COMPARISON OF NEIGHBORING SEMIRIEMANNIAN GEOMETRIES by HENRY HO-KONG LING B.Sc. The University of Calgary, 1999 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming to the required standard THE UNIVERSITY OF.jfelTISH COLUMBIA July 2001 © Henry Ho-Kong Ling, 2001 In presenting this thesis in partial fulfillment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for exten-sive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of Mathematics The University of British Columbia Vancouver, Canada Abstract Given a convergence sequence of metric tensors gn g on a manifold M , in what sense does the corresponding geometries (M, gn) converge to (M, g)l This question is exam-ined by comparing the nature of the straight lines in each (M, gn) with the nature of the straight lines in (M, g). The cases of semi-Riemannian geometry Riemannian geometry, and Lorentz geometry are considered in succession. The relevance of this question to prob-lems in general relativity is emphasized, and some potential applications of our results to questions concerning, for example, topological censureship, and singularity theorems are indicated. ii Table of Contents Abstract ii Table of Contents iii Acknowledgement iv Chapter 1. Introduction 1 1.1 Introduction and Statement of Principal Results 1 1.2 Outline 3 Chapter 2. Selected Facts of Semi-Riemannian Geometry 5 2.1 Scalar products and semi-Riemannian manifolds 5 2.2 The Levi-Civita connection and geodesies 8 2.3 The particular case of Lorentz manifolds 11 Chapter 3. Convergent Sequences of Metric Tensors 15 3.1 The choice of topology on the set of semi-Riemannian metrics 15 3.2 Sequences of semi-Riemannian metrics 20 3.2.1 Existence, uniqueness, and continuity theorems in differential equations 20 3.2.2 Comparing neighboring semi-Riemannian geometries 23 3.3 Sequences of Riemannian metrics 26 3.3.1 Curves in Riemannian manifolds 26 3.3.2 Comparing neighboring Riemannian geometries 29 3.4 Sequences of Lorentz metrics 33 3.4.1 Causal paths in a Lorentz manifold 33 3.4.2 Length and distance in a Lorentz manifold 36 3.4.3 Comparing neighboring Lorentz geometries 40 Chapter 4. Conclusions and Future Research 44 4.1 Conclusions 44 4.2 Future research 45 4.2.1 Further elaborations 45 4.2.2 Some examples and applications • 46 Bibliography 47 Appendix A. Notations and Conventions for Calculus on Manifolds 49 iii Acknowledgement I am grateful for the financial support of NSERC and the University of British Columbia. It is my pleasure to thank my supervisors, Richard Froese—for several helpful discussions and his much appreciated support, and Donald Witt—for all his guidance and for suggest-ing the topic of this thesis. I also wish to thank my parents, Richard and Judy Ling. Thank you Arthur, Joseph, Matthew, and Deborah. iv Chapter 1 Introduction 1.1 Introduction and Statement of Principal Results Einstein's theory of spacetime and gravitation, commonly known as general relativity, has served as one of the prime motivations for the study of Lorentz geometry. In this the-ory, spacetime is mathematically represented by a Lorentz manifold (M,g), that is, M is a smooth (usually 4 dimensional) manifold and g is a Lorentz metric. The law of gravity in general relativity is Einstein's equation, which expresses the influence of matter on the structure of spacetime. Ricg is the Ricci tensor of g, sg is the scalar curvature (also known as the Ricci scalar), and T is the stress-energy tensor that represents the energy, momentum, and stresses of matter. Supplemented with a particular model of matter (i.e. given a specific form for T), Einstein's equation can be regarded as a second order differential equation in the unknown g. With the complexity of Einstein's equation, simple explicit solutions are an essential guide towards understanding the general features of solutions. One way to obtain new solutions of Einstein's equation is to join together pieces of simple known solutions. For example, let us consider the Schwarschild metric in 'comoving coordinates' 9s = -dt2 + AM. . i —(r-tr1 2/3 dr2 + 2/3 d n 2 , where dQ2 is the metric for the unit 2-sphere in R 3 . This metric is defined on the manifold M i that is the product manifold of the unit 2-sphere and U = {{t,r) £ l 2 : r > 0,r-t > 0}. This is a solution of Einstein's equation in vacuum (T = 0); it represents the spacetime of 1 Chapter 1. Introduction 2 a black hole of mass M . Now consider the spatially flat Friedmann solution g F = - d \ 2 + 9 A 2 l 2 / 3 (dR2 + R 2 d n 2 ) . 2 This metric is defined on the manifold M 2 that is the product manifold of the unit 2-sphere and V = {(A, R ) e M2 : A > 0, R > 0}. This solution represents a homogeneous and isotropic dust-filled universe (T = pdA2, where p = l/(67rA2) is the density of the dust at time A). We observe that the surface r = 1 in M i (with metric induced by gs) is isometric to the surface R = 1 in M 2 (with metric induced by gp). Now let M ~ be the region of M 2 for which 0 < R < 1 and let M + be the region of M i for which r > 1. In-this case, there exists a prescription, due to Israel [12], for joining together (M~ ,gp) and ( M + , gs) at their isometric boundaries so as to obtain a new Lorentz manifold (M, g) with M smooth and g of class C 1 . The resulting Lorentz manifold (M, g) represents the spacetime of a collapsing spherical ball of dust of uniform density. On account of Einstein's equation, we see that sg is discontinuous at the mutual boundary of M ~ c M and M + c M so that g must not be C2. This 'cut and paste' technique has been applied to obtain solutions representing collapsing shells of matter [12] or impulsive gravitational waves [20]; these solutions are generally continuous but not Cl. Such solutions are valuable for understanding the math-ematical nature of Einstein's equation and for understanding the physical implications of general relativity. On the other hand, another approach towards understanding the Einstein equation is to directly prove theorems concerning the solutions under general assumptions concern-ing the global structure (causality conditions) and the nature of the matter source (energy conditions). The most famous results of this approach include the singularity theorems, the positive mass theorem, and the laws of black hole mechanics. For convenience, the metric is often taken to be smooth in these discussions. With this assumption of smoothness, we have the situation where these powerful theorems sometimes do not refer to the types of useful examples mentioned above. One way to remedy this situation is simply to examine the proofs of theorems to see if the smoothness conditions might be weakened. Another, perhaps cheaper, method to address this issue is to argue as follows. Given any nonsmooth explicit solution of the Chapter 1. Introduction 3 Einstein equation, we argue that there exists a smooth approximation that retains certain essential geometric properties of the original. This smooth approximation then serves as a bridge between the nonsmooth explicit solution (which is often given in terms of relatively managable functions) and the powerful theorems. The key issue is whether or not the smooth approximation really has similar geometric properties as the original. This ques-tion of comparing 'neighboring' geometries is the main problem that we try to address in this thesis. Our problem of comparing geometries may be restated in the following fashion. We are trying to formulate precise versions of the statement 'Given that a sequence of metric tensors {gn} on M converges to a metric g on M, the geometry of (M,gn) converges to the geometry of (M, g)'. Clearly, such a description is not very meaningful unless one specifies what is meant by the terms 'converges' and 'geometry' in the above statement. Of the many different aspects of geometry, we choose to focus on those properties that are related to the notion of a 'straight line'. For C1 metrics, this means that we are concerned with the behaviour of geodesies. Since geodesies are solutions to a differential equation that involves the first derivatives of the metric, we expect that we need to choose a notion of convergence that controls the behaviour of these first derivatives. We antici-pate that this result should hold for metrics of any signature, so we formulate our result not only for Lorentz manifolds, but for general semi-Riemannian manifolds. For metrics that are continuous, it may not be meaningful to talk about geodesies, so we fall back on a different notion of a straight line. For Lorentz manifolds, we will con-sider that paths that are maximizers of Lorentz length, in analogy with the minimizers of Riemannian length in Riemannian geometry. In our discussion, we will first establish a comparison result for minimizers in Riemannian geometry to serve as a guide for the more difficult case of maximizers in Lorentz geometry. 