Algebraic invariants of the Riemann tensor for class B warped products by Kevin Santosuosso A Thesis submitted to the Department of Physics in conformity with the requirements for the Degree of Master of Science Queen's University Kingston, Ontario, Canada February, 1998 Copyright @ Kevin Santosuosso, 1998 National Library Bibliothèque nationale I*I of Canada du Canada Acquisitions and Acquisitions et Bibliographic Services services bibliographiques 395 Wellington Street 395, rue Wellington Ottawa ON KI A ON4 Ottawa ON KIA ON4 Canada Canada Y0ur iUe vom ruference Our Na Notre ?&-Ce The author has granted a non- L'auteur a accordé une licence non exclusive Licence allowing the exclusive permettant à la National Library of Canada to Bibliothèque nationale du Canada de reproduce, loan, distribute or sell reproduire, prêter, distribuer ou copies of this thesis in rnicroform, vendre des copies de cette thèse sous paper or electronic formats. la forme de microfiche/film, de reproduction sur papier ou sur format électronique. The author retains ownership of the L'auteur conserve la propriété du copyright in this thesis. Neither the droit d'auteur qui protège cette thèse. thesis nor substantial extracts fiom it Ni la thèse ni des extraits substantiels may be printed or otheiwise de celle-ci ne doivent être imprimés reproduced without the author's ou autrement reproduits sans son permission. autorisation. This thesis examines invariants polynomial in the Riemann tensor for a class of space- times known as class B warped products. These spacetimes may be descnbed by the line element where Cl is a twedimensional manifold with the coordinate chart (sa}and C2 is a two- dimensional manifold with the coordinate chart {x"} and one of Ci and C2 is Lorentzian while the ot her is Riemannian. Alt hough algebraically special, t his class of spacetimes contains many of interest, including ail spherically, plane, and hyperbolically symmetric metrics. The set of invariants used to examine these spacetimes is the set of six real and five complex algebraic Riemann invariants proposed by Carminati and McLenaghan aug- mented by a further cornplex algebraic Riemann invariant suggested by Zakhary and McIntosh. It is demonstrated that the set of invariants composed of the Ricci scdar R, the first two Riemann invariants ri and 7-2, and the second Weyl invariant w2,which must be real in this case, forms a complete set of invariants for class B warped products. The set is complete in that there exist no relationships in general between these four invariants and that any other invariant of the Riemann tensor may be constructed as polynomials in these invariants, and furtherrnore that for each further algebraic specialization of the Riemann tensor that is a subclass of class B warped products, the set {R,ri, r2, w2} gives the correct number of invariants corresponding to the number of degrees of freedorn in- herent in the specialization. The explicit relationships (cailed syzygies) between the set {R, ri, Q, wZ} and the other invariants in the augmented set of Carminati-McLenaghan invariants are given, and a preliminq discussion of the implications of the existence of this minimal set and the syzygies is presented. Statement of Originahty The contents of this thesis are solely the work of the author, and of the author in collaboration with his supervisor, Dr. Kayll Lake. Some of the work presented in Chapters 2 and 3 is review, while the rest is a collation of results both old and new that are relevant to the work in this thesis. Portions of Chapter 4 have been submitted for publication. The theory of gravity as deueloped by Einstein has become a beautifil fossil. Its jate is to be worshzpped by the lesser mortais who wdl harnmer away ut the more tedious detaikr as the rare loopholes are slowly plugged. 1 would like to thank Dr. Kayll Lake for providing me with the opportunity to learn about the most intriguing field in the realm of physics. It has been a pleasure and a privilege to work with someone so knowledgeable and enthusiastic about his work. Special thanks &O go out to my various colleagues around the office. Denis Pollney has been a constant source of valuable information (even hom Southampton) and can aiways be relied upon to provide me with a different spin on a subject (no pun intended). Peter Musgrave always patiently answered whatever stupid questions 1 had, f?om general relativity to GRTensor. The entertaining discussions 1 have had with Nicos Pelavas have helped me corne to a better understanding of physics, while our Doom sessions have helped my ego. Also, my vocabulary would be severely lacking were it not for the assistance of Brandon van Zyl. Thanks are also due to the secretaries in the physics office, Margaret and Terry, for their friendly assistance in fighting the red tape throughout the years. And hally, 1 would like to thank my parents, Antonio and Alma Santosuosso, for their constant and unquestioning support. Contents Title i Abstract i i Statement of Originality iii Acknowledgement s iv 1 Introduction 1 1.1 Outline of this Thesis ............................. 