , but restricted his consideration to invariance under homothetic motions. Glirsey [32] developed a 2X2 matrix formalism in view of obtaining a synthetic expression for conformal transformations as well as for spinor wave equations of Dirac type. Among the first to consider the application of the conformal group to elementary particle physics was Ingraham [40] in 1954. Regarding the conformal group as a group of proj ective transformations on a 5-dimensional manifold of spheres in Minkowski space [41] (point transformations of the conformal group are identified with null spheres , but non-null spheres are also required to form the domain of definition of physical fields ), he develops a theory in which free elementary particles maintain a state of uniform velocity under all motions of the group . This contradicts Einstein 's relativity theory , and Ingraham suggests that one may therefore design an experiment to choose between the conformal and Minkowski for physics . Over the last fift een years an increasing amount of attention has been given to the use of the conformal group as a dynamical (internal symme�j) group in quantum physics , with activity in this area currently being quite intense. In reference [42] we give a list of some of the original and review papers not already referred to explicitly in this section. The books by Ferrara ,

Gatto & Grillo [43] anG. by Barut & Brittin [13] provide fairly recent summaries of the use of the conformal group in quantum physics. Both books contain fairly extensive bibliographies. An earlier paper by Kastrup [44] reports critically on the applications of the conformal group up to 1962. As mentioned before , global considerations are not developed in this account and, apart from their appearance in the discussion on causality to follow , will remain outside the score of this dissertation. We remark in passing, hm;ever , that a very recent treatment of zero rest-mass wave equations under the action of global conformal

transformations L.:1 been given by Post [45]. ,.., � • "- I

Cau al ty s i . u y made i o u The notion of ca s alit has been r g ro s by h a s l as follows : Hawking & Ellis [1] w o define local c u a ity w a convex no a l neighbourhood lf U is rm of a connected C space-time (Hausdorff , ed paracompact ) manifold M endow with a Lorentz metric g and ar2 a if p,q points in U, then signal can be sent in U between nd q i f and on l if a ur P a y p ,q can be joined by c1 c ve lying entir·ely in U, vector and either whose tangent is everywhere non-zero is timelike or null . metric at up This definition determines the g p fa to a conformal ctor [ 1]. The conformal factor may be determined

if one assumes that energy and momentum are conserved locally i.,e.. the presence of matter fields there exists an energy-momentum in V tf,nscr on the open s t U tha f! TIJV e E M such t = ('1. ; \1 For a Hawking global trea·tment , & Elljs lo ,. ri;.;, : 2 :-t �s_<:: lity condition : us l if and only if !lo is ca a the-r>E' -. -_• c __, cbf>'"' c rv s in �acelike u e M. They assume that �l .1.S non- -.,.-

''·- ce _ �pter of re er n give a number · .. b f e [1] of c:0ndit .c •w. _,---. ·.· or near violations of �· iclations causality . f' �"· ?.. LJ ... ' ·., n . ..:...... ,-:: ..:

� • . . r:" , Unl. r . ' - 'i • a. CdUSa1 Space-t :;.. ;::c, :.. s .;OC.e.!_ ' S ve.cse t_" ,- a recent In .. , very l•c.,., <::r' [1�7] ��.:w_ki�:_:: : ��·i. •. new for stron ly I • ·1·C"! "' topclogy ? g ca.u_s a.� ,. .-;dC "'· group of is confm·m::J · •. ··· -·. . 1orp •. l.sm P the group of r- : ;- ,�L · -:,

·' .nkowski spa.ce . The=:; claim is intt'1·rtv�l�,' r t:· : P more "' �-- -

nd physica__ the topo.co�y proposec1 by � ',,,_ .,- ,.._ec.ule , a than Zcf! ' " �3 � ':!r , 9o bel_:_ ntrodu es finer topolo y -.-;h ici1 1" ;·, [ 49] i c a g t.:; e': :: � ;: :1.'�­ .::; 5. cal th n the P-topology of Kin ·:Jc:_, a Hawking , g & HcCa:rthy . Zeeman [50] s owe that the largest h d connected broup or -.-:-;:-ace-time transformat ions ;-1h ich preserves the ca usal relat:ion betHeen v s t Weyl Poincare group + pair:.; of e e nt is he group (= dilations). c o al group violates causality , since a always be �hus the onf rm SCT can changes the timelike o spacelike separation of an arbitrary found which r into a space:ike t l k separation pa tr of events or ime i e respectively . ha�. a geometrical p-coof of eeman 's theo::'em . R•.:ccnt ly Nanda [51] given Z examples in which s l of thz Rosen [52] bives two rever a ernpor< 1 ,l ordering of ever�ts is achieved by a SCT. He ar·gues that

ty violat ion invalidate the group causaU shoul not in itself conformal of This based on his =:1s a sytnmetry group _;;hyslcs . view is interpretatioP 2 8. of space- time transformations as leaving both the manifold and the unaffected , the transformations serving only to map the world-lines and events of a physical process under consideration . According to this non- conformist view , violation of causality means that the causal relations among events of a physical process might differ from the causal relations among corresponding events of a conforrnally-transformed physical process .

Schroer & Swieca [53] have resolved the conflict with causality in quantum field theory by showing that on� is , in general, dealing there with representations of the universal covering group of the conformal group . Caus ality is essentially of topological character , and the only satisfactory way to interpret the conformal group physically in the large is with respect to the topological properties of the manifold on which it acts . The causality violating property of the conformal transformations is usually avoided by restrict io� to infinitesimal transformations . To put this on a more rigorous footjng,

l: Go [54] pointed out that SCTs can be considered on Minko...-s:-: : space M only as a local group of local . They ea� je defined as a global transformat:on gro·.1p on compactifica.t ion of �: :..r:.:-:o;.;ski space [55], but then no glo�a.i. causal structure can be de:<'i;:ed [56]. "'

.. Hmvever, by considering the universal H of 4 4 c (compactified M ), the local causal structure on M can r..ade global. c �E: That is, the four-fold covering group su(2,2) universal o:- -rhe 4 "' conformal group of M acts on E as a transitive group of tra�sformations preserving the causal ordering of events, and allows a coh�ormal metric g on M which is invariant under SU (2, 2 ) . It turns out that the reason for the causality violating property of the (global) finite special 4 conformal mapping on M is that this mapping is not a home::-::-.orphis· : �om 4 4 M onto M . The space (M,g) has no constant curvature anc so admiL� 4 only 7 Killing vector fields , in contrast to Minkowski space M which admits 1 0 . In the above sense M is a physically acceptable non-compact Lorentz 4-manifold having the conformal group as its infi�itesimal and globa.'t transformation group . 29 .

2. 3 Conformal Group in Relativity & Gravi tation . The general relativity theory of Einstein is invariant under the infinite-parameter group of general coordinate transformations

' l-l x = also called the general covariance group . A remarkable recent development was the recognition by Ogievetsky [57] in 1973 that the action of this group can be reduced to alternating actions of its two finite-parameter subgroups , namely , the SL(4,R) and the conformal group . Specifically , he proved that any generator of the general covariance group can be represented as a linear combination of repeated commutators of the generators of SL(4,R) and the conformal group . That is , the infinite-dimensional algebra of the general covariance group is the closure of the finite-dimensional algebras of .!'-''= special linear and conformal groups .

In a more recent paper [58] Borisov & Ogi evetsky deduced Einstein's field equations from the requirement of invariance under the affine and conformal groups realized non-linearly (the realization becoming linear on the Poincare subgr·::>up ), in the same way that tr1e equations of SU( 2)XSU( 2) chiral dynamics are derived in the theory of non-linear realizations of chiral symmetry [59]. They maintain tha� the

analogy between the theory of the gravitational field and the essentially simp ler theories of non-linear realizations of internal symmetries (unitary , chiral , etc. ) suggests new ways of searching for connections between the theory of gravitation and the theory of elementary particles , although they do not discuss any of these "new ways". Freund [60] has generalized these arguments to the case of a conformal graded Lie algebra (orthosymplectic algebra ) Hi.th a non- linear realizat ion over the imbedding superspace. This has possible applications in elementary particle physics , In 1936 Page [61] developed a "new relativity" theory which was invariant not only under coordinate transform:1.tions between two observers moving relative to each other with constant velocity , but Also with constant acceleration. He did not realize that this was just an extension of Einstein 's in flat Minkowski space, based on the conformal group instead of the Poincare group . This was

pointed out by Eng.strom & Zorn [62] who showed that Page ' s theory involved the determination of all coordinate transformations which . 2 2 2 2 2 2 preserve the null l1ne element ds dt - dx - dy - dz 0 a = c = ,

problem solved " by Li e [::_-,] ll':.,· <1 . 30.

Page 's theory was physically meaningful if all measurements were local; distances and times were to be determined by the radar principle (attributed principally to Milne [63] ) , &nd differed from point to point in the space-time manifold. Rest masses were no longer constant ln Page's theory . Robertson [64] claimed that Page 's kinematical theory could be regarded as a special case of his own (Robertson 's) more general theory [65], ar.d that Page 's treatment should lead to the usual classical expression for the ponderomotive force. In reply , Page pointed out that he and Adams [61] had shown that the "new relativity" may lead to a "very different electrodynamical equation of motion" . But just what this difference is is not very clear . Robertson also demonstrated that the relation between two equivalent observers in Page's theory is formally the same as that between two free observers in general relat ive motion in the de Sitter universe of general relativity theory . Bourgin [66] has also criticised Page's theory . Hill [67] joined the fray in 1945 when he considered the problem of determining those coordinate transformations for which the uniformly accelerated motion (hyperbolic motion) of a particle is transformed into another mot ion of the same type , for bot� [;ewtonian and special relativistic mechanics. The symmetry grou� of the characteristic differential equation of hyperbolic motion is the conformal group of Minkowski space. Hill also applied the theory of uniformly accelerated motions based on the conformal group to the so-called "clock problem" of general relativity and found two solutions , both at variance with the us ual theory . The kinematical properties of the conformal transformation group have continued to cause problems in interpretation. As recently as 1971 Laue [68] gave exam?les to show that one of the SCTs transforms world lines of massive particles at rest into world lines of massive particles with different constant accelerations . It has been noted earlier (Section 2 .2) that conformal symmetry is broken in. the presence of massive particles unless one hypothesizes that mass transforms under dilations and SCTs according to m f(x)m , o = c where m is the (usual ) rest mass of the particle , m is a new conformally 0 e invariant mass, and f(x) is a scale factor ��presenting the conformal transformation. Some authors have interpreted m as a rest energy 0 (rather than rest mass ) containing the potential energy associated with

the porit ion (x) of the part icle in an apparent gravitational field 31 .

("apparent " because we are talking about conformal transformat ions in flat Minkowski space ). Fulton , Rohrlich & Witten [29] asserted that such apparent gravitational fields are constant and homogeneous .

This assertion is false because of the property of SCTs whi ch takes world-lines of massive particles at rest into Horld -lines of particles with different constant accelerations , as emphasized by Laue . An earlier example of Rohrlich [69] supports this conclus ion .

A belief which had become firmly established (see e.g. [70]) was that no radiation field is generated by the linear uni formly- accelerated motion of a point charge . This was repudiated by Rohrlich [ 7:] . �1 der a SCT the coulomb field of a charge at rest is mapped into the field of a uniformly accelerated charge , as shown by Haantjes

[ 72]. This latter field contains radiation .

By restricting the interpretation of the term "equivalent observers" in general relativity to mean those observers ....-hose (local ) coorcinate sy stems are connect ed by a group of transforrr.ations which corresJond to physical reality , e.g. Lorentz transformat ions ,

Ue:-�o & Takeno [ 73] cons idered two kinds of "equivalent o:::: se�ver" . In the.::.r first paper , the class of observers of the "second kind" is ch�acterized by their coordinate systems being connected only by the three-dimens ional rotation group when they ar e at relative rest ; when these observers are in relative motion , it was shown that The group of trar.sformations connect ing them contained the SCTs . An extension of eno this theory was given by U & Takeno in the second paper . Meksyn [74] reminded us that transformations from rest fraces to accelerated frames are, in general , singular on certain surfaces .

There is a misprint in his paper , so the transformation of his example is spelt out here. The frame (x ,y ,z ,t ) is transformed to the 0 0 0 0 frace (x,y,z,t) moving with constant relative acceleration by a 1� wx (1+2wx) cosh wt - 1, 0 =

= y, Yo z = z, o k wt = (1+ 2wx) 2 sinh wt , o 2 where w = and is the velocity of light . This transformation ale c 2 becomes singular on the x = -c /41, as can be seen by comput ing the Jacobian determinant of the transformation . 32.

The interpretation of the SCTs as connecting coordinat e frames with constant relative accelerations has been contested by Kastrup [42] on quantum mechanical grounds. The main objection appears to be that the group velocity of the wave packets formed by the eigenfunctions of the Hermitian operators (of the Lie subalgebra of SCTs ) has the same form as that of plane waves , whereas the phase velocity shows the hyperbolic structure usually related to accelerated motions . In quantum mechanics it is the group velocity , not the phase velocity , which describes the motion of particles. Thus if, as he desires , the conformal group is to span both microscopic and macroscopic physics , Kastrup suggests that the interpretation of an accelerated motion be dropped. Another obj ection is that the Schout en-Haantjes postulate [37], that mass transforms locally under dilations and SCTs as on page 30 of this chapter , is at variance with the quantum me chanical view that 4-momenta, and therefore rr.asses , are non- local quantities. Kastrup argues that "the forr.,al local transformation of a non-local quantity has the conse�uence that no conservation law exists , et least not in the usual sense , as one can see from the Klein- Gordon equation with non-vanisr.ing resT r..ass" . This apparent conflict over the kinema�ical in�erpretation of the conformal group seems unresolvable at presen� . As long as one beli eves that relativity can be quantized, there should be a5reement over physical interpretation of the transformations of the conformal group if this group is to be accepted as a symmetry group o� physics .

Imbedding tec:-tniques have been studied extensively in relativity theory (a fairly recent discussion is given in [75]). �e note in passing that among the best results available are : (1) Friedman ' s theorem [76]: An n-dimensional Riemannian space 2 � with line element ds = dx dxv can always be locally and �v isometrically irnbedded in an m-dimensional pseudo-Euclidean space with line element 2 1 2 P 2 p+1 2 p+q 2 ) ..• ) ds = (dz + + (dz - (dz ) - ... - (dz ) , 1 where m = p+q = 2 n(n+1) and neither p nor q is less than the number of positive or negative eigenvalues of respectively . �V In the case of the Riemannian spaces of general relativity 4 and has signature -2) it is known that the imbedding (n = �v pseudo-Euclidean space (a) has dimension 10 at most , (b) must be 33. at least 6-dimensional to imbed a non-flat vacuum space , (c) must have at least 5 dimensions to irnbed a non-vacuum solution . ( 2) Nash's theorem [77]: A -time can be globally imbedded in a pseudo-Euclidean space of at most 46 dimensions . The present ceiling is higher for a non-compact space-time .

Imbedding techniques have been used , for example by Sigal and Ingraham , in the application of the conformal group to relativity. See also Maia [78]. According to Sigal [ 79.] a field theory is conformally invariant if solut ions of the field equations are mapped into solut ions of the field equations by the conformal group , assuming that the field equations and fields are tens or densities under a general change of coordinates . This assumption puts restrict ions on the transformation properties of terms in the equations under the Minkowski s�ace conformal group . In particular, it im?lies that all terms in the equations must have the same scale dimension . Sigal clai�s that it also makes questionable the introduction of conformally i:Y>ariant masses la Schouten Haantjes [37]). To support his claim �emonstrates (a & he that , under his definitio�s an� assumpt ions , the massive ::�ein- Gordon e�uation lS not conformal y invariant . Making use of the local isomorphism between SU( 2,2) and

S0(4,2) , Sigal examines conforr., al invar iance (in his sense) by studying the relationships between geometrical structures and field e�uations in the flat 6-space and an arbitrary 4-space conformal to ��inkowski space . He then studies the imbedding of Minkowski space e;uations in the flat 6-space. He shows that , in terms of 6-space structures , there are only two ways to break conformal invariance , correspon�ing to the addition of mass-like terms and to coupling terms between the covariant derivatives of each geometry . Ingraham's [41],[80] "conformal relativity" deoands that physical laws be invariant in form under the full conformal group . He treats the conformal group as a group of projective transformations in "sphere space" . The 4-dimensional space-time events are represented as nul l spheres in a curved 5-dimensional manifold , described by six homogeneous coordinates. At each point of this manifold there is a local flat 5-dimens ional proj ective geometry . Equivalent observers are defined by the property that they are related transforr.:ationally 34 . by the conformal group i.e. point transformations between observers carry null curves into null curves . (This is an extension of the concept of equivalent observers introduced by Page [61],

Ueno & Takeno [73], and Hill [67 ].) In each local space there is a quadric, and the group of transformations preserving this quadric (the quadric gr oup) induces a group of point transformations in the local 4-dimensional space-time via the imbedding equations . The intersection of this last group and the group of conformal point transformations in the 4-space is called the physical group of the theory . The quadric geometry and physical geometry of Ingraham 's conformal relativity denote the set of all geometric definitions and properties invariant under the quadric group and physical group respectively . The principal theorem of his theory is : The quadric geometry and physical geometry coincide, and the physical geometry is the conformally flat metric 4-space . This result had been established very much earlier by Klein [26] in 1872, so one wonders what the force of Ingraham 's theory is . He claims that his (Ingraham's) theory is a un��ied field theory in the sense thaL it describes force fields which co�?rise several mesons along �ith the gravitational and electromagr.etic fields .

The Kaluza relativity [81] and Einstein 's general relativiLJ- are subtheories , according to Ingraham , and he also extends his theory to what he terms "spinor relativity" . But !aub [82] disagrees with Ingraham 's claim because "the theory of a perfect electrica2-ly neutral fluid moving in its own gravitat ional field does not seem to be contained within this theory" . Ingraham does not appear to have replied to this criticism.

Takeno [83] has studied conformal transformations wh ich transform spherically symmetric (s.s. ) space-times into s.s. space­ times . By definition a space-time is s.s. when its metric tensor �V is form-invariant under the 3-dimensional group of space rotations . The metrics di scussed by Takeno do not necessarily satisfy Einstein's field equations . The main theorems are : (a) Let be any s.s. space-time with metric tensor which is s 0 �v not conformal to the space- time with metric SIIA 35.

2 2 2 2 2 2 � ds = C(r,t)dt - A(r,t )dr - B(d8 + sin e d ), '1: -�� where B is a constant . �f S ( �v) be related to s by 2f(xa) 0 1: = e then is s.s. iff f = f(r,t). �v �v ' S ...

(b) The metric of any conforrnally flat s.s. space-time �s reducible

to the form 2 2 2 2 2 2 2 2 ds = A ( >..dt - (dr r + d8 + r sin 8d� ) } , 2 2 where A= A(r,t) and).,=[a( t) + r b(t)] , and a,b are arbitrary functions of t.

Takeno also determined the groups of infinitesimal coordinate

transformations which leave any given s.s. metric conformally invariant .

He obtained the specific forns of the conformal Killing vectors (CKVs )

for the Robertson-Walker metric in the form

2 2 2 2 2 2 g(t) 2 2 -1 ds = dt - f (dx dy dz ), f = e (1 r /4R ) S (L): + + + , 2 2 2 2 where r = y ana' "[;,, = constant . No...- the Robertson-Walker X + + z ' metric is conformally flat , so the CKVs are just those of flat

r-: inkowski space . But Takeno saw fit to obtain the CKVs directly

from the R-W metric T�is was followed specific determin- S ( L). by a ation of CKVs in general s.s. space-times with me trics of the form

2 2 2 2 2 2 . 2 2 � s1(0): ds = C(r,t)dt - A(r,t)dr r d8 - r s�n 8 d ; 2 2 2 2 ds sin S ;;; 2D(r,t)dr dt - B(t)(d8 + 8 di> ), dB/dt "f. 0 ; I ( �): 2 2 2 2 2 2 ds = C(r,t)dt - A(r,t)dr - B(d8 sin e di> ), S + II : o. B const . > Note : At each point of the space-t ime s 11 the tangent space is composed of two 2-dimensional subspaces

�hose metrics are 2 2 v Cr,t) : = C dt A dr 2 ' 2 2 2 V (8,i> ): -- B(d8 sin e di> ). 2 + The 2-space V (8,�) has constant curvature -1/B . When the 2-space 2 v Cr,t ) also has constant curvature , the space-time is said to 2 s11 be of class s 11A. The theorems proved by Takeno will not be restated here .

The following table , adapted from one given by Takeno , indicates in

the final column the space- tillieS of part icular interest because they

contain CKVs which are not Killing vectors . The notation is as defined 2 2 2 above , plus ds = dt - A(t) dr non- constant 2-curvature. " 2 2 } · ds� = C(r) dt - dr 36 .

Dim . of No . CY:Vs Space-time Remarks con formal not typ e group Killing vect .

S(L) Conformally flat 15 5 S (O) (i) Conformally flat I 15 5 (ii) Conformal to S A 6 $ 3 II ( iii ) Static, A=A(r) , C=f(r)g (t) 4 0 2 r h(t) 4 1 (iv) A = f(r)g(t), c = (v) Others 3 0 S ( ) ( i) Con formally flat I 6 15 5 ( ii) Conformal to S 6 $ 3 IIA h § (iii ) BID = e fu-, 4 1 (iv) Others 3 0

S (i) Con-formally flat II 15 5 (ii) s -. not conformally flat 6 r 0 lr. 4 V (iii) S( or 2) "' J 1) S(J (iv) Others 3 c

In s (6) (iii) arbitrary functions of t, § 1 u c::� v are r a:: :: , h = h(W), W = = udr U(r) , vdt = U-t: ·:, U J = V=J V( t).

Ignoring the conformally flat cases , we see that the onl:,- s.s. me-:-ri.cs admitting conformal trans forcations which are not motions are s (o)(ii), (iv) and (ii i). The exterior 1 s1C�)(ii), SchKarzschLc space-time is of class s (o) and is static, so possesses CKVs 1 no other than Ki lling vectors .

Any s . s. geometry admits a shear-free null hy� e'surface .

The same will be true of any conformally related geometry ;(1ich is s. s. , since the vanishing of shear is a conformally invariant rroperty [118].

Derry , Isaacson & Wini cour [84 ] showed that the only "regular" conformally s.s. solut ion to the vacuum Einstein field equations is the Schwarzschild solution. "Regular" here means asymptotically flat plus the following restriction : "The shear-free null hypersurfaces of the s.s. geome try form a regular (sic! ) diverging family of hypersurfaces Kith topology

1 such that consecutive members do not intersect in s2 X E be neighbourhood of future null infinity" . Their proof did not: ap;- ly to N . P etrov type vacuum space-tlmes . 3 7 .

Conformally flat spaces . These admit the full 15-parameter conformal group . Well-known conformally flat spaces are: (1) The Schwarzs child interior solution [85]. (2) The de Sitter space-time [86]. (3) The Einstein space-t ime [87]. (4) The Robertson-Walker metric [88]. T (5) The Friedmann space-times [89] . Conformally flat spaces are amongst the easiest to deal with on account of the vanishing of the Weyl tensor in these spaces (Weyl [26],1918) . and Brinkmann Schouten ( 1921) , Schouten & Struik ( 1921) , ( 1923 ) [26] were early workers on ques tions of conformal flatness . In particular, Schouten proved that a Riemannian n-space is conformally flat iff

R M g M g M g M = ( 1 g S6 - � o - 6 py 0 ) aSy6 + S6 ay t ay a a M - M = 0 ( 2 ) and y ay aS ; ;13

for some symmetric tensor For n > 3 , (1) implies ( 2 ) . Ma s· Levine ( 1936 ) [90] determined the necessary and sufficier1t condition for a conformally flat to admit a group of motions , in V n terms of the rank of a certain matrix involving derivatives of the

conformal factor in the metric of • He also proved that every V n conformally flat space with metric of the form

2 a 2 a 2 ds = e (dx ) /f(�) , R = e (x ) , e ±1 a a a = admits the rotation group as a group of motions ; if other groups of motions are admitted by such a space, then f(R) = aR , or the space has constant curvature and 2 = f(R) ( aR + b) , where a and b are constants. Other results given by Levine are : If a conformally flat space admits a simply transitive group of translations , the space is flat .

A given simply tra sitive g o p G · will be a group of motions of a n r u r conformally flat space iff G is a subgroup of the 15-parameter r conformal group. In a sequel Levine ( 1939 ) tabulated the 20 types of subgroups of the conformal group that can serve as a group of motions of a conformally flat Riemannian n-space into itself , and gave the form of the metric of each of the admi ssible spaces .

• • . ' • space-time conforma lly flat re:1rose , in J" "' • il 'l)' i :,;o�ropic is [ + -r , .. , .. ..., [ ':' ,---.-,10 ,...\' . n . 5 P() (Cnr.i o f B .,..,., .,. �'· . 1.�:-::: )] 38.

Later (1950) Levine proved that a conformally flat , non-flat Riemannian space M can admit at most one (linearly independent ) field of parallel vectors , which if non-null implies that the universal covering space M of M is the product of a one-dimensional Riemannian space and a (n-1)-dimensional Riemannian space , and hence M has constant curvature (see also Yano & Nagano [91]).

In their 1969 paper Katzin , Levine & Davis [20] considered curvature co llineations (see Section 1.3) and proved that if a conformally flat Riemannian n-space (n > 2), which is also a non-flat Einstein space , admits a curvature collineation (CC), then the CC must be a mo tion . In a subsequent paper these three authors (1970b) began the task of determining all conformally flat Riemannian n-spaces (not Einstein spaces ) which admit a CC. This paper was devoted to

CCs which are not conformal motions (Conf M). Levine & Katzin (1970) [ 20] continued the story for the case when a CC is a Conf M. They proved inter alia that there are no non-flat conformally flat spaces which admit a symmetry which is simultaneously a CC and a proper Conf M with lir a non-null vector , Hhere the function � is given by (1.15). They also'�' ; f..! determined the nature of the various non-flat conformally flat spaces whi ch admit a S Conf M and a null field of parallel vectors for each of the three canonical forms of the metric [92]; the group of S Conf Ms was specified in each case.

Conformally flat s?aces admitting special quadratic first integrals (SQFis) i.e. covariant-constant symmetric tensor fields

h , were the subject of two papers by Levine & Katzin [92]. Among f..! V their results was the following : A conformally flat Riemannian n-space C which is (i) flat , or (ii) of non-zero constant curvature , n admits (i) n(n+1), (ii) one SQFI respectively . If a C of non­ % n constant curvature admits more than one linearly independent SQFI , then it admits exactly two . Canonical forms of the metric were obtained for these latter C n Ruse [93] mentioned conformally flat 4-spaces as a very special case in his discussion of a quadratic complex of lines in an (n-1)-dimens ional projective hypersurface of an n-dimensional Riemannian space . 39.

Narlikar Karmarkar used 4 independent scalar & [94] 1 invariants of the curvature tensor in a Riemannian v 4 to establish the necessary and sufficient conditions that a spherically symmetric v 4 be conformally flat .

Taub [7] gave the first proof of the theorem : A

Ri emannian V (n 3) admits a n+1)(n+2)-parameter group of n � � infinitesimal conformal transformations iff the V is conformally flat . n Sasaki [95] had previously established part of this result .

Adati [96] studied Riemannian spaces V admitting a family n of totally umbilical hypersurfaces , when the hypersurfaces are conformally flat . Also included was the proof that when a conformally flat V (n 3) admits a terse-forming vector field the n > cr, a hypersurfaces cr(�) = const. are of constant curvature . Matsumoto [26] (1951) proved that a positive definite conformally flat n-space (n � 4) with non-constant curvature is of imbedding class one iff a certain matrix is of rank � 2 and certain inequalities are satisfied ; the matrix and the inequalities involve

only g and R � · A new proof of a theorem of Brinkmann [26] (1923) �V a �Q yu was given, namely , the imbedding class of a conformally flat space is at most two.

Verbickii [97] developed a criterion, involving the second fundamental form of a positive definite Riemannian V (n � 4), n for the V to be conformally flat of imbedding class one . n Vranceanu [98] had proved that in a sufficiently small neighbourhood of a Riemannian subspace V , the metric of the enveloping n Euclidean space E may be written in terms of the fundamental tensors N of the first , second and third kinds and the torsion of the V . n Blum [99] generalized this result to the case where the enveloping space is conformally Euclidean . 40.