1.2 Outline Chapter 2 gathers some preliminary definitions and basic notions in semi-Riemannian ge-ometry. In sections 2.1 and 2.2, only the bare essentials of semi-Riemannian geometry are Chapter 1. Introduction 4 discussed; the remaining section 2.3 deals with Lorentz manifolds in general and the the-ory of causality in Lorentz geometry in particular. The main ideas of this thesis are presented in chapter 3. In section 3.1, we specify the topology on the set of metric tensors on a manifold that is most suited to our needs. Next, our three main sets of results, corresponding to the cases of semi-Riemannian geometry, Riemannian geometry, and Lorentz geometry, are presented in section 3.2, 3.3, and 3.4 re-spectively. These three sections are structured in the same manner: we first collect some known facts and establish some needed results, then we present our result on the compar-ison of geometries associated with a sequence of metric tensors. Finally we summarize the work presented in this thesis in chapter 4 and discuss the possible future directions of this work. We outline some problems in which our results can be readily applied. Our conventions and notations for the calculus on manifolds are set out in appendix A . Rather than presenting this information in the style of a glossary, we briefly outline a standard approach to calculus on manifolds in our conventions so as to illustrate our usage of the notation. Chapter 2 Selected Facts of Semi-Riemannian Geometry 2.1 Scalar products and semi-Riemannian manifolds The familiar Riemannian manifold can be simply described as a manifold with an inner product assigned to each tangent space in a continuous manner. From this perspective, there is a natural generalisation of the Riemannian manifold to the semi-Riemannian man-ifold, in which the role of the inner product is replaced by the scalar product. Definition 2.1 A scalar product on an m-dimensional real vector space V, is a symmetric bi-linear mapping b : V x V -» M that is nondegenerate (i.e. b(v,w) = 0 for all w e.V implies v = 0). A scalar product on a real vector space is a generalization of the inner product in that the positivity condition is not assumed (i.e. it is not assumed that b(v, v) > 0 for all v e V). This also suggests a classification of scalar products according to 'how many' vectors have a negative scalar product with itself [19]. We say that b is negative definite on a subspace W C V if b(v,v) < 0 for all v E W. Let be the set of all subspaces W on which g is negative definite. Definition 2.2 The non-negative integer 1 = maxW£N {dim(W)} is called the index of the scalar product b. A n alternative description of the index of b can be obtained from the following construc-tion. Take any basis of vectors {ei, ei... em} of V and form the m x m matrix B = [bij], where = b(ei,ej). This matrix is symmetric and can thus be diagonalized; in other words we may assume that we have chosen the basis {ei... em} such that B is diagonal. 5 Chapter 2. Selected Facts of Semi-Riemannian Geometry 6 Now since b is nondegenerate, none of the diagonal elements of B can be zero. So we have essentially proved the following lemma. Lemma 2.3 Given the scalar product b on V, there exists an orthonormal basis {ex ...em}ofV such that the matrix B = [6(ej, e^ )] is diagonal and each diagonal element has either the value 1 or - 1 . It can be shown that the number of ones and negative ones appearing in the diagonal matrix B is independent of the choice of orthonormal basis. In fact, the number of negative ones coincides with the index of b. One of the most common examples of a scalar product that is not an inner product is the Minkowski scalar product n on the vector space W1 defined by "m—1 T](v, w) = ^2 . 1 = 1 for any arbitrary vectors v = (vl,v2 ...vm) and w = (wl,w2 ... wm) in M m . This scalar product has an index of 1. The pair (Mm,r?) is usually called the m-dimensional Minkowski space. From the above lemma, we see that any m-dimensional vector space with a scalar product of index 1 can be identified with Minkowski space. For a general scalar product b on V, we may define := \b(v, v)\1/2 to be the extent of the vector v with respect to the scalar product b; v is a unit vector iff \\v\\b = 1, and each vector in an orthonormal basis is a unit vector. Unless b is positive or negative definite, it is possible for non-zero vectors to have zero extent, so the map || • ||f, is not generally a norm on the vector space. Furthermore, the scalar product b and its associated map || • ||(, may not satisfy the Schwarz inequality \b(v,w)\ < ||v||(,||i/j||f,. As an example, consider the 2-dimensional Minkowski space (E 2 , rj). The vector v = (z, 1), \z\ < 1, has the extent \\v\\,j = (1 — z2)1!2. So we may form the unit vector v = (z/(l — z 2) 1/ 2,1/(1 — z2)1/2). Clearly, w = (0,1) is also a unit vector, and the scalar product of v and w is V(v,w) = l/(l-z2)^2. As a result \rj(v, w) \ can be made arbitrarily large by taking z sufficiently close to 1, and the Schwarz inequality is not satisfied. Chapter 2. Selected Facts of Semi-Riemannian Geometry 7 In a vector space V with scalar product b, a vector v may be classified as one of three types, according to whether b(v,v) = 0, b(v,v) > 0, or b(v,v) < 0. In Minkowski space (Mm, ri), a vector v is timelike iir](v,v) < 0, spacelike if r?(v, v) > 0, and null if r/(v, v) = 0. The set of all null vectors v consists of all vectors lying on the cone (v1)2 + (r;2)2 + • • • + [vm-1)2 - (vm)2 = 0; we call this a null cone H. Pictorially the timelike vectors lie 'inside' the null cone. The set H \ {0} is the disjoint union of two half cones Hi — {v e H : vm > 0} and JV2 = {« £ H : vm < 0}. Similarly, the set of all timelike vectors T is the union of the disjoint sets Ti = {v e T: vm > 0} and T2 = {v € 7": vm < 0}. The sets H\, Hi, T\, T2 can be described in component-independent language. For any timelike vector t we define the sets jV 4 + = {v € H : rj(v, t) > 0} and Hf = {v € H : r]{v,t) < 0}. Similarly, define Tt+ = {v € T : rj(v,t) > 0} and Tf = {v e T : r)(v,t) < 0}. It is not difficult to verify that Hi = Hf, H2 = Hf, T\ = Tt+, and Ti — Tf whenever tm > 0. On the other hand, Hi = Hf, H2 = Hf, 71 = Tf, and T2 = Tf whenever tm < 0. As a result, any timelike vector allows us to define a 'time direction' as follows. Given any timelike vector t, we define the future null cone to be Hf and the past null cone to be Hf. Similarly, we define the set oi future timelike vectors to be Tf and the set of past timelike vectors to be Tf. Essentially, this says that we define the future null cone to be the half cone into which t points, and a future limelike vector to be a vector that points into the same half cone as t. Having discussed the basic facts of scalar products, we proceed to the definition of semi- Riemannian manifold. Now on a manifold M , any symmetric nondegenerate tensor field g of type (0,2) defines on each tangent space TPM a scalar product gp. A metric tensor is usually defined as such a tensor field which satisfies an additional condition. Definit ion 2.4 A metric tensor g on a manifold M is a symmetric nondegenerate tensor field of type (0,2) such that gp has the same index for all p e M. The index of g is the index of gpfor any pG M. Chapter 2. Selected Facts of Semi-Riemannian Geometry 8 With the concept of a metric tensor, the definition of a semi-Riemannian manifold can be concisely stated as follows. Definition 2.5 A Ck semi-Riemannian manifold is a manifold M together with a metric tensor g of class Ck. Note that in our conventions, the differential structure of the manifold M is always taken to be smooth. 