Algebraic Invariants of the Riemann Tensor for Class B Warped Products

Algebraic invariants of the Riemann tensor for class B warped products

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

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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 spacetimes 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 augmented 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 inherent 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 coordinate 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 important 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. It is t hese physically important symme- tries that motivate our investigation of this class of metrics, which are known as class B warped products.

1.1 Outline of this Thesis

In chapter 2, the background on invariants constructed from the Riemann tensor is detailed. After some mathematical preliminaries that set the ground for our discussion, the construction of Riemann invariants is investigated and some important aspects of invariants and sets of invariants is discussed. It is assumed that al1 readers of this thesis will be faniiliar with tensor analysis; this rnay not be the case with such topics as spinors, bivectors, and NP tetrads, and so a brief discussion of these topics is presented which, it is hoped, is sufficient to give the reader a basic understanding of the different ways in which we may construct the same invariants of the Riemann tensor. This is followed by CHAPTER 1. DVTRODUCTION 3

a background on the different sets of invariants proposed as complete. It is £iom two of these sets that we obtain the set of invariants we will use to investigate class B warped products. Chapter 3 details the classification system of warped products and briefly presents some important results for the physics of class B warped products. This chapter also includes a theorem that provides a way of distinguishing when a metric is a class B warped product. In chapter 4, it is first demonstrated that there can be only four independent invariants of the Riemann tensor for class B warped products. The syzygies of the set of invariants we use are presented and the set {R,TI, rz, w2) is proven to be complete for class B warped products. Some prelirninary ideas on the implications of these syzygies are also discussed. Al1 of the calculations performed in this section were evaluated using GRTensorII [GRTensorII]. Chapter 5 is a summing up of the important results obtained, with a brief discussion of the possibilities of furt her investigation.

1.2 Notation

Throughout this thesis, except when dealing with tetrads, we use the Landau and Lifshitz spacelike convention, so that the signature of four-dimensional spacetime is +2. By convention, the signature of tetrads (and specifically the NP tetrads we present here) is usually taken to be -2, and we will keep this convention. Repeated indices on any quantities (tensors, spinors, etc.) follow the Einstein summation convent ion. The metric tensor is represented by g,b, the Riemann tensor is represented by Rabcd7and the Ricci tensor is represented by Rab. Geometric units are used such that G = c = 1. For the most part, the letters representing indices are chosen in the following fashion to avoid confusion: lowercaçe Roman letters from the front of the alphabet indicate tençor indices, lowercase Roman letters fiom the middle of the alphabet indicate tetrad indices, CHAPTER 1. LIVTRODUCTION 4 and uppercase Roman letters indicate spinor indices. Although this has not always been possible, it shouId be clear fiom the context what the case is. The symbol 6; is the familiar Kronecker delta symbol such that 6; = 1 if a = b and

= O if a # 6. The Levi-Civita symbol cabcd or CAB is the alternating tensor, such that

E. = O unless al1 of its indices are different and E = 1 for even permutations of O, 1,2,3 or

0,l and E = -1 for odd permutations. The operations of symrnetrization and antisymmetrization of indices are represented by parentheses (. . . ) and square brackets [. . .] respectively. So, for example, we have Chapter 2

Invariants of the Riemann Tensor

There bas been a great deal of debate for many years as to the nature and value of invariants constructed from the Riemann tensor. Among other things, one would suspect that invariants would help distinguish spacetimes, but it was discovered long ago t hat t hey are not useful for this purpose [Narlikar and Karmarkar 19481. It is believed that Riemann invariants are helpful in identi@ing scalar polynomid singularities in a spacetime, but invariants Say nothing about the singularity itself other than whether it is a removable singularity or not (removable via a coordinate transformation) [Tipler et. al. 19801. Fur- ther debate centers around defining a cornplete set of invariants- in fact, the definition of completeness itself has altered throughout the years [Penrose 19601, [Sneddon 19961, [Caxminati and McLenaghan 19911, [Zakhary and McIntosh 19971. In this chapter, we discuss invariants at a basic level and briefly discuss attempts to obtain a compIete set. CHAPTER 2. INVARIANTS OF THE RlEMAMV TENSOR 6

2.1 Mat hematicd Preliminaries: Groups, Rings, and Invariants

Let G be a nonempty set. Suppose that for any elements a and b of G there exists a uniquely determined element c of G, called the product of a and 6 and written as c = ab. We cd1 G a (multiplicative) group if it has the following properties:

ii ) For any elements a, 6 E G there exists uniquely determined elements x, y E G satisQing ax = 6 and ya = b.

Note that condition (ii) is equivalent to the two foUowing conditions:

iii ) There exists an element I (called the identity element of G)such that al = a and la =a for al1 a E G.

iv ) For any element of a E G there exists an element a-l E G (called the inverse

of G) such that aa-' = 1 and a-'a = 1.

A ring is somewhat similar to a group. For a ring A, there are two operations that are defined, called addition and multiplication, which send an arbitrary pair of elements a, b E A to elements a + b (addition) and ab (multiplication) in A. These operations must satisfy:

ii ) (a + 6) + c = a + (6+ c).

iii ) (ab)c= a(bc).

iv ) a(6 + c) = ab + ac.

v ) (a + 6)c = ac + bc. CHAPTER 2. INVARIANTS OF THE RIEMANN TENSOR 7

vi ) For every pair a, 6 E A, there exists a unique element c E A such that a+c=b.

Note that, as before, condition (vi) implies that there is an identity element of A for addition, called the zero element and denoted by O. Cornbined with the distributive laws, the zero element also satisfies the equation a0 = Ou = O for any a E A. There may or may not be a multiplicative identity element 1 such that al = la = a. If there is, the ring is cailed a unitary ring. Furthermore, if we have the additional property

vii ) ab = bu

then the ring is called a commutative ring. Suppose, then, that we have a commutative ring A and a group G, and G acts on A; that isJ

i ) each element a of G defines an automorphism a + au, where a,au E A, and

ii ) ~(aa)= (ra)a for r,a E G and a E A.

Then an element a of A is said to be G-invariant (or it is simply called an inuariant, when the group is known and the invariance of the element is known to be related to that group) if oa = a for al1 a E G.

2.2 Invariants of the Riemann Tensor

2.2.1 Tensors

Consider the set of elements obtained ty contracting the Riemann tensor with itself in any combination to form scdar quantities. For example, R~~~~R~~~R~~~R~~~~R~~~~, R ~aCd abcd + R~~"R~~~~R~~~~,and R~~~R~~~~R~~~~ ~~~j~ Rjtef are al1 elements of this set. If we add the number O to this set, then it is trivial to show that we obtain a commutative ring under the standard operations of addition and multiplication. (We could also add 1 to the set and obtain a unitary commutative ring.) CHAPTER 2. LNVAWTS OF THE RIEMANN TENSOR 8

The active Poincaré transformations- that is, rotations in space, translations in space time, and providing a uniform motion- clearly form a group, as do the passive Poincaré transformations, which are transformations of the coordinate space, ie. recoordinati- zations of the underlying space [Penrose and Rindler 19841. Now the active Poincaré transformations exist independent ly of coordinates, while the passive transfomat ions do not. However, an active transformation may always be framed in terms of coordinates. It is convenient to do so in this case, for then it is trivial to demonstrate that scalars are invariant under the Poincaré groups. For example, if we have two separate coordinate systerns 2" and sa,then

Although we demonstrate this property for a scalar constructed via contractions of the Riemann tensor, the same property hoIds, of course, for any scaIar whatsoever. So for any scalar S we have the relation 3 = S and we see that scalars do not change under coordinate transformations. Returning to our original set of elements obtained by contracting the Riemann tensor with itself, which is simply a set of scalars, we have groups (the active and passive Poincaré groups) which act on a commutative ring (the scalars) arid do not change the elements of the set. We therefore cal1 these objects invariants, and, in particular, these ones obtained from the Riemann tensor Riemann invariants. These invariants are not the only class of invariants that one may construct fiom the Riemann tensor. It is also possible to obtain invariants by contracting derivatives of the Riemann tensor, such as the quantity v~R.~~v~R~~~~.The property of Poincaré invariance also holds for these quantities, and they are cded diflerential invariants. The order of a differential invariant is the maximum number of derivatives of any of the components of the invariant (see below). An invariant of order zero, that is, an invariant CHAPTER 2. INVARJANTS OF THE RIEMANN TENSOR 9

constructed without taking any derivatives of the Riemann tensor at dl, is called an algebmzc inuariant. Since we will only concern ourselves with algebraic invariants, we shd simply refer to them as invariants without qualifying them as algebraic. In four dimensions, the Riemann tensor may be decornposed into the trace-free Ricci tensor Sa(= Rab - a R~~~,the Weyl tensor Cabcdand the Ricci scaiar R = ka:

Like the Riemann tensor, we may obtain Ricci invariants via contractions af the trace-kee Ricci tensor and Weyl invariants via contractions of the Weyl tensor. We rnay also obtain invariants by contracting the trace-fiee Ricci tensor with the Weyl tensor; for example, R~~R~~c~~~~.Invariants of this nature are called mized invariants; invariants that are not mixed are sometimes referred to as pure invariants. Now as a consequence of this decomposition, we may actually express any Riemann invariant as a polynomial in the Ricci scalar and Ricci, Weyl, and mixed invariants. So to investigate any invariant of the Riemann tensor, it is sufEcient to examine the Ricci, Weyl, and mixed invariants, which are sometimes referred to as Riemann invariants as well. Since we know that the trace-free Ricci and Weyl tensors contain different information, besides guaranteeing t hat examination of the invariants constructed from t hese tensors will contain different informat ion, the invariants t hemselves will have the added convenience of being simpler to deal with than invariants constructed from the full Riemann tensor. Each tensor that takes part in such a contraction to produce an invariant is called a component of the invariant. The degree of an invariant is the total number of tensors contracted together to form the invariant. Note that it is only the number of contracted tensors in each term of the invariant that sets the degree. For example, the invariant c~~,&~~s~~c~~~~+ c~~~~s~~s~~c~~~~would be of degree four. In some cases, it is convenient to describe the degree of an invariant in a way that differentiates the trace-free CHAPTER 2. i2VVARLANTS OF THE EUEMANN TENSOR in

Ricci tensor fiom the Weyl tensor. This wilI be discussed later on in the following section.