Using properties of an imbedding in a flat 6-dimensional space, Stephani [100] found all solutions of the Einstein equations for a perfect fluid or an electromagnetic field, which are conformally flat. These solutions include the metrics of Bertotti [101]. Space-times of imbedding class one have also been studied by Barnes [102] who showed that a class one perfect-fluid space-time has at least one of the properties (a) conformal flatness, (b) geodesic flow , (c) it admits a 3-dimensional group of isometries with 2-dimensional spacelike paths . The conformally flat nature of gravitational fields in perfect fluids has also been discussed by Obozov [103] who proved that geodesic flow lines imply the conformal flatness of the space­ time (cf. Barnes ). Sigal [79] generalized his discus sion of conformal invariance to the case where the 4-space fielc e�uations are imbedded

in a conformally flat 6-space (see also page 33 o� this chapter ). ·[104] Kloster , Som & Das investigate� t�e class of conformastationary vacuum metrics ; these are t�e sTa�ionary metrics

whose background 3-space is conformally flat . They found that there

are only three such metrics , one of which is the �: solution, which belong to the Papapetrou-Ehlers class [105]. Their proof that there cannot be any metrics outside of this class was faul�j, and this possibility still remains open. The initial-value problem of general relativity is that of constructing a complete set of Cauchy data on a spacelike hypersurface for Einstein 's equations . These data are subj ect to

initial constraints. O'Murchadha & York [106] treated the special case of conformally flat metrics on the initial spacelike hypersurface. 41.

McLenaghan, Tariq & Tupper [107] employed the Newman­ Penrose formalism to obtain a derivation of the most general conformally flat solution of the source-free Einstein-Maxwell equations for null electromagnetic fields . The metrics are of the form 2 2 - 2 ds = 2q 2 du dr - 2 dz d , (u)zzdu + z which are the conformally flat members of the exact plane-wave family of solutions of the Einstein-Maxwell equations . Peters [108] solved the equation of geodesic deviation ln conformally flat space-t imes in a covariant manner . This enables one to express , say , the derivatives of the parallel propagator in terms of other geometrical quantities , independent of any particular coordinate system. The solution is given by Peters as an integral equation for general geodesics. Zund [109] gave an examp le of a conformally flat space-time with recurrent curvature ��ich represents pure radiation in the sense of Lichnerowicz [110]. Levine Zund [111] ext ended this T�e� &

discussion and gave , in p2'-:icular , the theorem : A conformally flc.t space-ti::-,e in '1-::-�icl-. -:-.ere is a null parallel vector field Hh icr is a gradient re�resents pure �adiation . Eall [112] p!"':)·,·e-:: that a null electrooagnetic field in a conformally flat space-tir..e must necessarily be expansion-free and twist-free. This resulT �so holds for null electromagnetic fields

whose repeated principal n·�- direction coincides with the repeated

principal null direction c� the Weyl tensor of a space-time of Petrov

type l\ . This was an ele���T extension of the work of Levine & Zund [111]. Based upon t�e paper of Dirac [32], a new form of conformally invariant waYe equation was studied by Sokolik [113] , and the gravitational int�action with respect to conformally flat space was investigated.

Pavelle [114] �'d Thompson [115] have discussed the Kilmister-Yang equations [116] (a K-Y space is Riemannian V which a n = satisfi es locally R � ) . Pavelle argued that conformally O.t-'� ; y ::;:. flat solutions should not �eo.y allowed;p (are unphysical). Thompson proved some theorems on co��ormally flat spaces , including : (a) For � 4 , every c ��ormally flat V with constant scalar n n

curvature is a K-Y space. (b) The class of conformally flat solutions 42.

1 3 of the K-Y eq uat ions is det ermined by solut ions p = Rp of o2 6 ' where R is the constant of v4 with metric

ds 2 = p 2 do2 , cto2 the Minkowski space metric, 2 When R 0, the conformally and o is the D'Alembertian operator . = flat spaces are determined by solutions of the wave equation .

Curvature collineations . A theorerr. due to Collinson [ 117] states that the only CCs admitted by an empty (�instein) space-time , not of Petrov type N, are conformal motions . Collinson also found the CCs admitted by the plane-fronted gravi tatior.al waves , and this work show ed that empty space-times of Petrov type N do admit CCs which are not conformal motions. He proved thaL the plane-wave metrics admit a 6-dimensional group of conformal motions consisting of a 5-dimensional group of motions and a 1-dimensional group of homothetic motions . In particular , he showed that the metric 2 ds = 2 du dv - 2 dz dz is a plane-fronted gravitational wave which admits the homothetic motion given by the homothetic Killing vect or ( ;<; ) ,_

y zo = uo + + o- 2 u z z z Collinson appears to have a sign wrong in his ex:: :::- ession for this HKV.

Katzin, Levine & Davis [20] (1970a) s:-.owed that every Riemannian V with an expansion-free, shear-free , rotation-free geodesic n congruence admits groups of CCs . In parti cular , the plane-fronted gravitational waves with parallel rays (pp-waves ) do . Geometrical and physical properties o� the Weyl tensor

were outlined by Pirani & Schild [118] in the spa�e-time of general relativity . In parti cular , they gave conformal<: invariant definitions of null geodesics and shear (see Chapter 4 and Sac�s [20�]) and gave an equation which related the propagation of she� along a null geodesic to the Weyl tensor.

The Goldberg-Sachs theorem ([209], see Chapter 4) is deservedly one of the most celebrated in general relativity theory . Very soon after its discovery the generalized Golc�erg-Sachs theorem

was established by Kundt & Thompson [119]. This states : (i) Any two of the following properties ir::;-ly the third : (A) The Weyl tensor C is algebraically special, abcd a with Debever vector k . (B) There exists a shear- free geodesic null congruence a with k as tangent . 43.

ea c d (C) v yn c 0 abc ;d =

c d or � c o if c is Petrov type III , abc;d = abcd

ea d or v c o if c is Petrov type N, abc;d = abcd

where V 0 and V is a null complex bivector . kb = ab ab ( ii) The properties (B)' (A)U(C) are conformally invariant . ( f..) '

This theorem was proved independently by Robinson & Schild [120] whc published their proof about five months later. Kundt & Thompson used spinors , while Robinson & Schild used tensorial methods to prove the theorem. The fact (ii) of conformal invariance was fully established by the latter t••o authors , while the former two merely noted it .

Szekeres [12:] s:udied conformal transformations o= a subset of the following �iercrchy of Ri emannian 4-spaces : � = 0 (a) C-spaces , characterize:. bJ-- C , aJ3y;fl J-spaces , = 0 , (b) Ffla �-'R y;iJ. (c) Einstein spaces , R� V = A.�v ' A. constant

(d) empty spaces , R� V = 0,

(e) flat spaces , R = 0 , �i3y where R and C are the �iemann and Weyl tensors . Szekeres R R �t->Y � t->Y obtained necessary and s�=icient conditions for a space to be conformal (c), to a space of type (a) , or (d), us ing spinors . J-spaces (b) were considered by Thompson [122-. [Kote : C-spaces and J-spaces are special cases of confornally symmetric spaces (see p. 22 ) and Cart an symmetric spaces [123] respectively.]

Debever [124] us ed the local isomorphism between the

Lorentz group and the 3- dinensional orthogonal complex rotation group

S0( 3,C) to develop a vectorial representat ion of bivectors , in particular of the curvature form and connection form. The formalism was applied to conformal changes of the metric, and in particular to specify the choice of conformal paraneter along isotropic geodesics . Included was a proof of the Pira,i-Sc�ild formula [118] on the propagation of the shear along a null geodesic . 44 .

The formalism developed by Debever was described more fully by Cahen, Debever Defrise They showed, for example , & [ 1 2 5]. how the formalism is related to the usual spinor representation. After a characterization of the Petrov types of the Weyl tensor , they introduced suitable canonical triads and computed the curvature invariants. The conformal theory developed in [124] was re-presented, and concise proofs of the Robinson theorem [126] for null electro­ magnetic fields an d the Goldberg-Sachs theorem were given. All homogeneous 4-space solutions of the vacuum field equations were constructed; these are the spaces on which a group of isometries acts trans itively . It was shown that there exist no homC..geneous spaces of Petrov types II, D, and Ill . They also studied spaces admitting "large" groups of isometries (when the stability group at each point of the space is not reduced to the identity ). Mielke [127] has reviewed conformal techniques as app lied to the initial-value problem of general relativity (see e.g. [106] ) ar.d has analy zed conformal vector fields on compact manifolds with constant scalar curvature . Hansen Winicour [128] found a conformal vector & Killi�g field on the manifold of traj ectories of the timelike Killing vector field of the Kerr solution [129], and showed that the cor.formal Killing vector field leads to a conserved quantity along certain null geodesics. Collinson French wrote the conformal equations & [9 ] Ki lling (1.15) and their first order integrability conditions (1.27), (1.34) and (1.35) in the Newman-Penrose formalism and proved the follo�ing two important theorems : ( 1 ) A conformal motion of non-flat empty space-time must be

homothetic, unless the space-time is Petrov type N with hypersurface-orthogonal (twist-free) geodesic rays . (2) For each Petrov type the maximum order of the group of conformal motions admitted in non-flat empty space-time is at most one greater than the maximum order of the group of isometries . For non-flat empty space-times of Petrov type I a result stronger than (2) is available : the maximum orders of the groups of conformal motions and isometries are equal . These theorems are improvements on the results of Shanks ( 950 ) Yano 26] (1951). Collinson French [ 2G] 1 and [ & also determined the maximal groups of isometries admitted by empty space­

times of each Petrov type which possess hypersurface-orth �onal r�odesic 4 5. rays with non-vanishing divergence ( i.e. expanding ) . Einstein [13G] used similarity solutions of his vacuum field equat ions ( such a solut ion admits homothetic motions ) to demonstrate that there are no gravitational solutions without singularities that represent particles of finite non-vanishing total mass.

Cahill & Taub [131] discussed similarity solutions of the Einstein equations for a spherically symmetric distribution of a self­ gravitating perfect fluid. It was found that the metric coefficients of such solutions depend essentially on a single variable ( the ratio of the radial coordinate and the time coordinate ) . The field equations then reduce to ordinaFJ differential equations . They also treated the problem of fitting a sirr� larity solution to another solution of the field equations across a shock wave ( hypersurface) . In a paper for the Synge festschrift [132] Taub obtained similarity solutions for self-gravitating perfect fluids with a particular equation of state, possessing plane symmetFj i.e. admitting a 3-parameter group of motions of the Euclidean plane . In contrast with the spherically s:,.-rr_-:-.etric case [131] Taub showed tha� a certain class of such similarirJ- �lane--times ca�not be fitted to a static plane-symmetr�c S?ace-time across a timelike shock . The results obtained were a?�lied to similarity solutions of the equations of special relativistic h�:drodynamics . Godfrey [133] classified static, axisymmetric vacuum metrics

( Weyl metrics ) according to the homothetic motions they admit . Of all types of collineation ( see Sect ion 1.3 ) the Weyl metrics ad�it only the two simp lest , name ly , ho�othetic motions and isometries . Godfrey discovered two then unknown families of space-times , none of which is asymptotically flat . T�is would seem to make them physically uninteresting, but many possess interesting horizons which Godfrey has investigated in detail. The Weyl metric may be written in the form [134] 2 2( -A) 2 2 2 - 2A 2 2A 2 ds = e V ( dr + dz ) + r e d� + e dt , where A and v are functions of r and z. It admits two orthogonal commuting Killing vectors K and K , one of which is spacelike and the 1 2 other timelike.. Godfrey determined all Weyl metrics admitting a group G of homothetic motions (n � 3 with this G of isometries as a subgroup . n ) 2 He divided the metrics into three classes : 4£,.

Class The symmetry group is G . There exists a homothetic I. 3 Killing vector ( HKV)

K = 6 + 2a�o + 2(a- 1 t o 1 2 � ) t ' where a is a constant , besides the two Ki lling vectors

A The functions and v take the form

= A a log r + z, 2 1 \) = a r az - r 2 + C c constant . log + 2 2 O ' 0 ( Mclntosh [141] has quoted this metric, but there is a misprint in his paper . ) Class The symmetry groups are ( i G , ( ii G , iii G , iv G . II . ) 3 ) 4 ( ) 5 ( ) 11 Each is characterized by the presence of one HKV a- 1 r + zo + ab�o ( ) t K2 = or 2 � + b_ 1 ot ' where a, b are constants . The dimensionality of the groups is made up

by the Killing vectors . The functions A. and \J take the form 1 2 2 A = b log r + -§-< a-b) log[z + ( r+z)1:2], 2 2 2 2 2 1 2,2 2 2 b log r + a -b log[z r + z 1:2] a log r + z + c , \J = %< ) + ( ) -§"< -n ) ( ) 0 where C is a constant . O ( i The symmetries are given by ....,K , K and K . The cases a = b, ) 2 1 2 = and a = 0,1 and b 0,1 simultaneously are excluded. This class

includes the C-metric ( see Ehlers & Kundt [135] ) given by a = , b = -1 or a b = admitting 2 ( = -1, 2 )

( ii Besides K , K and ....,K there is third Killing vector ) 1 2 2 G

1 Here a = b, but a i -1, 0, 2, 1, 2. The metrics are Levi-Civita's cylindrically symmetric static fields [136].

( iii Besides K , K , K and K' there is another Killing vector which ) 1 2 3 2 takes different forms depending upon values of the constants a and b: - a = b 1, K = g.\0 zo = 41 z g.l ' = 1 a b K = to + g.\o , = 2' 4 2 g.� t = a = b 2, K = to + zo . 43 z t 47 .

All metrics in this case are Petrov type D and have been dis covered by Levi-Civita [ 136 ] , Petrov [ 8] , and Kasner [ 137 ] .

(iv) When a b 0 and a 1 the metric is that of flat Minkowski = = = b = space , wh ich admits the maximal group of 1 0 Killing vectors and 1 proper HKV CY-2) . Class The symmetry group is G and comprises 4 Killing vector·s . Ill . 4 The metri cs are the Schwarzschild metric (A1) and three others closely related to it (A2, B1, B2). All are Petrov type D. The notat ion in [ ] parentheses is that of Ehlers & Kundt 135 . The ray congruences of the metrics A1 , A2 and the C-metric are hypersurface-orthogonal and diverging ; those of B1 and B2 and hypersurface-orthogonal and non­ diverging. Godfrey also gives the Bianchi type for each of the metrics listed above . Sigal [ 138] proved that the only vacuum Einstein space which admits a timelike proper (i.e. non-isometric) homothetic mot ion with hypersurface-orthogonal traj ectories is flat . [ ] Eardley 139 discus sed the nature and us es o� self-similarity (homotnetic transforma�ions space-time into itself) general o� a �n relativity . Amongst ot�er results he showed that the evolution equations of the initial-value pro�leu. preserve a self-s imilarity of initial data ; he seems to have been unaware of the work of York [ 140] in this area. His main result prior to an application of self-siQi larity to cosmological models (see Section 2.4) is:

Each space-time (M,g) with non-trivial (i.e. non-isonetric) homothetic group H of dimension n and isometry sub�uo c H has n o • - - G m - n the properties : (1) The commutator of two homothetic vectors in is a Killing vector H n in G and n- 1. m , m = (2) Either (a) (usual case) (M,g) is conformally related to anoth�r space-time with isometry group such that (H,g) Gr H and dim G (p) = dim H (p)-1, where H (p) n-CGr , m n n denotes the orbit p under the action of H,et c. ; of EM n or (b) (exce�tional case) (M,g) is a (vacuum or non-vacuum) plane-Kave space-time.

The part of this result due to Eardley appears to De his listing in an

Appendix of the exceptio�al rlane-wave space-times of (�)lr). 48.

Part (1) is due to Yano [26] (1951) and Collinson & French [9], and (2) is due to Defrise-Carter [26] (1969, 1975). Mclntosh [141] studied properties of homothetic motions in general relativistic space-times with particular reference to vacuum and cosmological perfect-fluid space-times . Using the formalism employed by Debney [142] he corrected and extended Debney 's results to the case when a homothetic bivector (HBV ) with components H [a;b] formed from the homothetic vector H = H arid interpreted as a test a electromagnetic field for any Killing vector field , was present . He also showed that quite strong restrictions are placed on the nature of the proper homothetic motions admitted by vacuum space-times . These are summarized in his theorem : Non-flat vacuum space-times can admit a non-trivial nomothetic vector field H on ly if such a vector field is non-null. H has either

(a) a non-null H�V ln which case �is not hypersurface-orthogonal ,

is not tangent to a geodesic, is shear-free aDd �as constant expansic:1 , or a null HBV in ...-�i ch case the space-time Petrov (b) lS necessarily

type Ill or :; .

Case (b) is containec within the result obtained by Mc_ntosh in a second paper [141]: If a non-flat vacuum space-time admits a �oo�thetic vector field H (trivial or non-trivial ) with an as socia�ec null nBV , then the space-time is algejraically special.

Other contributors to the application of the con�ormal group

in general relativity and gravitat ion are Buchdahl [143] on a set of equations derived from a conformal invariant which possess solutions which are conformally-Einstein spaces ; Popovici [144] who obtained conformally-invariant gravitational and electromagnetic field equations from a variational principle based on a part icular Hamiltonian ;

Erez & Rosen [145] whc displayed explicitly a conformal mapping between a given static axially-symmetric metric and the Schwarzschild (exterior )

solution; McLenagh an & Lerov [146] on conformally recurrent space-times ; Scheurer [147] on a 5-dimensional space-time admitting the conformal

group ; Schnirman & Oliveira [148] on conformal invariance of the equations of motion in curved spaces ; Soleimany [149] on generalized 49. conformal transformations of space-times and the construction of boundaries for past and future timelike infinities ; Agnese & Calvini [150] who investigat ed the consequences of conformal invariance of the ormal matter Lagrangian; Englert , Gunzig , Truffin & Windey [151] on conf invariance with a dynamical symmetry breakdown; Ross [152] who produced a scalar-tensor theory of gravitat ion with field equations "conformally equivalent to the vacuum Einstein equations" ; and

Barut & Komy [153] on conformally invariant action-at-a-distance electrodynamics .

2.4 Cosmological Applications . By 1940 spat ially isotropic, homogeneous relativistic cosmolo�y had become well established , resting on the �oundation laid by Einstein [87], and with important contributions fro:.. de Sitter [86] , Friedmann [89], Lemaitre [154] , Milne [63] , Robertson , ��Crea, Tolman , Eddington , and �cVittie [155].

Robertson [155] introduced the metric, na�e � �or him and

Walker , in 1929 an� developed the kinematics of a coss=:ogical theory with this space-ti�e metric in 1 9 35. In his critica: a??raisal of

Page 's relativity [6::.. ], Robertson showed [64] ho;,.. r.::.s }-:::.:-�el7latical theory of 1935 could be seen to contain Page ' s relativiry.

Wave geometry was the subject of attention o� the Hiroshima school earlier this century , and was applied to cosr.olo&.:; . The basic ideas of wave geocetry as developed by Sibata, Take2o , :�imaru Iwatsuki [156] are the adoption of a "mi crosco})ic involving & r..etric" a set of Dirac matrices and a spinor (to be inter�reted analogous ly y. � - l to the Dirac wave functions ) which is required to satisfy a certain constraint equation . The theory is applied to cosmology by requiring that this constraint equation for � be completely integrable. Sibata showed that under these conditions the constraint equation took one of two forms . Takeno o�tained the solutions of each of these forms and showed that the only types of universe allowed by the �ave geometry are the Einstein and de Sitter universes . Itimaru com�uted the mass of the universe in terms of the integral of the time com?Onent of the velocity vector of the cosmological flui d. Later Takeno [83] (1955, 1966) determined the infinitesimal conformal transformat ions admitted by the Robertson-Walker metric. This metric is conformally flat and so admits the full conformal group Minkowski space-tiDe . of But Takeno calculated tlw c���nrmal Ki lling 50.

Vectors directly from the F.-W metric. Schild Infeld & [157] believed that cosmologi cal theory

" . , , o link between special an e eral relat i vit an t- ., d g n y , d ae1opteu · was different both in spirit and i n et .:tr. anproc.... _ ·eh d ail state C'rt e in Ro e t to the of tl le a._, ...... ·· .., riz d very elegantly b r s o n' s

Pev. Mod . Phys . article of 1933 [155j. The Robert son model involved rr.etric expressed comoving coordinates ; the problem of the mot ion o. in

of the fundi3.mental particles (galaxies ) disappeared and the model was c��aracteri zed by the curvature of the 3-spaces of constant cosrnological In the Infeld- Schild cosmology (1945) the metric was

�.!Iterpre ted as differing from the Minkowski metric only by a gauge

factor which det ermined the behaviour of clocks and measuring rods i t to point . frorr: po n The metric was taken to be

2 2 r ,_ . .- y

structure of a 3- space did not enter the pictu ::.· :..: the ..c• t-:

· of the fundament al particles b the l· ±" 'r:ot::.on e camE: cha:." .-.<.. '�i· model. They showed that three fundament<:' ., �-��-, types of ':•r• .: :. 2

·. re possible : ' :; 1on v1 e moti on, Givergent and (III) rest . correspon ded at least general , ...·;e re ;_ ;:;.::-;d , in exar:t J.�.- : " · . =- J<. L)•"' --:.

The typ{':: mo model ·. (I) t ions ocr�urred in a w:;_-,. 1, "'� ,,,��·:.:•:; ·.' invariant under spatial rotations . type ( II ) ...... o e L r n·�:/. ·:::;- c:r•1Sforr.l c< · .. c.. •s spatial ra la i inversions . t•rc, e III ---. t ns t ons ;::;·. (l Le�ween the Robertson and is Infeld- Schild models su�nar�.�e� rtS fo__ lows : Robertson Infeld-S-:hild 3-space curvature type of fundamental. lf\otion 1 I oscillatory -1 I I convergent-divergent 0 III rest.

h1 Robertson coordinates the velocity of light is a function of posi t:i.on

�nd direction, in Infeld-Schild coordinates it is constant. Although the Infeld-Schild models are metrically equivalent to Robertson ' s ,

·e h�!"� is a topological differe;1ce . 51.

Infeld & Schild appear to have been unaware that [ ] Schouten & Haantjes 31 noticed earlier that Maxwell's equations take the same form in vth Minkowski and conformally-Minkowski space.

In a second paper (1946) Infeld & Schild addressed themselves to the question of whether the form of Dirac's equation for an electron depended on the choice of their conformal metric function F( t,r) ; their conclus ion was that it didn 't. Glirs ey [ 158] produced a theory of gravitation in conformally flat space-time . This theory is well at odds with existing experimental evidence, because it predicts that there is no bending of light rays and that the perihelion advance is half that of general relativity . Gursey used this theory to build an expanding steady-state universe with continuous creation of matter. [ ] Littlewood 159 pcinted out that the most general system ln which Einstein ' s velocity-addition formula is valid is a Riemannian with signature ±2 subject to an arbitrary conformal transformation. v4 If sp,,ce-time is conformally flat with zero scalar curvature , Little11ood showed that the wave equation for the gravitational potential 2 !TlUSt r/J = 0 iJ in Einstein's theory be replaced by 0 e , where e is the conformal factor of the metric. Littlewood applied his theory to a uniformly expanding world model and gave an upper liQit to the maximum [ 0] speed of recession of a galaxy as 7/8 the speed of light . ?irani 16 showed that Littlewood 's theory leads to a perihelion advance 1/6 that of general relativity, with opposite sign. This makes the theory untenable. In two later papers [159] (1955, 1956) Littlewood employed his method of conforrnal transformations to determine the properties of uniformly expanding world models with zero pressure , first when the perfect cosmological principle was assumed, and second when the restricted form of this principle was assumed. To conform with Einstein's theory an anisotropic cosrnological term had to be added to the equations in the first case. No new solutions were obtained , the only interest being in the simple method used to solve the cosmological equations. Bhattacharya [161] investigated an expanding conformally flat , zero pressure , perfect-fluid universe. 52.

[162] Tauber & Weinberg discussed the significance of a statistical mechanical theory of gravitational equilibrium of masses in connection with possible general relativistic effects in white dwarf stars. The assumption of local dynamical isotropy restricts the velocity field � of a perfect fluid. The flow pattern is such that successive 3-dimensional spacelike hypersurfaces normal to � must differ essentially only by local constant magnification i.e. there exists a subgroup of homothetic motions transitive on these 3-spaces. [163] Ehlers , Geren & Sachs examined the gravitational field generated by a gas whose one-particle distribution function obeys Liouville's equation (Boltzmann equation without collision term) assuming the distribution to be locally isotropic in momentum space with respect to some timelike velocity field � . and the gas to be irrotational if the rest-mass of the particles is zero. Thus their methods and results are extensions of the work of Tauber & Weinberg [162]. They (Ehlers et al) showed that the model is either stationary or a Robertson-Walker model. Their conclusion is that in general relativistic cosmology the restricted cosmological principle and the Weyl geodesic postulate "can both be considered as consequences of the apparently weaker postulate of an isotropic distribution of peculiar velocities" of particles near each event in the universe. Among their results is the following , due originally to Ehlers [164]: Let � be the tangent vector to a congruence of timelike curves , such that v� � = -1. Then the curves are the orbits of a 1-dimensional (local) group of conformal mappings of space-time into itself iff the gruence is shear free and satisfies con 1 - = V VA = V� = �;A -- ev V� U, v� .v vv ,�' 3 �· � - -2U where �v = e �V is the new metric under the conformal mapping , and e is the expansion. _li = u� The generators of the group are X e v . 0, If 8 = the mappings are isometries. Developing these investigations further , Trlimper [165] showed that Liouville's equation implies that the locally ellipsoidal distribution function in momentum space depends only on a quadratic form in the 4-momenta, whose coefficients are a Killing tensor in the case of non-vanishing particle rest-mass, and a conformal Killing tensor in the case of rest-mass zero particles. He suggested that 53.

cosmological models of Bianchi type I can be described in terms of ellipsoidal momentum distribut ion functions whose ellipsoidal tensor

is built out of the Ki lling vectors associated with the spatial

homogeneity .

Hoyle & NarliJ.:ar [ 166] (1964,1966 ) developed a conformal theory of gravitation based on the idea of particle interaction

"at-a-distance" , the equat:ions of motion (not geodesics , in general)

and the gravitational equat:ions being obtained from a variational

mass-action principle. �n:ike the Eins tein equat ions which determine

the 1 0 components of the me:ric tensor completely , the Hoyle-Narlikar

gravitational equations , o� ;.rh ich only 9 are independent: , do not give a complete determination o� �v - one function is left undetermined. This function may be taken be the conformal factor in the metric to 0 � . 2 mapp1ng g = 0 consistent with their requirement of �v � �v �v ' conformal invariance of t!-,e :;;ropagator (scalar wave ) equation for

the interaction between �� � ?articles , plus the adoption of the Schouten-Haantjes device [�-] that mass transforms accorcing to -1 - Jr. = 0 m m. The Hoyle-�:=.::-likar gravitational equatior,s for a smooth

fluid approximation reduce to Einstein's equat ions for a special choice

of o. However , th9 smoo�� =luid approximation is not valid in the neighbourhood of a particle , as they showed explicitly .

Arguing strong::.- =or physical theories to be conformally

invariant , Hoyle & NarliKc: -166] (1972a) app lied their �irect-part icle conformal theory of gravit:=.� ion to cosmology . They discussed the

Friedmann models which have a Robertson-Walker metric with 3-spaces of

constant curvature k. T��s metric is conformally flat , so by a suitable conformal transfc��tion the geometry of these world models

can be made Minkowskian . :-:owever, in the usual relativistic cosmology

it is not possible to emrl :.- this geometrical simplification because

the physics changes ; Eins�ein' s equations are not conformally invariant.

Hoyle & Narlikar's argument is that , because their gravitational equations are conformally invari ant , •hey can exploit the conformal transformation

of the metric in these models . Doing so, they find that although the

cases k = ±1 are spat ially homogeneous in the Robertson-\\alker frame they are not spatially ho=2£eneous in the Minkowski frame . Spatial

homogeneity is , however , ?�eserved under a conformal transformation

from the k = 0 Robertson-�a-�er frame to the Minkowski fra�e . More

surprisingly , the singulc.::-:.-ry ("origin of the univers�" ) of the Friedmann models in the F.v:e:-tson-Walk er frame does ::y�,ically no� :::' ::-e ; 54.

ln the present theory . Hoyle & Nar likar take the view that the singularity is only a mathemat ical construct , due to an unfortunate

choice of the conformal frame ; there is a second half to the universe which appears when the Minkowski frame is used. Both halves

contr ibute to the mass function m of the theory , and so both appear

to be required in this cosmology .

