Copyright is owned by the Author of the thesis. Permission is given for a copy to be downloaded by an individual for the purpose of research and private study only. The thesis may not be reproduced elsewhere without the permission of the Author. The. Con6oJunai. G!toup & A thesis presented for the degree of Doctor of Philosophy in Mathematics at Massey University WILLIAM DEAN HALFORD January, 1977. THE CONFORMAL GROUP & EINSTEIN SPACES ABSTRACT This thesis presents (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 expanding and/or twisting congruence of null geodesics , which locally possess a homothetic symmetry as well as a Killing symmetry (isometry). Unless the space-time is Petrov type N with twist-free geodesic 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 field 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 geometry 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 isometries . 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 Riemannian manifold with symmetric metric tensor 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 <iE)�y == £x{�y} �Sy� Wl g { (£ g ) + g ) - <.f g ) } , ( 1.13 ) = % x w ;S <fx � ;y x � ;1-1 ) 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 cs + c.r c.s + cr .