2.2.2 Spinors

In 1960, R. Penrose [Penrose 19601 developed an alternate method for dealing with prob- lems in generai relativity that does not use tensor analysis. Instead, quantities called spinors are taken to be the essential elements of the geometry. Here we will briefly present enough background on spinors and spinor algebra to obtain the quantities we shall use instead of the Ricci and Weyl tensors to obtain the same invariants that one may construct fiom these tensors. ' Spinors, like tensors, are defined by transformation properties. There are two classes of spinors of rank 1, cA and where the normal and dotted indices serve to denote different classes of spinors. Both are compIex vectors in a two-dimensional space that transform as

where a;'B and ?FAg are complex conjugate matrices with unit determinants:

A 10 Bl = 1aABl= 1.

Spinors of higher rank transform analogously to tensors. For example

E~B-- a A Ca-B &'.

For two spinors of the same class, their determinant

coq1 - c1qo (2.10) is invariant tc; unimodular transformations, so we may defin antisymmetric metric

€AB for the space such that

~ASC"S~

o or a more detailed treatment of spinors and spinor aigebra see [Penrose and Rindler 19841. CHAPTER 2. INVARLANTS OF THE RIEMANN TENSOR 11

is invariant. Cornbining (2.11) with (2.10) reveals that EAB is, in fact, simply the tww dimensional Levi-Civita symbol. Similady, we have the Levi-Civita symbol for dotted indices. These are the quantities we shail use to raise and lower spinor indices (as we use the metric tensor for raising and lowering tensor indices). They are applied as

noting the order of the spinors and the indices on the spinors as well. Of course, it will be useful to be able to obtain tensors from spinors and vice versa. This is accomplished by the relat ionships

and

where ouABand

and since

with (2.15) we have

b LACebi) = gab~~~h~ce- (2.20) *1t turns out that from a relation defined between the coordinates in Minkowski space and a cornplex two-dimensional vector that motivates the discussion of spinors, the naAhare the the identity matrix (for a = 0) and the Pauli spin matrices (for a = 1,2,3) multiplied by a normalization factor of $; [Chandrasekhar 19921. CHAPTER 2. WARIANTS OF THE =MANN TENSOR 12

So we can now relate tençors of arbitrary rads with their spinor equivalents. For example,

Thus we have the correspondence

With this correspondence we may develop the spinors that we will use to construct the same invariants we constructed with tensors. We will do this by breaking down the spinor obtained £rom the Riemann tensor

The antisymmetry in the flrst and last two indices of the Riemann tensor gives the relation

w here

and

The antisymmetries of the Riemann tensor also give us the relations CHAPTER 2. INVARIANTS OF THE RZEMANN TENSOR 13

while the interchange symmetry Rabcd = Rednbis equivaient to

Now (2.32) implies that QAAsd corresponds to a real tensor aaband (2.30) implies that

If we then raise C and c and contract with A and A we obtain

where we define3

So then we have

since, for any spinor 4,4B, lc AB = Z4~CAB (2-37)

and (2.29) implies that xABCBis antisymmetric in AC. Rom (2.26), with (2.36) and the real nature of we may calculate the Ricci tensor

3The letter A is also typicdly used to denote the cosmologicai constant. This leads to an occasionai abuse of notation in the literature where spinors and the curwture components are investigated for spacetimes with a nonzero cosmological constant. It should be noted that the two are not related (except as the natural consequence of a nonzero cosmologicai constant being included in the Einstein equations). CHAPTER 2. INVARIANTS OF THE RJEMANN TENSOR 14

which we may write as

Since we know that Gaa = 0, for a four dimensional spacetirne we obtain

which relates A to the Ricci scaiar R. So the trace-& Ricci tensor is given by

Because of this correspondence, we cdQABriB the Ricci spinor. We may also isolate the totally symmetric part of XrlBCD.With (2.291, (2.31), and (2.37), we have

and so with (2.36) we obtain

where we define

In the decomposition of the Riemann tensor, gABCDcorresponds to the Weyl tensor and so is called the Weyl spinor. As we did with the Riemann tensor in the previous section, we have decomposed the Riemann spinor into t hree independent components: the Ricci spinor, the Weyl spinor, CHAPTER 2. LlWARlANTS OF THE RIEMANN TENSOR 15

and A, which corresponds to the Ricci scalar. And as before, we may construct Rie- mann invariants out of these objects. For example, the Ricci invariant constructed via tensor contractions, sabsab7is equal to the Ricci invariant 4~ ABAB~A*'B. Sirnilarly, we may construct equivalent Weyl invariants and mixed invariants from the Weyl and Ricci spinors, and so a~yRiemann invariant may be expressed as a polynornial in the Ricci and Weyl invariants and A. As mentioned in the previous section, it is sometirnes useful to describe the degree ' of an invariant in a way that differentiates the invariants based upon the the differences between Ricci and Weyl. For spinors, we consider three quantities: the total number of Ricci spinors, Weyl spinors, and conjugate Weyl spinors used to construct the invariant, If these numbers are given by n, ml and respectively, then we rnay Say that an invariant is of degree (n,m, m). If we are not concerned with this distinction, then we simply Say that the degree is n + m + 5.

2.2.3 NP Curvature Components: Tetrads, Dyads, and the Ricci and Weyl Scalars

When creating a tetrad to represent spacetime, we set up at eacb point of spacetime a basis of four four-dimensional contravariant vectors {e(il'}, where the parentheses denote the tetrad indices and unparenthesized indices correspond to the tensor indices. To form a covariant vector, we contract with the metric tensor

We define the inverse, {e('),}, of the tetrad according to the relations CHAPTER 2. INVmNTS OF THE RTEMANN TENSOR 16

Rom the tetrad and its inverse we define a constant syrnmetric matrix q(i)b)according to

As a consequence, we have

Given a tensor, then, to obtain its tetrad form we project it onto the tetrad to obtain its tetrad components:

Although the number of possibiIities for a choice of tetrad to represent spacetirne are large, it is often usefd to choose a tetrad of nul1 vectors I, n, rn, and Eï, where 1 and n are real and rn and 5 are complex conjugates of one another. This is known as the Newman- Penrose formalasm, and such tetrads are called NP tetmds.[Newman and Penrose 19621 They satisfi the following conditions of ort hogonality

as well as CHAPTER 2. INVARIANTS OF THE RlEMANN TENSOR 17 since the vectors are nuii. In addition to these requirements, the vectors are often chosen to be normalized4 so that

in which case the hindamental matrix v(i)b)is given by

where we now have the correspondence e(lla = la, e(21a= na, e(3)a= ma, and e(4)a= S. As with a coordinate systern, we may subject a tetrad to a Lorentz transformation at some point and extend it continuously through all of spacetime. There are six degrees of Ereedom amilable to rotate a tetrad, corresponding to the six parameters of the group of Lorentz transformations. For the NP tetrads, it is usud to consider three different classes of rotations:

i ) rotations of class I which leave la unchanged.

ii ) rotations of class II which Ieave na unchanged.

iii ) rotations of class III which leave the directions of la and na unchanged and rotate ma and m" by m angle 9 in the (ma,*)-plane.

The explicit transformations for each class are given by

1 : la 3 la,ma + ma+ala, 73' + m"+a'la, and na + na+a*ma+aEP+aa*la,

II : na + na, ma + ma+bla, 3' -, W+b*la, and la + la+b*ma+lrm"+bb*na,

4~otethat normalization is a choice, not a requirement. Since it is convenient for us that the tetrad be normaiized, we wiIl choose them to be so. CHAPTER 2. MARIANTS OF THE RLGMANN TENSOR 18

III : la -+ AIa, na + A-lna, ma -t eierna, and mD -t e-'%Y,

where a and 6 are complex functions and A and 0 are real functions of the coordinates on the manifold. Note that class 1 and II rotations correspond to translations, and class III rotations correspond to the familiar Lorentz boosts and rotations. We now use the NP tetrads to generate scalars that correspond to the independent components of the Riemann tensor and fkom which we may construct invariants. First we decompose the Riemann tensor into its tetrad components5

where R, and Cijtl denote the tetrad components of the Ricci and Weyl tensors ad R = = 2(R12- RJ4)is the Ricci scalar. Again, because of this decomposition, it is sufficient to examine the independent components of Ricci and Weyl tensors to obtain the independent components of the Riemann tensor. We define the ten independent tetrad components of the Ricci tensor in terms of the following four real and six complex scalars:

or the remainder of this section, for brevity we will omit the parentheses around tetrad indices when there is no ambiguity as to which indices are being considered. CHAPTER 2. LNVARLANTS OF THE RIEMANN TENSOR 19

The {Qab) are called the Ricci scalars (to be noted as difIerent than the standard Ricci scalar R; in this notation the place of R is satisfied by A). They satisfy the relation

and so redy there are ody three independent complex scalars that give rise to six pieces of information in general. Now, making use of the trace-free nature of CGkl and its cyclic identity [Chandrasekhar 19921, we find the ten independent tetrad components of the Weyl cornponents and represent them with five complex scalars:

These {Qu) are called the Weyl scalars. Together, the Ricci scalars, the Weyl scalars, and A comprise a set of scalars that are called the curuature cornponents, a name which arises fiom their relation to the Riemann curvature tensor. It is important to note the distinction between the use oi the term 'scalar' for scalar polynomial invariants and the Ricci and Weyl scalars. The Ricci and Weyl scdars are called scalars because they are scdar functions; that is, they map the coordinate space {xa} to a one-dimensional function (which may be complex). Scalar invariants use the term 'scalar' in reference to the definition of a scalar in tensor analysis- a zero rank tensor which is invariant to coordinate transformations. As is demonstrated Iater, not a11 of the Ricci and Weyl scalars are invariant to coordinate transformations. This is why the CWAPTER 2. INKARIANTS OF THE MEMANN TENSOR 20

Ricci and Weyl scdars and A cannot be used as a complete set of invariants themselves even though, as a set, they contain al1 the information that any invariant codd possibly contain. As we have just seen, we can now express the various components of the Riemann tensor by use of these scalars. In fact, we can do the same thhg with the Ricci and Weyl spinors through use of a dyad basis (or simply a dyad). This is an orthonormd6 basis A, 1 a,b, A, A = 0,l) that we set up at each point of the spacetime. This basis acts as a basis for spinors andogoudy to the tetrad basis that we set up for tensors. Typically, the two basis spinors iOIAand il)" are written as the special symbolç oA and L", respectively, and we shd adopt this notation. We may choose the dyad to be normdized.' The dyad then satisfies the orthonormality condition

The dyad and their cornplex conjugates then determine four nuU vectors; we may relate these vectors to the NP tetrad via the relations

With this correspondence we may express the Ricci and Weyl spinors as being generated

'~eare not required to choose a normalized basis; we make this choice purely for couvenience. 7~gain,this is a choice that is convenieut for us to make. 1s is perfectly acceptable to choose a non-normalized dyad. CHAPTER 2. INVARIANTS OF THE RlEMANN TENSOR 21 by the Ricci and Weyl scalars and the dyad [Carminati and McLenaghan 1991]:8

So whether using tensors or spinors as the generators of the invariants we will be able to express any invariant as a polynomial in the Ricci and Weyl scalars. For example, we have

This type of representation will prove to be particularly useful for obtaining a complete, independent set of invariants for the metric we will consider. To rely on them, however, we must recall that, unlike the invariants, the individual Ricci and Weyl scaiars are not al1 invariant to coordinate transformations, although one may form combinations of them that are. Rotations of the tetrad of a particular class may change some of the scalars. Al1 three classes of rotations will have an effect on individual scalars [Chandrasekhar 1992).