In another paper (1972b ) Hoyle & Narlikar, instead of starting with the Friedmann models and proving their consi�tency with

the direct-particle theory characterized by the mass function m( X) ,

started by introducing a constant A such that the indivioual part icle 2 mass was Am( X) . As suming that A is related to th� nur.o er of particles

giving rise to the mass field, a new conformal cosmological model was obtained which involved continuous creation of matte� . The

interesting thing about this new model is that the creation lS

concentrated in active local ized "centres" , rat�er than uni formly as [ 1c7]. in the now disfavoured Steadj'-State theory [ 168] c Segal pro?osed a space-time � ��i h is en ext ens ion ,... [ ] of the one considered by Vej len 12 . Br·iefly , '-' deno"Les the

4- dimensional manifo ld of al� pairs (t,u) , where t is c real nur.0er 1 2 3 � . = ( 4 ) . . . 4 . . an d u u ,u ,u ,u 1s a po1nt on t h e sp h ere � 1n -c1�ens1ona l 4 Euclidean space . Minkowski space M is imbedde � in E i� a caus ality- ,... preserving manner . r� admits the full 15-parar.:eter co:::=o�oal group whi ch is causality-preserving since M is a covering space of co::-.:;Hctified [ ] [ ] 4 Minkowski space ( 28 , 54 , and see Sect ion 2.2 ). Loca�ly 1 an d M

are indistinguishable , but on the cosmological scale di:':'erences arise.

Segal claims that his world model "provides a satisfacto�y explanation

for the cosmologi cal red-shift and eliminates the appare�t need to

hypothesize the expansion o:= the universe" . �e also cla ir.s thaT his

theory resolves the controversy about the smallness in size of quasars

relative to their energy output .

Barnes [169] investigated flows of a perfect fluid in which

the flow lines form a timelike shear-free normal (twi st-free ) congruence.

Such a flow restricts the space- time to be Petrov t)�e I and either

static or degenerate (PetrO\· type D or conformally flat ). He proved

that a non-degenerat e perfect fluid field admits a conformal Killing

vector (CKV ) field parallel to the flow (cf. Mcl ntosh [141])

Trump er [170] had proved that an expand ing vacuum space-time which

admits a hypersurface-orthogonal CKV is either conforn3lly flat or static.

Barncs arrived at an i�ed:ate generalization o� this result : 55.

A non-degenerat� perfect fluid space-time in which the fl�1 lines form a norrr.al shear-free congruence is static. Barnes listed all cor�orrrellj flat prefect fluid space-times with such a flow ; these in clude a Friedmann universe, the interior Schwarzschild solutioD , the Stepanyuk metric [171] , the Einstein universe, and the de Sitter universe , plus models wh ich contain no

Killing vectors . f� l static conformally flat perfect fluid fields with non-negative de:�::i ty >: ere shown to be spherically symmetric. The Petrov type D fields witi the present flows admit at least a 1-parameter group o� isometries with spacelike traj ectories ; they include the exterior Sc��arzschild solution with cosmological constant, the Levi-Civita metrics [13s], the expanding spherically symmetric shear-free perfect-�l�c �lo�s with uniform density considered by t�!e metrics of Faulkes [173] anG. Thom:t :.m & Whitrow [:7L], Nariai [174] , and ge:�e�aliz���ons of the vacuum B- and C-Qetrics of Jordan , Ehlers ����� Barnes showed that a ?etrov type D & �:7�] . vacuum metric whic� a����s a s�ear-free normal congruence of timelike curves is static. ::-:�s :::-es·..:...:...�s incluC.e those of God�e:: � :33].

Ea.rdley _:� J ce::.:.:�ed a spatially self-simil� cosmological = mo del to be a space--:: �::-.e ....- :-. :. c:-. admits a simi larity grou:;: ::-:3 ( group of homothetic traps ::��-a�io�s ) transitive on certain space_ike only hypersurfaces . Sue":-. a s; =.ce-time generally admits -:":-.e isometry group G H and is spatially inhomogeneous (u:�less H ). 2 c 3 �:-.e�e::c:'-2 G 3 = 3 Closely parallellin,; of Ta� , Heckmann �:-. e ,.- c :::-:-: & Schi..ic:-:ir:6 ,

Ellis & MacCallum , Cc��.:.�s � ::-:awking [176] , Eardley constructed all spatially self-sir.�lac cos=c�o6ical space-times , putting them into classes and B 3ianchi types and A (incl��.:.�� I, II , VI 0 , VII0, VIII IX) which are the homoge�eo".l.S co ::.els (H G ), and classes and which 3 = 3 C D admit one proper hono��etic co tion. Noteworthy is the �act that the homogeneous Bianchi �'-::-e c: ::: ::: and IX cosmologies do not admi t homothPtic motions. He pointec out that only the simplest kinds of matter are allowed in a self-siC.:.lar s?ace-time e.g. dust, electroca�netic field , photon gas ; mixt ure s o:: these are not permissible, in general . This restriction can be so=e��at 'relaxed by allowing , for examp le , matter to be "self-similarly (see [ 1 3 1 ] and [ 132]) - TatD ' s plane­ s::oc:...:e-.:1

symmetric similari�· so_ut lc�s [132] are , in fact , special cases of Bianchi types 1 1 an:: 56.

Eardley discussed the homothetic nature of the following space-times : (a) Minkowski . The simi larity group is the Weyl group H G 11 � 10 . (b) Robertson-Walker. (i) k = 0 . The metric 2 2 2 2 2 2 ds = R (t)(dx + dy + dz ) - dt n admits a homothetic motion if R � t , n constant, given by the homothetic Killing vector

2 For n � 3 this metric is the solution of the Einstein equations for hydrodynamic matter with equation of state p = (y- 1)p , = 2h "Other equations of state generally break self- y n). similarity e.g. the 'hot big-bang ' model of the universe is asymptotically self-similar before and after onset of matter dominance , but not during" . The symmetry group is H G , 7 � 6 where H is transitive on the 4-dimensional space-time . 7 Mclntosh [ 141] has also remarked that for arbitrary R(t) the hypersurface of homogeneity t = const . admits the homothetic motion given by the homothetic Killing vector

= xo + zo . K c·o + X J 'j Z These k = 0 Robertson-Walker models are special cases of self-simi lar cosmologies of Bianchi types iii, and v �I . f fv f h (ii) k 1 0 . None of these Robertson-Walker models admit exact non-trivial homothetic motions , except for the unphysical equation of state p = -p/ 3. At sufficiently early times (close to the "big bang" ) the models are asymptotically self-similar. At lar�e times the k = -1 models are close to

being Minkowskian and �o are asymptoti cally self-similar . (c) Kasner . These vacuum space-times with metric 2p i i 2 ds 2 = -dt 2 + t ( dx ) , p. = 1 p 2 L: L:• = L: ' l. . i i l. l. where the p are constants , admit a hornothetic group H G , i 4 � 3 the homothetic Killing vector being i =to + z:; C 1-p.) x .. K o t . l. l. l. 57.

A wide class of space-time singularities, investigated by [177] Belinskii, Khalatnikov & Lifshitz are approximated by the Kasner metric sufficiently near the singularity . Such singularities are therefore asymptotically self-similar . (d) Heckmann-Schucking. These anisotropic dust unive�ses with metric 4 I 3- 2p 2 i � i 2 -dt + t 2p (t+t ) �(dx ) t = const., = I 0 , 0 where the p. are constants satisfying the same co�straints as in J. the Kasner metric, do not admit homothetic motions - their symmetry group is a G of isometries . However , for t >>t and 3 0 t t they are a?proximated by Kasner and dust Rojertson-Walker << 0 models respectively , and so are asymptotically se��-similar .

Eardley also constructed the ADM [178] Harr���onian action principle for vacuum self-similar cosmologies , and warne � that such an action principle somet imes leads to wrong field equatio:-,s , just as in the homogeneous cosTlX)logies wl·1 ich have only a Killing s:.-=:-.etr:,' . He proved that the correct field e�uations were obtained ��c 2 self- similar cosmology iff the space- tioe is of (his) class A or a s·..::: -::�ass of (his ) class D, thus generalizing result of MacCallum Taue a & ::7;]. Milton considered a scalar-tensoc �heory proposed & ;;g [180� by Schwinger [181] and shoNed , by subjecting Schwinger 's �agrangian function to conformal transformations, that his theor:,- ;.; =.s in many ways similar to the Brans-Dic�e �182] theory . For example, =��h theories may be characterized by a time-varying gravitational "cons�=-":�" , and the theori es coincide when the scalar field is weak . App l:: .:. :::� Schwinger's theory to Friedmann world models , Minton Ng obtained scalar-dominated & a cosmology in which the acceleration parameter q 2, a: �hich differs 0 � .. � considerably from the corresponding Brans-Dicke cosmolo�: . Hilton & Ng suggested that their model could provide a resolution o� Lhe cosmological "missing mass" problem. On the other hand, the matter-dominated model in the Schwinger-Milton-Kg cosmology gives predictions identical with the Brans-Dicke one . [85] Glirs es & Gurse\· showed that the Schwar::schild interior solution is the only solut ion of Einstein ' s equations fo� a spherically­ symmetric perfect-fluid distribut ion Hith non-negative ;-r-cssure wh ich is both static and conforoally flat. It is remarkable that they did not acknowledge that Buchdahl [85], in a widely availa2--= a:·t icle , had 58. dealt fully with this matter some years before . The same criticism [85]. can be levelled at Rao & Patel Two points perhaps worth mentioning about the Glirses & Glirsey paper are : (i) they gave explicit conformal transformation equations for expressing the Schwarzschild interior solution in each of the Einstein and de Sitter forms ; (ii) they remarked that a physical model of the interior of a star consistent with causality cannot have the Schwarzschild interior metric as a metric , and so cannot be conformally flat (see also [183]). [184] Chang & Janis studied conformally invariant scalar radiation fields , electromagnetic fields , and non-conformally invariant Klein-Gordon and gravitational radiation fields in Friedmann cosmological backgrounds. Their purpose was to find under what conditions back- scattering of the waves (i.e. the presence of wave-tails ) would not occur; the multipole radiation is then said to be "characteristic" . They found that characteristically propagating waves are possible fo� both conformally invariant scalar fields and electromagneTic fields in any Friedmann universe. This confirmed expectations because the Friedmann geometry is conformally flat , and so the behaviour of such radiation in a Friedmann model should be similar to the behaviour of the corresponding radiation in a Minkowski world . On the other hand , characteristic propagation of Klein-Gordon and gravitat ional radiat ion fields was found to be possi�le only for special Friedm��n worlds , which nonetheless include two physically important cases , namely , those world models in wh ich 0 and p = p/3, where p is the pressure and p = p is the density. This result should be compared with the earlier work [185] of Kundt & Newman wh ich suggested that the presence of matter

would lead to the presence of wave-tails. Chang & Janis used [10] perturbation methods employing the formalism of Newman & Penrose and Hawking [186], achi eving a simplification over previous methods used by Tauber [187]. Other contributors to the application of the conformal group in cosmology are Gi.irsey [ 188]; Lopez [ 189] who obtained "exact solutions to the conformally homogeneous model universes constructed by Edelen" (no reference available); Stephani [190] who used Debye potentials to get all solutions of the source- free Maxwell vacuum equations from a single scalar equation, with applications to conformally flat cosmological models , plus extens ions ; and Caves [191] who compared recent observations with cosmological theories which are either conformilllv flat or have conformally flat spacelike sections . 59.

CH APTER 3

For the remainder of this work I shall consider only 4, Einstein vacuum spaces of dimension which satisfy the field equations R = 0. ( 3. 1) 1-l'V The purpose of the study is to see what conformal symmetries these spaces admit. As we shall see, the spaces selected for investigation (3.1). are in fact a subclass of those represented by equations I shall be concerned with those vacuum spaces which admit proper homothetic motions and which are algebraically special, possessing a diverging and/or twisting geodesic shear-free ray congruence. There are good reasons why one should study such special spaces , and this chapter is devoted to setting the scene for such a study .

3.1 The Backdrop. The simplicity of the vacuum space-times of general relativity make them obvious candidates for a first loo� into the real A world. huge �-n ount of effort has gone into the s-rudy o� the groups of motions (isometries ) which these space-times admit . �ndeed, the exploitation of the group theoretical method has been a major reason for the success in finding large numbers of metrics satisfying Einstein 's equations. Such successes have led many investigators , �s we observed in the last chapter, to look for more general symmetries in the vacuum space-times, and this search has been spurred on by such deeply held convictions as the conformal invariance of physical 3.3 theories (see also Section below). [117] Collinson proved that the only curvature collineations (CCs ) admitted by a vacuum space-time not of Petrov type N are conformal motions. He also found that Petrov type N vacuum spaces do admit CCs which are not conformal motions i.e. they admit more general types of symmetry . [26] (1925) Brinkmann determined all Einstein spaces which can be conformally mapped non-trivially (i.e. non-homothetically ) on Einstein spaces. Indicating his scorn for the homothetic case, Brinkmann omitted the qualification "non-trivially" from his results, [135] as did Ehlers & Kundt in their version of Brinkmann' s theorem: 60.

A vacuum field can be mapped conformally on another vacuum field iff both fields admit a covariant constant vector field i.e. iff both fields are pp-waves. Schouten ([21] , p.314) rendered the following version of Brinkrnann 's theorem: If a special Einstein n-space (i.e. one satisfying equations (3.1)) is not conformally Euclidean , then for n = 4 it is impossible to map it conformally on another special Einstein space, and added the footnote: It is impossible to map it (i.e. a special Einstein space) conformally on a non-special Einstein space. Ehlers & Kundt [135] gave an example to show that Schouten's version is false. It is easy to see why Schouten's proof fails. For , in the case of a homothetic motion ( � = const . in equation (1.14)), the Ricci identities (1. 31) for X = � . a become

iJ. 0, R r:<-, !6 , = (3.2) a..-r iJ. lS which trivial since � = const. Furthermore , in a vacuum RiJ. = ci-laPy and so (3.2) and the first order integrability condition aPy ( 1. 35) give £. iaSy = o . The whole of Schouten's argument , which hinges on the non-triviality of (3.2), breaks down at this point. Thus it appears that the question of whether or not a vacuum Einstein space-time admits a homothetic motion re�ains wide open. We know that the connections in two homothetically related vacuum spaces are given by 2� �V = �v' �V = e �V ( � const. ), J(la l3y = ($laSy =

,..., 0 3 .3) = = · ( and RI-!V �v On the face of it , investigation of the Riemann and Ricci tensors is fruitless in providing an answer to the question. 61.

Suppose, however , that a vacuum Einstein space admits a homothetic motion. Then setting $ = 1 and choosing a coordinate system so that the components of the generator � of the homothetic motion are such that � = 6� (x4 = t), equations (1.15) imply that = e t h (x 1 ,x2 ,x3 ) ( ,v = 1,2,3,4), �v �v � > o . where h44 = E = ±1, E�v With the metric in this canonical form it is easily shown that Ricci tensor R is independent of t, and equations (3.1) can in �V principle be solved for the h This is, however, a formidable �V task, and an alternative procedure which adapts a coordinate system to the algebraically special ray congruence of the space-time is followed, as described in detail in the next chapter. Fortunately , we are aided a great deal in ourta sk

by the results of Collinson & French [9] who proved A conformal motion of non-flat empty space-time must be homothetic, unless the space-time is Petrov type N with hypersurface-orthogonal geodesic rays . Their proof was based on the use of the Newman-Penrose spin coefficients. A direct proof, which to my knowledge has not appeared before , is as follows : Suppose a vacuum space-time is mapped conformally onto itself by 2r/> = e = �(xa) 1n· gener al . �v ' r/> "" Then equations (3.3) hold iff r/> - r/> r/> , = ( 3. ;�v ,� v + 4) where 1 v 2r/> K - u � � - R e /2n(n-1) ( 3. 5) 2 g "" '�"" 'v and L is defined in (1.32). Now, in accordance with the theorem �V of Yano [26] (1951) (see p.18 of Chapter 2), an Einstein space with R 1 0 cannot admit a proper homothetic motion (and a fo rtiori a proper ,..., R 0 conformal motion), so we are taking = = R. Then from (3.5) and (1. 32) we have K 1 J-LV = 2 r/> ,�r/>,v and = o . L�v 62.

From (3.4) and these last two results we obtain for a vacuum � = 0 = �'�C '� �a �y ��y (3.6)

If � 1 constant , this is just the condition for the space-time to be Petrov type N or conformally flat (see Table 1, Chapter 4, p.76). � The gradient '� is the Debever vector which is tangent to a hypersurface-orthogonal congruence of geodesic rays , as may be seen from the argument given by Eisenhart ([5], p.114). The argument used

in the above proof breaks down when � is constant , leading us back to equation (3.2). Since a conformally flat vacuum space-time is flat , we have proved Theorem 3.1. Another result due to Yano [26] (1951 ) and also given by

Suguri & Ueno [26] (1972) and Eardley [139] is the following : The commutator of two homothetic Y�lling vectors is a Killing vector.

A short proof of this result , based on the theory of the Lie derivative, is as follows : Let � , � be two scalars corresponding to the tomot�e�ic motions 1 2 ..... generated in a space-time by the homothetic Killing vecto�s K1 , K2 . Then, by equation (1.15),

Therefore

i.e.

= 0

and hence )., a constant , ( 3. 7) where = 0.

Theorem 3.2 asserts that any vacuum space-time admits at most one independent proper homothetic motion. This has also been pointed out by Mcintosh [141]. Hence we need concern ourselves with only those space-times which admit one proper homot�etic otion together with a group of isometries. 63.

Theorems 3.1 and 3.2 together tell us that , in looking for higher symmetries of non-flat vacuum spaces, we can restrict our attention to those spaces wh ich admit at most one proper homothetic motion , unless the space is Petrov type N with twist-free geodesic rays. Trumper [170] showed : (a) If an eX?ansion-free vacuum field admits a non-null hypersurface-orthogo�al conformal Killing vector (CKV), then the CKV is a Killing vector . (b) If an expanding vacuum field admits a hypersurface-orthogonal CKV , then the space-time is either static or conformally flat . (If static, it must admit a timelike Killing vector v;h ich is also hypers·.ll'face-orthogonal. ) Mcintosh [141] showed that if a vacuum fiel� admits a yroper homo�hetic

vector field K, then K must be non-null. r: 7 has a non-null homothetic bivector , then K is not hypersurface-orthogonal, is not tangent to a geodesic, is shear-free and has constant expansion. If K has a null homothetic bivector , then the space is algebraically special and, if non-flat , is necessarily Petr�v type III or N.

Petrov type N vacuum S?aces are :;:;erhaps the most

interesting and mathematically t�n�alising o: all. The only type N vacuum fields which admit proper co:1:'ormal r.::J":lons are the hypersurface­

orthogonal plane-fronted parallel (??- ) waves (Brinkmann [26],

Ccllinson [117], Ehlers & Kun�t [13:] , Thon�s8n [192]). A subclass of the plane-fronted gravitational �aves , the ?p-waves have all been determined by Kundt [193] ; they are charact�ized by the presence of an expansion-free, twist-free, null geodesic shear-free ray congruence. Some pp-waves which admit homothetic motions are known e.g. Mcintosh [141] has cited the pp-wave 2 ds 2 = 2 du dv - 2' (�,C ,u)du2 j dC j ,

2 2 U- - where U = I x -Y 1 ' f. 0 , J'X. = x+iy, �;, which admits the homothetic Killing vector = xo + yo + 2vo K X y V

For type N vacuum spaces with eh?anding and/or twisting rays, the symmetries present depend on whethe!' the ray congruence is (i) twisting , or (ii) twist-free. In case (i) Collinsbn [194] has shoHn that there exists at most one Killing vector in the space. He

was unable to integrate the field e;uations. In case (ii) Held [195] 64. proved that the metric admits at most two isometries - the Killing vectors are necessarily spacelike in the asymptotically flat region.

Collinson & French [9] have given examples of such metrics . The Weyl metrics which Godfrey [133] found to admit homothetic motions could be among the metrics to be obtained in later chapters , as could be the Kasner metric (Section 2.4, page 56 ).

3. 2 The Plot . Against the backdrop described above, we see that the type of vacuum space within which we can perform our conformal motions is very much restricted. Indeed, for Petrov type I spaces the conformal motions must be homothetic, and the dimensions of the maximal group of homothetic motions and the maximal group of isometries are equal. There is a little more latitude in the case of algebraically special spaces, and it is this case which will be investigated in the remainder of this work . Specifically, a systematic search for algebraically special vacuum Einstein spaces with a diverging and/or twisting s�ear-free null geodesic ray congruence, which admit homothetic motions will be undertaken. The results of this research fill part of the gap in our knowledge of symmetries of vacuum Einstein spaces . Such a systematic search has not been done before. There remains the problem of finding all Petrov type I vacuum spaces which admit homothetic motions, and the pro�lem of determining those pp-waves which admit (a) proper conforQal motions , and (b) homothetic motions . This last hole must be filled by future research.

3.3 The Players: their physical characteristics. The principal players are the homothetic Killing vectors. They determine -the symmetry , along with the Killing vectors (if any). The physical importance of homothetic motions has been noted by

Gobel [49], Einstein [130], Cahill & Taub [131], Taub [132], Eardley [139],

Mcintosh [141], Winicour [196] , and more generally by Hoyle & Narlikar [166] (1972a). A few points will be made briefly here. 65.

It is well known that all space and time meas urements can be made in the same kind of unit, say length L. All physical measurements are in terms of dimensionless numbers , which are formed by combining physical quantities with dimensionality Ln at some space-time point P. Changing the unit of length by a factor A does not affect the formation of dimensionless numbers at P. The factor A may depend upon the point P in the space-time manifold, or it may be a constant. Either way , physical measurements may be compared via dimensionless numbers at the same space-time point and the physics is conformally invariant . The use of dimensionless variables in classical continuum mechanics to produce "similarity solutions" of a given problem is well known. The usual technique consists of exploiting the symmetry to reduce the system of partial differential equations to ordinary ones by assuming a solution in which the dependent variajles are essentially functions of a single independent dimensionless variable. This same technique should be of use in obtaining solutions of Einstein's

(vacuum or non-vacuum) equations which are conformally oY , in part icular , homothetically invariant . Eardley [139], in applying t�s technique to cosmology , has said: "One may hope to discover new :'acts about cosmology and singularities by building new models that presume self­ similarity (homothetic invariance) from the outset" . Scale invariance i.e. invariance under a uni�orm change ln length scale in space-time, is of considerable interest in elementary particle physics because of its relation to deep inelastic scattering , as we noted in Chapter 2. The Weyl group is the transformation group concerned, and it has also served as a gauge group for relativistic theories of gravitation (see e.g. [197]). These are a few reasons why one can feel con:=ident that an investigation of conformal and homothetic symmetries will help towards a better understanding of physical theories. It is to be hoped that the present work will contribute towards that end. 66.

3.4 The Props. The framework of the theory and the formalism used to develop the argument is an extension of the work of Debney, Kerr & [198] [199], Schild and Kerr & Debney and is the subject of the next chapter. Before the performance begins , it is emphasized that the play and players are local. To enable the production to extend globally would first require it to win approval locally and then would necessitate finuing more time and more durable props , at least. Let the curtain rise. 67.

CHAPTER 4

This chapter sets out the formalism that will be used ln the rest of the work . The main references are the papers of

Debney, Kerr & Schild [198], Kerr & Debney [199], and the thesis of

Debney [200]. Similar formalisms have been developed by

Cahen, Debever & Defrise [125], Wilson [201], and others.

4. 1 Tetrad Formalism. The tetrad formalism has been used in general relativity for a long time (see e.g. [5], [10], [21], [202], [203]) so the following presentation is not completely detailed. Take a 4-dimensional Riemannian manifold M with a metric g expressed in local coordinates � ( � = 1,2,3,4) by the tensor � (xa), and with signat e (+ + + - ). Let T denote the tangent v �.. p space at p E M and let Tp denote the dual (cotangent ) space. In terms of the coordinates � at p the holonomic (natural) basis for T is · p (o - T T�: f ala�] . he dual basis for p is (d ). � l Let (e a a= 1,2,3,4) be any anholonomic basis for Tp i.e. to each point p E M there is attached a tetrad of vectors e a a l = * Let (� a 1,2,3,4) be its dual ln T p , defined by the inner product (4.1) '#� If x E T p and f E T p , we have a X = X e a , a f = f a !=: . We adopt the convention that Latin indices a, b, c, ... will always refer to components of geometrical objects with respect to an anholonomic

A, �. .. basis (tetrad), while Greek indices v, . will refer to components with respect to a holonomic (coordinate) basis . Vector symbols are not singled out typographically by underlining , unless confusion will result in this not being done. a Each of the bases {ea ), {e ) may be expressed , in terms of local coordinates at p, by (4.2)

a !": (4.3)

� 0> \\here the component: c are C functions on M. a ' 68.

We may rewrite the orthogonality relations (4.1) as b e 11� ( 4. 4) a · 11 The inner product of two vectors e and � is defined by v a g = (e ,e ) = g e 11e = e 11e (4. 5) ab a b 11v a b a bll and inversely ab b b g = (e a ,e ) g11 v a � = ( 4. 6) = € 11 . V ab . . The g ' g w1ll c h ange f rom one po1nt of M to another, in general. ab V, • • b · · · The tetrad components Ta ... o f any Tensor Tl1· .. are computed as follows : b 11 b V ... (4.7) . .. 8 T T e •. a ... = a v 11 · and inversely V . , . Tb ... (4.8) T = ..• 11·.. a v Tensor indices are raised and lowered by � ' � and tetrad indices ab v by gab , g The directional derivative of a tensor T ... in the direction of the tetrad vector e a is . ... =< ' .. 11 . . (4.9) T oT = e T . ... , a a . . . a .. , 11 where the comma denotes partial differentiation. The tetrad components V. .. T • of the covariant derivative 11· . ;y are b . . . b y V . . ' T = e l1� e T ( 4 . 1 0 ) a ...; c a�v c 11. .. ;y b .. . d ... = T + a ... ,c �de T a .. . (4 11) + . where the rs are the Ricci rotation coefficients defined by (4. 12) or, equivalently ,

- € e. 11e V (4.13) r = ' abc a�J;Vb c where d r = g r and a be ad be 69 .

Using (4.3) together with the rotation coefficients and the tetrad components R a of the curvature tensor, the connection a bed a . forms and the curvat ure f orms o0 are gl.ven y w b b b a c ,· (4.14) w b = = ra b ce (4.15)

where A denotes the wedge product of differential forms . The Cartan structural equations a a c de + UJ c A r:: = O, (4.16) d.ua a c 1 a (4. 17) b + wcAwb = 2e b a a a relate the forms UJ b and e b to the vectors E: . The metric on M is 2 v = a b ds = �vdJ-ldx gabE: A E: (4.18) and this gives

= In the case of rigid tetrads for which dgab 0 , that is g� = covariant constant over the whole of M, this gives (4.19) so that there are only six independent connection forms and therefore only six independent curvature forms. We shall adopt a system of rigid tetrads for the present work . Definition (4.14 ) and property ( 4 .19) give - = r - ( 4. 20) rabc = rbac ra[bc] + b[ca] rc[ab]' where we have used the convention that round and square brackets denote symmetrization and skev>-symmetrization respectively , thus : T 1 T ) (ab) = -§-< ab + Tba

and then recursively � T (a ...a ) = {T ...a 1 m m a1(a2 m) T 1 - m 1 a ...a· J {T - T • (-1) [ 1 m m a= a • ••a a a ...a + • • + - T J ). 1 [ 2 m J 2 [ 1a3 m J am [ a1 ...a m_1 The ra[bc] are determined from definition (4.14) and equations (4.16). 70 .

a Then the components of the curvature tensor R bed are determined from (4.15) and (4. 17), where for any vector u a the Ricci identity

r 1 m u = u (4.21) a;[bc] 2 � abc m defines the curvature tensor . We shall effi?loy a complex null tetrad (see e.g. [200], p.6)

{e = {m, m, n, k} , (4.22) a ) where m, m are complex conjugate null vectors and n,k are real null vectors . The bar abo·;e a symb ol denotes complex conjugation. The vector m may be defined from a pair of real orthogonal unit spac �ike vectors p and q by J2m = p+iq . The dual tetrad is

a - • {€ } = { m, m, k , n } With 0 1 0 0 1 0 0 0 ab gab = 0 0 0 1 = g (4.23) 0 0 1 0 the following orthogonc�ity relations obtain:

o, J!lk� = nflm1-! = nt-lni-l = k�m� = ml-!n� = - (4. 24) � = 1. kl-!n � = m mt-l Also ab = e e g (4.25) ""1-LR · v al-l bV so that the metric on "' in terms of this null tetrad is

= 2€ + 2 (4. 26) {2 € l- 4

= 2min + 2:1:...: ' b where = . € a gab� The set of ?roper (no space reflections), orthochronous (no time reversal) Lore�tz transformations of the null tetrad which leave unchanged the dire;::tion of one of the null vectors , say e4 = k, 71. is, �: = "i': -iB -iB = = y ), (4.27a) m e 1 e (e1+ye4 ) e ( m+ }: -A -!: �'; = n = e3 e ( e3 ye1 - ye2 - yye4 ) -A - - = e (n - ym - ym - k)' (4.27b) A A yY -!: �"\ = = ) k = e4 e e4 e k, (4.27c where A and B are real numbers and y is a complex number. If y = 0 equations (4.27) describe a timelike rotation in the (n,k)­ = plane and a spacelike rotation in the (p,q )-plane , where J2m p+iq. For A = B = 0 equations (4.27) describe a null rotation about the vector k. Thus the general trapsformation (4.27) is a non-commutative �roduct of these two types of transformation. Null tetrads have been widely used in general relativity.