2.2 The Levi-Civita connection and geodesies One fact that survives the generalization from Riemannian manifolds to semi-Riemannian manifolds is the existence of the Levi-Civita connection. The criterion used to select this special affine connection from the set of all affine connections on the manifold is the same as that used in Riemannian geometry. Roughly speaking, the Levi-Civita connection on a semi-Riemannian manifold is the unique affine connection that is compatible with the metric tensor in the sense that parallel transport of vectors preserves the scalar product. The precise result is displayed in the following well-known theorem [19]. Theorem 2.6 On a Ck (k > 1) semi-Riemannian manifold (M,g), there is a unique torsion-free affine connection V on M that satisfies the postulate of metric compatibility: for any C1 vector fields X, Y,Z, Z(g(X, Y)) = g{VzX, Y) + g(X, VZY). This connection is called the Levi-Civita connection of(M, g). If there is more than one Ck (k > 1) metric defined on M , then each metric will have a corresponding compatible torsion-free connection; for any particular metric g on M we will use the phrases 'the 5-compatible connection V and 'the Levi-Civita connection V of (M,g)' interchangeably. The differentiability class of the ^-compatible connection on M is determined by the differentiability class of g. This is most easily seen by considering the situation in any chart. Chapter 2. Selected Facts of Semi-Riemannian Geometry 9 In a chart (x, U) the components of the ^-compatible connection are given by the relation m v{d/dxJ)(d/dxi) = J2FUd/dxk)-k=l The components Yk- of the g-compatible connection are also called the Christoffel symbols of g. There is a well-known expression for the Christoffel symbols in terms of the components of the metric tensor. More specifically, if g = Y X j = i 9 i j ( x ) d x i ® d x j t n e n t n e Christoffel symbols of g are 1 m Tij = 2^gkl {dj9il + di9lj ~ dl9ij) • {21) where the matrix G^1 = [g1^] is the matrix inverse of G = [gij]. This follows directly from applying the metric compatibility postulate to the coordinate vector fields d/dxl, d/dxf d/dxk and solving the resulting algebraic equation for the functions r f •. Incidently, since any affine connection is fully determined by giving its components in every local chart, the preceding discussion gives us the main parts of a 'bare-handed' constructive proof of theorem 2.6. Now if the Christoffel symbols of g are of class Ck in every local chart, then we say that (/-compatible connection V is of class Ck. So, the relationship between the order of differentiability of the metric and the associated connection is this: a Ck (k > 1) metric tensor g o n M gives rise to a unique g-compatible Ck~l connection on M. A geodesic of (M, g) is simply a curve a m M which satisfies the geodesic equation V a ' c / = 0, where V is the ^-compatible connection. Clearly, for this equation to make sense everywhere on M , we require that (M,g) be at least C1. Geodesies may be alternatively described in terms of a particular vector field on TM called the geodesic spray of (M,g). The geodesic spray arises naturally from the following local considerations. Let (x, U) be a chart of M and (z = (x o n, x),TU) the associated chart of TM. By directly writing out the geodesic equation in the chart (x, U) we see that it is equivalent to the first order ordinary differential equation c'(s) = F(c(s)), where the map F : z(TU) ->• M 2 m is given by F(u,o = ((,-r(u)(t,0), Chapter 2. Selected Facts of Semi-Riemannian Geometry 10 for all («, £) G z(TU) ~ x(U) x Mm, and ( m m m \ Er*» '^> E T ^ ) ^ ••• E • i , j ' = l i , i = l i , j = l / Here, the j?*- are the Christoffel symbols of g in the chart {x, U). To each chart (z, TU) of T M , there is a corresponding map F . By virtue of the transformation law of the Christof-fel symbols, we can patch together these local maps F into a global vector field on the manifold T M ; this vector field is the geodesic spray Y. With Y defined in this man-ner, we see that F is the representation of Y in the chart (z,TU). Thus we may write Y = £ 2 ™ F i ( z ) d / d z i or equivalently F = z o Y o z~x. So we see that if cr is a curve in TM satisfying c r ' ( s ) = Y(a{s)) for all s (i.e if a is an integral curve of Y), then a = 7r O a is a geodesic of (M, 5). Conversely, if a is a geodesic of (M,g), then a = a' is a curve in T M satisfying c r ' ( s ) = Y(a(s)). The standard theorems of existence and uniqueness of geodesies can therefore be gathered immediately. A l l of this is simply a convenient way of speaking about geodesies as solutions to an ordinary differential equation using chart-independent language. In fact, let us make use of the following fundamental theorem on vector fields and flows. Theorem 2.7 Let X be a C k (k > 1) vector field on a manifold M. Then there is C k map 4> : D(X) -> M, where D(X) is an open subset ofR x M, such that for each p e M, f/>(-,p) : ' i n-(f)(t,p) is an integral curve ofX. The map 4> is called the flow ofX. A proof of this theorem is given in Lang [15]. Now for a C2 semi-Riemannian manifold, the geodesic spray Y is a vector field on T M of class C 1 . So the flow 4>Y of Y is a C 1 map. Define the 0 C T M to the the set of all v in T M for which </>y (1, v) is defined. It can be shown that 0 is open in T M (for example, see Lang [15]). Then we may define the exponential map expft;] : 0 -» M by exp[{/](v) = 7r O 4>Y(l,v) The restriction of exp[<?] to the tangent space T p M is usually written as expp[<7]. Since the projection map is smooth, we see that exp[g] and expp[g] are C 1 maps. Furthermore, it may be shown that expp[<7] is nonsingular atp, so by the inverse function theorem, expp[c;] maps some neighborhood U of the zero vector in Chapter 2. Selected Facts of Semi-Riemannian Geometry 11 TPM diffeomorphically into M. We call expp\g](U) a normal neighborhood of p. In fact, at any p 6 M there exists a convex normal neighborhood of p, that is, a normal neighborhood Np of p such that any two points in Np can be connected by a geodesic whose image is contained in Np (for example, see Hicks [10]). The existence of convex normal neighborhoods will prove to be a useful tool in certain proofs. 2.3 The particular case of Lorentz manifolds By the above definitions, we see that a Riemannian manifold is a semi-Riemannian mani-fold whose metric tensor has an index of 0. Since the basic properties of Riemannian man-ifolds are so familiar, we do not discuss them here in any detail. Instead we skip to a discussion of Lorentz manifolds. Definition 2.8 A Lorentz manifold is a semi-Riemannian manifold ( M , g) with g of index 1 and M of dimension greater than or equal to 2. Given a Lorentz manifold ( M , g) and any point p £ M, we can find an orthonormal basis of TPM, {ei . . . em} such ' o i f ; / j 9p(ei,ej) = <J -1 if j = j = m 1 otherwise The existence of an orthonormal basis allows us to identify the pair (TPM, gp) with the Tri-dimensional Minkowski space. More precisely, we map TPM onto R m via the isomorphism Y^ILi v%ei ^ (w1, u 2 . . . vm). Then we have gp(v, w) = r?(r/w, <f>w). In the same way that a Riemannian manifold locally resembles Euclidean space, we see that an m-dimensional Lorentz manifold locally resembles the m-dimensional Minkowski space. In fact Minkow-ski space itself can be regarded as a Lorentz manifold by the follow construction. Take M — W"1, then we identify TM with R m x W1 and define the metric tensor g by g^v) (v,v) = viZiQ whenever v = (u, £) G TM. Definition 2.9 A Lorentz manifold (M, g) is time-oriented if there is a continuous non-vanish-ing timelike vector field T on M (i.e. gp(Tp, Tp) < Ofor every p e M). Chapter 2. Selected Facts of Semi-Riemannian Geometry 12 At eachp, Tp defines a time direction and the vector field T allows us to continuously define a time direction at every point in M. From this point on, we assume all Lorentz manifolds are time-oriented. A future timelike curve is a piecewise Ck (k > 1) curve a such that a'(s) is a future timelike vector for all s where a'(s) is defined. A future null curve is a piecewise Ck (k > 1) curve P such that (3'(s) is a future null vector for all s where /3'(s) is defined. A future causal curve is a piecewise Ck (k > 1) curve 7 such that ^f'(s) is either a future null vector or a future timelike vector for all s where j'(s) is defined. We define past timelike, null, and causal curves similarly. A n aspect of Lorentz geometry that has no counterpart in Riemannian geometry is the study of causality or causal structure of Lorentz manifolds. In short, the causal structure of a Lorentz manifold is essentially based on the question of which points can be connected by a future timelike or causal curve. Causality will play an important role in the next chapter of our discussion, so we now consider some of the definitions and results in more detail. Given a Lorentz manifold (M, g), we define two relations <C5 and < f l. We write p <9 q iff there is a future timelike curve a : [a, b] ->• M from p to q (that is, a(a) —p and a(b) = q). We write p <g q iff either p — q or there is a future causal curve 7 from p to q. Note that the relations <C9 and < g depend on the metric tensor j o n M a s well as the time direction T. We also define the sets Ig(p) = {q G M : p <<, q} and I~(p) = {q E M : q <£g p}, which are respectively called the chronological future ofp and the chronological past of p. We may define Jg(p) (the causal future ofp) and Jg(p) (the causal past ofp) similarly, using the relation < 9 in place of < s . We write Jg(p,q) := Jg(p)nJ~(q) andlg(p,q) := 1+(p) f) I~ (q). Clearly, the relations <^g and < g are transitive. That is, p C 9 q and q < 9 r p < 9 r, p<g q and q<gr p <gr. One may show that in a continuous Lorentz manifold (M, g) the sets l£(p) and Ig(p) are open for any p G M (for a proof of this, see Kriele [14]). On the other hand, the sets Jg(p) and J~ (p) are generally neither open nor closed. It turns out that if we place some causality Chapter 2. Selected Facts of Semi-Riemannian Geometry 13 conditions on the Lorentz