For example, consider the orthonormal dyad {oA,iA}. The most general change of dyad which leaves these two nul1 directions unchanged is

where X and p are arbitrary nowhere zero complex scalar fields. Since orthonormality requires oaiA = 1, we have the relation p = A-l. Fkom (2.81), (2.82), (2.83), and (2.84) we rnay obt ain the corresponding te trad transformations

--1 a 4 -1--1 la ct xXP,ma ct XX m ,m ct x-~XW,na H X X na. (2-89)

or brevity, we do not include the vector symbol if the same vector is repeated twice; ie. we mite

OAOBLCLD as OABLCD. CHAPTER 2. INVARLANTS OF THE RIEMANN TENSOR 22

If we t hen set

we may express (2.89) as the boost

combined with the spatid rotation

These are the clâss III tetrad rotations, and the scalars constructed from these rotated tetrads rnay be transformed themselves. Under a (not necessarily normaiized) class III rotation, then, for some scalar rl we have the transformation

We then set p = r' - r and q = t' - t and cal1 q a weighted scalar of tgpe {P,q}.9 Equivalently, we Say that q has a boost wezght of 3(p +q) and a spin weight of $(p - g). A scalar with a nonzero boost weight will not be invariant under boosts. Similariy, a scdar with a nonzero spin weight will not be invariant under spins. A scaiar with zero boost weight (spin weight) will be called boost neutml (spin neutml). The boost and spin weights of the individual Ricci and Weyl scalars may now be calculated to determine whether they are invariant under boosts or spins. Their weights are simply given by the forrnulae:

A : (0, O), art: (2 - 2r, 2 - 2t),\Er : (4 - 2r, 0). (2.95)

Multiplying two scalars toget her produces anot her scalar whose boost and spin weight is simply the sum of the individual scalars' boost and spin weights, respectively. The

'If we relax the normaiization condition, then, in generai, p # A-' and (2.93) will not contract to (2.94). In this case we dlq a weighted scalar of type {r',r : t', t). CHAPTER 2. INVARIANTS OF THE RIEMANN TENSOR 23

Riemann invariants must, of course, be invariant to transformations. So considering the invariants to be constructed fiom the Ricci and Weyl scalars as mentioned above, it is not difficult to see that although individual scalars rnay have nonzero boost or spin weights, in constructing an invariant they must be multiplied together in a way that each term in the invariant is boost and spin neutral and thus leaves the invariant unchanged by rotations.

2.2.4 Bivectors

In addition to tensors and spinors, it is also possible to construct invariants by considering bivectors. The use of bivectors to construct invariants is advocated by Sneddon [Sneddon 19961. It is not clear that the use of bivector notation provides a benefit. While it is true that in the bivector notation the Ricci and Weyl spinors become 3 x 3 matrices wit h complex vaiued components (and therefore have been subject to extensive classicai analysis), the procedure for constructing d invariants of a given order may not be as simple as the corresponding spinor version [Pollney 19961. We will not use the bivector notation in this thesis, but for completeness we shall sketch an outline for the procedure here. A bivector is an antisymmetric second rank tensor, that is,

w,b is called a bivector of rank 1. wabcd would be a bivector of rank 2, with the antsym- metries

The dual of a bivector is defined as CHAPTER 2. LNVmNTS OF THE RlEMANN TENSOR 24 where

where eabcd is the Levi-Civita symbol and g is the determinant of g,e and is negative definite. As a consequence,

If wSab= i-, Wab is called self-dual; if w*,~= -auab,wab is cded antz-self-dual. Now in the same way that spinors and vecton are related by aa,&, a self-dual bivector w,b can be written in terms of a rotor 4" by

where TAab is antisymmetric and self dual in ab. We may obtain the covariant components of the rotor +A by the relation 4A = (1~4~4~~where the quantity

acts as the metric to be used in rotor space. Then wabwab= and we may obtain the rotor fiom the bivector by the relation

For any rotor dA we ako have its cornplex conjugate #"or which the bivector 47a4ab#À w il1 be anti-self-dual. We may now obtain the Ricci and Weyl tensor equivalents in the rotor space. They are

. . -.A 3 With these objects and their complex conjugates I' and we may construct invariants in the same rnanner as before. so then, for example, we have sabsba= 4rAgrBA. CHAPTER 2. MANANTS OF THE RIEMANN TENSOR 25

2.3 Independence, Completeness, and Syzygies

As we have demonstrated, invariants are expected to provide a coordinate-kee source of information about the spacetime. Upon constructing a set of invariants, the question then arises as to whether ail of the elements of the set provide some unique information or not. For example, consider the set of invariants {R,sabsab, c~~~~c~~~~ R~~~R"~'~). Certainly there is no duplication of invariants in the set, and R, sabsab,and Cabcdcabcd ail contain different information. But from the decomposition of the Riemann tensor (2.5), we observe t hat

and so R,~~~R~~~may be written as a combination of the other three invariants. So any informat ion obtained Erom cdculating R~~,~R~~~~cannot be independent of the ot her t hree. R~~~~R~~~~ t herefore provides us wit h no extra information not contained in the set (R,sabsab, c~~~~c~~~~} and so its addition to this set provides no real value. This idea motivates the investigation as to whether an invariant in a set is in some way independent of the other invariants in the set and so contains some unique information or whether it is dependent and so its inclusion adds no information to the set. We now define two kinds of dependence for a set of invariants I = (Il,..., In]:

i ) A set of invariants I is said to satisfy weak dependence if there exists a polynomial P satisfying

ii ) A set of invariants I is said to satisS strong dependence if there exists at le& one polynomial P such that one can write

for any 1 5 j 5 n. CHAPTER 2. WARIANTS OF THE RlEMANN TENSOR 26

Here we shali refer to any strongly dependent set simply as not independent, while what we cdindependent shall depend on which definition of dependence we wish to use. In our terminology, a set that is weakly dependent may be cded weakly independent, while a set that is not dependent at ail may be called strongly independent. Thus, accord- hg to our definitions of independence, our earlier set {R, s~~s~~~C,MC-, R,~R~~~~} is not independent. If we remove R~~~~R~~~fiom the set, then we obtain an independent set since it is not possible (in general) to obtain a polynomial P such that P(R,S,~S"~, c~~~c~~~) = O. For the most part, we will be satisfied to cail a weakiy independent set independent and note weak and strong independence only when it is necessary to make a particular point. In deterrnining the independence of a set of invariants, we establish the relative usefulness of the invariants, that is, in the most general case each inwiant provides a unique piece of information that cannot be obtained from the other invariants. However, deter- mining independence does not address the question of whether the set contains enough invariants to obtain the mazimum amount of information available fiom examining invariants of the Riemann tensor. For example, consider the set of invariants with only one element, the Ricci scalar R. This set is trivially independent, but it is also clear that it does not contain the maximum amount of information available, since we know that in the most general case there is information in the trace-hee Ricci tensor and the Weyl tensor that is not contained wit hin the Ricci scalar. In some sense, then, t his one-element set lacks completeness. Motivated by the desire to obtain a set of invariants that will provide us with as much information as possible under every possible circumstance, we will define what we mean by a complete set of invariants. In the Literature there are a few different definitions:

i ) A complete set of invariants is one which can be used to construct all other invariants as polynomials of memben in the set. [Penrose 19601

ii ) A complete set of invariants is one which possesses a number equal to the CHAPTER 2. WARLANTS OF THE RIEMANN TENSOR 27

number of degrees of fkeedom inherent in the system and, for each aigebraic specialization, has a number equal to the number of degrees of fieedom inherent in the specialization. [Sneddon 19961

iii ) A complete set of invariants is one which, for each algebraic specialization, has a number equal to the number of degrees of freedom inherent in the specialization. [Carminati and McLenaghan 199 11, [Zakhary and McIntosh 19971

To be able to compare the different definitions of completeness, we must first define what we mean by a syrygy. Consider a set of invariants which is complete (regardless of which definition is used) and which are, in the most generai case, independent of each other. An algebraic specialization of the spacetime is a specialization of the spacetime brought about by restricting the form of the metric in some way (by imposing a symmetry, by restricting a function within the metric, etc.). In four dimensions, in general, the Riemann tensor has twenty independent components, and the metric tensor has ten independent components. However, the freedom in the coordinates removes the independence of sixteen of t hese components and Ieaves the generai four-dimensional Riemann tensor with only fourteen degrees of Ereedom. Upon algebraic specialization, the system will lose further degrees of fieedam and there will be equations relating these invariants, since we now have at least fourteen invariants but less than fourteen degrees of freedorn. These relations are called syzygies [Thomas 19341. Since, in general, the Riemann tensor has fourteen degrees of freedom, regardless of which definition is used a complete set of invariants for the general Riemann tensor must contain at least fourteen elements. For definition (i), the number of independent invariants depends on the definition of independence. If strong independence is required, then the set may have no more than fourteen elements. However, if only weak independence is required, then the number can be greater than fourteen since invariants which are related by high order syzygies are considered to be independent. (See the set of Penrose in the next section, in which he finds twelve weakly independent invariants for Einstein-Maxwell CHAPTER 2. INVARIANTS OF THE =MANN TENSOR 28 in spite of the fact that there are oniy nine degrees of freedom.) For definition (ii), there can be oniy fourteen invariants in the most general case, but for definition (iii) it is possible that more than fourteen would be required due to syzygies. In this case, weak independence must sufiice for independence requirements of the set. The requirements for completeness for definition (iii) may seem odd at first glance- after dl, since we know that there are only fourteen degrees of fieedom for the Riemann tensor, any set of more than fourteen invariants must necessarily contain some form of dependence. Therefore one would assume that tbese 'extra' invariants could be considered to be removed fiom the set to obtain a set of fourteen invariants. This rnay not be possible, however, since the additional invariants which rnay not be required in one specialization rnay be required in another. The reasoning behind this is subtle and is most easily understood by considering a simple example. Suppose for some algebraic specialization we have a set of three Riemann invariants (ri,Iz, 13} but t hat only two of the curvature components {a& Qa, A} are independent. Since any Riemann invariant rnay be constructed as a polynomial of the curvature components, the fact that we have only two independent scalars implies that we rnay have only two independent Riemann inwiants, for one rnay not construct three independent objects as polynomials of Iess than three different independent objects. So there must be sorne syzygy for our set of invariants such that