A more detailed description, together with discussions on null rotations , may be found e.g. in [10] and [204] - [208] and references cited therein. Using the complex null tetrad , the six independent c8�>ection forms wab from whic� the rotation coefficients are determined c:e . JJ42 , w 12 + w 34 , w 31 anc their complex conj ugates w 41, -w 12 + w 34 , l132 . .:-. e independent curvature equations (4.17) are = 1 8a b + A ( + ) R A € (4.28a) dJJ42 w 42 w 12 w 34 2 42ab ' 1 a b d( + ) + au A = + R A E: (4. 28b) w 12 w 34 42 UJ 31 �R12ab 34ab )E: ' 1 a b + ( + ) = R /\ 8 . (4.28c) dJJ31 w 12 w 34 A w31 2 31ab8 c :�e independent components of the Ricci tensor R = ab = �a R abc are

= R R24 4212 - R4234 (4.29a) R44 = 2 R4214 a:1d R12 = R1212 + R3412 2 R 4231 = R34 R1234 + R3434 2 R 4231 (4. 29b) R33 = - 2 R3132 '

w�ere R44, R12, R34, R33 are real and R11, R14, R13 are the complex conjugates of R22, R24, R23 respectively. 72.

4.2 Congruences . To expedite this study of algebraically special vacuum spaces we shall orient the null tetrad and choose coordinate systems bearing in mind the Goldberg-Sachs theorem [209]: A source-free gravitational field is algebraically special iff it admits a shear-free null geodesic congruence. The terms used in this theorem are defined below. Here we note two significant points: (1) we shall choose one of the tetrad vectors (e4 = k) to be tangent to a null geodesic congruence , (2) the Goldberg-Sachs theorem applies to any field that is conformal to a vacuum field [120], [209]. Much of the present knowledge of null congruences and their prop erties can be attributed to Ehlers [210] and Sachs [204] and may be summarized in the Ehlers-Sachs theorem (below). Since the theory is well presented in the literature (see also [5] pp.97 ff. , [207], and [211] - [213]), only an outline is given here for completeness. Consider a space-time filling set of curves whose a a equations are x = x (yS ,w), a = 1,2,3, where the individual curves are given by y[:s = constant , and w is an affine parameter along each curve. Let the null vector

k� = 0 1-1 be tangent to one of curves. Then l-1these v (ok /ow) + l-l v o, k k = A k kcr = �-t ;v vcr where Al-l is the affine connection, is the condition for the curve to vcr be a (null) geodesic. This condition is preserved under conformal transformations of the metric. Then the complex rotation coefficient r424 vanishes: V r =- - k m-� = 0 , (4.30 ) 424 �-t ;V which implies that its complex conjugat e does too , r414 = o. (4.31) (The complex conjugate of a real geometrical object is obtained by performing the permutation 1,2,3,4 � 2,1,3,4 on the tetrad indices.) The geometrical properties of a null geodesic congruence can be visualized as follows [204], [207]: 73.

Think of the congruence as a bundle of light rays . Insert a small plane circular disk into the bundle at right angles to the rays . On a nearby plane screen, also perpendicular to the rays , the shadow of the disk appears as an ellipse , and all portions of the shadow hit the screen simultaneously . The shape,size and orientation of the shadow depend only on the location of the screen, and not on its world velocity. If the screen is an affine parameter· distance t,w from the disk, then the shadow is expanded , rotated and sheared relative to the disk by the respective amounts 6t,w , wt,w and l a l t.w , where e = � rate of expansion �2 k ;j.l k w = } (4. 32) rate of rotation (twist ) (�2 k k� ;v 2 [� ;v] 2 k2 = e } rate of shear l al (�2 k �;v - • (� ;v) This is the Ehlers-Sachs theorem. 6, w, a are referred to as the optical scalars . Let u = constant be a family of null hypersurfaces in M. v u, u, = 0. The differential equation of the family is � V Therefore j.l the vector k = u, is null and is normal to these hypersurfaces. But , � � being null, it is self-orthogonal an so lies in the hypersurface to which it i� normal. Thus a single null hypersurface u = 0 determines within itself a null geodesic congruence . Conversely, given a congruence of null geodesics , these curves lie 1n a 1-parameter family of null hypersurfaces u = constant to which k 1s normal iff � a = (4.33) f(x )u,i-1

A congruence of null geodesics with tangent vector ki-1 satisfying condition (4. 33) is called hypersurface-orthogonal or normal. It can be shown that any congruence of null curves which is hypersurface-orthogonal is always a geodesic congruence. The result

where is the alternating symbol, shows that (4. 33) holds iff 0 w = i.e. a null geodesic congruence is hypersurface-orthogonal iff it is twist-free . 74.

From (4. 32) it follows that the optical scalars 6, w, a are determined by the null geodesic congruence (i.e. by specification � of k ) alone, up to an ambiguity in sign of w and an undetermined

phase in a. The vectors m, n which complete the tetrad do not play a part in determining the optical scalars . The completeness relation (4.25) and the null geodesic = Property k �;v kv 0 = k� ;v � give k = 0� 0� a m� m � v v = (m � + m m + k na + n ka )k �(m + m �; � a ;f3 � � � � a ; f-' v v k n� + p + v �k ). Then

- k �;V (pm� m V + am� m V + (a+�)k �m V + Tm �k V + yk� k V )

+ complex conjugate

-� V = = � = - complex divergence p k �;Vm m r421 6+iw' ffiv complex shear a = k rrfl = -r �;v 422' V k �- a+P = �;v n m = -r432' (4.34) v T = k� ;v rrfln = -r 413' v n�n y+y = k �; v = -r433 The notation used here is chosen to accord with that used by

!�ewman & Penrose [ 10]. It can be seen from (4.34) that the optical

scalars e , w , a are invariant under a null rotation given by (4.27) A with = B = 0. For a null geodes ic congruence with r424 = 0, the connection form w 42 contains all the optical information: a w42 = r42 a€ =

= (4.35)

We shall return to this equation when setting up a coordinate system . 75.

4.3 Con formal tensor. Petrov classification . Let R be the Riemann tensor, R be the Ricci ��� �V = � �� 1 scalar, = R - be the tensor, R = �a be the curvature s �V �V �V R traceless form of the Ricci tensor . The Weyl conformal tensor is defined by [ ] ] _u v _u v 2� �8v � (4.36) a� a� + v [a �] + ! v[� Ca �u�Jv R C = K 6 and possesses the symmetries � - - o - v c - - c c - c (4.37) c [�v][a�] a�v ' �[va�] - ��� � These statements also apply to tetrad components . Integrability conditions for conforr.al transformations , involving the Weyl tensor, have been discussed in Chapter 1. Here we are concerned with types of this tensor. The algebraic and geometric study of the Riemann and Weyl tensors has done much to clarify the structure of gravitational fields . In particular , it has advanced the understanding of gravitational radiation fields and has become a convenient tool in the search for new exact solutions of Einstein's field equations (see e.g. [135], [213]). Shlers & Kundt Pirani Earlier work on the geometry of the Riemann tensor may be found in references [21 3] - [216]. Petrov [8], [217] gave, using matrix methods , the first systematic account of the algebraic classification of the Riemann tensor for Einstein spaces. The physical significance of the Petrov scheme for [218 . [219], gravitational radiation was recognized by Pirani ] Kerr [220], Goenner & Stachel Petrov and others have discussed the classification of the Weyl and Ricci tensors from the point of view of symmetry groups . Penrose [221], following Witten [222] , developed a spinor method for classifying the Weyl tensor according to its Petrov type. Further contributions to the algebraic and geometric study of the Riemann and heyl tensors and the use of the Petrov classification in general relativity up to 1971 may be found in references [175] and [223]-[227] anQ the literature cited therein. An exhaustive set of references , including more recent ones , is not given here. 0 (4. 36) For vacuum spaces R and gives C = R �v = �va�R ���R ' so the algebra and geometry of the Weyl and Riemann tensors is the same in such spaces. Debever [216] (see also Debney [200]) showed by tensor methods that in every vacuum space-time there exists at least one and 76.

at most four independent vectors � t 0 which satisfy the algebraic relations 0. k� = (4. 38) 11 These � are called Debever vectors and determine the principal null directions of the Weyl tensor. The Petrov type is given according to the partitioning of these Debever vectors ; the higher the multiplicity of coincident Debever vectors , the greater the degree of specialization of Petrov type. Table 1 is adapted from that of Pirani 's [207] and displays the relationship between Petrov types : Debever Petrov Algebraic partition type relation

[ 1 1 1 1] I c 11 \1 = 0 k[a i3]l1v[yko]k 1< [ 2 1 1] II 11 \1 c1311v[ykoJI< = 0 [2 2] D } 1< [ 3 1] \1 0 III cP\..!\1[yko l = y [4] c 0 r: 131-.lvyk = 0 c 0 131-.lvy =

Table 1

0 Other notations for types D, N are Id , IId respectively. Type corresponds to a conformally flat space. Type I is called the algebraically general type. Any space with a Weyl tensor of type

II, D, III or N is called an algebraically special space. Penrose [221] gave the following scheme (the ) for the increasing order of specialization of Petrov types : I

II D

III/� -----� N ------� 0 77.

Sachs [204] pointed out the existence of five complex quantities which completely characterize the Weyl tensor. Referred to our null tetrad they are 5 ) C( = 2 R 4242' 4 C( ) = R R 4234 + 4212' 3 ) 1 ( = R + R - R + R, ( 4.39 c 1234 3434 34 6 ) 2 ( ) R + R c = 1231 3431 ' 1 ( ) 2 R c = 31 31

The necessary and sufficient condition for the tetrad k = vector e4 = to be a Debever principal null vector is c(5) 0. The vacuum spaces are algebraically special as follows : ( 5 ) = ( 4 ) o, (3) o, Type II or D, c c = c � ( 4.40a) 4 3 2 Type < 5 ) = < ) = < ) o, < ) o, 4 .. 40b rn , c c c = c t- ( ) ( 5 ) = < 4 ) < 3 ) < 2 ) = o, < 1 ) t- o, Type N, c c = c = c c ( 4.40c) ( ( 4 ( 3 ) ( 2 1 Type 0, C 5) = C ) = C C ) = C ) = 4.40d = ( 0. ( ) t- Goldberg & Sachs [209] showed that a vacuum metric with 0 Cajcd is Petrov type D iff there exist two independent shear-free null . . (3 ( 1 (2) C ) C ) = = geodesic congruences , 1.e. 1f � 0, C 0. Hence type D metrics are completely characterized by 4 ( 2 1 ( 5 ) C ( ) C ) = ( ) C ( 3 ) 0 Type D , C = = C = 0 , f. . ( 4.41)

Under a proper , orthochronous Lorentz transforrr�tion ( 4.27) of the null tetrad which leaves unchanged the direction of e = k, . 4 ( 1 the C ) transform as ( [200], p.33 )

(5) * = 4(A+iB) ( 5 ) C e C ' 4 ) 2(A+iB ) (5) ( 4 ) C( * = e [yC + C ] , (3) * 2 (5) ( 4 ) (3) c = y c + 2yC + c , ( 4.42 ) 2 -2(A+iB 3 (5 2 4 3 C( ) * = e ) [y C ) + 3y C( ) + 3yC ( ) ] , (1)* -4(A+ iB ) 4 5 ) 3 4 ) 2 (3) ( 2 1 C = e [y C( + 4y C ( + 6y C + 4yC ) + C( ) ] ,

and under a conformal change of metric 2iJ I = e ( 4.43 ) �v �v ' . = C ( ) where iJ �( xa ) in general , the 1 transform as [ 120] 7 8 .

( 5 )' -4� ( 5 ) c = e c ' ( 4 )' - 3� < 4) c = e c ' ( 3 )' - 2� < 3 ) c = e c ' (4.44) ( 2 )' -� ( 2) c = e c , ( 1 )' ( 1) c = c .

The transformation equations (4.44) show at once that algebraic degeneracy of the Weyl tensor is a conformally invariant property .

4.4 Coordinate sys tem. Field equations.

The coordinate system used by Kerr & Debney [199], developed by Debney, Kerr & Schild [198], [200], will be used in this investigation of the homothetic symmetry of vacuum Einstein spaces . The null tetrad (e ) (m,m,n,k) is chosen so that e k a = 4 = is tangent to a shear-free null geodesic congruence in the algebraically special vacuum. That such a congruence exists is guaranteed by the

Goldberg-Sachs theorem. Then we have = -r 0 geodesic property R 424 = ( ) = 0 ( shear-free pro?erty ) . We further demand that r 1 0 o = -r422 42 1 i.e. the rate of expansion e and the rate of twist w do not both vanish , e + iw t- o . P = From (4. 3 5 ) we then have r . (4.4 w42 = -p� 2 + 423€ 4 5 ) Under a transformation (4. 27) of the null tetrad, transforms as r423 follows : : v �· �·ile · �: 8 e 3 4\.l;V 2 A iB -1-1 -A - - e � e ( m y-.KU e ( nV - -V ym-V - y-Vk ) ;v + ) ym - y iB = e r < 423 + YP )' - where we have used (4.24) and k kl-1 0. Choosing p we = y = �423 ; V . get r* = 0. Thus by a tetrad1-! transformation we can make r vanish. 423 423 In order to preserve this condit ion we are henceforth allowed only tetrad transformations with y = 0; specifically

:� -iB m = e m,

-.": -A n = e n, (4.46)

:': A k = e k, and where A n arc rC';: 1 nu··_:- (' :'.· . 79.

Setting r = 0 equation 4.4 ) reduces to 423 ( 5 ( 4.47)

For vacuum spaces the field equations Rab = 0, together with 4) = ( = 0 3 the degeneracy conditions C(S) c and ( 4. 9), show that the only non-zero component of R is R see equation 4.29b)). Therefore 42ab 4231 ( ( ( 4. 28a) becomes ( 4.48)

Hence ( 4.49)

, This is the condition for the existence of two complex functions C � such that ( 4. 50)

Under a tetrad transformation 4. 46) transforms as ( w 42 *a iB 1 �·· = r-:: = e A.....l" e w 42 42aE: 421 E: A+iB = e ( 4. 51) w42 +i .� = -e A B+ -,- ( 4. 52) G�· , rewembering that r r = r 0. Taking 0 we 422 = 423 424 = A+ iB+� = further restrict the allowed tetrad transformation , but this enable us to write , after dropping the asterisk, = 3 w • ( 4. 5 ) 42 -d{; Then ( 4.47) gives

1 = - 1 p ( 4.54) e: �2 = � . We take C , C as two coordinates in the space-time , where C is the complex conjugate of C . The other two coordinates u,v of our coordinate system are both real and are introduced in the definit ions u, 1, u, 4. ) 3 = 4 = o, ( 55 = € = k du + + d , 4. 6) 4 0� o C ( 5 ' -1 -1 and e ) , or vtfi ) , 4. 7) V = R ( p p = ( ( 5 where p is the complex divergence and is defined by - /]. i Im( fu) ( 4. 5 8) 11 = = -6 · 80.

The operator D and hence its complex conjugate D in the last equation is defined by ( 4.59 )

Upon using p = 6+1w we find - 1 2 2 !:,. = i Im( p ) = -1w/( 6 +w ) . Thus the Debever vector k is hypersurface-orthogonal iff !:,. = 0 ( from the well known result on page 73) . Hypersurface-orthogonal spaces belong to the Robinson-Trautman clas s [22 8]. (C In the ,(, , u, v ) coordinates the metric takes the form

( 4.60 ) where the metric vectors are m ( v+ d C , e: 1 = = t:,) n dv - Re{[( v-t:,. � Dt:,. ]dC} e:3 = = 2 ) + + Re D 0 + , ( � )e:4 ( 4 . 61)

The functions 0, � . 6 are independent of the coordinate v, and � is referred to as the "complex mass". The dot used in (4.61) denotes differentiation with respect to u, thus :

dJ/ou ; o o = uo.

The reader is referred to the paper by Debney, Kerr & Schild [198] for details of the derivation of the fi eld equations , which are - D�-! 3 0 4.62a = �. ( ) Im( �-DDD O ) = o, ( 4. 62b ) 2 o ( �-DDD l o DO . 4.6 c u o ) = u 1 ( 2 ) The independent components of the Weyl tensor are 3 3 < ) 4.6 a c = IJ.P , ( 3 ) 2 - 2 3 < ) -(Do terms if < ) 0 , 4.6 b c = u UJ)p + ( = o c = ) ( 3 ) ( 3 ( 2 < 1 ) ( o o D ( terms 0 if ) ) 0 . 4. 6 c c = u u 0 ) p + = c = c = ) ( 3 ) 81.

The Weyl tensor therefore has the following Petrov types :

Type II or D � 11 0. (4.64a) 1 Type III � 11 = Do D o 0. (4. 64b ) o, u -1 Type N � 11 = 0 = Do D o , o o DO-l (4.64c) u u u o. Type 0 � 11 = Do Do= o o D o = 0 (4.64d) u u u �·= It can be shown that the conditions for Petrov type D are 2 Type D , 3 11 o o D = (Do D 0 , u u 0 u ) 4 D 11Do Do = 3 11 DDo o , u uD 2 4(DI1) + 311no D o (DDo- DD o) (4.65) u 2 = 3 11 DD 11 9 11 o D , - u 0 DI1(DDo - DD o ) = 11D(DDo- BD o) .

4.5 Group of Allowed Transformations .

We shall now adapt the coordinate system to the cas e of proper homothetic changes of the vacuum metric 2{1$ �\) -+�\) I = e �\)' (4.66) where !IS is a constant . Cons ider a � from one connected manifold M

to another Write = �( p ) for each point p M. Let e ) M. q EM E { a be a basis in T (M), the tangent space at p E M, and let {�)be a - p a . bas is in T (M) , the tangent space at q �defines the linear q EM. map ��·: , �... : T (M) ..... T .. p q (M) .

a * * - Let ) , �)· be bases in the cotangent spaces T (M) , T (M) at p EM, {e {e p q * q respectively . induces the linear map � , EM � * �": �·= T (M)- ..... T (M) � q p by the requirement that the inner product ( , ) is preserved , thus :

a · � a -a a , )(p ) � ,e )(p ) = .�.. : e )(q) = ""'a, -..... )(q) o . (4.67) (e � = ( � e b (e b (e � = b

Let {e ) = {e- ) be a new basis in T and let {e' ) . a �a q (M) , a be a basis in T ob tained from {e ) by a Lorentz transformat ion q (M) . a which leaves the direction of e4 unchanged. Then the transformation

ation . R. P. Kerr & G. Weir , Private communic 82. equations (4.46 apply since we are demanding that 0 in the ) r423 = manifold. We have -iB .. -i&- e I e e e -r/Je e 1 = 1 = 1 -A .. el e e e -r/Je e -tr- ( 4 .6 3 = 3 = 3 8 ) A "' A- e I = e e -r/Je e 4 e 4 = 4 where A and are real B fu..... dt'oi'\ S. a a 1: - Also let } { e� } be a new basis in T (M and {� = q ) a introduce the basis {e1 } in r* M) by q < a b a ( el , el ( o ( 4 .69) ) q ) = b Define the commutator [X,Y ] of two vector fields X and Y on M by

[X,Y]f X(Yf) Y Xf = - ( ) for any function f on M. In particular

[ e ,e ]f = e ( e f ) - eb( e f) ( e 1-!Ve e 1-!Ve '1-! )f, v a b a b a = a b '1-1 - b a (4. 7'J) upon using ( 4 .2) .

c The structure constants C are defined by ab c c c C = , [ e , = C b (e e. ] ) - a a n b a which implies ( 4.71 )

This last equation is equiva_ent to the Maurer-Cartan equat�o�s c 1 c a b d£ = - • 2 c ab€ 1\ €

Equations (4.70) (4 7 1 give and . )

Multiplying by e we get dv

e e e V = ( dV ,I-1 di-! ,V ) a1-1 �

1 i.e. = cdab - 2 83.

Now

=

Hence the rotation coefficients and the structure constants are related by 1 (L�.72) r b = c d[a ] 2 dab Using (4.23) we obtain 1 c c - c ) . (4.73) rabc = 2( abc + bca cab

Under the mapping � we require e ,e ] [ e , e ] . �:·J a b = �:': a �:': b - Define, at q E M,

slnce � is constant , - - 2� --, 2� c e e = e e = �:': [ a , � J �... .. ( C ab c ) - 2r/r.::c e �e = C ab c

-c where c Therefore = ab ?.c 1/r:::.c ... e,.. = e - e c ab c C ab c and so, at q EM, - r/:Jxc = e C (4.74) ab

I f , , are the rotation coefficients on M r'abc rab c rabc defined with respect to the bases {e' ), {e ), {� ) respectively , a a a (4.73) (4.74) then using and _at q EM, we have 1 ...... ,.. . = C + C ) rabc "§"< abc �ea - cab -� (4.75) = e rabc 84.

Hence ... = ...c = -r/r.::. r/r;:-c r (q). (q) e ra b c (q).e � (q) abc € ·'· - 1 = W'

·'· _ 1 = -1 d(cp.. C (p)) -d(C OCfJ)( q) = a:(q), (4.77) where is a differentiable function on M. If c' is a differentiable function on M, we have also, by the reasoning of Section 4.4, that

' = (4.78) w 42(q) -d�' (q) and (4.79)

4. Tl1erefore , from ( 4. 77) and ( 7 9) , we have A+iB et:' = e <( which implies c ' = �<�), ( 4.80) where = e A+iB The function r is thus coupled to the tetrad through 1 �1 =eA. (4.81)

Define functions u, v, 0 on M by - 1 � -1 u = u 0 CfJ-1 V = V 0 cp o = o ocp (4.82) and also define (4.83) � 4 = du + od� + oar u' o' Let ' be functions on M such that ' 'd ' . (4.84) e 4 = du' + o'dC' + o ' Then because of t/J fr-- = e e � 4 we have ( 4.85) 85.

Now u1 = u1 (( , C', �, ';/) and if we restrict the function u1 by requiring

u1 = 1, uI ' - 0 (4.86) , 3 4 - we get -.. - -::,. -::,. 1 1 1 = d du1 � dC + o� dC + � -;; or oc a-;; Then (4.80) and (4. 85) give

(4.87)

Hence

which has the solution 16 U1 q = e 1 (4.88) ( ) � I(�+ S) (q), - " w�ere S(C,C) is a real function on M. (4.87) Equation �so gives ou' e 161 _ � 2 I u - - � o'( so that 16 - 1 e l ,..... --.- 1 -1 (,.....u + ,..... S ](q), (4.89) O' (q) = � I � [o - s - 2 � � ) r r r ur h�ere we have used (4. 88) The complex divergence relative to bases {e1 and {� a J a J is given by

_ � I r..J �I I I V ( q ) : ) e e ( q p - 4 ( - � 21 � �;v 2 1 )

i� � - i (e16 e IYv Ce-r/J e e He. -r/J e &--e V (q = € �;) 2 1 ) ) A e -16 e r"" (q = - 421 ) 16 e , (4. 90) = - 1�""' liJ < q ) "' - 1 c where = • p p 0 cp Let v' be a function on .....M defined by v' = 1 - 1 Re p • 86. Then , using (4.90 ), = �, - 1 --1 vi (q) Re ( e iz, p ) - 1 -- (4.91) = e �, tc', v(q).

Let t:,1 be a function on .....,M defined by

-1 • = (vi 1 ) PI + t:. Then -1 t:,l ( q ) = ( p 1 - vi )(q ) � -1 .....,_ 1 - ....., = e , � , (p v)(q) t I ( q ) = e t:,(q), (4.92) t:. i>l� .....,�- 1 ""' ' ....., -1 where !:::. = t:. 0 � and we have used (4.90 ) and (4.91). t'V -1 - �CC , C , Let � = � o � , where u) is the comp lex mass function introduced in Section 4.4. Under the map cp we have

cp�·c_c3) c 3) �P 3 = c = which defines cC 3) i.e.

At q EMwe have (see [198], e�uation (3.17)) < 3) + ) e = 2P

{ 3)1 = "(3) c c (see also (4.42) with y = 0). Hence 3 2 3 <1-1' p1 )(q) = e - i>

Now let us view � as a mapping, not of one manifold into another, but of the manifold � into itself, � � .... M. Then we can interpret the trans�ormation equations (4. 80) , (4.88) , (4.89), (4. 91) , (4. 92) and (4 . 9�) in the following ways :

0. ( i) � = identity map (q = p), � = This corresponds to a change

of coordinates at the sar.� point p in M. 0. (ii) � 1 identity map (q dist:.-.ct from p, in general ), r/J i This corresponds to a pr��er homothetic motion , where the coordinate system is "drc.u;ed along" by � i.e. the same

coordinates are used at � and q. The symbols with a tilde are to be identified with tr1� symbols without .

Collecting togethe� tje group of allowed transformations (4.66) under a homothetic change of cs��:c and a tetrad rotation, we have

I = ( 4.96) u er/J I�c l

v' = e � 1v, r/Jl C 1- ( ) where s ' ,, is a real functio!:., and -1 - 1 - - + S)], o.' = e r�;l� I 4> [ s. --2�\:4>\: (u , c 0. "' 2

= - 3 (4.97) 1-l' il4>-l u' ' "' 1 ' = e r/JI 1 - t:, 4>c t:,.

The tetrad vectors rr�1sform as follows : e = e- � 1 � r/J I4>, I C e1 , - - 1 e; = e r/J �� I e , (4.98) , 3 - e • e� = e ��� 1 4 , 0 h homo-::Cety When

4.6 Local one-parameter groups . Homothetic Killing vectors .

(-o , o ) Let I0 be an open interval of R and let U be an open set of M. A local 1-parameter Lie group of local (infinitesimal) M M transformations of is a mapping (t,p) - �t (p) of I0 X U into which satisfies the following conditions [ 2]: (i) For each t E 10 the map �t : p - �t(p) is a diffeomorphism of U M. onto the open set �t(U) of (ii) If t,s and t+s are in I0, and if p, �s (p) are in U, then � (p) = (� � )(p) � (� (p)). t+s t 0 s = t s The group properties are completed by noting that (� 0 � )(p) = (� 0 � )(p), (� ) -1(p) = � (p), t s s t t -t and � (p) is the identity. 0 Each local 1-parameter group of local transformations �t induces a vector field X defined on U cM as follows : For every point p E U, the vector Xp is tangent to the curve = A(t) = �t (p) at A(O) p. This curve is the integral curve of X. a If (x ) are local coordinates so that the curve A(t) is given a para1uetrically by x ( t) and the vector X has components If,then this curve is locally a solution of the set of differential equations

Conversely , it can be proved [ 2] that every vector field X on M generates a local 1-parameter group of local transformations . Geometrically , the diffeomorphism �t : U-M takes each point p E U a parameter distance t along the integral curves of X. The tangent vector field X to the integral curves A(t) of the group M �t on is defined by

X f ( 4 .99) p = t= O where f is a differentiable function on M. (xa) M, If are local coordinates at p E and (x'a ) are local coordinates at q = �t(p) on A(t), then

= • X 13 89.