manifold, we obtain some very mteresting statements concerning the relations < 9 and < g . In general, there are many different causality conditions that have appeared in the literature. For our purposes, we consider only two of these conditions. To formulate our first condition, we need to define the following concept. Definition 2.10 A Lorentz manifold (M, g) is strongly causal at p e M if given any neighbor-hood U of p, there is a neighborhood V (Z U of p such that J(q, r) c V whenever q and r are in V. Note that in the above definition, q and r need not be distinct points. This definition allows us to define what is meant by a strongly causal Lorentz manifold. Definition 2.11 A continuous Lorentz manifold {M,g) is strongly causal if it is strongly causal at every p € M. Lemma 2.12 On a strongly causal Lorentz manifold (M, g), p <g q and q <g p iffp = q. Proof. Assume (M, g) is strongly causal and p <g q and q <g p but p ^ q. Then there is a future causal curve fromp to q and a future causal curve from q top. Joining these together, we have a future causal curve a from p to p. So for any neighborhood U of p that does not contain the image of a, no neighborhood V C U of p can ever contain J(p,p). • From the above lemma, we see that in a strongly causal Lorentz manifold, the relation < g is a reflexive partial ordering. It also follows that the relation < 9 is an irreflexive partial ordering in a strongly causal Lorentz manifold. In a general Lorentz manifold, an open neighborhood of p of the form I(q, r) is called ah Alexandroff neighborhood of p; the Alexandroff topology is the topology generated by Alexandroff neighborhoods. In a strongly causal Lorentz manifold, every neighborhood of a point p G M contains an Alexandroff neighborhood of p. So the manifold topology and the Alexandroff topology are equivalent in a strongly causal Lorentz manifold; in this sense the causal structure of a strongly causal Lorentz manifold determines its topology. The second condition we will consider is that oi global hyperbolicity. This condition plays a particularly important role in Lorentz geometry ([6], [9]). Chapter 2. Selected Facts of Semi-Riemannian Geometry 14 Definition 2.13 A continuous Lorentz manifold (M, g) is globally hyperbolic if it is strongly causal and J(p, q) is compact for all p,qeM. In a globally hyperbolic Lorentz manifold, the relation < g is particularly well-behaved. Lemma 2.14 Let (M, g) be a continuous, globally hyperbolic Lorentz manifold. Then the sets J+ (p) and J~ (p) are closed. Proof. Choose an arbitrary p e M and let q G Jgip). So there is a sequence {qn} in J + ( p ) converging to q. Now the set Ig{q) is open and non-empty, so choose r G L+(q); then we also have q G Ig(r). Since Ig(r) is an open set, we can assume w.l.o.g. that the sequence {qn} is in Ig(r) C Jg(r) also. So we have a sequence {qn} in J£{p) fl Jg{r) converging to q. Since ( M , g) is globally hyperbolic Jgip) fl Jg~(r) is compact, so we must have q G Jgip) D Jg~(r) also. Thus q G J^ip). Since q G Jgip) was chosen arbitrary we see that Jg'ip) = Jfip), and therefore J+ (p) is closed. Likewise, J~(p) is closed. • Corollary 2.15 T/ze relation <g, as a subset ofMxM, is closed. Proof. Let { ( p n , qn)} be a sequence in M x M such that for each n , p „ <g qn. We wish to prove this implies p <g q. Suppose, to the contrary, that p ^g q. So there is a neighborhood Np ofp that is disjoint from Jg(q). Choose any r G J~(p)niV p ; thenp G Ig{r) C J + ( r ) . N o w since J+ (r) is an open set containing p , there must be an integer N such that every pn with n > JV is contained in /^"(r). Thus J£(pn) C •//(O for every n > N. Since 7VP is disjoint from Jg{q), we see that <? must be disjoint from Jg{r). However, each qn G Jg{pn) C J£{r), and since Jg~(r) is closed (by the above proof), we must have g G J^~(r), a contradiction. • So in a continuous, globally hyperbolic manifold, the relation < 9 is a topologically closed reflexive partial ordering. Chapter 3 Convergent Sequences of Metric Tensors In this chapter, we will present three sets of results which correspond respectively to the cases of serni-Riemannian geometry, Riemannian geometry, and Lorentz geometry. As we have already mentioned, our goal is to compare the geometry given by a metric tensor with the geometry given by 'nearby' (in some topology) metric tensors. Of the many dif-ferent aspects of geometry, we choose to compare the 'geometry of straight lines'. For semi-Riemannian manifolds, the 'geometry of straight lines' refers to the nature of its geodesies. For Riemannian manifolds in particular, we have in addition the notion of a minimizing curve, and for Lorentz manifolds, we have the notion of maximizing paths. We generally restrict our attention to curves that are mappings of a compact intervals, and paths that are compact sets in M ; we do not discuss issues such as geodesic completeness. Our results are formulated so that we consider a convergent sequence of metric tensors, and compare the corresponding 'straight lines' of each geometry. To begin our discussion, we first specify the topology of the set of semi-Riemannian metrics. 3.1 The choice of topology on the set of semi-Riemannian metrics Let Sj(M) be the set of all continuous semi-Riemannian metrics of index 1 on the T r i -dimensional manifold M . We define the compact-open topology on Sx{M) using the meth-od found in Hirsch [11]. Suppose that g e 5x(M). Choose any chart (x, 17) of M , any compact set K C U, and any 0 < e < oo. Now we define a 'weak subbasic neighborhood' (Hirsch [11]) of g, N(g; (x, U),K, e), to be the set of all h G Sx(M) such that m sup V \gij(x(p)) - hij{x(jp))\ < e, 15 Chapter 3. Convergent Sequences of Metric Tensors 16 where g and h are locally given by g = £™=1 gij(x)dxi ® and / i = Ey=i hij{x)dxi <g> ete-?. The compact-open topology on 5j(M) is the topology generated by the collection of all sets of the form N(g; (x, U),K, e), for arbitrary g, (x, U), K cU, and e. A neighborhood of g is any set containing the intersection of a finite number of weak subbasic neighbor-hoods of g. The convergence of sequences in Si(M) with respect to the compact-open topology can be conveniently characterized as follows. A sequence {gn} in Sx(M) converges to g in the compact-open topology iff for any chart (x, U) and any compact set K c U, {{gn)ij} converges to gij uniformly on x(K) for all i, j. This is clear from the definition of the compact-open topology. This also demonstrates that convergence in the compact- open topology is much like uniform convergence on compact sets. We may describe convergence in the compact-open topology in the following chart-independent manner. Since M is paracompact, we may assume without loss of generality that there exists a continuous Riemannian metric h on M. The following result is very useful to our subsequent discussion. Lemma 3.1 Let h be any continuous Riemannian metric on M. Suppose the sequence {gn} in Sj(M) converges to g in the compact-open topology. For any compact set K c M, there is a sequence of positive real numbers {Sn} converging to 0 such that \9x(v) (V, v) - (9n)n(v) (">«)•!< 8nK(v) (v,v) for every v e TK. Proof. Let h be any continuous Riemannian metric! Suppose the sequence {gn} converges to g in the compact-open topology, and let K be compact. So there is a finite covering of K by compact sets {Ba} such that each Ba C Ua for some chart (xa, Ua)- Now let B be a typical member of the collection {Ba}, and suppose B C U for some chart (x, U). We define / i n := SUp \9p{ip,ip) - (9n)p{t,p,Zp)\, P6B,||{||=1 where ||f || is the Euclidean norm of f e M m and | p = C ( d / d x % f o r a l l P e B - W e a l s o Chapter 3. Convergent Sequences of Metric Tensors 17 define A := inf M£P>£P)-Since B x § m _ 1 is compact and g - gn is continuous, ^ > 0 is finite. And since h is a continuous Riemannian metric, A > 0 is finite. If we define e = YALI d%i ® dx\ then we see that \9ir(v)(v,v) - (gn)w(v)(v,v)\ < nnen(v){v,v), and Xen{v){v,v) < h v^){v,v), for all v 6 TB. It follows that • \9ir(v)(v,v) - (gn) v^)(v,v)\ < (nn/X)hnM(v,v), for all v € TB. Now \9p(tp,ip) ~ ( S n ) p ( f p , ip)\ m < E \9ij(xiP)) ~ (9n)ij\ • \C£3\ < E I5«(«(P)) - (9n)ij\-i,j=l i,j=l Thus fj,n < s w p p e B YTj=1 (a;(p)) - (gn)ij I• Since {c/n} converges to 3 in the compact-open topology, we see that /xn approaches 0 as n 00. And since the compact set i f is covered by a finite collection of sets {Ba}, this construction can be repeated for each B a and it is clear that there exists a sequence of real numbers {6n} converging to 0 such that \9*(v)(v,v) - (gn)ir(v)(v,v)\ < 5nh v^){v,v), for all v e TK. • For C 1 metrics, we can define the C 1 compact-open topology in a similar manner. More specifically, let SX(M) be the set of all C 1 semi-Riemannian metrics of index X on the man-ifold M. A C l weak subbasic neighborhood of g G