This relationship may be considered to remove one of the invariants from the set, since given any two invariants the third may be constructed. So the question now becomes: which invariant to discard? The standard answer to this question will typically rely upon complexity arguments- we wish to keep the invariants that are simplest to calculate, that have the easiest form with which to work, etc. Let us assume that II and I2 satisfy these requirements. So via the syzygy (2.110) we reduce the set of invariants to {Il,12}. Now suppose we specialize the original specialization even further, that is, we obtain CHAPTER 2. INVARLANTS OF THE ELI%MANN TENSOR a subclass of our original speciaiization. Suppose that the two cunature components that we found to be independent are stili independent, but now, simply because of the form of the scalars and the way in which the invariants are constructed, Iz = O identically. Now let us examine ouset of invariants. From our original specialization we reduced {Il,12, 1') to {Il,12} via syzygy (2.110), and t his syzygy must still hold because we are in a subclass of our original specialuation. But now, since 12 = O, OUT set of inkants is reduced to {Il}, since I2 contains no information. Thus we now have an incomplete set of invariants- we know that there are two pieces of information because of the independent curvature components, but we only have one invariant left that provides us with any information. It is not now sd'ficient to consider with hindsight that one should have removed I2 to retain the set {Il,13), since a different subclass of the original specialization might leave I2 as nonzero but reduce ri or I3 to zero identically. From this example one rnay see the problem encountered in attempting to arbitrarily reduce a redundant set of invariants. Certainly there will be syzygies, and these syzygies rnay be considered to remove invariants hom the set due to dependence. The problem lies in deciding which invariant to remove. For in one specialization one invariant rnay become identically zero but in a different specialization that invariant rnay contain information while a dgerent invariant becomes identically zero. It might be argued that if the syzygies are known for the general case, then those syzygies might be used to find the new relationships in a specialization. However, there is no guarantee of this, since a specialization might reduce a particular syzygy to the identity O = O in which case no information rnay be obtained. Therefore it rnay be necessary to retain a set of invariants in general that contain more than fourteen invariants and then reduce the set to the number required by independence only for particular algebraic specializations. Even in these cases, however, it will be necessary to ensure that the set that remains cannot be made incornplete due to further specialization as above. CHAPTER 2. WARlANTS OF THE RIEMANN TENSOR 30

2.4 Candidates for a Complete Set of Riemann Invariants

Over the years there have been a number of sets of invariants that have ben proposed as complete. Most of these have ben demonstrated to be incomplete. At this time, we have three sets that have been proposed to ùe complete that have neither been proven complete nor incomplete. We examine these sets here and from them generate a set that we shall use in chapter four to investigate the special case of class B warped products,

2.4.1 Early Sets

The first set of invariants of the Riemaan tensor proposed to be complete is the set of V. Narlikar and K. Karmarkar [Narlikar and Karmarkar 19481. Interestingly, of al1 the early sets proposed, the Narlikar/Karmarkar set is the only set that has not been proven to be incomplete. However, the set is deficient in that it is of unnecessarily high degree. Narlikar and Karmarkar define the tensors CHAPTER 2. INVARIANTS OF THE RIEMANN TENSOR 31 and, with these, define the fourteen invariants

This set as originally given by Narlikar and Karmarkar is actually defective in the sense that it is possible to define a nonsingular spacetime for which the definition of invariant K3 includes a division by zero. The flaw can be removed by defining

ins tead. It is of historical interest to note that Narlikar and Karmarkar note that for the general spherically symmetric line element

there are only four independent components of the Riemann tensor: R13~3,R3434, RL414, and R2323.They take this to imply (correctly) that there are onIy four independent CHAPTER 2. n\TVARIANTS OF THE RIEMANN TElVSOR 32 invariants for spherical symmetry. They propose that the set {Il,12, 13, Ji) suffices to give a complete set in this case, although they do not attempt to prove this. As we will demonstrate later, this set is deficient in the sense that one could not use it to construct all possible invariants of the Riemann tensor. There were a number of other early candidates for complete sets, given by J. Géhéniau and R. Debever [Géhéniau and Debever 19561, L. Witten [Witten 19591, A. Petrov [Petrov 19691, P. Greenberg [Greenberg 19721, G. Sobczyk [Sobczyk 1981], and G. Sneddon [Sneddon 19861. Al1 of these sets have been proven to be incomplete. J. Carminati and R McLenaghan [Carrninati and McLenaghan 19911 have demonstrated that the Géhéniau/Debever set contains a syzygy that reduces the set to at most thir- teen invariants. They also state that the set of Witten and the equivalent sets of Petrov and Sneddon do not yield the required nine independent invariants in the general perfect fluid case, with a demonstration that the Petrov and Sneddon sets contain at most five independent invariants in this case. Sneddon [Sneddon 19861 has demonstrated that the equivalent sets of Greenberg and Sobczyk contain at most twelve independent invariants and t herefore is incomplete as well. Penrose [Penrose 19601, [Penrose and Rindler 19861 has generated a complete set of invariants (complete in the sense of definition (i)),but only for the Einstein-Maxwell field. The set he constructs is

(2.134)

(2.135)

(S. 136)

(2.137) (2.138) if strong independence is required. If the criterion for inclusion is weak dependence, then an addi t ional invariant CHAPTER 2. LNVARLANTS OF THE RIEMANN TENSOR 33 is required. This invariant is related to the others by the syzygy

For the Einstein-Maxwell field there can be at most nine degrees of &dom in the Rie- mann tensor. Since only strongly independent invariants contribute to the degrees of fieedom, N does not contribute, but this still leaves ten invariants which mut be in some way related. Since the phase of $AB is undetermined by QAs~kfor the gravitational field, Penrose then takes the invariants {I,J, IKI, ILI, IMI, KZ,LM, MF}. Rom this it may be seen that the problem of specifying ail the degrees of &dom is different fkom the problem of finding a set of invariants from which all others can be constructed.

2.4.2 The CM Set

In [Carminati and McLenaghan 199l], in addition to demonstrating that almost al1 previous candidates for complete sets were in fact incomplete (in the sense of definition (iii)), Carminati and McLenaghan constructed a set of sixteen invariants that they show to be of lowest possible degree and complete in the Einstein-Maxwell and perfect 0uid cases. This set is weakly independent, and its elements are CHAPTER 2. ZNVARIANTS OF THE RIEMANN TENSOR 34

EFAB QASCO*~~EFQ

where C*abcd= 1 (COw- iCabCd)denotes the self-dual Weyl tensor and cabcdits complex conjugate. This set, the elements of which are referred to as the CM invariants, contains

SU real invariants R, ri, rp, rg, mg, and md; wl, w2, m~,ma, and m5 are complex. The set is constructed so as to be of lowest possible degree. It is common to consider specializations of the Riemann tensor according to its Petrov type, that is, according to the eigenvalues of the Weyl tensor [Kramer et. al. 19801. When we do t his, the Riemann tensor loses some degrees of keedom. The number of lost degrees of freedom depends on the Petrov type. For type 1, which is the most algebraically general case, there are 14 degrees of fieedom, while there are 12 for type II, 10 for type III and type D, 8 for type N, and 4 for type 0, for which the Weyl tensor is identically zero. There therefore must be enough syzygies between the CM invariants for each Petrov type to remove the extra degrees of fieedom as t hey are lost. Most of these have been discovered, CHAPTER 2. INVARIANTS OF THE RIEMANN TENSOR 35 and are trivial- the invariants simply reduce to zero identically. For Petrov type 0, the known syzygies are [Pollney 19951

for type N,

for type D,

for type III,

and for type II,

For types III, II, and 1 there are still some unknown syzygies.10 In each case the syzygies should remove the dependence of two more invariants to reduce the set of invariants for each Petrov type to the necessary degrees of Çeedom. As the Xernann tensor is specidized further, more syzygies of the CM invariants may be discovered as degrees of freedom are lost. In dernonstrating that their set is complete for perfect fluids and Einstein-Maxwell fields, Carminati and McLenaghan find

------'O~hesesyzygies will not be easy to discover. For example, N. Pelavas [Pelavas 19971 has demonstrated that any further syzygy for Petrov type III must be in the form of a polynomial of degree greater tban 12. CHAPTER 2. INVARLANTS OF THE RlEMAMV TENSOR 36

the syzygies for their set in these special cases. For perfect fluids, they note that in general,

For the Einstein-blaxwell case, in general

while for the nul1 case these reduce to

They also note for certain Petrov types for each specidization some syzygies which correspond to some of the syzygies detailed above which, as Pollney has demonstrated, are actually true in general. Although Carminati and McLenaghan do not attempt to prove that their set is complete for each Petrov and Segré type, they do believe that this is the case. The completeness of the CM set in general has neither been proved nor disproved. Some further work on the CM set was done by Haddow [Haddow 19951, who examined the interpretation of the CM mixed invariants in terms of collineations of principal nuU directions of the Ricci and Weyl spinors.ll In doing sol he replaces rnq with another real

"Consider the Ricci and Weyl spinors to be constructed of four 1-spinors a~,PA, TA, and b~ in the CHAPTER 2. RVVARlANTS OF THE HEMANN TENSOR 37 invariant

where Tabcdis the Bel-Robinson tensor,

In spite of the fact that M4 is of higher degree than m4 (six as opposed to five), he makes the replacement because it is convenient for him to use to extract certain relevant geometric information. For example, when the principal null directions of the Ricci spinor are aligned (ie. QAsAB = 7A~B~A"yb)then M4 takes on the square of Penrose's second Weyl covariant, Q:

A B C il2 Md = IQABEFQ~~CDT7 7 7 I .