Putting f = xa in (4.99 ) gives

( a ) ( p) = o x' a ' (4.1 ) flCp) = Xx lot ] 00 � t= O where fl(p) denotes the components of X relative to a coordinate basis ( ) at p, so that o� � X = :f< p)o = lo�� (4.101) p � J [ t=O M - M A diffeomorphism � : is a conformal symmetry if the ,e E M metric g(ea b ) at p is related to the mapped metric g( e e ) at �* �1: a '�* b q = �(p) EMby (4.102) for some non-zero differentiable function � on M. If the local 1-parameter group of diffeomorphisms � - t generated by a vector field K is a group of conformal motions (i.e. for eac� t, the transformation �t is a conformal symmetry ), t�e vector field K is a conforrnal Killing vector field.

Suppose now that K is a homothetic Killing vecLor (HKV ) i.e. Lhe �in (4. 10 2) is constant , and that the local 1-pararneter grou? �t generated by K is de�ined by the coordinate transformation a xa-x' , where (xa ) = C C, �. u, v) and the x'a(x�,t) are given by C ' = HC ;t), �(t) u' = e l 4> -C.,� ;t) j [ u + S(� ,� ;t)] , (!;..103) 1:, !ll(t) - v' = e l 4>-C�;t) 1-1 v, \;, according to (4. 96). That is , we are now visualizing the homothetic motion as a coordinate change xa - x' a rather than a point transformation w�ich drags along the coordinate system. Since x'a(t=O) = xa , we have 4>CC ;o) = C , S(C,�;O) = 0 = �(0 ). Then, by (4.100) , the components of the HKV are

1 : [�:] - a ( t:o o J< : �;: J t: C), 90.

= au + u Re(a.c ) + R, 4 l K' = lov� = + t � 1•,1-lv 0 J c, ��J �0( ]] [ t=O [ t=O L t=O - av - v Re(a ), o C where o�t - a (real constant ), [ J t=o los RCCO �t ] - L t=O and we have used the result

+ = Re(:.,) . � C � tC -J t =CJ

�ence , by ( 4 .101 ), we have

( ..,. . 104) as the form of a HKV which generates the allowed local 1-parame�e� group of local homothetic transformations , in the sense of SecT��� 4 .5. ,., The form of K will change under an allowed tr�1sfor��:ion on M to ' - a 1 ' 1 1 u' + ·.-' K - 0 I + 0.'0? + Re(a r l ) ( u O l - V 0 I ) + R 0 I + a( 0 I c I ) C b 1 ';, u V U U V ..-�ere C1 , u' , v1 are related to C, u, v by equations (4.96). To find the transformation equations for a and R we need

= 6 - 1 0 � o , + e l (, l o + 1 �"�' (u1 o - v1 ) , ' , , � sc ul 2 ul o V1 � l 0 u = e l 41c ou' , 6 1 - 1 0 = e 1 cp O l· V ' v These are substituted in ( 4.104) and the result compared ..-ith the primed expression for K, giving ( .105) and (!.;.106) 9 1 .

Using these transformation equations it is possible to write K 1 0 in simple form . For example , given a HKV K with � we can transform R to zero by solving o K's - (Re

K = or +or + a(uo + vc ), '.:> b U V where we have dropped the primes on the new coordinates. On the other hand , if � = 0 the form of K in ( 4 . 104 ) reduces to

K = Ro + a(uo + VO ). U U V * Introduce a new function u through

* au = au + R.

Then, after dropping the asterisk, we :-, ave K = a(uo + vo ). u \' 0 fo� Since a 1 a proper ��=�thetic motion a1 � it is not to ze�o , possible to transform a non-zero a. He thus arrive at the following two canonical forms of the r:::-;; (we are ac: liberty to set a = 1 ) : ,...,. (i) K = 0{; + ().,. + uo + vC 1 u \" ' "" ( 4. 107) (ii) = uo + va K'2 u V These two forms are mutually exclusive . :·�oreover ,

in accordance with Theorem 3.2, page 62,

Of interest in la�er chapters is the form of the finite transformation equations (FT�s ) corres�onding to an infinitesimal homothetic motion generated by a HKV of the form ( 4.1 08) where m, p,q are constants. The FTEs are given by equa ion ( 1.1 7 ) , and, for ,...,K as in ( 4. 108) are � m �·: �; {; ·· = b C P q ( 4. 1 09 , u = b u, V = b v ' ) where b (=et ) is a real constant . Co�sequently , a metric which admits the HKV ( 4. 108) is homothetically inv�iant under the coordinate changes ( 4.109) . 92.

4.7 Homothetic Killing equations . Expression (4.104) for K is the form which a HKV generating

the local 1-parameter group of local transformations on M, consistent with a Lorentz rotation of the tetrad as well as the homothetic change of metric in a vacuum Einstein space M, must take. However , a vector of the form (4.104) is not necessarily a HKV ; in order that it be so, K] it must satisfy the homothetic Killing equations [ (1.15) with X=

£K�v = ��v ' = M. where 4� � ;� is a constant function on It can be seen from (4.57 ), (4. 58) and (4.61) that the only

quantities appearing in the �v are � and 0 (and derivatives of Q). We need, th�refore, to find the homothetic Killing equations which will

involve only the functions 0 and � for a HKV of the form (4.104). It is convenient to use the following definition [ 229] : Let yA(x) denote the components of the geometrical obj ect at a point with local coordinates �. Under a mapping � of the ma�ifold, let

the transformed geometrical ob ject be denoted by y� . A symmetry of the local geometrical object is defined by y' x' ) = y ( x' ) (4. 110 ) A ( A i.e. the difference between the original components yA(x' ) at the � point x' and the transformed components y' A(x') at a point that is � mapped onto x' by the mapping � is zero. Now let the manifold mapping be a local 1-parameter transformation �t generated by the HKV of (4.104). Then (4.110 ) defines a homothety and we nave 1 ( x' ) = ( x' ) 0 0 (4.111)

and ll1 ( x' ) = ll (x' ) , where the transformation equations (4. 97) obtain. We have x' = x' (x,t ) and

' (x' ) . 0 , ��ot �, o < x' 0 = o � t = O

o �' (x') - ll(x' � = 0 . ot r J t=O Explicitly , these are

t) - 1 , - 1 1 o e Ill< l c t) C - � � �( (�;t) I �r ( ;t)(J( x) - s

Using the results

c ; t = e < ; t = , it[•, c � t=o [•t c � t=o (le

O� O (C ;t) = Re(Cl ) , Q C � t=O C

a (real constant), [��]t =O

o C . t (C , t) Ot � < C ; � = tC ; = R , where R = R, c t=o � (: J t=D C

0 (x' = 'Kn , t G l = 0 =o ��:� 0�" 1=0 #iJ.( = x' ) Jt=O KiJ. , the homothetic Killing equations (4. 112) and (4. 113) become - 1 (I) (K - a) 0 + 1 a - a ) + a u + R = 0 (4. 114) < c c n 2 �C ' and -§- (Ill) 5)

The commutators [o a)o , u ,'KJ = (Re(ac ) + u R-o o ,'KJ = a-o- Cuo - v ) (4. 117) ea, + u +!2 a-­ U ' c c ' cc V 1 o [ D,K] = Cl D - - a v c 2 cc V are useful in obtaining the integrability conditions. Differentiating (4.114) with respect to u we get o 1 1 uc'Ko) - a O. + a - ci- ) 0 +-2 a = 0, -¥ c c cc where 0 "" ClJ2, and then using ( 4.117) this becomes

. 1 = CII) 'KO. (2 Cl" 0 . ( 4 .118) + Cl + 2 If we substitute (4. 118) back into (4.114) we get ( I) in the alternative form RO R = (I) (K - a Ho - u 0 ) + �ClC u 0) + + 0. (4.119) ' Differentiating (4.118) with respect to u we get R (IVd) <'K + a( + e(aC ) + a)O = 0. (4.120) Differentiating (4.118) with respect to C gives R = b ) + ) ) 0 . 4.121) (IV cK 2 e ( ac ) ( D 0 ( Differentiating (4. 115) with respect to u gives (IVa) (K + 4 Re(a, ))� = 0. (4. 122) Differentiating (4. 114) with respect to C and using the fact that

�� is real, we get ...;, - (IV c) ( K Re( ) - a)l! 0. (4.123) + aC =

We have now got all first order integrability conditions. Continuing in like manner would give higher order conditions . For easy reference, the full set of homothetic Killing equations and their first order integrability conditions are now collected together: - . (I) (K - a)( 1 o, O - uO) + i

a' = (4.146) and = � T' e c0[T - (a0+a)A + KA], (4.147) •• where the form of Kin (4. 147) is given by (4. 145). These two transformation equations are obtained in exactly the same way as (4. 105) and (4. 106) were in the (C ,C ,u,v) system of coordinates. The two mutually exclusive canonical forms of K in the present coordinate system are (i) ( 4.148) (ii) K = a0(sos - ror ) + a(sos + ror). (cf. (4.107)). For the homothetic motion represented by (4. 144) we have

o l/\' ) - /\ ' � ( x' ( x � = 0 ' ut-::. J t=O ' p I o, -e0 G p ( x ) -e( x 'j - ot t=o

' at L'm ' ( x ) - m ( x' >] = 0 ' o t= 0 MASSEY U�V US1Tt LIBRAA't 100. from which follow equations (I), (II' ) and (III ) below . The other equations (II), (IVb ) and (IVc) follow from these by differentiation . Alternatively , they can all be obtained directly from (4.124) by using the commutators [os ,K] = ( a0+a)os ' [< , = a. o + T o (4. 149) \ 'KJ c c (; s ' a-o- [o ,J

for the HKV given by (4.145). Thus we have the homothetic Killing equations and their first order integrability conditions in the cc ,c ,s,r) coordinate system :

(I) K = o, ( + a.C - a0 - a)A + TC (4.150 ) (I ,..., 1 I) < K + a. , )pc + 2 a.cc = 0 , (4.151) K (II' ) p + Re

The field e;uations (4. 62) become , in the present coordinate system, = 0 , (4. 156a) me (2) 0, R = (4.156b) ,, - (2) Im(m) = e 2p d cc - 2 R d. (4.156c) 10 1. CHAPTER 5 Spa c.v., 2, wU:h One HKV and 3 oJt 4 Killing Vec..toM

Although they could not solve some of the field equations , [ Kerr & Debney 199] found, in principle, all algebraically special vacuum Einstein spaces with non-vanishing complex divergence, which admit 2,3 or 4 Killing vectors , with 4 the maximum number for a non- flat space. Their results, summarized below, are used as the starting point in this chapter. The metric functions only are given.

Case I. (4 Killing vectors ) o = A = id0C, 11 = m = mo , non-zero real constant, = id = constant , real, b. 0 do p = 0 , R( 2 ) = 0 .

Case II. (4 Killing vectors ) C C A = -id0 / R0 < C - R0 ) , m = m0 , m0 real constant -- Schwarzschild metric, mo complex constant - NUT metric, d = = = real constant, do i ( m0-m0 ) /4R0 -p - e = 2) = cc Ro , R( Ro = real constant .

Case III.(3 Killing vectors )

1\ = m = d = 0 , -2p 2 - 3 e = j< C + C ) ,

Case IV. (2 Killing vectors ) 2p 2 -2 1\ = i'"'? e [- �2 Im(m0 )R + C 1 + C 2 (2 1 og R + R 0 R ) 2 + 2 -2 ] , + C 3 ( R R0 R ) m = m0 = complex constant , but we may take either Re(m0 ) = 1 or Im(m0 ) = 1, -2p d = e Im(/\C ), where R = I cl and R0 , c1 , c2 , c3 are real constants, -p R 2 (2) and e = C� - R 0 = - R0 , R = Ro . 1 02 .

Case V. ( 2 Killing vectors ) -5/ 2 . J1 3 ' --- A = 1 ( c0x s1nh 2 m= m0 = complex constant , d = e-2pim(A )' C - p 3 ( ) e 2 _- 32 x , R 2 = x = rb + r� ' where c0 is an arbitrary real constant .

Case VI . (2 Killing vectors ) 3 - 1/a.O --+ 1 CC o A = A.0 a. m =

d = Im(A )' C ( ) p =O, R 2 = 0 , where Re (a.0 ) = 1 (a.0 invariant ), A.0 is a complex constant, and m 0 is a complex constant which can be made real if the rema:�ing coordinate freedom is used.

The following cases were unsolved by Kerr & Debney :

Case VII.(2 Killing vectors ) 1 iS A= i(C + C)- L(8), C = pe = r3/2 m �0� ' k d = < C + C)2 D(8), D = D, 2 - 3 (2) - p R r e 2 = j

Case VII I. (2 Killing vectors )

0 = O(u), 3 = 1-l - 1-1 o o No solut ion .

Case IX. (2 Killing vectors )

0 = O(t), t = u/Im(C), 3 )l = u- \i(t). (Note error in [199] , p.2817). No solution.

We shall now try to determine whether any of the spaces represented by the above cases also admit a HKV . In cases I - VII there is a Killing vector of the form

= -p K o s = e o u so that we may use the (C,C,s,r) coordinate system to see whether the space will also admit a HKV of the form (4.145). If sue� a HKV is present , the additional equations to be solved i.e. satis�ied by the functions A, m and d, are (4.131) and (4.150 )-(4.155). The procedure is to use these equations to determine the functions a(C), T(C,, ) and the constant a0 of the H�� .

Case I. (4 Killing vectors ) ( p = 0 , R 2) = 0 and (4.154) is trivial. (i) Since m= m0 (real constant ), equation (4.153) gives for m0 � 0 3a0 - a = o . Since d = d0 (real constant), equation (4.155) gives for d0 � 0 0 a 0 - a = . 0 is Hence a = , and there no HKV for m0 � 0 , d0 1 0 .

is trivial. (ii) If m0 � 0 but d0 = 0 , then (4.155) Equation

(4. 153) gives 3a0 a = 0. (4. 152) implies a = a0C + t3, Re(a0) = ao , where t3 is a constant . There is enough coordinate freedom left 104. to transform a to

Equation (4. 151) is automatically satisfied , and (4.150) implies = T T0 (real constant ). Hence we have succeeded in finding a HKV of the form where But since a HKV is determined only up = to a Killing vector , and because K os (and therefore K = T0os ) is already in the space, we may take the HKV in this case in the form ..... K = 4a S ( 5. 1) �oco, + 3io'o' + OS + 2a ror , where 3 Re(a0 ) = a. The metric which admits this HKV is Petrov ty�e D:

1 2 = 2 - -1 2 2 dT r dCdC + drds + m0r ds . ( 5. 2) This metr ic can be written in the form

( 5. 3)

11 = on putting ,r, w = s-CCr. In these coor�inates the HKV is = K' 3TJo + 3'11o- + 4 w o + 2r0 (5.4) Tl T) w r This solution is , in fact , the Kerr-Debney solut ion ([199], eQuation (5. 10)) with d0 = 0. (Attention is drawn to a misprint in their equation (5.10)). = = By writing ./2UJ z+t, ./2 r = z-t, ./2 T1 x+ iy the metric (5.3) becomes 2 2 + 2 2 2 dT = dx dy + dz - dt 2 m 2 2 2 o x y 2( xdx+ydy ) --- dz dt + Cdz dt) (5.5) + z t { + - 2 - + z t } ' - (z-t) - which is manifestly of Kerr-Schild type [230]. This is also clear because QO = 0. There is a plane of singularities z-t = 0 so the metric is that for a nullicle [231]. It is , in fact , one of the Debney-Kerr-Schild solutions ([198], page 1852) of the form

+ qY where P = pYY + qY + c, �(Y) = F- + (qY+c)(C-Yv) - (pY+q)(u+Y� ) 0, 105.

-1 = Z - Pfy ,

t/>, � are arbitrary functions of Y, � = 0 in vacuo ,

m,p ,c real constants , q complex constant. IdentificatioD with the metric (5.3) is achieved through

- - - - Tj/r, c r. , u LU , V r, y - 1 t/> = G, z = z = I' m - mo , = 1' 6) p -::. = 0, c = p = 1. ( 5. Comparing (S.c) with [198], equation (5, 80c) , we fin� we have the Debney-rerr-Schild case (c).

= (iii) When m = ? , the space is flat , as can be' see� immediately G - ,., 0 V

from ( 5 . S).

Case II. (4 Y.il:��� vectors ) bot!-. zero constants. ( i) m0 ,d0 ne =:- Equations (-.::�) and (4.155) imply a= 0.

Hence the:r10: �s ::: :> :1:-:v .

= (ii) m0 non-zerc �����an� , c0 0. �s Equation (L.:::) t�e zero identity , while (4.15-J gives a0 = 0. Then (4. 153) �;:ies a= 0 .

Hence there �s �o HKV .

This resulT c2�irms the well known fact that the 5c��arzschild ��e metric do� ��� admit a HKV, and also proves tha� same is the true of �::._� r:-.e�ric.

= = = d = d 0, which implies that ( iii) mo o 0 E=�ves 1\ m = =O=ooDO =Do D O u u u which is co=:���ion (4.64d) for a flat space.

Case III. (3 Kil_i�g vectors) = For 1\ = m = c 0, equations (4.155) and (4.153) are trivial. . ( 2 ) -:: S1nce R = �+�r , e�uat1on. ( 4.154 ) g1ves. with solution a = --a CC+ib0 ), b0 real constant . a There is still a 1:=-:e� transformation in C left to tra�s=orm to 106.

the farm a. = - 2a0C. - 2p = 3 Then (4. 152) with e �( C+C ) is satisfied identically , as is (4.151). The remaining equation (4.150) gives T = T 0 (real constant ). Hence we have a HKV of the form K = 2a ( + o ) + a (so - ro ) + T o + a(so + ro ) - o Coc C c o s r o s s r ' but because K = o s is already present in the space we take the HKV in the form K 2a + ) (a + a)so - (a a)ro ( 5. = - 0(CoC CoC + 0 s 0 - r ' 7) where a0 and a are arbitrary real constants. The metric which admits this HKV is 1 2 3 2 - - 3 - - 2 2 dr = 2 r < C+C) dCdC + drds +

By writing BC = 3(x+iy) we can write the metric (5.8) in the form 2 2 - 3 2 2 3 2 dT = r x ( dx + dy ) + 2 drds + 2 x ds , ( 5. 9)

which is the metric of Kerr & Debney ([199], equation (5. 19) -but note the misprint in the line above their equation (5.19)). In the real coordinates of (5.9), the HKV takes the form � 2a + 0 K = - 0( xox + yo y ) + (a0 a)sos - (a0 - a)ror ' (5.1 ) where ao and a are arbitrary (a;iO).

Case IV. (2 Killing vectors ) The space is flat unless m0 ;i 0.

Equation (4.154) gives a0 = 0, which together with (4.153)

implies a = 0. Therefore, there is no HKV . 107 .

o , o. ( 2) m0 1 R0 = Equation ( 4.154) is the zero identity , while (4. 1 53 ) gives . 3a0 - a = o ( 5. 11 ) -p With e = C�, equation (4. 152) gives

with solution (5. 1 2)

where a0 is a complex constant. Equation ( 4 . 1 51 ) is satisfied identically . For R0 = 0 , and choosing Im(m0 ) = 1 as we may do, we have 2p e [- c c 1og c r7J 1 1\ = l·r::. ]:.2':-'"'..'' + 1 + 2 (r?::.::.) + 3"- '::, (5. 3 ) so that = 2p 1\(, le. [ c " (5. 14) 2 - '"' 1 and (5. 1 5)

Then d 3C + 2 log(C ) ] C ' =(,de = eL:;:Ii-::: 1 - 2 C2 (, and so Kd = < 2a )Cd . a0C + a0� - 0 , Using this in equ2�ion (4.155) gives

2( ) - a + a] + 2 c - 3 + 0 c2log

Using (ii) with (5. 11) gives a = 0 . Thus there is no HKV.

When (i) obtains , ""e have from (5. 13 ) 1\ " .- - 1 c 1 - l tJ';,Q tJQ = 3 - ( 5 . 17) ' 2' and from (5. 14)

1\- = 0' � d = 0. ' 108.

Then

and substituting in (4.1 50 ) gives C TC = (a0 + a - a0 )A whence

Using (5. 11) we get finally 4 -- T = i�[3 a log(C/� ) - aoC + aocJ. (5.18) Thus we have obtained a HKV which is , along with the two Killing vectors K = o , (5.19) 1 s a symmetry of the space with metric 2 2 r CCC) - dc d' - 1 -1 = + drds + iSCC dC c dC )dr - 1 - 1 -1 2 0 + Re(m )r [ ds + i SCC dC -C dC)j , (5.2 ) S 0 where = c3 -%is an ar� itrary real constant , and �0 is an arbitrary = 1. complex constant with Im(m ) 0 The HKV is = C 3a C-a )Co a)Co ii3[4a logC C/0 - ]o 'K 0 c + <2(l0c - c + 3:t0C + 3:l0� s 2 + 4aso s + aro r , (5.21) S wJ-,ere and a are arbitrary real constants, and a0 is an arbitrary complex constant . The metric (5.20 ) is PeLrov type D and can be put ln the form + + iS(11/Ti) d( f1/11)dr �2 dT 2 = d11dTi d w dr -1 - 1 - 2 + Re(m0 )r [dw + + 11dii)r - rilr dr Cild11 2 iS(11/Ti)d (Ti/11 ) 22) + ] ( 5. by writing C - 1 r

Case V. (2 Killing vectors ) (2) -2p 2 3 R c = 3 x = ' -t = x, e . 2 2 - -3/(4x ), Since o D n = P = we have u cc Pc 0 , 2p 0 , o uo u D O = but Do u D O = e # so that , by (4.64d ), the space cannot be flat . There are two i o , = 0 , a possibilities: ( i) m = r., 0 (ii) m = mo where mo is complex constant . Equation (4. 154) gives a -t a -t 2a o 0 (C-t� ) = with solution b0 real constant. But in the presence of the two Killing vectors - (5.24) �< � = Hor o1 ) L. , ... there is still a linear tra::1s:'ormation available on C , so we use this a freedom to write in the �o�c a 2 - r (5.25) - - ao"' . Then, as in Case III, equc:�o::1s (4. 151) and (4. 152) are satisfied identically . Now - I 1 5 7 s.:...:- ,:-. . Ji3 ( x- x ) J--3 c x 2 -2- 0 -t 2 0 2

- 2 d p Hence = e Im(/\C ) 1 3 � = x -� J ,., r:::-;;1 10c sinh (x-x0 ) 6 uv 0 �2 l - ( )x (5.26) 9 Im mc � l· 1 - 3/ 2 3 = d = x [ ::- � x s . n h Jf3-- ( x-x ) afi3c x cosh �( x-x ) <\ C 12 L � 2 0 c 0 10c �(x-x ) I rn(c ) x-� . -t 0 sinh 0 -t 18 0 ] Then so that 110.

Hence (4.155) is satisfied iff

a c = o , ac = 0 o o o and either ( 1) a - 3a = 0, Im(m ) 1 0, 0 0 or ( 2) a - 3a 1 o , Im(m ) = 0, 0 0

or ( 3 ) a - 3a = 0 = Im(m ). 0 0

= If c 1 o , then a 0 and there is no HKV . 0

co�st:ant , If c = 0, then A real 0 constant , m = m0 = comp lex

-1 d = - 2y x .

o, o. (1) a- 3a = Im(m ) 1 0 0

If a = 0, then a = 0 and there is no HKV . 0

If a 1 0, then (4.153 ) is satisfied identically for m 1 0 . 0 0

= Equation (4. 150) reduces to TC 0 wi th solution

T = T (real constant ). 0 Thus we arrive at the me tric

m o 2 + {x +Re ( _ )}x. , (5.27) 1 r- 2iy x

where - - 3 - 11. = ds + ix (ydC - ydC ),

y 1 0 is an arbitrary real constant .

This metric is Petrov type II and ad mits the HKV ,..., K = (5.28)

where we have taken account of the presence of � = o ln the 1 s c . spa e a0 i� an arbi trary real cons tant . 111.

The fTEs corresponding to this HKV are read off from (4.109) and take dT - dT -': , where �·:2 = -3 2 d'f b dT , b real constant . If we put 2.C = x+iy , r = 1.w , y = A, the metric (5.27) takes the form 2 - 3 2 2 -2 2 2 -2 dT = 3x (w + A x ) { dx + dy ) + 4( d1J + Ax dy )K m x Re 0 2 (5.29) ( wx-iA ) } x. ' 3 X. = ds - Ax - dy , where 4A = 3 Im(m0 ) � 0 is arbitrary , real, constant . The HKV admitted by (5. 29) is

K = xo + yo 2so wo (5.30 ) X y s w

Now m0 is a real constant , and equation (4.153 ) gives = o, ( 3a0 - a)m0 m = so that o o. Then = 3 -1 o . 1\ = m = d 0 � 0 = - 2 x u, � = 6 = From (4.150) we get T = 0 , so that {; = T T 0 (real constant). Thus we arrive at the Petrov type III metric 2 2 -3 2 dT = 3r x dCdC-:- + 2drds + 2xds , (5.31) which admits the HKV = (a - a)r (5.32) K 2a0({;oC + Co� ) - (a0 + a)sos + 0 or ' where we have taken account of the fact that the Kill�ng vector K1 = os is already in the space. The constants a0 and a in (5.32) are subject to the restriction a i 3a0 , a i 0, but are otherwise arbitrary. This restriction is the only difference

between Case III and the present case. Writing 8{; = 3(X+iY), we can put the metric (5.31) in the form of (5.9), with the HKV in the form = x (a -a)ro K 2a0( ox + Y<\-) - (a0+a)sos + 0 r ' ( 5. 33) where 3a0 i a i 0 . 112.

( 3) a - 3a 0 = 0 = Im(m0 ).

Again mo is a real constant , and equation ( 4.153) gives ( 3a0 - a)m 0 = 0 , so that m0 is arbitrary , but not zero because , if it were zero,

we would have 1\ = m = d = 0 and flat space. Equation (4.150) now reduces to TC = 0 so that T = T0 (real constant ). Thus we arrive at the metric (4.129) with

1\ = 0 ' m = m0 , arbitrary non-zero real constant , d = 0 ' -2p- 2 3 2 e = 3 X R(' ) = X : -.,r +r-., . Written in full, this Perrov type II metric is

2 = 2 x 2(x -1 2 dT 3 r -3d', d", + 2�ds + + m0r )ds , ( 5. 34) which admits the H� = 2so - ro (5. 35) K' C<\ + - s r ' where we have accounted �or the presence of the Killing vecTor K 1 = o s in the space. x Putting = C+�, y =�-� brings the metric (5.34) into the form 2 3 2 x - 3 x 2 - 1 2 d T = � r ( d - GY. 2 ) + 2dr d s + 2 ( x + m0r ) ds , (5.36) admitting the HKV

K = XO + y' - 2so - ro . (5.37) x 0y s r Alternatively , by putting 8C = 3(X+iY) we can write the metric in the form 2 - 3 2 2 2m0 2 T 2 = d r X (dX + dY ) + 2drds + (�2 X + r )ds ( 5. 38) and the HKV becomes K = Xo + Yo - 2so - ro • (5.39) X Y s r 113.

Case VI. (2 Killing vectors ) p = 0 and equation (4.152) give Re(aC ) = a0 , with solution

If m0 is taken to be a complex constant, there is enough coordinate freedom left on C to transform a to ( 5. 40)

- 1 - 3/ao Since m = 2m ( 1 - 3a. and p = 0 , we have i D 0 so that 0 0 )C \ 0 = = = 0 � o ououD 0 0 DouD 0 and th<: only non-flat possibility is m I m0 I . Then (4.153) gives (5.41) Subtracting this from its COQ?lex conjugate gives

w�ere ;..:e have used Re(a0) = 1 and Re(i30 ) = a0. Substituting back in (5.�1) gives a = 0 , so there is no HKV.

Case V ::: J:. (2 Killing vectors ) (2) s·1nce R = c + ' = x, equation (4.154) gives

a + + + o. a 2ao< C 0 = Just as in Case III we obtain

( 5. 42) 114.

Then (4.151) and (4. 152) are satisfied identically . m = 3/2 For �0C there are two possibilities : m O either (i) 1 0 � �O 1 ,

(i) m 1 0 . Then (4.153) gives a= 0, so there is no HKV .