2 1.2 Notation ..................................... 3 2 Invariants of the Riemann Tensor 5 2.1 Mathematical PreIiminaries: Groups, Rings, and Invariants ........ 6 2.2 Invariants of the Riemann Tensor ....................... 7 2.2.1 Tensors ................................. 7 2.2.2 Spinors ................................. 10 2.2.3 NP Curvature Components: Tetrads, Dyads, and the Ricci and Weyl Scaiars .............................. 15 2.2.4 Bivectors ................................ 23 2.3 Independence, Completeness, and Syzygies .................. 25 2.4 Candidates for a Cornplete Set of Riemann Invariants ........... 30 2.4.1 Early Sets ................................ 30 2.4.2 TheCMSet ............................... 33 2.4.3 TheZMSet ............................... 37 2.4.4 A Full Set of Invariants of Degree Five ................ 39 2.5 The Value of Invariants ............................ 40 3 Class B Warped Products 44 3.1 The Classification of Warped Products .................... 44 3.2 Invariant Characterization of Class B Warped Products .......... 46 3.3 Physics of Class B Warped Products ..................... 49 CONTENTS vi 4 Invariants of Class B Warped Products 53 4.1 The Curvature Components .......................... 55 4.1.1 Class BI Warped Products ...................... 55 4.1.2 Class B2 Warped Products ...................... 59 4.2 Syzygies of the Riemann Invariants for Class B Warped Products ..... 62 4.3 A Complete Set of Invariants for Class B Warped Products ........ 64 4.4 Consequences of the Syzygies ......................... 66 5 Conclusions 72 Bibliography 74 Curriculum Vita 77 Chapter 1 Introduction It is certain that physics must be independent of whatever coordinate system is used to describe it. It is this property of independence that was the motivation for formdating the equations of physics via tensor equations, which must hold true regardless of any coordinate system in which one might wish to express them. Einstein's theory of gravity relates the p hysicd description a spacet ime, contained in the energy-momentum tensor Tab,to the fundamental geometry of a four-dimensional manifold described by the metric tensor gab, fiom which all the information that can possibly be known about a manifold can be obtained. From the metric tensor one may generate the Riemann curvature tensor, the basis of Riemannian geometry, in which is contained al1 the information about the manifold, and sol the spacetime. From this tensor rnay be constructed scalar quantities which are invariant to coor- dinate transformations. This would seem to be of great value- it was, after all, the principle of physical invariance to coordinate transformations that motivated the tensor description of physics in the first place. Yet the true value, if any, of these quantities, which are called invariants, has not yet been discovered. Their coordinate invariant nature would seem to be an important source of information, but what this information might be is not truly known. They are of some value in identieing certain kinds of singularities within a spacetime, but otherwise an understanding of their usefulness is elusive. CHAPTER 1. LlVTROD UCTION 2 In spite of this, considerable attention has been given to finding a complete set of invariants for the most general Riemann tensor possible- loosely speaking, to finding a set of invariants that provides as much information as can be obtained for every possible form of the Riemann tensor. Many sets have been proposed, most of which have ben proven to be incomplete, and the few sets that have not been proven to be incomplete have not been proven to be complete, either. In keeping with our la& of understanding of the invariants thernselves, it appears that the difficulty in finding a complete set of invariants mirrors the difficulty in understanding what the value of invariants really is. It is somewhat sirnpler to investigate more specific spacetimes that are algebraic speciaiiza- tions of the general Riemann tensor and find a complete set of invariants for that case, and that is the approach that we will take in this thesis. The class of metrics which we will examine, while dgebraically special, is a very im- portant class of metrics, since it contains wit hin it al1 four-dimensional semi-Riemannian spaces wit h a two-dimensional subspace of constant curvature- t hat is, al1 spherically, plane, and hyperbolicalIy symmetric spacetimes.