2.4.3 The ZM Set

The final candidate for a complete set of invariants is one put forward by E. Zakhary and C. McIntosh [Zakhary and McIntosh 19971. This set, the elements of which are called the ZM invariants, contains seventeen invariants and is virtually identical to the CM set (up to constants), replacing one real inrariant with a complex invariant instead. The correspondences between the ZM invariants (in ZM notation) and the CM ones (in CM notation) are

same way that we might form a fourth rank tensor £rom the outer contraction of four vectors. We cal1 these the principal spinors of the Ricci or Weyl spinors. Then an example of a collineation would be

*ABCD = QAOBQCQD (which would correspond to Pecrov type N). CHAPTER 2. INVARTANTS OF THE RlEMANN TENSOR 38

The real invariant rnq is replaced by a cornplex invariant

The reasons for replacing m4 with M are unclear. Zakhary and McIntosh do state that in constructing the ZM set, the presence of TiBCZ,is undesirable, but they do not give an explicit reason why this should be so. It is possible that rn4 was considered but discarded as unnecessary since the number of degrees of £&dom for each case was satisfied by the other invariants, in which case m4 must always satisfy some syzygy with the other invariants in the CM set. However, it has also been suggested [Pollney 19961 that the procedure that Zakhary and McIntosh use to construct their set is incompIete because it appears that rnd could never be constructed using their rnethod. If this is the case, then it is not possible to make any clah as to whet her the ZM set is of lowest possibIe degree.

But even if m4 can be constructed via their procedure, it is still not clear that their method is complete, for while they construct tensors Erom the Ricci and Weyl tensors they do not mention how these particu1a.r tensors which they construct could lead to al1 the tensors that could ever arise. Although Zakhary and McIntosh claim that their set is complete in the sense of definition (iii) from the results of their attempt to demonstrate that the ZM set reduces to the correct number of invariants required by the number of degrees of freedom for each Segré and Petrov type, for the more complicated Segré types a number of apparently independent invariants is found which is larger than the number of degrees of freedom CHAPTER S. WARLANTS OF TNE RIEMANN TENSOR 39 in the Riemann tensor. It should be the case that there are enough syzygies among the invariants to account for this discrepancy, but these syzygies have not been found. Futbermore, the criteria for choosing independent members of the set for each type is not given, which makes it difncult to be convinced that the set is, in fact, independent.

2.4.4 A hl1 Set of Invariants of Degree Five

Presented with these candidates for a complete set of Riemann invariants, instead of choosing a particular set we approach the problem fkom a different angle. In the interest of simplicity, we discard the Narlikar/Karmarkar set as of unecessarily high order. Now neither the CM set nor the ZM set contains invariants of degree higher than five. So instead of choosing a particular set, instead we attempt to construct a set that contains al1 the possibly independent invariants up to order five. A notation has been developed by Pollney [Pollney 19961 to attempt t his. This notation graphically represents contractions among spinor indices. To begin, one draws the Ricci and Weyl spinors which make up an invariant as points on a graph. Contractions over indices are then represented as lines connecting the spinors. The shape of the 'web' thus produced will then determine if the invariant is possibly independent, that is, if two invariants are drawn in this way and their graphs cannot be deformed into one another without cutting one of the links, then it is possible that they are inequivalent. In this manner the problem of finding a full set of invariants for a given degree becomes one of finding al1 the topologicafly invariant graphs of the invariants. For Iow degree, the graphs are trivial and it is simple to find a full set of invariants. As the degree increases, the graphs become increasingly complex and a proof of completeness would rely on a combinatoric analysis of the number of possibie graphs that can be produced at a given degree. The complete andysis up to degree five is given in [Pollney 19961. It turns out that the set of possibly independent invariants is given by the union of the CM and ZM sets. In accordance with this, we will choose this set as the set of invariants to investigate the CHAPTER 2. INVARIANTS OF THE RlEMANN TENSOR 40

class B wqed products. We adopt the CM notation dong with those invariants and to this set we add another complex invariant rn~equivalent to M:

Thus we now have a set of 18 invariants S = {R, ri,r2, r3, wi, w?, mi,rnz,rn~, m4, ms, m6}, of which six are real and twelve are complex.

2.5 The Value of Invariants

It is natural to suspect that invariants of the Riemann tensor should play some important roIe in general relativity. What this role should be, however, has yet to be fully realized. The fact that the invariants are invariant ta coordinate transformations suggests that they might aid in distinguishing spacetimes, but it was known very eady that they cannot do this [Narlikar and Karmarkar 19481. As it has been demonstrated, the line element

generates no nonzero Ricci invariants, and, being conformally fiat, may generate no Weyl or mixed invariants either. Recently, a larger class of metrics with line element

has been demonstrated to have this property as well [Koutras and McIntosh 19961. Thus it is clear that these spacetimes, which are not flat, are indistinguishable fkom Minkowski space if only the invariants are examined. Not even the inclusion of differential invariants (regardless of their order) will overcome this problem, and so the equivalence problem cannot be addressed via the examination of invariants. CHAPTER 2. LNVARIANTS OF THE MEMANN TENSOR 41

Invariants are, however, of some use in the study of spacetirne singularities. They can be used to detect scalar polynomial (sp) singularities, which are the endpoints of at least one curve on which a scdar polynomial in the metric and the Riemann tensor takes on unboundedly large values [Tipler et. al. 19801. In this case, at least one of the invariants will also become unbounded. For example, for the Schwarzchild metric

there are two singularities, at r = O and at r = 2m. Since this metric is vacuum, the

Ricci invariants are identically zero, but the two Weyl invariants zui and w2 are not. Their values are

Examination of these invariants reveals that while the singularity at r = O is an sp singularity, the one at r = 2m is not; it is merely a coordinate singularity, and can be removed, for example, by a transformation to KrusM coordinates {u, v, 9, $} [Kruskal 19601. As has been demonstrated, the Ricci, Weyl, and mked invariants are necessary and sufficient for locating an sp singularity [Siklos 19791. However, even here invariants will fail to distinguish shell crossing singularities12 from more significant singularities. Invariants might be of some use when examining certain physical properties of a spacetime prescribed by a particular energy-momentum tensor Tab.nom the Einstein equations, we have

"Consider, for example, a sphericdly symmetnc dust solution. Specify at some initiai time the density and velocity of mattex as a function of a radial coordinate R. Then consider the matter to be describeci by spherical 'shells' labetled with a comoving coordinate r, taken as the value of R at the initial time. For a certain stable set of initiai conditions it tunis out that the analytic form of the solution breaks down at a stage where becomes zero. This is cdled a shell cmssing singularity [Clarke 19931. CHAPTER 2. INVARIANTS OF THE RIEMANN TENSOR 42

The Ricci invariants, then, may be expressed in terms of contractions of the energy- momentum tensor. So then we have [Fevens 19933

and if we make the choice of a certain energy-mornentum tensor we obtain expressions relating the inwiants to the prescribed physical quant ities (such as the energy density, pressure, etc.). While this might be of limited use in and of itself, it may be more significant if syzygies are found between the Riemann invariants. For then in removing a generally independent quantity we may be able to use the syzygy to find a further relationship between the physical objects in the energy-momentum tensor. That is, by restricting a metric to a certain symmetry and examining a physical specialization (perfect Buid, or anisotropic pressure, or anisotropic pressure with a heat flux, etc.) the syzygies (if any ) inherent in the spacetime may determine relat ionships between physical quantities such as energy density and heat flux that must be obeyed in that case. The Weyl invariants may also be related to physical quantities. If the electric and magnetic components Eab and Hab of the Weyl tensor are defined in the usud way:

where ua is a unit timelike vector, then the Weyl invariants may be expressed as [Santosuosso et. al. 19981 CHAPTER 2. INVmTSOF THE RIEMANN TENSOR 43

Again, these relationships may be of some significance if syzygies are found for a particular algebraic specialization of spacet ime, as t his would impose restrictions on the electric and magnetic components of the Weyl tensor. Furthermore, invariants might be used to identify certain properties t hat a spacetime possesses. We already know t hat the invariants cannot totally distinguish spacet imes, but we might still be able to use the invariants for this purpose in a limited fashion. For example, it has been demonstrated that for the generaI perfect fluid case, ml = O identically. Thus we know that if we evaluate the invariants and determine that mi # O then the spacetime cannot represent a perfect fluid. Unfortunately, this rnay not be a sufficient condition for a perfect fluid, in which case having ml = O tells us nothing more than that the spacetime might be a perfect fluid. Nonetheless, these possibilities are worth investigating, and might provide a quick test of certain properties of spacetime. Chapter 3

Class B Warped Products

Consider a semi-Riemannian manifold (M, g) which we construct as the manifold product of two distinct manifolds (Mi,gi) and (M2,g2) so that M = Ml x M2. However, instead of constructing the metric g on M as simply gl8g2, suppose we take a smooth function

O : Ml + !R and form the metric for M as g = gi 8 e2agz. Then we cal1 M a warped product manifold and denote the manifold product as M = MI x, M2. Ml and M2 are known as the factors of the warped product manifold. The function a is known as the warping /unction. When one of {MI,M2} is Riemannian and the other is Lorentzian and M is four- dimensional, (M,g)corresponds to a spacetime M with metric g. We then refer to this manifold as a warped product spacetime or, more simply, a warped product.

3.1 The Classification of Warped Products

Choosing either of {Ml,Mz} as the Lorentzian manifold wiii have an effect on the nature of the spacetime, as will choosing the individual dimensions of Ml and M2. Ac- cording to the notation as developed by J. Carot and J. da Costa, we now set up a classification system based upon these considerations [Carot and da Costa 19931. CHAPTER 3. CLASS B WARPED PRODUCTS 45

Class A. We have a class A warped product when one of {Ml,M2) is of dimension 1 and the other of dimension three. Class A warped products can be broken down into two subclasses:

i ) Al: dimMr = 1 and dim M2= 3.

ii ) A*: dimMl = 3 and dimM2 = 1.