(ii) m = 0 . Equation (4.153) is satisfied identically , so we are left with equations (4.150 ) and (4.155) to determine d, A and T. Now C = peiS , so that

Co - Co- = ' ' Therefore ,.., + + + K = -2a0pop a0(sos - ror ) + Tos a(sos ror). Since it is known that +C k2 k2 d = < c ) D < s ) = < 2p cos s) D < s ) k = P 2H s >, where f(S) = (2 cos S)k2 D(S), equation (4.155) gives ,.., 1,-2f ( K + a 0 - a) d = - ap ( S) = 0 so that

either a = 0 and there is no HKV,

or f(S) = 0 , which implies D(S) = 0. Then the equation for D( S), namely , - 2iS - 3/2 2 2 d 2D Im[�o( 1+2e ) ] = 3 cos e dS2 is satisfied. This leaves the equation in 1(S) to be solved: (d1/dS) - 21cosec 2S = 0. ( 5. 43) The solution is 1 = C tan S, C real constant . Hence 1 1 A = i(C+� )- .1(S) = � iCp- secS tanS - 2 = C( C -C> < C+0 ( 5. 44) = - A. Equation (4. 150) now becomes T o C - (a0 + a)A = ( 5. 45) which, Hith its complex conjugate, implies 115.

so that T = T(C-C ). Substituting back in (5. 45) gives = - - -2 T' C(a0+a) (C-O < C+C) , where the prime denotes different iation with respect to (C-C ). This last equation holds iff = (1) a 0 + a = 0 , or (2) C = 0 , or (3) a0 + a = 0 C, and then we have T = T 0 (real constant). Taking these three possibilities in turn , we have : c (1) ao + a = 0 , .,. o. Put BC = 3(X+iY). Then 4 1\ = 3 l C X- ;,

m = 0 = d, -2p 9 3 (2) � e = 32 X ' R = 4 X and the metric (4.129) is 2 2 - 3 2 2 -2 dT = r x (dX +dY ) + 2drds - 2CX YdY dr 3 -2 2 + "2 X (ds-CX YdY ) , (5.46) where C is an ar�itrary real, non-zero constant. This Petrov type III metric admits the HKV ,_ K = XoX + Yoy + ror. (5.47) (2) a0 +a¥ 0 , C = 0. The metric is (5. 8) of Case III,

admitting the HK\' (5.7) but with the restriction So this is a degenerate case, whi c� can be expressed in the form (5.9), (5.10) with a0 i -a i 0 . ( 3) a0 + a = 0 = C. Again a degenerate case with me�ric (5.9),

but the HKV in this case is restricted to the form + K = XO y 0 + ro . ( 5. 4 8) x y r

Cases VIII and IX. (2 Killing vectors )

These are the case II metrics of Kerr & Debney ([199], p.2817). -p They do not admit a Killing vector of type o = e o , so �e must use s u a(C), the (C,C ,u,v) coordinate system and look for solutions R({;; ,C ) of the equations (4. 124) which are rewritten here for convenience, with the HKV in the form (5.49)

K The homothetic i l li ng equat io�s and their int errQhi lity co�ditions are 116.

+ 1 O O = 'K - a)(O-uO) ¥a eo u ) + R + E 0, ( 5. 50 ) c c ac ) c ,_. . + KO + a 1 = 0, ( 5. 51) ,o 2 ace = c'K + 3 Re(aC ) - a)f..l 0 , (5.52) = c'K + 4 Re(aC ) )fi 0, (5. 53) = ( 5. 54) c'K + 2 Re(aC ))(D 0 ) 0, ,_ = ( K + Re(aC ) - a)t:, 0, (5.55) O (K + aC + Re(aC) + a), = o. ( 5. 56)

Case VIII. (2 Killing vectors )

The functions 0 and f..! are known to the extent that they are functions of u only : = 0 O(u) , ( 5. 57)

f..! = - -3 = f..! (u), f..l00 flo complex constan�, ( 5. 58) where (5.59) and o. (5.60 )

The two Killing vectors .,.,-jich give rise to equations ( 5. 57 ) - ( 5. 60 ) are = K = + 0-:: , '' Ha - o ). (5.61 ) 2 c 1 <\ � c If we can solve equations (5.50) - ( 5. 56) for a and R subjec� to .... lC (5.57) - (5.60 ), with a f. 0 in K, then we will have found a r:-:et or metrics which admit a Hi

= o, (I) a (;, R = ,_

K = ea, + ea + uo - vo + a(uo + vo ) ' (5.62) , u V u V

= xo + yo + ( a+l )uo + (a-1)vo = x+iy . X y u v ' ' (II) a = i(;, R = 0, ,_ + K = H co - eo-) a(uo + V ) ' (5.63) , c u V

= xo - yo + a(uo + VO ) . y X u V 117.

(Ill ) a = a0 (complex constant ) , R = 0, a K = a0oc + 0o� + a< uou + vov), ( 5. 64)

Taking each of these possibilities in turn , we have firstly

(I) Equations (5.50 ) and (5. 51) give - a)o = o . K (K With as in (5.62), this is

(a+ 1) u dudfl - a 0 = 0 (5.65) with two solutions , depending on the value of a: a/(a+1) either 0 = C u , a 1 -1, (5.66) or 0 = 0 , a = -1, (5.67) where C is a complex constant . But ,_ ,_ --3 !-! -4--- --1--- K!-1 = K(�00 ) = - 3 0 0 KO = - 3 �0 K 0 so that (5.52) gives

1 (3 0 - K 0 + a - 3 )I-! = 0 . (5. 68) = �ither (1) 1-l 0, so thaT = ( i) �0 f. 0 ' 0 0 ' (ii) or �O = 0 , 0 f. 0, or (iii) 1-lo = 0 = 0;

,_- a - or (2) KO = (1 - 3)0, � f. 0, and substituting its complex conjugate back into (K - a)O = 0 gives 3 a = 4' 0 f. 0 , � f. 0 ;

or (3) � = 0 = <'K - 1 + %)0 so that , using K - a)o = 0, - 4;>o.( � = 0 = ( 1 o, Then (i) o f. 0 = 0, a arbitrary , non-zero, 1-l 3 or ( ii) = o, 0 , a = · 1-lo 0 f. 4

Also, 2!::, =DO - DO = 0 for both (5.66) and (5.67), so that any r.etrics will have a h)yersurface-orthogonal principal null ray congruence. 118.

Case VIII (l)(1)(i). Condition (4.64d) is satisfied, so space is flat .

= 0 = = Case VIII (I)(1)( iii). � 0 (� � 0). 0 Likewise, flat space.

= = 0, 0. Case VIII (I)(1)(ii). � �O 0 i = If a -1, then (5.67) requires 0 = 0 , so we have a contradiction and there is no solution . a/(a+1) t- 0, If a i -1, then 0 = C u ' c = - -1 and 0 cc 0, so that the field equation (5.59) reduces to (5.69) Now "'2 '' = 2a(a-1) 2 Q -2/a t- ( �· ) c ( ) ' a -1, (a+1)2 C and substituting this into (5.69) gives a(a-1)(2a-3) = 0, a t- -1, so that a = 1 or a = 3/2, since a t- 0 for a proper homo thetic motion .

= k c t- 0. a 1: 0 = Cu 2, Condition (4. 64d) is satisfied , so space is flat .

= -·3 = ,... 3/5 t- 0. a 2. 0 cu ' c The space is not flat and possesses the metric

2 2v2 dCd�? [ 3 -2/5v ( CdC - - - -4/5 dT = + 2 dv - s u + CdC ) + CCu e4 J� 4 ( 5.7 0 ) where 3 5 dC € 4 - du + u 1 (C + �d� ), and C is an arbitrary non-zero complex constant.

This Petrov type Ill metric admits the HKV = 2Co vo K c + 2Coc + 5uou + v (5.71) besides the two Killing vectors (5.61). 119.

1/5 = If we put C = x+iy, w u then the metric (5.70) becomes 2 2 2 2 - 2 2 2 2 -4 5 ) e 1�... , dT = 2v (dx + dy ) + 2[dv - 3w v(c1dx - c2dy ) + � c1 + c2 w 4 4 (5.72) where

c and c1 , 2 are arbitrary real constants, not both zero. The HKV is now ,_ K = 2xo + 2yo + wo + vo (5.73) X y W V 3 0, 0 , = Case VIII (I)(2). �f. 0 f. a 4 From (5.66) we have 3 7 = 1 , c 0 Cu f. o. = Then OVC 0/C and field equation (5.59) becomes 5 2 2 - 1� c c - 5 o-\l 2 (o[oco )"J'} · = [Co )"/, (5.74) 0 + while field equation (5.60) becomes 3 3 3 2 2 2 (4 c c - o- 2c- c o[o

Calculating 0/ C 2 " ( ) 0 and substituting these into (5.74), we get 2 5 343 16C c o , c 0. �0 + = f. Hence 9/7 - - c 2 - 0. � � = c ) u = �343 c f. (5.75) Field equation is satisfied for this 0 and �· We have arrived at the Petrov type II metric - 3 -4/7 - - 2 2 [ · dT = 2v dCd' + 2 dv - 7 u v(CdC + CdC ) 4 - ____ 9/7 1/7 _ 1 - <21 - , ( 5. 76) + 343 cc- u u 4CCv � 4� 4 where

and C is an arbitrary non-zero complex constant . 120.

This metric admits the HKV = 4C 4 'K oc + Coc + 7uou - vov ( 5. 77) besides the two Killing vectors (5.61). C = = 1/7 = 7 Putting x+iy , w u C 2 Cc1+ic2 ), the metric (5.76) can be written in the form

2 2 2 2 3vw-4 dT = 2v (dx + dy ) + 2{dv - (c1dx - c2dy ) 2 2 -9 2 2 -1 + (c1 + c2 )w [ 3w - 7(c1 + c2 )v }:: 4}£4, ( 5. 78) where

and c1 , c2 are arbitrary real constants, not both zero. The HKV is now K = 4xo + 4yo + wo - vo . (5.79) X y W V

Case VIII (I)(3)(i): = 0 = 0, 0, a arbitraF; 1 0. � �O �

Condition (4. 64d) i� satisfied so space is flat .

3 (I)(3)(ii). � 0 = 0 1 0, = Case VIII = � ' a O 4 E�uations (5.74) and (5. 75) must hold if there is a solution. 0, 0 However , (5.74) gives C = contrary to the condition � 0. Hence there is no solution .

(II) Equations (5.50) and (5.51) give

(K + i-a)O = o . (5.80) (5.63), Since 0 = O(u), and with K as ln the last equation is + 0 au dOdu (i-a)O = ( 5. 81) with solution = (a-i)/a 0 C u , (5.82) where C is a complex constant ...... = --1 -- Frorn ( 5. 58) we have K � -3 � 0 K 0 so that (5.52) gives (5.83 ) 121.

0 , Either (1) 11 = so that 0, ( i) t- o, 0 = 11o (ii) = t- 0, or 0, 0 11o = 0 or (iii) = 0; 11o

..... - a- 0 , or (2) K0 = --;:;v 0, 11 t- and subst ituti ng its complex conjugate back into (5.80) gives 3i - 4a = 0. Since a is real, there is no solution in this case ;

0 or (3) 11 = = cY. + �)0. ( 5. Putting Y.0 = - � 0 in 80 ) leads to only one = 0. possibiliry' , namely , 0 Having dealt with ( 2), He are left Hi th (1) (i) Conditions (4.64d) are satisfied, so flat space . Like�ise , flat (1) (i�� ) space. SujstiLutin� (1) (i�) (5. 82) into field equation (5. 59) gives = is 4ia + 2 S. Since a real , there is no solution . (3) Con��tion (4.64d ) is satisfied, so flat space.

(Ill ) Equatio�s (5. 5C ) and (5.51) give

a)O = cK - 0. (5.84)

Since 0 = O(u) , and Kith K as ln (5.64), equation (5.84) is

u-clD -0-- 0 dl:. with so_ution = Cu, C 0 complex constant . Hence ooDO = O Do DO. (5.85) u u = u NoH (5.58) and (5.52) ioply + = 0. dK' n a 0)11 ( 5. 86) Either (1) 11 = 0 , and then (5.85) gives flat space; -KO a 0. or (2) = - -;:;-0 11 t- ;) , 0, Substituting in (5.84) gives a = so there is

no E:-.\' ;

,..., a or (3) 11 = 0 = (k + -;:;-)0, unJ tl.cn (5.85) gives flat sp ce . 0 a

This concludes the c:scussio� of Case VIII. 122 .

Case IX. (2 Killing vectors ) From Appendix 2 we see that the two Killing vectors = K1 o' + oC = ox ' K = eo + o + uo ( 5. 87) 2 , � c u VO V = xo + yo + uo vo , X y U V where C = x+iy , and the HY.V = K a0, + ao + Re(a )(uo - vo ) + Ro + a(uo + vo ) , , u v u u v can all be present in the space when, and only when,

(I) a = C-1, R = 0 , K + a(uo + vo ), ( 5. 88) 1 U V

(II) a = C, R = O, ,_ K = K + a(uo + vo ), (5.89) 2 U V

(III) a = 1, R = 0 , ,_ K = Y1 + a(uo + vo ), (5.90) U V

(IV) a. = 0 = R, + K = uo VO (5.91) U V There is effectively only one case to consider. For, since K1 and K2 are already present in the space, we can take the HKV to be (5.91), the = constant a being absorbed into K (equivalently , choose a 1). We shall choose (IV) to represent the situat ion. The homothetic Killing equations and their integrability conditions (5.50 ) - (5.56) are the� : = (i< - 1)(0 uO) 0 , (5.92) .....K O . = 0 , (5.93) ( ,.,K- 1 )1-1 = 0 , (5.94) 'Kt1 = o , ( 5. 95) K(D o) = 0 , (5.96) (K - 1)6 = o , (5.97) 1)j (i< + = o . (5.98) We shall also require the equations for K1 and K2 corresponding to (5.92), (5.93) and ( 5 . 94) . They are: 1 23 .

= 0 ( 5. 99) K 1 (0 - uO) , 0 00 K 1 0 = , (5.1 ) 0 K1� = 0 , (5.1 1) and 0 02) K2(0 uO) = , (5.1

= 0 (K2 + 1)0 0, (5.1 3) (K2 + 3)� = o . (5.104) Equations (5.99) and (5.100) give 0 K 1 0 = 0 (5.1 5) while (5.102) and (5.103) give Kfl = 0 (5.106) and (5.92) and (5.93) give

- (iZ 1)0 = o. (5.107) Using (5.91) the last equation is (uo u - 1)0 = 0 , implying O=uf(y), where we have used (5.105) as well. Substituting this in (5.106) gives -1 f = Cy , so that 0 = Cuy-1 , (5.10 8) where C is a complex constant . Equations (5.91), (5.94) and (5.101) give (uo u - 1)� = 0 , and � = ug(y). -4 Substitut ing this in (5.104) gives g = Ay , and so � = Auy -4 , (5.109) where A is a complex constant . The constants A and C are related through the field equations (4.62). Equation (4. 62a) becomes - -4 A ( 2C + i ) uy = 0. (5.110 ) Field equation (4. 62c) is satisfied iff � � C 3). A = 2 C(2 i - C)(4i - (5.111) Using this value of A the remaining field equation (4. 62b) is satisfied identically. Equation (5.110 ) thus holds iff � C C o �2 C(2 i- C)(2 + i)(4i - 3) = 0 1 . or 3 . C = , 2 � 4� . 124. c = 0 = 0 : 0 = �and condition (4. 64d) gives flat space. c = 1 . 0 -1 = = 1 . � 0 and condition (4.64d) gives flat space. 2 �: 2 � uy -1 1 - c = 3 . 0 = 3 = - �. uy 0 . D - D 0 . 4 � : 4 � Also /1 = t< 0 0) = Do 0 Since uD 0 t- the space is non-flat , and has the Petrov type III metric = 2 2 2 1 - 2 1 d1 2 2v (dx + dy ) + 2[dv + � vy- dy - � y (du -%uy - dy )] X 3 -1 X (du - 2 uy dy ) (5.112) admitting the HKV (5.91). = If we put 2w 3 log y, the metric becomes 1 2 2 2 4w 3 2 -2w 3 ] dT = 2v (dx + �e / dw ) + 2[dv + vdw % e Cdu - udw ) X

X (du - udw) . (5.113)

This concludes the discussion of Case IX. 0heorem 5 . 1)· The non-flat algebraically special vacuum Einstein spaces with non-zero complex divergence which admit 3 2, or 4 Killing vectors and one HKV are given in the following table:

l�o. of Case .Metric Petrov type Killing vectors HKV

(5.3) Kerr- D 4 I(ii) Schild (5.4) III ( 5 . 9) III 3 (5.10 ) IV(2) (5.22) D 2 (5.21) V(1) (5.29) II 2 (5.30 ) �·: v < 2) (5.9) III 3 (5.33)* V(3) ( 5.38) II 2 (5.39) VII (ii)(1) (5.46) III 2 (5.47) *VII(ii)(2) (5.9) III 3 (5.10 )�': '':VII ( ii ) ( 3) (5.9) III 3 (5.48)* 1 VII I (I )( )(ii ) (5.72) III 2 (5.73) VIII (I)(2) (5. 78) II 2 (5.79) IX (5. 112) III 2 (5.91)

* degenerate cases

All of the above contain a hypersurface-orthogonal principal null congruence except Case V(1), metric (5.29). 125.

CHAPTER 6 Spae� w�h One HK V and One K�g Vecto�

From Theorem 3.2 we know that the commutator of two HYVs is a Killing vector. This theorem and its proof can be specialized to give the result }..K ' CK',K J =

.... where K is a Killing vector and K is a HKV. This restriction on the geometry is discussed further in Appendix 1, where it is shown that for a given Killing vector the HKV must take a specific form . The ����se of this chapter is to determine those vacuum spaces which admit o�e HKV and one Killing vector. We shall consider the Killing vector in each of the canonical forms = = (i) K a (ii) K a + o­ s X c in turn, using the results of Appendix 1 to write down the form of the associated HKV .

6.1 The space admits K = o ( 6. 1) ,.,. s and K = a0(sos - ro r ) + a(sa s + ro r ), ( 6. 2) w:1ere the cc,� ,s,r) coordinate system may be used owing to the presence of Kin the form (6.1). Equation (4. 152) gives = ao o . Absorb the arbitrary constant a into K (equivalently , choose a = 1) so that K = + so s ro r . (6.3) Equations (4. 150 ), (4. 153), (4. 155) and field equation (4.156a) give /\ = m = d = 0 . (6.4) Equations (4. 151) and (4.154) are satisfied identically , as is field /\ = equation (4. 156c). By (4. 130), Cl implies 0 p ,...u, = ';,

so that Petrov type III metrics will exist if - D

where p(C,0 is to be determi�ed by the remaininr; field e-1uation ( 4 . 15Gb ) . Otherwise the space is flat . 126 .

In discussing the field equation (4. 156b ) we shall consider ( 2 (2) separately the possibilities R ) = 0, R = non-zero constant , (2) R � constant.

(2) Case 6. 1 (I ) : R = 0.

In this case = 0. The coordinate freedom available is given in pCC Appendix 1, equation (A1.11). In Appendix 3 we show that by an allowed coordinate transformation (4.134), (4.137) we can transform p to zero. = o 0 In the new system of coordinates (4. 130) gives 0 1\ so that u DO = . = Hence, by (4. 64d ) the space is flat since � m = 0 also.

1 (I ( 2) = Case 6. I ) : R R0 (non- zero constant ) . From (4. 137) we find that R(2) transforms as (2)' -2 (2) R = C0 R ( 6. 5) l under the allowed group of transformations . By choosing c� = IRo (2) ±1. 2p we can use (6. 5) to make R = The 2-metric e d(;dC is then that for a sphere or pseudosphere and so the coordinates can be chosen (see Appendix 5) so that -p e = (;� - R0 • (6.6) p = o p ) = o But then pCC � so that D(pC(; � and the s?ace is flat . (2) Case 6.1 (III): R i constant. (2) -2p In this case R = e p C = 2Re{F((;)}for some analytic function (; (; -2 F of . Now (6.5) implies that , if we set (;1 = c0 F((;), then R ( 2 )' = <:' + C' . Dropping the primes and working now in the new coordinates gives -2p - e p C - (6.7) (; - (; + (;. The coordinate freedom left is, as may be seen from (6. 5), 2 r ' = c- ( ' + 1.. e ) ' � o 'b o = s' c0s, (6.8) ' -1 r = c0 r, where e0 is an arbitrary real constant and we have used equation (A1.11) of Appendix 1 with a0 + a i 0 . 127.

Differentiating (6.7) with respect to C gives - -1 = 2p P p p = C ) e CCC - 2 C CC C+C Pcc and so for all p

Hence, if we can solve (6.7) for p, we have non-flat spaces admitting homothetic motions . Unfortunately , the only solution of (6.7) known is

e 2 - 3 -2p = � C+O . (6.9) Substituting (6.4) and (6.9) into (4.129), we have the Petrov type III hypersurface-orthogonal metric dr 2 = 3r2CC+C )- 3dCdC + 2drds + 2CC+C )ds2 . = Putt�ng BC 3(x+iy) this sim?lifies to the Kerr-Debney [ 193] metric 2 2 -3 2 2 3 2 dT = r x ( dx + dy ) + 2drds + 2 x ds , (6.10) which is the metric ( 5. 9) of Chapter 5. . It is, in fact , in·,.rariant not just under a G1 of isometries but under a 3-dimensional group of isometries with Killing vectors

= - K3 -2(xo x + yoy ) + s� s r�r In this sense, we have here a degenerate case. The HKV admitted here, though, is (6. 3) which is (5.10) with the = restriction a0 0, a = 1. Other solutions are theoretically possible, but aHait the discovery of further solutions of equation (6.7) .

6.2 The space admits K = os (6.11) and (6.12) where a(C) is non-zero. We could have chosen the canonical form (A1.7) of Appendix 1 for K, but this Hould have restricted the coordinate freedom available. At this stage we prefer to have the full coordinate freedom. = Since - [K,K] (a0 + a)K we shall split the analysis according to whether (a0 + a) is zero or not , and whether the 2-curvature R(2) is constant or not .

( 2) = = R constant R0. 128.

From (4.154 ) we get a P = 0 . We shall consider ln turn the = 0 0 possibilities a0 1 O , a0 0 .

(I): Case 6. 2 R 0 = 0.

Just as in Case 6.1 (I) we can transform p to zero. The coordinate freedom left is a linear transformation in C. with complete freedom still in s and r (see (A3.3)). The homothetic Killing equations and their integrability conditions (4.150 ) - (4. 155) are now + a a)/\ = 0 (6.13) (K aC - 0 - + \ , = o ace , (6.14)

) = Re(aC ao , (6.15) ,... (K + 3a0 - a)m = 0 , (6.16) = Ro 0, (6. 17) ,... ( K + a 0 - a)d = 0. (6. 18) The solution of (6.15) is

� = where a0 , O are constants and Re(a0 ) Performing a coordinate transformation (A3.3)

we can transform a to the form (6. 19) where ao is an invariant (constant ) and primes have been dropped from the new functions. From field equation (4.156a) and equations (6.16), (6.19) we obtain = c -1 m((;) NC , c = (a - 3a0 )a0 , (6. 20) where N and c are complex constants. The third field equation (4.156c) reduces to (6.21) Define � = � (0. (6.22)

Then the general solution of (6.21) is (6.23) 12'.!. where �is an analytic function of C . This implies t\' = 2c13 + � + \;c , where A(C,C ) is a real function , and so -2 - t\ = ' 13 + '� + \ · (6.24 ) We now have the option of either (a) transforming AC to zero and finding a particular integral for t\ from (6.18) by using the coordinate freedom on s (equation (4.142)), and then determining T(C,C ) from (6.13); or (b) transf )rming T(C,C ) to zero by means of (4.147) and determining A from (6.13) and (6. 18). See Appendix 4. We take option (a) in this case. By means of a transformation (4. 138) we can quit the last term of (6.24) and write -2 - t\ = ' 13 + '�· (6.25) where we simply require a particular solution for � from (6.18). Now (6.20) and (6. 22) give c+1 13(C) = No{; + MO , where M0 ,N0 are complex constants and 2(c+1)N0 = N. Hence

where B = 2 Now make a transformation s - s' = s + B. After dropping the prime we are left with (6.26) and (6.27) It remains to determine �· This is done by solving (6.18), which is < o a0Cdc + a0Cctc + a0 - a)d = or, through (6.23) and (6.26), aoc�, + (ao - a)� = complex conjugate, = remembering that Re(a0 ) a0 . This implies aoc�, + (ao - a) � = c (real constant) 1 30 . so that where D is a complex constant . Remembering that we seek only a D particular solution for �. we choose = 0 . Then = -2 c+1 -1 - A N0C C + C(a0 - a + a0 ) cc. (6.28) Equation (6.13) now gives TC = - CCC . This require3 C = 0 since T is real, and hence T = 0 . (6.29) Summing up , we have arrived at the metric (4. 129) with

p = 0 = R( 2), = c+ 1 A NoC\ ' c (6.30) m = 2(c+1)N 0C , - N c c d = iCC C 0C - N0C ), where a0 , N0 are arbitrary com?lex constants with Re(a0 ) = a0 and where a0 + a i 0 and a i 0 . Since T = 0 we can absorb the constant factor a into the HKV i.e. effectively a = 1, and so = aoc 1 - 3ao , a0 arbitrary , and the HKV admitted by the me tric (6.30 ) is (6.31) = o D = 0 Since A 0 and u 0 , condition ( 4.64d ) tells us that space is flat iff m= 0 , that is, iff N0 = 0 . Otherwise this metric is Petrov type II.

o Case 6. 2 ( II ): a0 = and either (i) R0 = 0 , or (ii) R0 1 0 . (i) a0 = 0 = R0 . Again we can transform p to zero, leaving C free up to a linear transformation, "ith s and r completely free. 131.

Choosing a = 1, the homothetic Killing equations and their integrability conditions reduce to 1)A - t = o (6.3 (K + aC TC , 2) o aCC = , (6.33) 0 Re(aC ) = , (6.34) CK 1 )m = 0 , (6.35) (K- 1)d = o . (6.36) The solution of (6. 34) is a = ib0C + a0 , (6.37) where bo ' ao are real, complex constants respectively . Either = 0 a = (A) b 0 , ao , (B) 0 or bo t . = In case (B) we can transform a to a ib0C, where b0 is an invariant , by means of the transformation c' = boC - iao· Without loss of generality we can choose b0 = 1. The linearity restriction on the transformation of C prevents us from doing better than obtaining a = iC for (B). o case 6.2(II)(i)(A): a0 = o = R0 , a = a0 t .

= a o + + To + SO + r . K oc aooc s S or Field equation (4. 156a) and equation (6.35) give Cla m(C) =Ne 0 (6.38) where N is an arbitrary complex constant. With p = 0 the field equation (4.156c) has the solution given by (6. 22) and (6.23). This gives A as in (6.24). Again, just as described in the paragraph following equation (6.24), we choose option (a) of Appendix 4 to find a particular solution for A. From (6.22) and (6.38) we get C la0 � = N0 e + M0 , where M0 , N0 are complex constants and 2N0 = a0 N. Hence Cla C 2 e 0 A =N0 + C q:>+ BC , = 2 where B :' Re( ��0�C ) and q:> js an analytic function of C. 1 32 .

Making an s-tran::formation s - s + B we obtain /a = C 0 ( 6 . 39) iJ N e 0 and 2 C a i o (G.40 ) A = N 0 C e + C�, where � is to be determined by ( 6 . 36) . From (�.23) and (6.36) we get a09C - � + 2i0S = complex conjugate, which implies (real constant ). This has solution where C is a COD?lex constant. We can eliminate D by means of the transformation s

Because we are seeking only a particular solution for �. we 0 . choose C = Then (6.41)

Equation (6.32) gives no� - 2 C \ = Na0 so that where E is a real constant which we can discard since it gives a term Eo s in the HKV , an d this is a constant multiple of the already present Killing vector. Hence C iao 2 -- �lci 2 Cia 'la � o o - o T = la0 1 { 0NCe + a0NCe - la0 l (N e +N e )}. (6.42) Thus we have arrived at the metric (4. 129) with ( 2 ) p = 0 = R , - 1 - A = NC<2 al., a U 0 (6.43) m = N e _ Cla0 = 1 i ( a C Ua0 d 2 0 - al., ) ( N e + N e ) ,

1 where ao , arc C'. r"bitrary complex constants, 1:-ut a,�.., t 0 . 133.