SX(M) is a set N l(g; (x, U), K, e) con-sisting of all h e ST(M) satisfying sup peK E \9ij(x{p)) - hij(x(p))\ + E \ dk9ij{x{p)) ~ dkhij{x(p))\ i,j=l i,j,k=l < e, Chapter 3. Convergent Sequences of Metric Tensors 18 where g = Y^j=i9ij(x)dxi <8> dxi and h - YT,j=\ hij(x)dxi ® dxK The C 1 compact-open topology is generated by the C 1 weak subbasic neighborhoods of this form. A sequence {gn} in SX{M) converges to g in the C 1 compact-open topology iff for any chart (x, U) and any compact K C U, {{gn)ij} and and {dk{gn)ij} converge uniformly on x(K) to gij and di-gij respectively, for all i, j, k. We also have the following useful result. Lemma 3.2 Suppose the sequence {gn} in S\(M) converges to g in the C 1 compact-open topol-ogy. Let [x,U) be any chart of M and (z = (x o TT, x),TU) be the corresponding chart ofTM. Also suppose Yn is the geodesic spray of (M, gn) and Y is the geodesic spray of (M, g) with respec-tive representations Fn and F in (z,TU). Then for any compact set K c TU, {Fn} converges uniformly to F on z(K). Proof. Suppose {gn} converges to g in the C 1 compact-open topology and let (x, U) be any chart of M. Suppose Fn and F are the respective representations of the geodesic sprays Yn and Y in (z, TU). Choose any compact set K C TU. Now 2m Y,\{Fn)i{u^)-Fi{u^)\ = Y, i=l /c=l m £ [(r^M-r^u)]^ < £ | ( r n )^ . (u ) - r^H | - | f | - l ^ l -i,j,k=l Since z(K) is compact, there is a constant A > 0 such that < A for all (u, £) e z(K), where || • || is the Euclidean norm on Rm. So, 2m m sup ^ K ^ r ^ o - ^ ^ O I ^ sup A2 l(r„)?>)-r*>)|. Now the Christoffel symbols T*- are continuous functions of the metric coefficients and their first partial derivatives dkg%j. Since {<?„} converges to <? in the C 1 compact-open topol-ogy, {gn)ij converges uniformly to g^ and dk {gn)ij converges uniformly to dkgij on xon(K). So the right hand side of the above inequality tends to zero as n —> oo. So, Fn converges to F uniformly on z(K). • We have several motivations for choosing the compact-open topology and the C 1 compact-open topology. One reason is that we are interested in curves with compact image or com-pact paths. So we expect that we do not need to control the behaviour of the sequence of Chapter 3. Convergent Sequences of Metric Tensors 19 metrics over the whole manifold to obtain our desired results. Another reason is that given an explicit sequence of metrics, it is relatively easy to establish whether a sequence of met-rics converges in the compact-open topology. For many concrete examples, we are given the metric coefficients for all the metrics in a global chart (or maybe an atlas with two or three charts). In this case, we need only check if the metric coefficients converge uniformly on compact sets. It is also relatively easy to construct sequences of metrics that converge in the compact-open topology. By contrast, it is difficult to construct sequences that converge in the strong topology, also called the fine Whitney topology. This topology has been used to establish some 'stability' theorems in General Relativity (for example, in Beem and Erlich [2], Hawking and Ellis [9], Lerner [17]). While the strong topology is quite useful in formulating certain abstract re-sults, most examples of sequences of metrics that one encounters does not converge in this topology. Roughly speaking, when the manifold is not compact, this topology has so many open sets that it is difficult to find a sequence that converges, and one usually has to consider nets instead (for example, if a sequence of functions on M converges in the strong topology, then outside of some compact set, the functions must eventually agree. See Hirsch [11]). A third reason that we choose the compact-open topology is that it fits well with cer-tain work in the literature. For example, there are theorems concerning the denseness of smooth complete Riemannian manifolds in the set of smooth Riemannian manifolds with respect to the compact-open topology [18]. Another example is the work of Geroch and Traschen [7] in which these two authors identify a class of metrics, called regular metrics, for which one can define a distributional Riemann curvature tensor. A property of these regular metrics is that any continuous, regular metric can be approximated by a convergent sequence of smooth metrics. The specific notion of convergence defined in [7] is similar to convergence in the compact-open topology (though we have not tried to prove that they are equivalent). Chapter 3. Convergent Sequences of Metric Tensors 20 3.2 Sequences of semi-Riemannian metrics Now we come to our first set of results on comparing semi-Riemannian geometries. Our main result, proposition 3.6, is easily anticipated, and it is essentially a matter of working out the details of the proof. The essential idea of this result is found in Beem and Erlich [2] (lemma 6.7 of chapter 6), though our presentation and assumptions differ slightly. This proposition is also sometimes alluded to as a small part of a larger discussion (for example, section 8.4 in Hawking and Ellis [9]). So we formalize this generally known result by going through the steps of the proof in detail. Perhaps the tedious nature of the proof makes it worthwhile to establish the proposition in a form that can be easily applied. 3.2.1 Existence, uniqueness, and continuity theorems in differential equations Let us first fix some definitions before continuing with our discussion. We say that a map / : (7 G l m -> satisfies a Lipschitz condition on K C U iii there is a constant k > 0 such that \\f(u) — f{u)\\ < k\\u — u\\ for all u, u G K. In this case, we also say that / is k-Lipschitzian on K; the constant k is called the Lipschitz constant of / on K. Finally, the map / is locally Lipschitz, if for any u eU, there is a neighborhood Nu such that / satisfies a Lipschitz condition onNu. Now we list two fundamental theorems in ordinary differential equations that will be needed for our purposes. The first theorem is a basic existence and uniqueness theorem. We quote a special case of a more general theorem stated in Birkhoff and Rota [3] (chapter V, theorem 6). Theorem 3.3 Suppose the map F : W1 -» Rm is continuous and satisfies a Lipschitz condition on Rm. Let u G R m and a eRbe arbitrary. Then there is a unique map c : R -» Rm satisfying c{a) = u, c'(t) = F(c(t))Jor all t G R. The continuity theorem below tells us how the solutions of two 'nearby' differential equa-tions compare. A more general version of this theorem is stated and proved in Birkhoff and Rota [3] (chapter V, theorem 3). Chapter 3. Convergent Sequences of Metric Tensors 21 Theorem 3.4 Let F : U - • Rm and G : U -> Rm be continuous on a domain £/ c Rm, and let F be k-Lipschitzian on U. Suppose c : [a, b] Rm and c : [a, b] ->• Rm satisfy the differential equations d(t) = F(c(t)) and c'(t) = G(c(t)), respectively, for all t e [a, 6] and suppose swpueu \\F(u) - G(u)\\ < e. Then Ht) - c(t)\\ < ||c(o) - c(a)||e*(*-a> + y[ek^ - l l . k While G need not satisify a Lipschitz condition on (7, there is a strong (exponential) de-pendence of the above upper estimate on the Lipschitz constant k of the map F. With these two theorems, we deduce a lemma that is sufficient for our needs. Lemma 3.5 Suppose F : U —> Rm is a continuous locally Lipschitz map on the open set U C Rm and suppose {Fn} is any sequence of smooth maps from U into Rm such that {Fn} converges uniformly to F on a closed ball K c U of radius r. Let c : [a, b] -> int(.rY) satisfy c(a) = u, and c'(t) = F{c{t)) for all t € [a, b]. Finally, suppose {un} is any sequence of points in U converging to u. Then there is a positive integer N such that (i)for each n > N, there is a unique map Cn defined on [a, b] for which Cn(a) = un, c'n{t) = Fn(cn(t)), and Cn(t) € K for all t 6 [a, 6], and (ii) the sequence {cn : n > N} converges uniformly to c. Proof. Suppose that {Fn} converges uniformly to F on the closed ball K C U; each Fn is smooth and F is continuous and locally Lipschitz. Suppose that c : [a, b) —>• int (If) satisfies the initial value problem stated in the lemma. Also suppose the sequence {un} converges ton E int(if). Both (i) and (ii) will follow from the following construction. We wish to eventually apply theorem 3.3 so we first extend Fn and F to functions defined on the whole of Rm in the following fashion. Since c([a, b]) is contained in 'mt(K), there is a closed ball K~ of radius r~ (centered at the same point as K) such that c([a, b]) C int(K~) C K~ C int (if). Chapter 3. Convergent Sequences of Metric Tensors 22 Now let A be a smooth bump function such that X(u) = 1 if ||u|| < r~ 1 > A(u) > 0 if r~ < \\u\\ < r, and X(u) = 0 if ||u|| > r. We define the function F by F(u) = X(u)F(u) if u £ K and F(u) = 0 if u £ (R m \ i f ) - Since F is continuous and locally Lipschitz on U, it follows that F is continuous and satisfies a Lipschitz condition on