For class AI, the line element can be written as

where c = iland cr, p,r = 1,2,3. {u} is the local coordinate chart in 1121 and {xf }?= 1,2,3 is the one in M2.In their respective charts, the metrics on Ml and M2 are gi = du @ du and gz = haLi(z7)dxQ@dxO respectively. Ml is Riemannian when É = +1 and Lorenztian when e = -1. For class A2, the element can be written as

where E = f1 and cr, /3,r = 1,2,3. In this case, {x'}~=~,~,~is the local coordinate chart in

Mi and {u} is the one in M2, while the metrics on Ml and M2 are g2 = h,,p(z7)dxa@dzP and gi = du @ du respectively. Again, M2 is Riemannian when E = +l and Lorenztian when E = -1. Class B. We have a class B warped product when dimMl = dim M2= 2. In this case the line element may be written as

where a, b = 1,2 and a, = 3,4. {x~}~=~,~and {xQ}a=3,4 designate the local coordinate charts in Ml and M2 respectively, with gl = gaa (xC)dxa8 dzb and g2 = ga8 (x7)dxaO dx@ t heir respective metrics expressed in t hose chosen coordinate charts. CHAPTER 3. CLASS B WARPED PRODUCTS 46

Class C. It is possible to have spacetimes that belong to both classes A and B. For example, the line element

describes such a spacetime. This class of spacetimes is denoted as class C. The importance of warped products in general is that their geometry is directly related to the geometry of the lower-dimensional manifolds from which t hey are construct ed. Since lower-dimensiond geometry is generally easier to study, this provides a simpler way of examining the properties of a spacetime. As an example, in the case where (Mi:gl) is Lorentzian, the causal structure of the waqed product can be deterrnined by examining the causal structure of (Mi,gl), a much simpler task. We will concern ourselves solely with class B warped products. As we shall see, this class corresponds to some of the most physicdly important spacetimes.

3.2 Invariant Characterization of Class B Warped Products

It is of use to have an invariant characterization of warped products, t bat is, the ability to determine if a given spacetime can be written as a warped product of a certain class via a coordinate transformation. In fact? it is possible to do this for both class A and B warped products [Carot and da Costa 19931, but since we will be analyzing the invariants for class B warped products only in this thesis, here we will only concern ourselves with class B. First we note that (3.3) may be written as

The line element ds2 is said to be confomally equzvalent to di2. di2 is then a metric CHAPTER 3. CLASS B WARPED PROD UCTS 47 expressible on the manifold (M, g) as

where unprimed indices run over {1,2) and primed indices run over {3,4). Such a metric is called locully decomposable, referring to the ability to decompose the manifold for which di2 is the line element as the manifold product of two distinct manifolds. In particular, this type of decomposition is cded a 2 + 2 decomposition, referring to the dimensions of each manifold in the decomposition [Hall and Kay 1988al. For a 2+2 locdy decomposable spacetime t here does not exist a global, covariantly constant, nowhere zero vector field. However, two global, iinearly independent recurrent nuil vector fields îa and ka such that îaka = -1 are admitted [Hd19911. These may dways be scaled so that the recurrence vector is paralle1 to one of them, say i":

where a is a smooth real function of the coordinates associated with the integrable distri- bution spanned by C and ka and the stroke denotes the covariant derivative with respect to the connection associated wit h the line element di2. Now define vector fields la and ka in (M, g) as la e-~pand ka e e-uLa (with the associated covariant forms 1. z eula and ka = euka so that laka = -1). Then [Kramer et. al. 19801

and

Then from the Newman-Penrose spin coefficients K, o,and w [Kramer et. al. 19801 spec- ified to 1, and ka, we cm see that both la and ka are geodesic (although non-&nely CHAPTER 3. CLASS B WARPED PRODUCTS 48 pararnetrized) , shear-free, and hypersurface orthogonal wit h expansions

If we wished, we could choose a different form for the vector fields la and ka. For exampIe, if we define La r e-*"i" and Ka i ko then instead of (3.10) and (3.11) we obtain

and

and again La and Ka are geodesic, shear-free, and hypersurface orthogonal, but now La is affinely parametrized. As before, their expansions are

We might instead a,fFinely parametrize Ka in the same fashion as above if we so wish. In any case, we can use the expressions (3.10), (3.11), (3.12), and (3.13) (or the al- ternative expressions (3.14), (3.151, (3.16), and (3.17), for example) to provide an invariant characterization of 2 + 2 warped products. We then have the two theorems [Carot and da Costa 19931: Theorem 3.1. Let (M,g) be a spacetime. Then

a ) if (M, g) is conformally equivalent to a 2 + 2 locally decomposable spacetime (M, g) with g = e26g then there exist nul1 geodesic, shear-tree, and hypersurface orthogonal vector fields la and ka (or, alternatively, La and Ka) on M whose covariant derivatives are given by (3.10) and (3.11) and their expansions by (3.12) and (3.13) (or, alternatively, by (3.14) and (3.15) and (3.16) and (3.17) respectively ) . CHAPTER 3. CLASS B WARPED PRODUCTS 49

b ) if there exists a function o : M + 8 and nul1 vector fields la and ka on M satisfying laka = -1 (or null vector fields La and Ka satisfying LaKa= -1) and their covariant derivatives are given by (3.10) and (3.11) (or by (3.14) and (3.15)), then the spacetime is conformal to a 2 + 2 locally decomposable spacetime (M, g), where g = e-2ug.

Theorem 3.2. Let (Mlg) be a spacetime conformally equivalent to a 2 + 2 locally decomposable spacetime (M, g) and let o and la and ka (or o and La and Ka) be the function and vector fields respectively whose existence is guaranteed by Theorem 3.1. Let hab = 21(,kb1 and hi gab - hab (or Hab = 2L[,Kb) and HA = g,b - Hab)- Then (M,g) is a warped product of class B if and only if either habob= O or h,bab = O (respectively ~~~o~= O or H&O' = O).

3.3 Physics of Class B Warped Products

Of course, the real importance in studying class B warped products is that contained within this class of metrics axe some of the most physically important spacetimes- namely, t hose wit h a twedimensional Riemannian subspace of cons tant curvature. Al1 spherically, plane, and hyperbolically symmetric spacetimes are contained wit hin the subclass of class B warped products. An evaluation of the Weyl scalars for class B warped products reveals that the only nonzero Weyl scalar is Q2. As a consequence, class B warped products are of Petrov type D in general (and may further specialize only to type 0) [Kramer et. al. 19801. The specialization of a spacetime to a class B warped product imposes no restrictions on the Segré type; al1 types are possible [Carot and da Costa 19931. However, some specializations of Segré type do impose restrictions on class B warped products. In generd, the components of the Riemann tensor evaluated for a warped CHAPTER 3. CLASS B WARPED PRODUCTS 51) product (that may or may not be of dimension four) are

where a,/?,. . . denote indices on (Ml, lh), A, B,. . . denote indices on (M2,*h), and

1~a876and 2~ABcDare the Riemann tensors evaluated on (Ml, 'h) and (M2,*h) respectively. From contracting the above, we obtain the components of the Ricci tensor:

where n is the dimension of M2. From these considerations, we have the following tlieorern [Haddow and Carot 19961: Theorem 3.3. Suppose that (M,g)is a class B warped product with Ml x, M2.In addition, suppose that at each point in M the subspace of the cotangent space spanned by the gradients of the eigenvalues associated with al1 Ricci eigenvectors can be spanned by the gradients of the eigenvalues associated with horizontal eigenvectors. Then M2 is necessarily of constant curvature. Using the above notation and denoting the Ricci scalars associated with (Ml,hl) and (M2,h2) as R and R respectively, from (3.24) and (3.25) we have CHAPTER 3. CLASS B WARPED PRODUCTS 51

Suppose that there are non-constant Ricci eigendues. Now for class B warped products there exists at Ieast one horizontalL Ricci eigendirection Za [Haddow and Carot 19961. From the above assumption its eigenvalue X may be assumed to be non-constant. So fiom (3.27) and (3.28) (that is, the block-diagonal structure of the Ricci tensor) it follows that X must be a root of the characteristic polynomial associated with Ragand therefore (3.27) a function on Mi,that is, it oniy depends on the coordinates x1 and z2. The same must hold for its associated horizontal eigendirection Za,that is, Zais tangent to Mi and its components are functions on Mi.Rom (3.28) it is seen that the (degenerate) vertical eigenvalue p is given by

According to the hypotheses of the theorem, p must be a (possibly constant) function of zLand x2. So since p and eu are functions of x1 and x2 only but * R is a function of x3 and x4 only, it follows from (3.29) that R is a constant. This is the case for non-constant Ricci eigenvalues; if al1 the Ricci eigenvalues are constant, then p is trivally constant and so, once again, we may conclude that 2~ is a constant. SO M2 mut necessarily be of constant curvature. Many physically important cases fall into this situation. For vacuum, A-termY2and electromagnetic null spacetimes, the Ricci eigenvalues are constant. In the case of electromagnetic non-null, the trace-free condition shows t hat the pair of eigenvalues are functionally related (and that one of them is associated with a horizontal eigenvector). For perfect fluids, the existence of an equation of state similarly shows that the conditions of the theorem hold, but it has been shown in an independent manner that the second factor of a class B warped product rnust be of constant curvature for perfect fluids in general [Hogan 19901.~So then we know that for vacuum, A-term, electromagnetic null and

'A vector .Ya is calted horizontal if 2h~~~"= O and verfiml if '~ABZ~ = 0. '~nthis section, A refers to the cosmological constant. 3~oganexpanded upon the technique of 1. Robinson [Robinson 19851, who demonstrated that for vacuum, class B warped products must have the second factor as a manifold of constant curvature. CHAPTER 3. CLASS B WARPED PRODUCTS 52 non-null, and perfect fluids, warped products of class B must necessarily be spherically, plane, or hyperbolicdy symmetric. As one might suspect, the Birkhoff theorem, of great physical signscance for spherically symmetric spacetimes, extends to class B warped products. To conclude this chapter, we present without proof an 'extended Birkhoff theorem' for warped products of class B due to Haddow and Carot [Haddow and Carot 19961. Other proob of extended Birkhoff theorems have been given by Goemer [Goenner 1970] and Barnes [Barnes 19731, while a more geometrical proof was given by Bona [Bona 19881. Theorem 3.4. Suppose that (M,g) is a class B warped product whose Ricci tensor is everywhere of Segré type {(1,1)(ll)} or some degeneracy thereof. Assume that the gradient of the warping factor does not vanish over a nonempty open set. It then follows that one can decompose as the disjoint union M = MK U MR U M' where MK and MR are open and a hypersurface-orthogonal non-nul1 Killing vector is admitted on a neigbourhood of any point of MK and a null recurrent vector field is admitted on a neighbourhood of any point of MR. The set Mi has no interior. Chapter 4