1\ = 0 and Since o u DO= u, condition (4. 64d) shows that the space = will be flat iff ti 0 , otherwise Petrov type II . The HKV is = ro ( 6 . 44) K aooc + aooc + Tos + SOS + r ' where T is given by (6.42).

= = Case 6.2(II )(i)(B): a0 0 R0 , a = iC.

= - S K i(Coc eo, ) + Tos + OS + ror Equation (6. 35) and field equation (4. 156a) give = NC -i m(0 , (6.45) where N is an arbitrary complex constant. With p = 0 the field equation (4.156c) has the solution given by (6. 22) and (6.23). This gives 1\ as in (6.24). Again we choose option (a) of Appendix 4 to find a particular solution for 1\ . From (6.22) and (6.45) we get 1-i P = ;;OC + MO , where = 2(1-i)N0 N and M0 is a complex constant which we can transform to zero through + 2 -2 s - s Re(M0CC ). �hen this is done we have (6.46) and 1\ = Noc2,1-i + c�. Using (6.23), equation (6.36) reduces to + '� + i(� - 0 ,�, �) = which implies = ccp, + icp iD, where D is a real constant . This has solution �( C ) -i = CC + D, where C is a complex constant . We can eliminate D through the transformation s - s + r�c . Remembering that under option (a) we need only a particular solution for �. we choose C = 0. Then and

\ = (G.47 ) 134.

= 0 From (6.32) we now obtain TC , so that = T T0 (real constant ) (6.48) Thus we have arrived at the metric (4. 129) with ( p = 0 = R 2), A = 1-iC2 NoC ' (6.49) . -i m = 2(1-1)N0C , = . -- -i - i d 1CC< N0C - N0C ), where N0 �s an arbitrary complex constant. The space will be = flat iff N0 0 , otherwise Petrov type II. Taking into account the fact that there is a Killing vector T0os already in the space, we have the follo�ing HKV admitted by this metric:

= K i(Co - �o ) + SO + ro . (6.50 ) c , S r -i = z = By putting C e so that iCoC oz ' the metric components (6.49) become p = 0 = R( 2), (Hi)z - 2iz - N oe 1\ (6.51) = z m 2 ( 1- i ) N 0 e , d = 1e. i( z-z)r- z z LN0e - N0e ] and the HKV is non = ro . K oz + o-z + so s + r (6.52)

= (ii) a0 o , As in Case 6.1 (II) we can choose coordinates so that (6. 53) ±1. where R0 = The coordinate freedom left is a bilinear transformation on C and complete freedom on s and r. The HKV is ofthe form a � = ao + o To a(so re ) C C + 8 + 8 + r

so that, for p as in (6.53), equation (4. 152) becomes 1 35. with solution ( 6. 54) where a0 and b 0 are constants , b0 real. There are t\-JO cases to consider :

(A) R 0 = -1 (sphere ), (B) (pseudosphere ).

-1. Case 6.2(II )(ii)(A): R0 = The form (6.54) becomes (6.55) Appendix 6 shows that it is always possible to reduce (6.55) to the form = 2 a a0C + a0 . (6.56) Change coordinates to (z,z ,s,r), where ( 6 . 57) Then 2 cos z 0 z and 2 a = a0 sec z. ow = K laol

- = K o z + o-z + a(so s + ro r ), (6.58) where we have redefined the constant a. Thus there is no loss of l l = generality in taking a0 1, so that a = -p = C 2 tan z, C = tan z, e 1 + tan z tan z, (6.59) and the remaining equations (4. 1 50 ) - (4. 155) become = o , (K + 2 tan z a)A (6.60 ) - o, (K a)m = (6.61)

(� a)d = 0 . (6.62) 136 .

The surviving field equations (4. 156) are

rn­z = 0 , ( 6 . 63) 2 z 6 m - m = 2i cos (z- )dzz- + 4id. (6. 4) In order to preserve the form ( 6. 58) of K, equation ( 4. 14 7) must be satisfied: <'K-a)A = 0, implying that A= -A = (eaz + e az ).f(z - -z), (6.65) where f = f is an arbitrary function of (z - z). Also (4.146) requires a' 1 a, = = implying that z' = z + c, where c is a complex constant . Hence the coordinate freedom left is z' = z + c, (6.66)

I -1 r = c0 r, where c0 is a real constant and A(z,�) is as in (6.E5). Equation (6.60) with (6.58) gives

az az 0 = (e + e )cos·z.h(z - �) , (6.67) where h is an analytic function of (z - z). Using the definition -2p 2id = e (/\- - i. ) c ' we get 2 az az- 2id = cos 6(e f - e f,) , 6 = z - z, (6.68) where + f( 6) = ah - he he . Therefore 2 az az- az- 2idz = - 4id tanS + cos SCe f8 - ae f + e f8 ). Substituting the last expression in (6.62) we find that equation (6.62) is satisfied identically . From (6.61) and (6.63) we obtain m (6.69) where m0 is an arbitrary complex constant . Calculating 2idzz - and substituting in (6.64) gives 137.

az az moe - moe 4 az- az­ = cos 8[2tan8( 2e fe ae f az az- az- az- + + f ae fe - e fee - ae fe e ee so that - (a = 4 fee + 4tan8 )fe - (4-2atan0)f mosec e . (6.70) = tanS , this Putting a = 2 (possible because ao 0 ) and X = becomes 2 2 " 2 - + 2 2 (1 + x ) f - 2(1 + x)(1 + x )f' - 4(1 - x)f = m0 (1 x ) , (6.71) where the prime denotes d/dx. 1 = I have no solution to (6.71) for m0 0 . If m 0 0 , then m = 0. But for pas in (6.53) we have o D = 2 = so that when m= 0 u O pCC - pC O , condition (4. 64d) is satisfied and space is flat. In this case, then, non-flat solutions are possible and derive from solutions of (6. 71) when �0 i 0 . However, I have not been aj le to solve this equation.

Case 6.2(II)(ii)(B): The form (6.54) becomes (6.72) Appendix 6 shows that it is not always possible to express this quadratic in canonical form. We need to consider three cases separately :

1 = Case 6.2(II)(ii)(B1): a0 o , bo 0 . a can be reduced to the form 2 a = ' - 1. (6.73)

Change coordinates to ( w ' w, where s, r)' 1 + w w c - 1 = ' + 1 � ' = 1 - w (6.74) It is stressed that this is a pure relabelling of the coordinates and is not an allowed transformat ion C - C' = � (C). for if (6.74) were to be interpreted as such a transformation, it would have to be sill)jcct to the constraint C}Uutions (AG.t.)of Appendix 6 in order to leave the form of p unchang(;d. 138.

Obviously these constraint equations are not satj cied when (6.74) is so interpreted. Then and so that the HKV becomes K = 2wow + 2wo- w + a(so s + ro r ) + To s . Since there is still complete freedom on the coordinates s and r, we choose option (b) of Appendix 4 to transform T to zero. Choosing a = 2 and absorbing a factor 2, we may now take

K = Wd + WO- + so + ro (6.75) w w s r The remaining equations (4.150) - (4.155) become -1 [K + 2(1 + w)(1 - w) 2}\ = 0 ' (6.76) c'K 2 )m = 0, (6.77) -- (K 2 )d = o . (6.78) The surviving field equations are m- = 0, (6.79) w w 2 m m = 2i(w + ) d - - 4id. (6.80) - w-w Equa�ions (6.76) and (6.75) give 2 w A= (1 - w) .f( e ), e = w/ ' ( 6. 81) where f is an arbitrary function of e . -2p Us ing the definition 2id = e (AC -AC ) we get - 2 - -2- - -2 ) id = ( W + W) ( HW fe - WW f e • (6.82) Therefore w -1 w 2 W - 3 w -2 idw = 2Cw+ ) id - Cw+ ) (2w t-8 + fe -2 -4- - 3 + w w fee + ww f ) - ee and substituting 1n (6.78) we get d = 0. (6.83) From (6.77) and (6.79) we obtain m = m0w, (6.84) where m 1s an arbitrary complex constant . Now (6.80) and (6.83) 0 · give m= m, so with (5.84) this implies m = 0 . (6.8 5) But then condition (4. 54d) is satisfied , so space is flat . 139.

= Case 6.2(II)(ii)(B2): a rv, 0 Choosing b 0 = 1 we have a = iC . (6.86) Then, using option (b) of Appendix 4 to transform T to zero, we may take the HKV to be K i C C o - eo� ) + a< so + rc ). ( 6. 87) = c s r The coordinate freedom left on s is s' = s + A, CA - CA = where C C + iaA 0 , giving -ia ia A =

a + i a)/\ = 0, (6.89) ,... (K a)m = 0 , (6.90)

= cK a)d 0 , (6.91) and the survi'T_;_ng field equat.:.ons (4. 156) are = 0 , ( 6 . 92) mC 2 2 C - m - m iC C 4id. (6.93) = 1 ) dcc Equations (6.90) and (6.92) give = ia rr. MC ' (6.94) where N lS an arbitra�· complex constant. Equation (6.89) is with solution C X = C - 1, (5.95) where f is a function to be determined by the remaining two equations.

Of these, equation (6.91) is satisfied identically upon using the de:'inition -2p 2 -ia ia 2id = e (/\- j\r ) = x (, f, - c f, ) ' (6.96) ( "' Hhere the prime denotes d/dx.

Substituting (G.9�) and (5.96) into the field equation (6.93) gives 140 .

-ia 4 C [N - x [(1 - ia)fu + (x+1)fm] 3 - 2x [( 2-ia)f' + 2(x+1)fu ] } (f>.97) = complex conj ugate. Denoting the expression in braces [ ) by E(x) , we can write (6. 97) as ia/2 (x+1)-ia/2 E(x) = (x+1) E(x) . (6.98) This equation is satisfied if (x+1)-ia/2 E(x) = C (real constant ). (6.99)

= 3 Put H(x) (1+ia)x f' -N. Then (6.99) reduces to

x(x+ 1)Hu + [ (-1-ia)x - 2]H' + (1+ia)H + C(1+ia)(x+1)ia/2 = 0 . (6.100)

I do not have any solutions H(x) when C f 0 . For C = 0 , (6.100) is a hypergeometric equation with solution 3 H(x) = kF(-1, -1-ia, -2, - x) + £ x F(2, 2-ia, 4, -x) (6.101) valid for l x l < 1, where k and £ are arbitrary (real) constants and F is the hypergeometric function. There is no solution eX?ressible = in finite terms (see Appendix 7) other than the trivial one H 0. Then 3 = ( 1+ i a ) X f' ( X ) N with solution = 1 -2 -1 f(x) B-2 Nx (1+ia) , where B is a complex constant. This gives

We can transform the first term away by means of the transformation s .... s + A , where = -ia ia A < C + C )B/ia, in accordance with (6. 88). Thus -1-ia = C 1\ N (6.102)

Summing up , the only solution found ln this case is the metric (4. 129) with e-p = � 1, (2) = 1 c - R ' as in (G.102), (6.103) N ia m = C ' 1 2 ) -1CC �-- 1)-1[ ( 1-iv)NC -ia . '-"'ia d = � i(1+a ( 1 +la ) J' -.. ] , 141. where N is an arbitrary complex constant . The space will be flat iff N = 0 , otherwise Petrov type II. The HKV admitted by this metric is (6. 87) . Other solutions are possible and are determined by solutions of

(6. 100) for C � 0 , and by solving (1+ia)x3 f' (x) = N + H(x), where H(x) is given by (6. 1 01). I do not have solutions of this sort.

a Case 6.2(II )(ii)(B3): 0 � 0 , In this case the best we can do with (6.72) is to reduce it to the form (see Appendix 6) a = 2 c + ib0C - 1. = Choosing b0 2 we can �Tite this as a = (6.104) Put C+i = -1/z. Then = 2� u::,.C z oz ' a = z -2 and , after transformin¥ T to zero as we may through o�ticn (�) of Appendix 4, we have = K 0 z + 0-z + a(sO s + rOr ). (6.105) The remaining non-trivial equations (4.150) - (4.15�) are

z -1 = c'K - 2 a)/\ 0 , (6.106) = c'K a)m 0 , (6. 107) = o, (i< a)d (6.10 8) and the surviving field equat:ions (4.156) are m- = 0 , (6. 109) z m = [ - ] 2 d m - 2i 1 + i( z-z) zz- - 4id. (6.110) Equations (6.107) and (6.109) give = m az m 0 e (6. 111) where m0 is an arbitrary complex constant . From e�uations ( 6 . 105 ) and (6.105) we obtain A = 2 az - z e f( z-z ), (5. 112)

where f is a function of c�-�) to be determined by the remaininr e�u�ti ons . 142.

From the definition of d we find (6.113) where e = z- z. Using this result and -1 0)2 eazf azf azf ) 2idz = -4(1+i0) d - (1+i (a o + e eo + e ee we find that (6.10 8) is satisfied identically . Substituting in (6. 110) we obtain m = 3 2i ) + ( 1 i0)(a f >). o (1+i0) { (afe + 2fee + feo + eee

Putting x = 1+i9, this may be \-Tritte n m 3 a 0 = x {2i(i f' - 2f0 ) - x(af• + ifm)), where if' = i(df/dx) = f0 . The equation ln f may be written as

[xf' + (2 - iax)�]· = (6. 114) with solut ion

= c 2iB - - 2 iC -1 iB -2 e iax f - - X + + -a X +- + Dx 2 -3 "o2a ) ) a \ a a (:� (6.115) where B, c, are D ar:t::"!:!'ar i j complex constants. On putting x = 1 + i(z-z) and s�stituting (6.115) into (6.112), we have an eh�ression for A . Summing up , we have the metric (4. 129), which ln (z, z, s, r) coordinates is 1 2 2 2 - z l-. 2 dT = (r + d )[1 + i{z-z)T 2 dzdz + [dr + i(dz.d - dzdz) y.. 2 + (1 + Re[m/(r+id)J}� , (6.116) where a az � = ds + e zf dz + e f dz , and the functions m and fare given by (6. 115) and (6.111) "ith x = 1 + i ( z-z ) . The function d is given by = [ z ] 2 az f- az 2id 1 + i(z- ) (e z - e f z ). (6.117)

me is Petrov tvpe = 0 This tric II, unless m0 when it is the metric of flat space . The HKV is (6.105). ( ) R 2 I constant Case 2 (III ) 6. .

We may take the 2-curvature I 2) n the or of Cas e R' i f m (�.7) E . 1 (:il), leaving coordinatE; freeJor.o (f; , g ), ful l freedom on the :.u-r nm-1 with :h"' coordinate according to the s i.e. s1 = c0(s + A), A arb� rQrJ , case . the only known solutic:J arguments employ ed in that Likewise, for p(C, ) is � (6.9) , namcl)· , = X ; ; �. Then ...... -2n -2c ,_ = e · K( e r ) - 2 l', p =

for 7 as in (6. 12) , so that ,_ = Kp (r�, 112)

Equation (4.154) gives

solut ion �d .th

wh 2re a The coordinat e e 1s real constant . transfor�& tlon 0 is allo·..: ed by ( C = + .ie0, which 6. 8), brings a. to

(C 1J J)

S�;Jy=·ti.tuti.ng in (6.118) gives Then e�u3t1o�s (4.151) this Kp = 3a0. are au(� (4.152) satisfied identically .

Since He still ave left the coordinate, h complete freedom on s we •.:rnploy option (b) App endix to trans form to zero. of 4 T becomes Then (6. 12) .... = -2a

a a)/\ = o, (5.121 ) (K 3 0 - (6.122) ( K + 3a0 a)m = o, - + (K ao a)d :: 0, t6.123)

ond tlte surviving field e;runtions ( 4. 156) are {G. 124) ITa ;: = 0, \. - 4 4i (6 125) t.1 r.: :: 3 i x' dcc - X d. 144.

Equations (6. 122) and (6. 124) give c m = moC ' (6. 126) where m0 is an arbitrary complex constant , and c is given by 2a0c = 3a0 - a. Equation (6.121) gives (6. 127 ) where -2aob = 3ao + a, ao(b-c) + 3ao = 0 . Using the definition of d we have then

- ·- 2id = e 2p(1\ C '\:) l l x -2cb+ c -2 b+ ;C = 1 \c £8 - c f8 ), e = c . (6. 128) We now find that (6.123) is satisfied identically . Using (6. 126) and (6.128), the field equation (6.125) is satisfied iff

f = C (complex constant ), � d = 0 , (6. 129) and m = 0. (6. 130)

2 -2 0 Now ouD 0 = pC(; - p(; = -( 3/4)X , so that o u o u D 0 = but -3 t- Dou D O = (3/2)X 0 . Hence we have arrived at the non-flat metric (4. 129) with (2) e -2p R = (; + C = X, (6.131) m = 0 = d, where C is a constant which can be made real by using the remaining coordinate freedom. In fact , from (4.138) we have, for C = c1 + ic2 ,

ic +l l A b Cb+ so that by taking = b+� ( C - ) we have transformed the imaginary _part of C away . This transformation does not alter the zero value of T� as can be checked by inserting the function A just determined into equation (4.147). 145.

The metric is b b dT 2 = 3r2CC + C )-3 dCdC + 2 drds + 2 c dr(C d� + c dC ) � b 2 + 2(C + C )[ds + C(C dC + C dC ) ] , (5.132) which is Petrov type III and admits the HKV (6. 120). However , we can reduce the number of degr�_s of freedom by one by noting that we can absorb the non-zero constant a in K (effectively, put a = 1). Then the metric 1s (6. 132), but now = - b ( 3a0 + 1)/2a0 , (6. 133) where ao is an arbitrary real constant , and the HKV is ,... K + {;o ) + 1 )so - 1)ro = - 2a0CCoC C ( ao + s (ao r (6.134) When C = 0 the metric (6. 132) degenerates into (6.10) [ or (5.9)], but the HKV still differs from (6.3).

= I a0 + a o

For proper homothetic motions to exist a � 0 , and this is so when

a + a = 0 iff a 1 o . Then we have in the space c 0 = K CS (6. 135) = and K ao, + ao' + Tos + 2aror ' (6.136) where a(C) is non-zero. Equations (4.150 ) - (4.155) reduce to K T = o, ( + aC )A + C (6. 137) - )p 1 o , (K + aC C + 2 aCC = (6. 138) K = p + Re(aC ) -a, (6. 139) (K - 4a)m = 0, (6. 140) (2) (K - 2a)R = o, (6. 141) (K - 2a )d = 0 . (6.142) There are two distinct cases to consider: 2) R ( (i) constaJ1t, ( ii) 1 constant . 146.

( 2) Case 6. 2 V)(i ) R = constant . (I : ( 2) 0 Equation (6.141) gives P = . Then , just as in Case 6.2 (1), we can trans form p to zero . \ole follow the analysis of that case step by step to deal with the present case. The details are omitted. We obtain the metric ( 4 . 129) •• i th

(6.143)

where a , N , are ar�itrar-J constants , a real , with 0 0 a.0 0 Re(a. ) = a 0 (since a cannot be zero ). 0 0 � = 0 If 1�0 the space is :'lat , o::herwise the metri c is Petrov type II. HKV The lS (6.144)

Killing where we have accountec for t�� presence of a multiple of the ..... vector o We now rea:ize t��L a factor a can be absorbed into K, s 0 that is , we may effect�vely a = -1. Then the metric is exactl p�� 0 � that of (6.30 ) with the HKV o:' (6.31) with a0 = -a. Thus the present case is a special case o:' Case 6. 2 (1).

Case 6 . 2(IV)(ii): constant .

Just as in Case 6.2(III), page 170 , we may take (2) - 2p R = = X, e C + C = this being the only kno;..-:1 solution for p(C ,C) . Then , just as on page 143 , 0 we obtain from (6.141) , remer.�ering that now a0 +a= ,

a. = 2aC (6. 145) and the equations (6. 13S) anc (6.139) are satisfied identically . Appe��ix Using option (b) o:' 4 to transform T to zero , and = 1 ch oosing a 2' we have = + + ro . (6. 146) K c c2, r C <,. 147.

Equations (6. 137), (6. 140) and (6. 142) become

,... + 1)/\ = 0 , (6.147) ( K ,... 2)m 0 (6. 148) (K = , ,... (K 1)d = 0 , (6. 149) and the surviving field equations are (6. 124) and (6.125).

Equations (6. 148) and (6. 124) give 2 , (6. 150) m= m0C where m0 is an arbitrary comp lex constant . Equation (6.147) gives 1 c- < C A = f CI ), (6. 151) where f is a function of C C /C ). Using the definition of d,

(6. 152) where the prime denotes differentiat ion with respect to the argument of the function. But from (6. 149), d = Xg(C /0 , /� where g is some function of ({; ). So, if (6. 152) is to be satisfied , then

f = c, d = 0 , (6. 153) where is a complex constant. The- (6. 125) and (6. 150) m = o. C i�?lY 0 Hence m = 0 . (6. 154)

Thus we have arrived at a non-flat metric (4.129) wit� -2p 3 (2) e = � x , R = C + C =X, -1 1\ = cc ' (6.155) m = 0 = d,

where is an arbitrary constant which can be made real by ng the C us i remaining coordinate freedom , by the same argument as follo�s (6.131), except here we take the function A to be A ic log(C /C). = - 2 The metric is 2 2 - 3 -1 -1 d1 = 3r C C+0 dCdC + 2 drds + 2 c drC C dC + c c� ) -1 2 C [ + dC ) ] , + 2(\:+ ) ds + C(C dC C -1 (6. 156)

where is an arbitrary real constant . This met ic typ e C r lS Fet�ov III

imd admits the Hl\V (6.14G). It degenerates to the me tric (6.10 ) 1 4 8 .

0 when C = , but even so the HYV is different from (6. 3). Putting BC = 3(x + iy ), the metric (G. 156) becomes 2 2 - 2 = 3 2 [ 9 2 dT r x (dx + dy ) + 2 drds + 2 C dr .d log t4-

6 . 3 We turn now to the case in whi ch the space admits a Y�lling vector of the form = K = (6.159) oc + 0' 0 x' C = 1 is form of where X + ly . In Appendix it shown that the the HKV may be taken as

= eo +eo + uo - VO + a(uo + vo ). K , , u V u v

(

See note at end of this section . 149.

The corresponding equations for the Killing vector !( are

K(O - = o , (6. 169) uO) KO = o, (6. 170)

K 11 = 0, (6.171) K � = 0, (6. 172)

-· (6.173) K(DO) = 0, K t� = 0, (6.174) KO = 0. (6.175)

As usual , the dot denotes differentiation with respect to u. Equations (6. 162) and (6. 163) give (K + 1)0 = 0 ' (6.176) while (6. 169) and (6. 170) give KO = 0. Using this result in (6.176), we get (yo 1)0 = o , o = O(y ,u). (5.177) y + -1 Putting y f(y,u) reduces (6.177) to f = 0, so that = f(u) 0 = y f and -1 (6.178) 0 = y f( u) . Equations (6. 164) and (6.171) give

( yo 4)11 = o, = 11 Cy ,u) , (6. 179) y + 11 with solution -4 = y g(u) . (6. 180) 11 The remaining equations (6. 165) (6. 168) and (6. 172) - (6.175) are all satisfied for this and The functions f(u) and g(u) must now 0 11 · be determined from the field equations .

and 11 as in (6.178) and (6.180), the operator becomes For 0 D

= - io n 1.2 y - nou and field equation (4.62a) gives

(3f = fg + + 2i )g 0. (6.181)

This cannot be integrated directly , but f anu e can be expressed in terms of a new complex function h(u) as follm�s : - 1 . h '3 f le . = 2l'/ h , g = - 8 h ( 6 . 1 82) It can be shown, in fact , that the only occasion on wh ich the field equation (6. 181) is immediately integl'.:lble is when a= 0 i.e. for an i sometry . 150 .

We next find 1 -2 DO =2y E(u) , (6. 183) where 2 • E(u) - if - 2ff = if + [ Cif) ] (6. 184) and = 1 y -2 E. (6. 185) Cl}:(1 2 The field equation (4.62c) gives the ordinary differential equation · ) E · = 2 4� - [ 3i(iE + f E + 2f(iE + f ) ] j E j , (6. 186) while the remaining field equation (4.62b) gives

4(g-g) = 3i(F + F) + 2(f r-f i), (6. 187 ) where F :::; iE + f E. (6. 188) By substituting (6. 182) in (6. 184) and (6. 186) we obtain a fifth order equation in the complex function h(u) , subject to the constraint (6. 187). The best I have been able to do with this situation is to spot some very special solutions which are rather uninteresting because they represent flat space. Noting that 1 -2 a o m = Y E (6.189) u u 2 . Do = 1 - 3 and uro - -2 y (iE + f s) ' (6.190 ) we observe that, by (4. 64d), the space is flat iff g = = 0, = 0 and E � f(i - 2�) constant . (6.191) Hence = f constant , g = 0 � flat space. [ It can be shown that the field equations are not satisfied when = g 0 , C complex constant.]

Note: At the beginning of this case we chose a = -1. This is a special choice, to make the ensuing equations easier to handle. As we have seen, the mathematics is even then not in good shape . Keeping a as an arbitrary parameter seems to make things worse! 1:,1.

The results of this chapter are summarized in the following theorem :

(Theorem 6. 1) The non-flat algebraically special vacuum Eins tein spaces with non- zero expansion and/or n: ist which admit one Killing vector and one HKV are given in the following table:

Petrov Hypersurface Metric HKV Case ype -orthogonal

�� 6. 1(III) (6.10) III ( 6. 3) Yes -.,";

6. 2(I) (6.30 ) II (6.31) No

6. 2(II)(i)(A) (6.43) II (6.44) !�o 6.2(II)(i)(B) (6.49) II (6.5 0 ) ,-_,o 6.2(II )(ii)(A) ? ( 6. 58) ,-,; 0

5.2(II)(ii)(B2) ( 6 . 10 3)+? II ( 6. 87) :;o

E.2(II)(ii) (B3) (6.116) II (6.105) ��0

6.2(III) (6.132) III (6.134) -:- es

6.2(IV)(i) (6.143) II (6.144) �;o

6. 2(IV)(ii) (6. 156) III (6. 146) -:- es

6.3 ? (6.1 6 0 ) ?

....': = degenerate case (admits 3 Killing vectors )

? = possible metrics but field equations unsolved . 1 52.

CHAPTER 7 P�v Type N Spac�

The type N vacuum spaces are treated separately because they are among the most interesting physically and mathematically . Only those type N spaces with the possibility of admitting homothetic motions are considered here . On account of the results of Chapter 3 the spaces can admit only one or two Killing vectors besides a proper HKV . All algebraic�lly special vacuum Einstein spaces with non-zero expansion and/or twist which admit 2 Killing vectors and one HKV are given in Chapter 5. From those results we have

v-a.c uum There are no Petrov type N spaces with non- zero 1\ exp&,sion and/or twist wh ich admit 2 Killing vectors and one HKV .

For spaces with only one �illing vector K we shall consider ln turn the cases where K assumes one of the canonical forms ( i) (ii) K = o X and add a HKV of the appropriate kind . For a type N space (4. 64c) requires

= o = ( 7. 1) 1-1 no u no, while the field equations (4.62) reduce to = DDD 0 DDD 0 (7.2 ) 2 ar::. -o nnn o = lo noj • (7.3) u u

for K in form (i) we can use the (�, C ,s ,r) coordinate system. Appendix 1 shows the allowed form for the HKV, but we do not need to concern ourselves with that here . For it is sufficient to note that , by (4. 130),

so that ooD =O. (7.1) u O This result coupled with guarantees that u the space is flat , by (4. 64d). Thus we arrive at

There are no Fetrov type N spaces which admit a HKV and a Killing vector of type For K in form (ii), Appendix 1 shows that we may taY.e the HKV in the form + uo vd + a(ud + vo ). u V U V

By choosing a = -1 this simplifies to

,_ ( 7. K = xo + yo - 2vo , 4) X y V where C = x+iy. The homothetic Ki lling equations and their integrability conditions are now just (6. 162) - (6.168) , with their Ki lling equivalents (6.169)­ (6.175).

Using (6.190 ), the type H condition (7.1) gives

it + ft = 0 , ( 7. 5) where E(u) is defined by E = if - 2ff , (7.6) -1 and O =y f(u) . ( 7. 7) Using (7. 5) and �= 0 , the field equation (7. 3), which is just (6. 186), becomes � o, tc£ + i + 2ff) = and by employing the definition (7.6) again, we see that this last er;_,_::;tion is the zero identity . Thus there is only one field equation, nar:-,ely (7:2), which can yield any information . It is just (6. 187), which now reduces to 3E - fE ( 3i+2f ) = 3:: (7.8) or 3E �(2if -E) = 3:S + E (2if+E).