the whole of M m ; furthermore, the maps F and F agree on i f " . Similary, we extend each function Fn to Fn. So each vF„ is smooth on W1; Fn and Fn agree on i f - . We also see that s u p „ e M m ||-Pn(w)|| = s u p „ g X | |F n(^)|| < oo. Now consider any particular n £ Z+. With the above mentioned properties of Fn, theorem 3.3 implies that there is a unique map cn \ K —> IRm for which Cn(a) = un, and c'n(t) = Fn(cn(t)) for allt 6 IL We may restrict the domain of cn to the interval [a, b}. We do this for all n £ Z + so that we obtain a sequence of maps { c n } all defined on [a, b] and satisfying their respective initial value problems. Also, since c([a, b]) C int(.rT~), we know that c satisfies c{a) = u, and c'(t) = F{c{t)) for all t £ [a, b]. Now consider the sequence {en} where en = s up u G R m \\F(u) - Fn(u)\\ = sup u 6 / c X(u)\\F(u) — Fn(u)\\. Since {Fn} converges uniformly to F on K, we see that {en} converges to 0. Applying theorem 3.4 we have the following estimate. sup \\c(t) - cn(t)\\ < \\u - un\\ek^ + £-^[ek^ - 1], te[a,b] K where k is the Lipschitz constant of F. Clearly we see that { c n } converges to c uniformly. To complete the proof, we note that since c([a, b}) is contained in int ( i f - ) , the above estimate implies there is a positive integer N such that for all n > TV, cn([a, b]) C mt(K~). As noted earlier, Fn — Fn on the set K ~~ C i f . So for every n > N, Cn is the unique map defined on [a, b] for which cn(a) = un, , c'n(t) - Fn(cn(t)),. andc„(t) £ K Chapter 3. Convergent Sequences of Metric Tensors 23 for all t G [a, b], and the sequence {cn : n > N} converges uniformly to c. • 3.2.2 Comparing neighboring semi-Riemannian geometries Lemma 3.5 is crucial to our proof of the proposition below. For the formulation of our result, we will use the notation Ck~ in the following manner. A tensor field T on a manifold M is of class Ck~ if it is of class Ck~l and if in any chart the (k - l)th partial derivatives of its components are locally Lipschitz functions. To compare different curves on the manifold M, we use the compact-open topology on the set of maps of the unit interval [0,1] into any topological space. For any compact interval [a, b] contained in [0,1] and any open set O in the topological space, we define an open set of the form N([a, b], O) where a curve a is in the open set N([a, b], O) iff a[a, b] C O. The compact-open topology is generated by sets of this form, for arbitrary [a, b] contained in [0,1] and arbitary O. Now we are ready to state and prove our result. Proposition 3.6 Let g be a C2~ semi-Riemannian metric on M, and let a : [0,1] M be a geodesic of (M,g) with o/(0) = v. Suppose {gn} is any sequence of smooth semi-Riemannian metrics converging to g in the C 1 compact-open topology. Then there is an integer N such that (i) for all n > N, an(s) := exp[gn](sv) is defined for all s G [0,1], and (ii) the sequence of curves in TM, {a'n : n > N}, converges to a' in the compact- open topology. Proof. Suppose the hypotheses in the proposition are satisfied. Let Y be the geodesic spray of (M, g). Y is a Cl~ vector field on TM because g is C2~. So in any chart of TM, the representation of Y is a continuous, locally Lipschitz map. For each n, let Yn be the geodesic spray of (M, gn). Since a is a geodesic of (M, g), so the curve a = a' : [0,1] ->• TM satisfies the equation a'(s) = Y(a(s)) for all s G [0,1]. The image of a is compact in TM, so it can be covered by a finite collection of open sets A such that each O G A is contained in the domain of some chart of the form (z = (x o 7r, X), TU) and B = z(0) is an open ball whose closure is contained in z(TU). Thus, there is a partition of [0,1], P : 0 = SQ < « i < • • • < s/ = 1, Chapter 3. Convergent Sequences of Metric Tensors 24 so that for each i, the segment c r ( [ s j _ i , Sj]) is contained some open set d £ A with a chart (zi, TUi) such that 0; C TiTj. Also ZJ(0J) is an open ball contained in T(7j. Now consider o\ = a\[so, si]. Let c(s) = z\ o CTI(S) and let F n and F be the representa-tions of Yn and Y in the chart (zi,TUi). The map c : [s0, «i] -> # i satisfies the equation c'(s) = F ( c ( s ) ) and c ( s n ) = - z i ( u ) , By lemma 3.2, { F „ } converges to F uniformly on B\. By theorem 3.5 there is a positive integer N\ such that (i) for each n > N\, there is a unique map cn defined on [ s 0 , «i] satisfying Cn(«o) = c'n(s) = Fn(cn(s)) and c n ( s ) G # i f o r all s G [s0, s i ] , (ii) the sequence {cn : n > Ni} converges uniformly to c. If we set o\n = 1 o cn : [SQ, si] ->• TM, then ain satisfies o[n(s) = Yn(oin(s)) for all s G [so, si] and the sequence { r j i n } converges in the compact-open topology to o\ : [so, s\] ->• T M . We continue in this manner for each i G {2,3...,(/ — l ) } i n succession. So for any i, we have a positive integer Ni such that for each n > iV,, there is a curve satisfying a i n ( s ) = Yn{oin(s)) for all s G Take JV = maxjiVj}, and for each n > N, join together the the curves r j j n , « G {1,2... /} to obtain a curve c r n : [0,1] ->• TM that satisfies Cn(0) = v and CT^(S) = l^(<Tn(*)) f ° r a H s G [0,1]. Furthermore, the sequence {en : n > N} converges to a in the compact-open topology. Thus if we define an := TT O a n , we have an(s) = exp[gn](sv) for all s G [0,1] and for all n. Accordingly, {a'n : n > N} converges to a' in the compact-open topology. • A nice feature of this proposition is that it says for any given geodesic of (M,g), for large n, the geodesic of (M, gn) with the same initial position and direction looks the same as the given geodesic. We shall see that our other results are not of the same nature; we cannot specify a given geodesic of (M, g), rather the geodesies of (M, gn) converge to some geodesic of (M, g). Note that with our available tools, the proof of the proposition depends crucially on the fact that the first derivatives of g are locally Lipschitz. The given proof fails without Chapter 3. Convergent Sequences of Metric Tensors 25 this condition. Indeed we do not expect the proposition to be true without this condition, since an ordinary differential equation with 'right-hand side' that is only continuous need not have a unique solution. So for a given initial point and direction, there may be two distinct geodesies of (M, g). In this sense, the sequence of geodesies of (M, gn) would not know to which geodesic of (M, g) it should converge. Chapter 3. Convergent Sequences of Metric Tensors 26 3.3 Sequences of Riemannian metrics Now we turn to the case of Riemannian metrics. Of the three sets of results, those for the case of Riemannian geometry are the most elegant. We consider 'straight lines' in the sense of mj^mizing curves (i.e. the 'path of least distance' between two points), and we show that the distance between two points on a surface changes continuously as it is stretched and deformed. 3.3.1 Curves in Riemannian manifolds To prove our main results on comparing neighboring Riemannian geometries, we need several small adaptations of the standard theory of curves in smooth Riemannian mani-folds to meet our needs in dealing with continuous Riemannian manifolds. First we briefly summarize the basic results in the study of curves in Riemannian manifolds We begin with the definition of the length of a curve. Definition 3.7 The length of a piecewise C1 curve a : [a, b] -> M in a smooth Riemannian manifold (M, g) is Lg(a) = / \\a/{s)\\gds, J a where \\a'(s)\\g = ^jg^a1'(s),a'{s)}. The extra tilde appears in our notation in anticipation of a forthcoming generalization. With this concept of length, the Riemannian distance function dg : M x M ->• R for a smooth (M, g) is defined as follows: dg (p, q) is the greatest lower bound of the set of lengths of all piecewise C 1 curves from p to q. One may show that the function dg is in fact a metric on M and that the topology defined by dg is equivalent to the manifold topology. Thus every smooth Riemannian manifold is a metric space. Naturally this leads to the notion of a minimizing curve. Definition 3.8 A piecewise Cl curve a : [a,b] ->• M from p to q is a mininizing curve of a smooth Riemannian manifold (M, g) ijfLg(a) = dg(p, q). Using arguments based on the calculus of variations, it can be shown (Aubin [1]) that any differentiable rnimmizing curve in a smooth Riemannian manifold (M, g) is a reparametri-Chapter 3. Convergent Sequences of Metric Tensors 27 sation of a geodesic of (M, g); in this case, every rrunimizing curve is equivalent to some geodesic. Furthermore, such a minimizing geodesic is smooth. Perhaps the penultimate theorem of basic Riemannian geometry is the Hopf-Rinow theorem which ties together topological and geometrical properties of a Riemannian mani-fold. Specifically, this theorem tells us that completeness of a smooth Riemannian manifold (M, g) (in the sense of metric spaces) is equivalent to geodesic completeness. Furthermore, completeness