Invariants of Class B Warped Product s

In this chapter we will investigate the invariants of class B warped products using the set of invariants described in chapter 2, that is, the union of the CM and ZM sets. As we will demonstrate, the set we use contains many syzygies in these cases, and we obtain definite complete sets for these warped products. For a class B warped product to represent a spacetime, one of the factors must be Lorentzian and the other must be Riemannian. As we shall see, while the choice of which factor is Lorentzian makes a difference in the calculation of the Ricci scalars and changes the form of the invariants, the relationships between the invariants, and so the resulting syzygies for this class of spacetimes, are unchanged. This will allow us to form the same complete set for class B warped products regardless of which factor is Lorentzian. To provide ourselves with a simple notation for distinguishing the two cases, we will denote class B warped products as class BI for the situation when the first factor is Lorentzian and as class B2 when the first factor is Riemannian. To begin, for computational purposes we wiil wish to obtain a form of the metric to be used in each case that is computationally simple yet is stili perfectly general. We are CHAPTER 4. INVARIANTS OF CLASS B WARPED PRODUCTS 54 aided in this regard by the 2 + 2 decomposability of class B warped products. It is not difficult to see that al1 two-dimensional metrics are conformally flat, whether Riemannian or Lorentzian [Nakahara 19901. First consider a twedimensional Riemannian manifold with the general line elemeut

We rnay rewrite this as

* where g = gzz9yy - gzy - Now according to the theory of differential equations, there exists a complex integrating factor X(x, y) such t hat

and

Shen ds2 = IxI-2 (du2 + du2). SO set 1 XI-2 = e2a and it is clear that ds2 = eZPis conformally flat. For a two-dimensional Lorentzian manifold, the procedure is similar. However, in this case we have two real integrating factors X(x, y) and p(x,y) such that

and

(&dZ + - = du du. 922

In this case, ds2 = A-' p-' (du2 - du2). NOW Xp is either positive or negative definite, so we may set IApl-l = e2~to obtain the form ds2 = fe2"(du2 - du2), which is again CHAPTER 4. INVARIANTS OF CLASS B WARPED PRODUCTS 55 clearly conformdy flat. Furthemore, for a flat Lorentzian line element we may make the fur t her t ransforrnat ion

which dlows us to write ds2 = fe2adwdz, where a is now a function of the coordinates w and z. So since every two dimensional rnetric is conformally flat, the following choices of line elements will sufEce to examine the invariants:

ds2 = -2 f (u, v)dudu + r(u, u)~~(B,4)2(d82 + dq52) for class Bi spacetimes and

ds2 = f (u, (du2 + du2) - Zr (u, ~)~~(8,#)dOd+ (4.10) for class BÎ spacetimes. There are many examples of class Bi spacetimes in the literature. Examples of spacetimes of class Bq ille considerably harder to End. It is known that the only physically interest ing energy-momentum types of class B2 spacetimes are non-nul1 electromagnetic, A-term, or vacuum [Haddow and Carot 19961. Haddow and Carot give one example of vacuum class B2 spacetime, for which the line element is

4.1 The Curvature Components

4.1.1 Class BI Warped Products

We will examine class B warped products first. To calculate the Ricci and Weyl scalars, we must first form an NP tetrad. Since we may freely boost or spin any tetrad corresponding to a particular spacetime we have many possible choices. To begin, we shall CHAPTER 4. RVVARIANTS OF CLASS B WARPED PRODUCTS 56 choose

There are only five nonzero curvature components that may be constructed from this tetrad- They are

a00 =

a11 =

'322 =

A =

where the subscripts denote partial derivat ives with respect to t Lat coordinate. We must now consider whether or not these scalars are independent. This is not necessarily the same task as deciding whether or not invariants are independent, since some of the remaining nonzero curvature components are not boost neutral and so are not invariant to coordinate transformations. alIlA, and @2 are al1 boost and spin neutral, but aooand Q22, while being spin neutrai, have boost weights of +2 and -2 respectively and so will change under boosts. Now we can be assured that alilA, and \E2 wilI remain invariant to coordinate transformations and so we will first examine their mutual independence. Since they are CHAPTER 4. INVAMANTS OF CLASS B WARPED PRODUCTS 57 composed of virtually the same terms, it will be simplest to prove their independence by defining some new quantities:

We can be certain that t hese three quantities are (in general) mutudly independent, for A is the only one that contains ru,,, B is the only one that contains derivatives of g(9, $), and C is the only one that contains f,,,. But we may construct A, B,and C frorn Qll, A, and G2:

Since one cannot construct three iudependent quantities from less t han t hree different independent quantities, and we know that A, B, and C are mutually independent, we may condude that al1,A, and \k2 must be mutually independent.

Now we return to the question of the independence of Boa and QS2- At first glance, from the same considerations we used to dernonstrate that the other nonzero scalars were independent, it certainiy appears that Qoo and $2, are independent. But since Goa and Qs2 are not invariant to boosts, we cannot use the previous method to prove th& independence. For example, if we boost the given tetrad so that CHAPTER 4. LNVAIUANTS OF CLASS B WARPED PRODUCTS 58

then we obtain

Now while the differences between the values of Goa and for the two different tetrads may seem like a minor discrepancy, it points to a larger problem. For example, consider

the effect of the coordinate transformation (u, v) i+ (v,u) on our original choice of tetrad.

Each of @ll,A, and q2 remain unchanged, but Goa and Q2* DOW become

Ooo and Q22 have now, in a sense, 'interchanged'. This interchangability suggests that they are, in fact, equivalent. This may be demonstrated by choosing for 1, and na the rat her unwieldy forms

and leaving maand unchanged. This form of the tetrad then produces

from which we rnay see that Goa and are equal and so are not independent. It is not surprising, perhaps, that this should be so, for since each term in an invariant must have zero boost and spin weight, we know that Qao and @22 must always appear in the combination QO0a22.This is because aO0has boost weight +2 and has boost weight -2 and these axe the only nonzero invariants with nonzero boost weights for this class CHAPTER 4. INVARIANTS OF CLASS B WARPED PRODUCTS 59 of metrics. Equivalently, we could deduce this Born considering the interchange of the coordinates u and v as above, since (except for a particuiar choice of tetrad) neither Qw nor is invariant to this transformation, but is. If we then treat as a new scalar, we determine that this is independent of Gtl, A, and !Pa, since it is the only scalar that contains ru,, or ru,. Therefore although we have five nonzero curvature components, only four of them are independent. The apparent fifth curvature component only contains information about how the tetrad chosen is boosted relative to the standard tetrad defined above.

4.1.2 Class B2 Warped Products

We shall examine class B2 warped products in the same manner as we examined class BI warped products. First of dl, we choose the tetrad

As before, this tetrad produces five nonzero curvature components. They are CHAPTBR 4. INKARIANTS OF CLASS B WARPED PRODUCTS 60

- recalling that 820= 002.Again, we must consider the independence of the scalars. We already know that ail,A, and iyz are boost and spin neutral. In this case, aW and @20 are boost neutral, but have spin weights of +2 and -2 respectively and so will change under spins. As in the previous case, since A, and q2 are boost and spin neutral, they will remain invariant to coordinate transformations and so we will first examine their mut ual independence. If we define

Ar-1 f (go94 - ~s.4~)+ f 293 (ru2 + ru2) 2 r2 f (4.45)

then we can be certain that these three quantities are mutually independent, for A is the only one that contains derivatives of g(9,9), B is the only one that contains f,,, or f,,,, and C is the only one that contains ru,, or r.,,. But, as before, we may construct A, B, and C from ail,A, and \ÿ2:

Since one canot construct t hree independent quantities fiom Iess t han t hree different independent quantities, and we know that A, B, and C are mutually independent, for class B2 warped products as well we rnay conclude that ail, A, and Q2 must be mutudly independent.

We now return to the question of the independence of iPO2and Q20. AS before, we note that under the coordinate inversion (u,v) e (v,u), which leaves the metric invariant, QO2 and are 'interchanged', which suggests that they are in fact equivalent. In the previous case we could demonstrate this equivalence directly by applying a boost to our CHAPTER 4. INVARIANTS OF CLASS B WARPED PRODUCTS 61 original tetrad. In this case, we a-ould be required to apply a spin to the tetrad so that it generated QO2 as a real scalar. Unfortunately, it is not possible to do this, for

Applying a spin eia to the tetrad would give instead a value of

but since

and

it is clear that no spin will suffice to make

4.2 Syzygies of the Riemann Invariants for Class B Warped Products

Recdl that it is possible to construct Riemann invariants fkom the Ricci and Weyl scalars. It is this construction that we will now exploit to determine the syzygies among our set of invariants. We know that for class B warped products we have only four independent quantities from which we may construct invariants, so for our set of eighteen invariants we know there must be at Ieast fourteen syzygies for warped products of class B. For class El1 warped products, our set of invariants reduces to CHAPTER 4, INVARIANTS OF CLASS B WARPED PROD UCTS 63 while for class B2 warped products, the set reduces to

The two sets are virtually identical. The scalar product @00@22 in the class Bicase is replaced by @02@20 in the class B2 case, and the invariants r2 and m4 have different signs as weli. The fact that class B warped products generate only four independent scalars ({A, @00@22,

To begin, we first note that since ik2 is always real, wl , w2, ml, ml, mg, and m6 must also bc real, which removes six invariants fiom the set immediately. Also recall that since CHAPTER 4. MANANTS OF CLASS B WARPED PRODUCTS 64 class B warped products are of Petrov type D (or O) we have the syzygies

where we have omitted my notation denoting complex conjugation due to the real nature of the invariants in this case. In addition, as noted origindy by P. Musgrave, [Musgrave 19961 for spherical symmetry we have the syzygies

which also hold in the more general case of cIass B warped products, since specializing the general class Biwarped product to spherical symmetry does not reduce the number of independent nonzero curvature components to less than four. Finally, we observe

and

288(q3 - r3 2 rl 2 ) - 141r3r22r1+ 90r3rL4+ 24~'+ 68i22î13- 9r16 = 0.