This field equation together •; i th the type N condition ( 7. 5) must furnish

all the information about f, and hence 0. !he only chance of success 0 i.e. in finding non-flat solutions, is \-ihen , by (6.1 89), we have 1:; f. . One possibility (not the general solution) is that the left side of (7. 8) is a real constant . Then (7.5) and (7.6) reduce to a single ordinary differential equation in E, namely ,

(7.9)

where C is a real constant . Putting z = E+C, this becomes 3... z• z + z• .... - 0 (7.10 ) 2 2",.! 2 - ..:l��"�'.! 3 • I have found only two part icular solut ions of this equation: 154.

7( 1): z = Case constant , = E = real constant . 0, flat . Then E = so space is

7( II ): z = ku = ku k Case Ae , � E Ae - C, and A real constant . i 0 N If A i 0 , then E and we have a type space. Putting w(u) = if in the defi nition (7.6) , we mu st now solve = ku 2wW + w Ae - c. (7.11) According to Karnke [232], this equation is one of the class investigated by Lernke [233j. Solutions for A 1 0 appear to be unknown in exact form. The differential equation (7.11) is also an Abel equation of the second kind (see Kamke [232], p.26), but this classification does not seem to help ln finding a solution . In summary , while I have no solution to offer as an e

In case the sp ace admits just one HKV (and no Killing vectors ) �e shall consider each of the canonical forms ,._ (i) 'K = uo vo (ii) K = U + V The non-trivial equations (4.124) are = o c'K - 1)(0 - u l) ) , (7.12) ,._ . K O = o, (7. 13) -;; CiS o ) = o, (7.14) - 1).6 = o, (i< (7.15) = cK + 1)0 o. (7.16)

= Case 7(111): K uo + vo . U V (7.13) Equation gives 0 = 0, and then (7.16) is satisfied automatically . Equations (7.12) and (7.13) imply 0 = uf(C,C ). o o 0, 7. 1) Then u u D 0 = and this to[_ether "'i th ( ensures that the space is fla t , by a ( 4. 64d ) . . Hence we h ve

Fetrov There are no vc1 cuum t:: i n�'tcin spaces of type N admit only wjt h non- :-. c�'.J L' �;pcms ion cmd/or t1-:i��t whi ch

<1 !!LV of th' form h = uo + v0 . U V

ku Case 7(1!): A ku A z = e ' � E = Ae _C, k and real constants. For a type N space we re quire E#O. Substituting this expression for E into (7.5) we find that the condition EiO is not met. So the only possibility in this case is a flat space solution .

In summary , while I have no solution to offer as an example N of a type vacuum space which admits the Killing vector K = o and X the HKV (7.4) , there is a possibility that such a space exists . For (7.9) is not the general solution of the field equation (7.8) and the type N condition (7.5). 1 55.

7(IV): + Case K = a, + a, + uau vav.

The sole non-trivial field equation (7.2) and the type N condition (7.1) give a high order nonlinear partial differential equation which is very complicated. Other than some very special solutions represent ing flat spaces , it has proved too difficult to solve . 156.

CONCLUS ION

The first three chapters of this thesis provide an introduction to the structure of the conformal group , and a comprehensive survey of the use of the conformal group in mathematical physics ; in particular, in relativity . These three chapters put together for the first time , as far as I am aware , the achievements of many people over the past 130 years . This review sets the stage for an application of a special case of conformal motions to relativity . The application is specifi cally this: a systematic search is made for algebraically special expanding and/or twisting vacuum Einstein spaces which admit homothetic motions. To do this, a new extension of the formalism used by Kerr & Debney [199j is develop ed in Chapter 4. The work of Chapters 5, 6 and 7 is all new . While many of the 23 metrics given in Theore1.s 5.1 and 6.1 have been discovered by others and have been known to adr.. i t a HKV , several are believed to be new, particularly the Case ;_;:;-:;:r and IX metrics of Chapter 5 and those Petrov type II metri cs Chapter The list of metrics is incom?lete in of E. the sense that in two cases the field equations could not be solved cor.;? letely . This lS pro�2ly a characteristic of the coordinates used the analysis. Another choice of coordinate system have in r.ay produced equations which we�e more readily soluble , or given metric forrrs which may have been s��ler in some cases than those given here. Hm·:ever , the value of the s�. steo used throughout this \o.""or:.C is that it is effective , and that it enaDles direct comparison with and the use of results obtained by Kerr & �e0ney . Although not enunciatei as a theorem , the result given ln Chapter 5, that the NUTme tric does not admit a HKV , is not without interest . The work of Chapter 7 re'uces the possibility of the existence of Petrov type N vacuum spaces wh ich admit a HKV to a narro" class which also admit one Killing vector of a special type. There is also a possibility that type N spaces admit a HKV (of a special �'pe) only . Some problems for future research are indicated in Chapter 3. The work which is currently being done on conformal Killing tensors [234] should thrm' further light cC� the area of relativity studied in this thesis. 157.

APPENDICES

Appendix 1

Given a Killing vector K in canonical form , the form of a HKV ,.., K will be determined by a special case of Theorem 3.2, namely ,

[K,K] = AK, A a constant . (A1.1) 3.2 (3.7) This is obtained from Theorem by replacing K1 in equation K' . by K and dropping the subscript on 2 Consider each of the canonical forms of K in turn :

= R(C ,C)o = - po ( 1) K u e u = 0 s . ( A1. 2) Usjng the CC,C ,s,r) coordinate system , we wish to determine the functions a(C) and T(C,C) in ao + ao + + + + ror) K = , ' ao(sos - ror) Tos a(sos ( A1. 3) when the geometrical constrair.� (A1.1) is applied. This constraint is explicitly + . (ao a)os = AO s ' ( A1 . 4) a A a + which is valid for all and T with = 0 a. Under a coordinate trans�ormation (4.142) with � = 0 which takes the Killing vector K into K' = bK, where b is a real constant , the functions a and T transform as = a.' �,a. ' (A1.5) A = = T' = c0(T - a0A + KA], A(C,, ) A, (A1.6) where ln this case ' a. = 0 = a. , = T1 = b, T = 1, a0 0 . Thus the coordinate transformations preserving the canonical form (A1.2 ) ,C) are (4. 142) with � = 0 and arbitrary ACC and Under these i(C), c 0 . trans formations we can always put the HKV into one or other of the two canonical forms ( A1. 7)

,.,

K a Cso - a(so = 0 ro ) + t ro ) (A1.8) s r s r by means of (4.146) and (4.147) with � = 0. Any subsequent 158.

K transformation mus t preserve the canonical form of both and !<". ' For K as in (A1.7), K-bY- if C = bC+c, where b and c are real, complex constants respectively . So the coordinate freedom left in - the presence of K and as in (A1.7) is = 0 s Y. C' = bC + c, ' 9) s = c0(s + A)' (Al. r' -1 = C0 r, where is an arbitrar; real constant and is a real function of c0 A the form (a ta)x 0 = e .f(y), (A1.10) where ' = x+ iy and f is an arbitrary real function . For -K as in (A1.8), the coordinate freedom left in the presence of is K = os and K = C' �C C ) , = c: Cs A), (A1.11) s' 0 + -1 ' r, r = c, J where Q is arbitrary anc = ( i) A is arbi trar:,: r,_;�e:-. a,.,+a 0, v (ii) 0 when a +a r A = o t- ._, ,

Cf course, we may no� �ls� to make T zero in (A1.3) i.e. the canonical form of is no so that ';e have the full generality K t des�red , of A(�,C ) to play with in t�e �ansformation equations (4.142).

� = x+ iy . (A1.12) In the (C,, ,u,v) coordinaTe system the HKV takes the general form = K ao + aor + Re(a.)(u� - vo ) + Ro + a(uo - VO ). (A1.13) -,r -, 1;, U V U U V The constraint (A1.1) is ex�licitly = a o + a o + )(uo vo ) + R o Ao , (A1.14) � c , , + �

Under a coordinate transformation (4.96) with �= 0 the Killing ' vector K = o goes to K = bd provided the transformation equations X X a' = �,a , (A1.17)

' A1 .18 R = I�, I [R Re(a, )S + KS], ( ) where s = S(C,C) = s, are satisfied for

a' = b, a = 1, R' = 0 = R. This requires C' � = = b{; + c, (A1.19) s = S(y)' (A1.20) where b, c are real, complex constants respectively . The form of the HKV is already restricted to (A1.13) with a and R as in (A1.15) and (A1.16). Any further coordinate transformation (4.96) made to simplify this form of the HKV is restricted by (A1.19) and (A1.20). By means of a transformation ( C' = C + e/A. ) , u' = u + S (y), (A1.21) v' = v, the transformation equations (4.105) and (4. 106) with � = 0 for a and R give a' = >..;.'' R' = R(y) (A.+a)S(y) + (A.y + Im e)S(y) = R(y) CA.+a)S(y) + A.y' S(y), where the dot denotes differentiation with respect to y. We can, if we wish, choose R' = 0 for we are always guaranteed a solution S(y) of (A.y + Im e)S - (A.+a)S + R = o (A1.22) when e and R(y) are known. Then, choosing A = 1, we can put the HKV in the form ,.., K = Co + Co- + uo - vo + a(uo + vo ) (A1. 23) {; {; U V U V simultaneously with the canonical form (A1.12) of the Killing vector K. However, the choice of the HKV in the form (A1. 23) severely restricts the allowable form of S(y) in the coordinate transformations (4.96).

In fact , the coordinate freedom left in the presence of K = o and K X as in (A1.23) is

C' = b� + c,

I l+a (A1.24) u = b(u + y ) '

' -1 v = b v, 160 . where b, c are real, complex constant s respectively . It r.&ay be preferable to leave the HKV in the more general form = + + + (A1. 25) K eo, + ,0, uou - vov + R(y)ou a(uou vov) with complete freedom of choice of S(y) in the coordinate transformations.

Appendix 2

HKV When there are two Killing vectors K1 , K2 and one Kin_ the space we must use the special form of Theorem 3 .2, as we diG in Appendix 1, to determine the form of the HKV associated with each o� the given Killing vectors . Specifically , we must consider the follawing nine cases :

[K ,K] = K , [K ,K] = , (A) 1 A.1 1 2 A.2K2 , A.1 A.2 t- o . = (B) [K1 ,K] = A.2K2 , [K2 ,K] A.1K1 , A.1, A.2 t- o . t- (C) [K ,K] = K = [K ,K] , o. 1 A.1 1 2 A.1 = V. 1,V. = K [ J o . ( I'; [ �-_., J A.2 2 r-2 ,i< , A.2 t- �.. ,..,-, = = o, 0. z : . Lr,1 ,KJ A.1K1 , [K2�K] A.1 �

(F) [K ,K] = o [K ,K] = 1 , 2 A.2K2 , A.2 t- o . = = i (G) [K ,K] A�Kr , [K ,K] 0, 2 o. 1 L 'L 2 A. (H) [K1 ,K] = 0, [K2 ,K] = A.l1' A.1 t- o .

,K = 0 = [ Y:1 ....J, [K2 ,'K]. Here we shall be concerned with K1 and K2 as in Cases \'::LII and IX of Chapter 5. Not all of the cases (A) - (I) will be workeJ in detail, since the same technique applies to all of them.

5): = + = 1 Case VIII (Chapt er K1 oC oC ox ' C = x+iy , (A2 . ) = = K2 i(o' - o '-) o y ' (A2.2) + + Re( )(uo + + + K = a.o, aoc a.� u - vo) Rou a(uou vo) · (A2.3) 1 2 (A) From Appendix v-2kn ow that [K1 ,K] = A.1K1 allOI\S the �orm (A1. 3) for K� with coordinate freedom (A1.24). Under such a transformation (' ·1.-,, ) •.P find the form of K2 preserved , and [h,... ,h] = �2 • So by

• � i � - 1 in this case we have the result that in the presence 1 6 1 .

K K - (A1.23), of 1 and 2 the HKV K may take the form with coordinate freedom (A1. 24) remaining, �..l,ff-.. ':f 1+-c.. -- k, cr.-, �1c..-:t

(B) [K = A. K iff 1 ,K] 2 2 R = 0, � R = R(y ) ' (A2.4) X = = A. � a. C (A2.5) 2i a. o, H A.2 +e), a., ' cc =

where e 1 a complex constant . Then . ( - d/dy )

= iff

= = 0, R R A.2 -A.1 and R = 0 (real constant ). Then -K must be of the form

,_ Kow K1 , K2 andthis K are all preserved under the coordinate transformation ' ( = bC + c, I u = b(u T s)' s = S(y) = s, (A2.6) v' -1 = b V. = = = By taking b :>..2 1, c e in (A2.6) and emplo:_,•ing (4.105) and (4. 106) we can transform R0 to zero, thereby reducing K to the form = eo ) + a(uo ) K HCoc , + VO U V (A2.7)

= + + ) xo yo a(uo vo . y X u V

This coordinate freedom left is (A2.6) with S = 0 , as can be seen by applying (4.106).

(C) From (A) above (see also Appendix 1, equations (A1.15) and (A1.16)), [K ,K] = :>.. K we have 1 1 1 iff a. = :>..1<: + e, R = R(y). (K ,K] = i + Then 2 :>..1.. 1 t-

Hence this case ic impossi"ble. 1 62.

Proceeding in the same way , we find and are all im ossible , and (D), (E), (f), (G) (n) p (I) [r: ,K] (K ,K] allows 1 = 0 = 2 K = a o a(uo vo ), (A2.8) a0oC + 0 C + u + v where ao is a complex constant . The coordinate freedom left ky N is (A2. 6) and S(y) = Ne , wh ere and k are real constants, k a/ m ). = I (a0

Case I X (Chap ter K = (A2.9) 5): 1 0' + oc = 0 x ' K = eo (A2.10) 2 , + eo, + uou - vov ' ,...K as before in (A2.3).

(A) Appendix 1 allows ,...K to assume the form (A1.23), which may now be written ,...K = r: 2 + a(uou + vo) . But then K ,K] = 0, contrary to the hypothesis 1 [ 2 A2 0 . So this case is impossi�le .

It is also found that (B) , (D) , (f) and (G) are impossible .

(C) Using Appendix 1, where equations (A1.15) and (A1.16) hold for ,K] K , we obtain [K1 = A1 1 . [K ,K] = ( - d/ dy ) 2 R)ou

e = -A 1 1 o and R(y) = ky , k real constant . ,... Choosing A1 = 1, K may now be written as K =eo, + �0, + uou - vov - (oc + oc ) + R(y)ou + a(uou + vo) .

By means of the coordinate transformation u' = u + S(y) we can now use (4. 106) to eliminate y ) Thus we may take R( o u . K = K - K + a(uo + vo ). (A2.11) 2 1 U V

The coordinate freedoG left is ( A1 . 24) . 153.

A x (E) It is easily seen that K as ln (A1. 23) of ppendi 1, with coordinate freedom (A1.24), satisfies the constraints. Thus (A2.12)

u (H) [K ,K] = < Ca - a ) o +

K = + + a( uo u + vo ) = K + a(uo + vo ). 0' 0' v 1 u v The coordinate freedom le�t is (A2 .5) with S = 0 . (A2.15)

In similar manner we find

(I) 'K = uo + vo , (A2.15) U V with the same coordinate freedom as in (H) above.

Appendix 3

P = o implies p = Re{F(C)}, (A3.1) cc where F is an analytic function of C· Under a coordinate transformation (4. 142) with � = 0 , the function p(C,C ) transforms according to (4.137). Putting F(C) = log f(C) this gives (A3.2) 0 . Then p' = 1£,4.

From (A3 .2) we see that � = 0 is preserved under further coordinate 1 = iA transformations if c0l�, �- 1, which implies �((,)= e O(c0(, + b).

Thus , having made p zero, the coordinate freedom left is

iP. 0 (,' = e ( c0C + b),

s = c0 ( s + A), (A3.3) -1 r' = c 0 r, wher·e c0 , A0 are real constants, b is a complex constant , and A( C ,C) is an arbitrary real function. In case 6.1(I) of Chapter 6 the freedom on the s coorcinate is restricted by A= 0, but in Case 6.2(I) the full freedom on s is available.

In solving the system of equations (4.150) - (4.155) t�e freeeom available on the s coordinate may be used (a) to reduce t�e �ask of <=) f��cing A to that of o�taining a particular solution , or to tra�sform t�e �unction T(,,C) in the expression for K to zero. (a) Putting = 2F ' �here is a real functio�, enajles AC + A(, CC F((,,C) the definition of the function d((,,C ), namely ,

A-= - r, = 2ide 2p (A�. 1) � c ·- be written as (A4.2) Integrating , 2P + A = Jide dC + FC h((, ), (A4.3) where h((,) is an arbitrary function of (,. Now, under an s-transformation (4.142), we have by (4.138) that A-,\' , where A' = A - \· By the Cauchy-Ko�aleski theorem [235] the equation

h((,) A = o (A4.5 re + - C ) A((,,C). can be solved for the real function 2 A' = 'ide PdC , ( :\4.6) which is a particular inte�ral of (A4.3). 165.

(t) On the other hand, if we choose to transform T to zero by means of (4.147), we need to preserve this value of T thereafter and this places the re �iction = (A4.7) KA (a0 + a )A on the real function A(C,C) in (4.142) , thereby restricting the coordinate freedom on s. It is clear that one cannot si�ultaneously perform (a) and (b).

Appendix 5

2-surfaces of constant curvature.

Consider the 2-metric 2 = 2 (A5.1) do e pdCd� , P = pC�.C). The 2-c'lrvature is ( A5.2)

(2) = R = (A5.2) Su�:;:ose R 0 constant . Then gives

so that • (A5.3)

Any coordinate transformation of the allowed form C' = �({:) preserves the form (A5.1) of the 2-metric provided that , by (4. 137), 1 = l - p' p + log{ c � l } . 0 , Hence F(C) in (A5.3) transforms as 2 = - 1r ) 4 r' e �r ' ) �- [r( �r ) 2' log � (A5. ) �, , C� GiYen p({:,C) i.e. given F(C), the Cauchy-Kowaleski theorem guarantees th3t we can find a solu ion �(�) which makes the right hand side of

(A5.4) vanish. Thus \ve can always transform F({:)to zero. Any = (A5.4), subseJ_uent coordinate transfor;:i3tion C' i({:) has to satisfy , by 1CE-.

which is just the vanishing of the Schwarzian derivative of i� that is�

(A5.5)

(A5.6) where the c. are complex constants . Hence the coordinate transformations l which preserve F(C) = 0 form the group of bilinear transformations (A5.6). Without loss of generality � then, we may take

-p -p 2 = (e = -e ( p ) 0 \c pCC C which implies� since p is real� -p e = aC� + BC + BC + c� (A5.7) Hhere a� c� B are constants � B complex. Substituting this in (A5.2) gives

= - R 0 B� ac. (A5.8) Now apply a bilinear transforoation (A5.6) and use (4. 137) to trans�orm (A5.7) to the form -p' e = �' C' -R0 � where R0 is the same constant as in (A5.8). [One such transformation C' = is C 0aC + B� with c0 the constant of (4.137).] Since the primed coordinate sysc:ec is used henceforth, we shall dro;: the prime and write (AS. 9)

The coordinate freedom left lS a bilinear transformation on C and com?lete freedom on s and r ln (4.142). If R0 = 1 the 2-surface is a pseudosphere� and if R0 = -1 it is a sphere. 167.

Appendix 6

Consider a bilinear transformation

+ _ a'' b 1 C ' + , ad - be 0 . (A6.1) - cC d

We choose to normalize this so that ad - be = 1. Then

2 • = ' + d ) (A6.2) � C ( cC Using (4.137) with p given by (A5.9), we find that this form for pis preserved iff 1 - + C = c� {Caa C C' ' - R0 R0cc)C' ' (ab - R0cd. )'' + - � (at R0cd) ' + bb - R0dd.) (A6.3)

C O = 1, a a R0cc = 1 , bb R dd = -Ro , 0 (A6.4) ab R0cd = 0 , ad be = 1.

Let

Then (A6.1) is S = AS' , (A6.P': ) while (A6.2) may be written as 2 �, = ( '/'�) . (A6. 2�': )

- 1 ' -1 1 -2 T ,T ,.., = � � �': (A6.3) becomes CO ' 2 S' (P PA )S' , ( A6. 3 ) 1 where det A = and -p -2 T e = 1 , 2 1 S PS. 1 1. The requirement det A = ensures that c0 = I -T For c0 = 1 and P = A FA, we have therefore a matrix A which keeps the matrix P in diagonal form. We want noH to see whether we can transform the quadratic

(AG. S)

where b0 , a0 are real , complex constants respectively , to a canvnical 168.

form using relations (A6.4 ). This will depend on the value of R0 and so we have two cases to consider

( i) R 0 = -1 (sphere ) ( ii) R0 = 1 (pseudosphere). In matrix terms this amounts( to expressing) 1 a. o 2 1'b o Q - % ib0 - a0R0

in canonical form under the transformation matrix A, since we can write - 2 T ( = ' S QS , 2 (A6.s�·� ) = '-2 s' TQ' ' 2 S if Q' is defined by Q1 = ATQA. (A6.6) So we wish to find, using (A6.4·) , a matrix A such that Q' is in canonical form.

Case (i): R0 = - 1 (sphere). Lquations (A6.4) are (satisf ied if we choose A) to be ip iq · e cos 6 -e Sln 8 A = (A6.7) - q -i . e 1 sin e e p cos e Then (A6.6) gives ' = a. 2ip 2 i(p-q) a e cos e + a e sin e cos e 0 0 . o (A6.8) 1 [ 2ip 2 i ( p-q) . + 2 1b e COS 8 + e S1n e cos eJ O _ and 1 p Hp +q) · . [- -H +q) . 1 2 2 ib' = a e - a e J S1n e COS e + b < COS e - sin 8). 2 o 0 0 21 O (A6.9)

o t- o. n n i 1 a = , b Choosing e = p = -q = - + - log 0 0 4' 4 2 <2 bo) _ reduces (A6.8) and (A6.9) to I 1 I = 0 so that ) a.o - ' b o -) ( _ 1 ' 1-i )(%b ) - ( 1+ i ) '2 . < 0 , ( 2b 0 ) Q' = A = % ( A6.10 ) (: : . {1-i) � (.!2 b 0 ) - ( 1 + i ) ( �b 0 ) � 169.

= 2 Then a' C' + 1. (A6.11) = In the original coordinates a ib0C . The transformation which takes a into a' is , from (A6.1) and (A6.10 ), 2 , , - %< 1+i) b = o ' (A6.12) + 2 ,, %< 1+ i) bo

= 1 . b = o . Choosing o p 1 log , q arbitrary , 0 a , = 2 a0

equations (A6.8) and (A6.9) reduce to ' - 1 b' = 0 so that ao (- , 0 1 0 a:-� o Q' = A = } (A6.13) ( ) k 0 1 . a 0 2 Then (A6.14) = 2 In the original coordinates a aoC + ao . The transformation ' which takes a into a is , from (A6.1) and (A6.13), (A6.15)

n Choosing a = 4' Then (A6. 8) and (A6.9) reduce to

I 1 - -� 1 - - � 1 - -� - � a a a a ) + a

C' - -2 = r (A6.18) + r '2 - 1/8 where r = (a0Ja0 ) . Thus , in the case of the sphere, it is alHays possible to express Q

in canonical forD, as in (A6.10 ), (A6.13) and ( A6 . 1 6 ). 170.

Case (ii): R0 = 1 (pseudosphere). Equations (A6.4) are satisfied if we choose A to be

A ( A6 . 1 9 )

Then (A6.6) is, in full, 2iP 2 - -2iq . 2 1 i(p-q) . h 28 (A6.20 ) a� = a0e cosh 8 a0e Slnh 8 + 2 l'b 0 e Sln and (A6.21)

Choosing 8 = 0 reduces (A5.20 ) and (A6.21) to � a� = O, b� = b0 (b� t 0 since cosh 28 1), so that

p arbitrary . (A6. 22)

C' Then a.' = ib 0 . (A6.23)

In the original coordinates a = ib0C . The transformation which takes a into a.' is, from (A6.1) and (A6.22), p arbitrary . (A6.24)

a = = 0 = a. , 0 t 0 , b0 0. Choosing 8 and p % i log 0 q arbitrary , --=------' = 0 (A6.20) and( (A6.21)) re duce to (a.0 1, b'0 = '>)so that 0 a 0 ' 1 o -'> Q = ' A = (A6.25) o o -1 a0

' = 2 Then a. , , - 1. (A6.26) a. = C 2 - - • In the original coordinates a0 a0 The transformation taking a. into a' is, fro� (A6.1) and (A6.25), 171.

In this case (A6.20) and (A6.21) show that b� and a� can never be zero. The best we can do is to choose e 0, = a ' a 1, = p % i log o q arbitrary , and then we obtain � = b'0 = b 0 so( that ) 1 ' % ib0 (A6.28) = , Q -1 . 1:.2 ibo Then (A6.29) 2 a a C C - a . In the original coordinates = 0 + ib0 0 The transformation taking a into a' is, from (A6.1) and (A6.28) ,

Thus , in the case of the pseudosphere , it is not always possible to express Q in canonical form. Q may be diagonalized in two cases, namely , (A6.22) and (A6.25).

f,:;:r::lendix 7

(6. 100) 0) The hypergeometric equation (with C = is put into standard form by writing z = -x, obtaining . z(z 1)H. [(a � y]H a�H 0, (A7.1) - + + + 1)z - + = where H = H(z) and the dot denotes d/dz, and

a � - 2 p = 1 = -2 + = ia, a + ia, y

a 1 , = -1 - y -2. (A7.2) giving = l3 ia, = Two independent particular solutions of (A7.1) ar e H = F(a, y, 1 � . z), 1-y H F(a + 1 - y, 1 - y, 2-y, 2 = z l3 t z) for l zl < 1, leading to the solution (6.101). Now it is known (see e.g. Forsyth [236]) that the quotient s of any two particular solutions of the equation = (A7.3) y + Iy 0, y = y(z), satisfies the equation = (A7.4) (s,::.) 2I , 172. where [s,z) is the Schwarzian derivative of s defined by the left hand side of equation (A5.5) of Appendix 5. In the case of the hypergeometric series for F(a , �. y, z) the value of I is 2 2 2 1 - v - + + + ). f-1 (A7.5) ( z-1)2 z( z- 1)

2 ( f-l = a From (A7.4) and (A7.5) we have a differential equation for s(z) . If this can be solved for s in finite terms , it follows that the hypergeometric series will be expressible in finite terms . There are only 15 separate cases in which this is possible , and unfortunately the present case represented by (A7.2) is not one of them. For we have 2 2 2 2 ' = f-l 1\. 9 ' = -a = v , and Schwarz's table [237] of the 15 special cases does not include these numbers , whatever value (zero not allowed) of the real constant a we choose. 173.

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W.D. Halford

ERRATA

Page 46, line 9: should read (Mcintosh [ 141] has quoted this metric. This

points up an error in Godfrey 's paper. ) Page 106, lines 8,9 ,10 and 11 : replace by but because of the presence of K =o and 1 s K = 2 (Co +Co- )-s o +ro in the space we may take 2 "'r C s r the HKV in the form ro . 'K = sos + r (5.7)

Page 106, lines 26 an d 27: replace by (5.10)

Page 107, after line 25 add : or (iii) a =c =c =o, 0 1 2

Page 107, line 26 : after "Using (ii)" insert "and (iii)".

Page 108, after line 21 add : But applying the theorem [K ,K ] = mK +nK (i=1,2) m,n constants i 1 2 leads to a0=0. Hence the form of the HKV may be taken as

= <:o +Co - 4so -2ro +4i�log(C/C)o • (5.21a) K "'r r.., s r s

Page 115 , after line 22 insert : However , it may be that a=a , when the matric (5.9) 0 admits the HKV = xo +yo -so . K y X S Page 152 , lines 29 and 30 (Theorem 7.2) : should read There are no Petrov type N vacuum spaces with non-zero .expansion a� d/or twist which admit a HKV and a Killing vector of t�pe K = o = R( C,C)o . s u

Page 160, second last line , after "preserved" add : 1+a l.' f y -� k' constant .

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