is also equivalent to the statement that (M,g) satisfies the Heine-Borel con-dition (i.e. every closed bounded subset of M is compact). A n immediate consequence is that every compact smooth Riemannian manifold is complete. The Hopf-Rinow theorem also has a well-known corollary which states that any two points in a smooth complete Riemannian manifold can be joined by a minimizing geodesic. This geodesic need not be unique, unless the two points are sufficiently close together. This is the extent of our brief summary. Now we introduce the additional refinements that will be helpful for our purposes. First of all, we note that the definition of the length of a piecewise C 1 curve makes sense for a continuous Riemannian manifold as well. For a continuous Riemannian manifold (M, g), we may use Lg to define the Riemannian dis-tance dg as before. Again, dg is a metric on M that gives the same topology as the manifold topology (a proof of this is found in Aubin [1]) and therefore every continuous Rieman-nian manifold is a metric space. It also makes sense to talk about minimizing curves on a continuous Riemannian manifold (M, g). However, for general continuous Riemannian manifolds, we do not expect to have something as powerful as the Hopf-Rinow theorem at our disposal. Our second refinement is to extend the definition of length from piecewise C 1 curves to continuous curves. Definition 3.9 The length of a continuous curve a : [a,b] -> M in a continuous Riemannian manifold (M,g) is Lg(a) = s u p ^ d dg(a(si-i),a(si)) where the supremum is taken over all partitions P : a = SQ < si < • • • < sf = bof [a,b]. This definition of length is compatible with our previous definition of length in the follow-ing sense. Chapter 3. Convergent Sequences of Metric Tensors 28 Lemma 3.10 If a : [a, b] M is a piecewise C1 curve, then L g ( a ) — L g ( a ) . Proof. This proof is similar to a proof found in Rudin [22] for the case of curves in R n . Let (M, g) be a continuous Riemannian manifold. First let us show that L g ( a ) = L g ( a ) when a : [a, b] -> M is a C 1 curve. Choose an arbitrary e > 0. Now since g is continuous on M and a is C 1 , the function ||a'| | e maps [a, b] continuously into R , and is therefore uniformly continuous. So there is a 8 > 0 such that |||c/(s)||9 — ||a'(s)||9| < e whenever \s — s\ < 6. Now choose a partition P : a = s 0 < si < • • • < */ = b of [a, b] such that mesh(P) = 6/2. So we have |||o/(s)||9 — | | C / ( S J ) | | 9 | < e for every s in the interval [SJ_I , si\. For convenience, let A s i = Si — S i - \ . Then rSi rSi d g ( a ( s i - i ) t a ( s i ) ) < \ \ a ' ( s ) \ \ g d s < / \ \ a ' ( S i ) \ \ g d s + e A S l JSi-l JSi-l < f {\\a'{s)\\g + e}ds + e A s i = d s(a(si_i),a(sj)) + 2eAsj. J Si-l Summing from over all values of i, we have L g ( a ) < L g ( a ) < L g ( a ) + 2 e ( b - a). Since e was chosen arbitrary, L g ( a ) = L g ( a ) . The proof for the case where a is piecewise C 1 now follows easily. Suppose that there is a partition a = s o < s i < - - - < S f = b such that each a>i := a \ ( s i - i , S i ) is C 1 . Each a; can be extended to a C 1 curve from [ S J - I , s;] to M . Then we can use the above argument to show that L g ( a i ) = Lg(oti) for every i. We use the fact that 2~l{=\ Lg(cei) = L g ( a ) and 2~2{=i L g ( a i ) = Lg(os) to obtain the desired result. • We note that with this definition of L g , it is possible for a continuous curve to have infinite length; these are sometimes called non-rectifiable curves. For the case of curves in R , these curves of mfinite length are functions of unbounded variation. Using L g , we may define another distance function d g ( p , q ) := infL g (oi) where the infimum is taken over all continuous curves from p to q. As expected, the distance function dg agrees with dg. Lemma 3.11 dg(p, q) = dg(p, q)for all p , q e M . Proof. Clearly, d g ( p , q ) < dg{p,q) for all p , q £ M . Now suppose d g ( p , q ) < d g ( p , q ) for some pair of points p , q e M . Then there must be a continuous curve a : [a, b] —> M from Chapter 3. Convergent Sequences of Metric Tensors 29 ptoq such that A = Lg(a) < B = dg(p, q). Now choose e = (B - A). By definition of Lg, we may find a partition a = s0 < si < • • • < s/ = b so that Yl{=i dg{a(si-i),a(si)) - A < e/3. By definition of dg, we may find a piecewise C 1 curve (3 from p to cy so that Lg(/3) — EtCi4(a(5i-i))a(si)) < e/3. SowehaveL 9(/3) < A + 2e/3 = 5 - e/3 < 5 = dg(p,q),a contradiction. Thus dg (p, c/) ^ dg (p, c/) for all p,q e M. • One of the advantages of using this definition of length Lg is that the same definition is used in the study of continuous curves in any metric space; this allows us to draw from the theory of continuous curves in metric spaces. The following lemma is easily proved with this approach (the details may be found in Kolmogorov and Fomin [13] in the section on continuous curves in metric spaces). Consider the set C(I, M) of all continuous curves mapping the unit interval I = [0,1] into a continuous Riemannian manifold (M, g). A metric pg is defined on C(I, M) by pg(a, /?) = sup s e / dg(a(s), fi(s)) for all a, (3 € C(I, M). Lemma 3.12 Lg is lower semicontinuous on C(I, M) with respect to the topology defined by pg. We say that a sequence of curves {an} in C(I, M) converges uniformly iff it converges in the topology defined by pg. The above lemma is equivalent to the statement that Lg{a) < liminf Lg(an) whenever {an} converges to a uniformly. Finally we quote the well-known theorem of Ascoli-Arzela (as it is stated in Roy-den [21]). This theorem will help us arrive at the main results of the next section. Theorem 3.13 (Ascoli-Arzela) Let T be an equicontinuous family of functions from a separable space X to a metric space Y. Let {fn} be a sequence in T such that for each x e X the closure of the set {fn(x) : 0 < n < oo} is compact. Then there is a subsequence {fnk} that converges pointwise to a continuous function f, and the convergence is uniform on each compact subset of X. 3.3.2 Comparing neighboring Riemannian geometries Now we present our main results. Proposition 3.14 Let Mbea compact manifold and let {gn} be a sequence of smooth Riemannian metrics on M converging in the compact-open topology to a continuous Riemannian metric g. Then for any points p,q£ M {dQn (p, q)} converges to d9(p, q). Chapter 3. Convergent Sequences of Metric Tensors 30 Proof. Suppose {gn} converges to g the compact-open topology. We apply lemma 3.1, choosing K = M, and deduce that there is sequence of positive real numbers {6n} con-verging to 0 such that for all n, \(9n)n(v){v,v) - gv(v)(v,v)\ < Sn g^v){v,v), for every v G TM. We may rewrite this inequality as (! - &n)g-K(v) (V, v) < (gn)-K(v) {V, v) < (1 + < 5 n f a , «), for every v G T M . So for any piecewise C 1 curve a : [0,1] —)• M (l-t>n)f7Q(s)[a'(s),a'(s)] < ( f f n ) Q ( s ) < (1+ ^ n) 5a(s)[a'(s),a'(s)], for almost all s G [0,1]. Thus, for sufficiently large n, \A - $n Lg{a) < L g n ( a ) < \/l + 6n Lg(a). By taking the infimum over all piecewise C1 curves from p to q, we obtain y/\ - 8n dg(p,q) < dgn(p,q) < y/\+8n dg(p, q), for sufficiently large n. Therefore, l i m ^ o o d9n (p, q) = dg (p, q). • Proposition 3.15 Let Mbea compact manifold and let {gn} be a sequence of smooth, Riemannian metrics on M converging in the compact-open topology to a continuous Riemannian metric g. Let p and q be arbitrary points in M. Then there is a sequence {an} of curves from p to q such that (i) for each n, a n : [0,1] ->• M is a minimizing geodesic of ( M , g n ) from p to q, (ii) there is a subsequence { a n k } of {an} that converges uniformly, and (iii) any such convergent subsequence of{an} converges to a minimizing curve of(M, g)from p to q. Proof. By hypothesis, M is compact and each g n smooth, so each { M , g n ) is a complete Riemannian manifold (by the Hopf-Rinow theorem). So for each n, there is a geodesic a n : [0,1] ->• M of (M, gn) from p to q, and we have proved (i). For a fixed s G [0,1], the closure of the set {an(s) : n G Z+} is compact, because it is a closed subset of a compact manifold M . Now since each a n : [0,1] -> M is a geodesic, we Chapter 3. Convergent Sequences of Metric Tensors 31 know that ||o4(s)|| f l„ is constant for all s E [0,1] and in fact equal to Lgn(an). Now observe that dgn(an(s),an(t)) < jls \\a'n{\)\\gnd\ = L9n(an)\s - t\. This observation together with proposition 3.14 gives us the following result. Given arbitrary e > 0 and s £ [0,1], there is a 6. > 0 such that for all n, dg(an{s), an(t)) < e whenever \t — s\ < 6. So {an} is an equicontinuous family. Applying the Ascoli-Arzela theorem, there exists a subsequence {ank} that converges uniformly (in the metric topology defined by pg) to a continuous curve a : [0,1] ->• M. This proves (ii). Finally (iii) is established as follows. Since the ank are minimizing curves and