4.3 A Complete Set of Invariants for Class B Warped Products

Having demonstrated the fourteen syzygies we knew were required to exist, we will now examine these syzygies to determine a complete set of invariants for al1 class B warped CHAPTER 4. INVARLANTS OF CLASS B WARPED PRODUCTS 65 products. The syzygies detailed above may be considered to remove certain invariants from the set via dependence. Wit h the intent of keeping the remaining set as simple as possible (ie. of lowest degree) we take the view that the syzygies (4.81) - (4.87) remove the independence of m2, m5, m4, ma, ml, ns,and r3, respectively. As an exception to our rule of simplicity, we choose to retain w;! instead of zq because of the way in which these inwiants are constructed. With wl = 6822 and w2 = -6423, one can see that because uil must always be positive it cannot be used to completely define w2; wi may only define wz up to sign. Knowing w2, on the other hand, will completely define wl, and so it is 2112 that we choose to keep. So from these considerations we now have the set {R, ri, 792, w2}as a candidate for a complete set of invariants. To be certain that this set is acceptable we need to test two further requirements. First we must be sure that there are no further syzygies among the candidate set in general. This is simple to prove in this case. Since R is dependent upon A done and no other invariant contains A, and the same is true for w2, which is dependent upon \E2 alone, we can be certain that there are no further syzygies involving these two invariants. Furthermore, upon examination of the construction of 7.1 = 2@00@22+ 4QlI2 and r2 =

6QO0a11@22 it is clear that neither of ri nor ra may be expressed as a polynornial in the other invariant in general. So these four remaining invariants are definitely independent. The second problem we must consider is the same problem as detailed in chapter 2 about the need, under certain circumstances, to retain a set of more invariants than the degrees of freedom inherent in the specialization of the spacetime. Again, we can be certain that R and w:! are acceptable choices, because R is dependent upon A alone and no other invariant contains A (even without considering the syzygies) and knowing

~2 completely defines \liz, the only underlying quantity used to construct it. So if R is reduced to zero by a further specialization, then we know that the degree of freedom described by A is lost in that specialïzation. The same is true for the case of w2 = O - this implies that q2= O and so ail Weyl and mixed invariant are reduced to zero as well. This leaves us considering whether we rnay use syzygy (4.87) to discard one of the CHAPTER 4. IWARTANTS OF CLASS B WARPED PRODUCTS 66

three Ricci invariants or whether we must keep ali of them to cover dl possible subcases of class B warped products.

In fact, we will never need rg to maintain a complete set of invariants for class B

warped products. For if r;>= O then we know that either Ooo@n= O or = O in which case we have lost a degree of heedom and so another invariant should be removed from

the set. And if rl = O then we must have OOo

degree of fieedom (or two, if any of the Ricci scalars are zero, in which case r2 = O as well) and should Iose an invariant fkom the set. So then we may be satisfied that under al1 possible dgebraic specializations of the general class B warped product the rernaining

set of invariants {R, ri, r2, w2) wiU give the correct number of invariants corresponding to the number of the degrees of &dom inherent in the specialization. Furthermore, we rnay be assured that any other invariant that could possibly be constructed may be constructed from the elements of this set, since there are only four degrees of fieedom and so no more independent invariants may be constructed. Thus, regardless of which definition of completeness one wishes to use, the set of real invariants {R, r 1, rz ,w2 } is a complete set of Riemann invariants for dl class B warped products.

4.4 Consequences of the Syzygies

Having discovered the syzygies of the Riemann invariants for class B warped products and having used these syzygies to obtain a complete minimal set of invariants for this class of spacetimes, the natural question is to ask what the existence of this set, or more specificdly, the reduction to a minimal set and the existence of the syzygies t hemselves, implies for spacetimes that fall under this class. The syzygies are certainly of value for cornputational purposes. If one wishes to examine the invariants for any class B warped product, it is sufficient to calculate only

R, rl, r2, and w2, for al1 the information to be gained fiom examination of algbraic Riemann invariants for these spacetimes is contained within these four invariants. If, for CHAPTER 4. LNVARIANTS OF CLASS B WARPED PRODUCTS 67 some reason, one wishes to have the actual form of any of the other invariants, then one has the fieedom to construct the invariants either boom their definitions or by using the syzygies, whichever is simpler. Understanding the physical benefit of the syzygies is somewhat more difficult. This difficulty is a consequence of the fundamental lack of understanding we have of what information the invariants can redly provide. As mentioned in chapter 2, the relation of the Ricci invariants to the energy-moment uni tensor may hold some benefit when syzygies between the invariants are known. However, this information may be difficult to corne by, as we shall demonstrate here. Before we do this, we shall examine the syzygies for the Weyl invariants, which provide somewhat less ambiguous results. The immediate consequences of the fact that both wi and wz are real are

and

So we know that Hab and Eab must be orthogonal, and we also have a relationship between the two. The syzygy (4.80) provides a further relationship:

The syzygies therefore restrict the possible forms of the electric and magnetic components of the Weyl tensor. To investigate the effect of the syzygies on the physicd quantities in the energy- momentum tensor it will be useful to construct Ricci invariants of higher degree than r3 and discover the syzygies between them and the lower degree Ricci invariants, which we know are guaranteed to exist in the case of class B warped products. We will construct CHAPTER 4. IRrVARlANTS OF CLASS B WARPED PRODUCTS 68

and denote them analogously to rl, r2, and rg. Then we have

and so on. The syzygies for class B warped products for these three invariants areL

and

To begin, we will look at the simplest energy-momentum tensor of possible interest- that of a perfect fluid. In this case,

'Finding the syzygies of the Ricci invariants is a purely aigorithmic procedure. Fust a possible degree for the syzygy is chosen, then al1 the possible combinations of the known independent invariants and the invariant for which one is trying to find the syzygy for that degree are combined. Expressing this combination as a polynomial in the Ricci scalars, the coefficients of any one of the Ricci scdars will form a Linear system of equations that may then be solved to find the numerical coefficients of the syzygy. If there is no solution, then the syzygy is of a different degree. CHAPTER 4. LNVARLANTS OF CLASS B WARPED PRODUCTS 69 where p is the energy density, p is the (isotropic) pressure, and ua is any normalized tirnelike vector (that is, uaua = -1). Then the Ricci invariants are (omitting the factors of 84

The procedure is then to substitute these relations between the invariants and the energy density and pressure of a perfect fluid into the syzygies (4.87), (4.97), (4.98), and (4.99), and look to find a relation between p and p. However, as might be expected from the form of the invariants for a perfect fluid, the syzygies return no vaiuable information in this case, for they al1 simply reduce to the identity O = O. Increasing the complexity of the energy-momentum tensor slightly does not improve this situation. For both the case of anisotropic pressure, where [Martens and Maharaj 19901

where pl and p;l are pressures measured ort hogonally to each ot her and na is a normalized spacelike vector (that is, nana = 1) such that uana = 0, and the case of a perfect fluid with a heat flux, where [Fevens 19931

where qa is a heat flux such that uaqa = O, each of the syzygies (4.87), (4.97), (4.98), and (4.99) simply return the identity O = O again and so no information on possible relationships between the physical quantities that describe the spacetime may be gained CHAPTER 4. WAARLANTS OF CLASS B WARPED PRODUCTS 70 from consideration of the syzygies in these cases. To obtain anything of use, we must consider a more complex physical description. The simplest energy-momentum tensor for which the syzygies provide any information is one with anisotropic pressure and a heat flux:

The Ricci invariants for this energy-momentum tensor are CHAPTER 4. ïNVARlANTS OF CLASS B WARPED PRODUCTS 71 reduce identically to O = O. However, syzygies (4.87) and (4.98) now generate polynomials in the quantities p, pi, pz, gaga, and n,qa. So, in principle, these polynomials define further relationships between these quantities. It is not known whether the information in these syzygies is independent- for example, it may be that in satisfiing (4.87), (4.98) is automatically satisfied, and so on (remember that we should also have syzygies for ri, rg , etc., for which the syzygies most likely do not reduce to O = O in this case). Regardless of whether this is true or not, (4.87) definitely defines at least one relationship between the physical quantities described by the energy-momentum tensor for class B warped produc ts. The problem is that these polynomials are of such high degree that knowing syzygy (4.87), for example, is not of great help if the actual relationship between one quantity and the rest is desired to be known. For example, (4.87) defines a polynornial that is of degree 12 in p, pl, and p2, degree 6 in qaqa, and degree 3 in naqa. Since, in general, we are not able to solve analytically for roots of polynomials of degree greater than Four, we will only be able to get an analytical expression for n,qa in terms of the other four quantities, and even then, the choice of root will not necessarily be clear (although it is possible the the weak or strong energy conditions may assist in this regard).

Nonetheless, in spite of t his shortcoming, the syzygies may have practical physical value. For whatever the form of the energy-momentum tensor may bel we know that for the cases where the syzygies do not reduce to zero identically there is at least one relationship between the physical quantities that describe the spacetime. Even when this is not the case, we are still assured that the invariants for which we have syzygies cm describe no physics not already described by the set {R, ri,r2, wz), and knowing merely this may be of value. Chapter 5

Conclusions

In this thesis, we have investigated the aigebraic invariants of the Riemann tensor for class B warped products. In doing so, we have discovered that we may construct a complete set of invariants for spacetimes of this class that contains only four elements- the Ricci scalar R, the first two Ricci invariants ri and rz, and the second Weyl invariant w*. This set is complete by any definition, since any Riemann invariant may be constructed from the dements of this set and any further algebraic specialization will produce the correct number of invariants corresponding to the number of degrees of fkeedorn inherent in the specialization. The elements of the set are also strongly independent. This provides a minimal set of invariants for some of the most physically important spacetimes, since spherically, plane, and hyperbolically symmetric metrics are contained as a subclass of class B warped products. It is interesting to note that the four invariants that comprise the complete set are actually constructed £rom five nonzero curvature components, yet since two of the curvature components must always appear as a product, the apparent fift h curvature cornponent is not truly independent of the others and merely contains information as to how the tetrad chosen to represent the spacetime is boosted or spun relative to other equivalent tetrads. While the computation of the invariants for class B warped products has certainly been made easier by the discovery outlined in this thesis, it is hoped that this is not the CHAPTER 5. CONCLUSIONS 73

only benefit that knowledge of this minimal set will provide. The value of invariants is not well understood. There is a fair amount of work that has been done on spherically symmetric spacetimes, principally because of t heir physicai relevance and relative simplicity. Perhaps by combining this set of invariants and the syzygies between this set and al1 other invariants that may be constmcted with what we already know about spherically symmetric spacetimes, a greater understanding of the nature of invariants might be gained. But perhaps the most surprising aspect of the existence of this minimal cornplete set of invariants for class B warped products is the possibility that, through syzygies, the invariants that are related to the physical quantities that specify the spacetime form relations t hat impose restrictions on the physical quantities t hemselves. This is equivalent to saying that imposing a particular form for the metric of a spacetime means that certain physical aspects of the spacetime may not be chosen freely. 1s t his the case? This is hard to determine at this point, sicce the equations that would appear to impose these restrictions are extraordinarily difficult to deal with due to their complexity. In the author's opinion, this surprising possibility would seem to be the most physically important consequence of the existence and nature of the syzygies of the Riemann invariants. It is hoped that the work in this thesis will lead to further research in this field of study, and, through this, that this work will lead to a better understanding of this problem. Bibliography

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