by propo-sition 3.14, we have lim Lg{ank) = dg{p,q). k—>oo By the lower semi-continuity of Lg, we also have Lg (a) < lim inf/^oo Lg (ank). This implies Lg [a) < dg (p, q) and therefore a is a minimizing curve of (M, g) from p to q. • Let us note the role of the compactness assumption in the above two propositions. In the proof of proposition 3.15, compactness of M is used only to guarantee that there ex-ists a sequence of curves {an} of interest, and also to ensure that we may use 3.14. So it is really in the proof of proposition 3.14 that compactness is exploited. The assumption of compactness of M serves two related purposes in the proof of proposition 3.14. First, compactness of M means that the first inequality in the above proof holds for all v E TM. In turn, this allows us to freely take the infimum over all curves from p to q in the last step of our proof. If M is not compact, then the first inequality holds only for v E TK for some specific chosen compact K containing p and q. In this case, one may not take the last step of above proof, as it is only possible to take the infimum over all curves contained within K. Though it is plausible that the distances dg(p, q) and d9n (p, q) may be obtained by taking the infimum over all curves contained within K only, further arguments are required to demonstrate that this is indeed the case. It might be possible to replace this compactness assumption with a combination of completeness of the gn, along with perhaps the condi-tion that the gn satisfy the Heine-Borel property, but our result seems to be most eleganty formulated for compact manifolds. Chapter 3. Convergent Sequences of Metric Tensors 32 A shortcoming of proposition 3.15 is that it tells us nothing about whether every min-imizing curve of (M, g) can be approximated by a sequence of minimizing curves of the (M, gn). So we see that while some minimizing curves of (M, g) resemble the minimizing curves of its neighbors, there remains the possibility that there exists a minimizing curve of (M, g) completely unlike any minimizing curve of (M, gn). Chapter 3. Convergent Sequences of Metric Tensors 33 3.4 Sequences of Lorentz metrics Here we attempt to obtain some results analogous to propositions 3.15 for the case of Lorentz manifolds. Again, we would like to formulate our results so that we have a se-quence of smooth metrics that converge to nonsmooth metric g; we would like to assume the least possible degree of differentiability of g that will still allow us to prove an interest-ing result. However, there are several complications that prevent us from directly carrying over the methods of the last section. First of all, one important fact that we need for the proof of our main result is the upper semi-continuity of the Lorentz length functional. We have not been able to find a proof of the upper semi-continuity without the use of con-vex normal neighborhoods. Another problem is that the existence of maximizing curves in a Lorentz manifold depends crucially on the causal structure of the manifold. The topology we have chosen on the set of metric tensors does not ensure that neighboring metrics have similar causal structures. So to obtain meaningful results, we must impose further condi-tions on the nature of the converging sequence of metrics. A l l of this means that our final result for the case of Lorentz manifolds are not as nice as the results for the case of Rie-mannian manifolds. In the remainder of this chapter, we will be using the facts of causality theory outlined in chapter 2 extensively. 3.4.1 Causal paths in a Lorentz manifold For what follows, it will prove extremely useful not to consider curves in a Lorentz mani-fold, but rather their images. Definition 3.16 A causal path in a continuous Lorentz manifold (M, g) is the image of a contin-uous curve 7 : [a, b] -> M with the following property. Given any s e [a, b] and any neighborhood O ofi(s), there is a neighborhood N of s such that whenever the points si < s2 are contained in N, there is a future causal curve from si to s2 whose image is contained in O. Our convention of considering a curve as a map, and a path as the image of a map is the opposite to that used in [24]. We write [7] to denote image of a curve 7. Note that if [7] is a causal path, 7 need not be differentiable. On the other hand a future causal curve is Chapter 3. Convergent Sequences of Metric Tensors 34 by definition piecewise Ck. So the image of a future causal curve is a causal path, but the converse is not true. Also note that the image of a past causal curve is also a causal path. While the definition of a causal path is somewhat clumsy, there is a considerably simpler characterization of causal paths in a strongly causal Lorentz manifold. Lemma 3.17 In a continuous, strongly causal Lorentz manifold (M, g), [7] is a causal path iff the curve 7 : [a, b] -> M satisfies j(s\) <g 7 ( 5 2 ) whenever a < si < s 2 < 6. Proof. Let (M, g) be strongly causal. (=>) Suppose [7] is a causal path 7 : [a, b] -> M. Choose an open set O containing [7]. At each s find a neighborhood JVS of s so that for any si < S2 in Ns, there is a future causal curve from 7(si) to 7 ( ^2 ) with image in O. Cover [a, b] with a finite number of these neighborhoods. Then for any si and S2 in [a, b], there is a future causal curve from 7 ( 5 1 ) to 7(s2). (<=) Now suppose 7 : [a, b] ->• M satisfies 7 ( ^1 ) < g 7(s2) whenever a < si < s 2 < b. Let s G [a, b] be arbitrary and choose any neighborhood O of j(s). Since (M, g) is strongly causal, we may find an Alexandroff neighborhood U C O of 7(s). Then N = j~l[U] is a neighorhood of s so that any points s\ < S2 in N can be joined by a future causal curve in U. So [7] is causal path. • The discussion in the remainder of this section borrows extensively from the recent work by Sorkin and Woolgar [24]. The only essential difference between our presentation and the one in [24] is that we have applied the techniques of these two authors to a different causal relation (specifically, Sorkin and Woolgar use their novel causal relation -<g rather than the traditional < g that we use here). Theorem 3.18 A subset S of a continuous, globally hyperbolic Lorentz manifold is a causal path iff it is compact, connected and linearly ordered by <g (that is, for every p,q e S,p <g qor q <g p). The proof of this theorem by Sorkin and Woolgar uses elegant order-theoretic arguments. We do not present a proof here, but we refer to theorem 20 of [24]. Our definition of a causal curve is the formally the same as [24] except that we use < g rather than their < g . Also, the statement of their theorem differs from our statement in that their assumption of 'K-causality' is replaced here by the condition of global hyperbolicity. However, the purpose Chapter 3. Convergent Sequences of Metric Tensors 35 of their assumption of if-causality is to ensure that -<g is a (topologically) closed reflexive partial order, and our assumption of global hyperbolicity serves the same purpose (i.e. to ensure that our < g is a closed reflexive partial order). Since reflexivity and closedness of -<g are the only properties of -<g that are used in the proof of theorem 20 in [24], we may repeat their proof here, essentially verbatim, while substituting the causal relation < g for the relation -<g. Now let us denote the set of all causal paths from p to q G M by Cg(p, q). Following Sorkin and Woolgar, we choose the Vietoris topology on Cg(p,q). More precisely, the Vi -etoris topology is a topology on the set 2X of nonempty closed subsets of a topological space X. For any finite collection of open sets {A0, A\... Af} in M, we define an open set in 2X of the form B(A0; Ai,A2,... Af), where S G 2X is an element of B(A0; Ai,A2... Af) if S (as a subset of M) is contained in Ao and intersects each A{ for i G {1,2... / } . Since the causal paths fromp to q are closed subsets of M, we may take the topology on Cg(p, q) to be the topology induced by the Vietoris topology on 2M. We list some properties of the Vietoris topology (these are proved in [24]). Theorem 3.19 The topological space 2X is compact and Hausdorff whenever X is compact and Hausdorff. Lemma 3.20 (i) In a compact space X, every Vietoris limit is a compact set. (ii) In a compact, Hausdorff space X, a Vietoris limit of connected sets is connected. These properties of the Vietoris topology allow us to give an elegant proof the following theorem (the proof is due to Sorkin and Woolgar [24]). Theorem 3.21 Let (M, g) be a continuous, globally hyperbolic Lorentz manifold. Then Cg(p, q) is compact in the Vietoris topology. Proof. Let (M, g) be a continuous, globally hyperbolic Lorentz manifold. Letp and q G M be arbitrary. Now J(p, q) is compact by hypothesis, and any causal path from p to q is a subset of J(p, q). Let [7n] be any sequence in Cg(p, q). Since J(p, q) is compact and Haus-dorff (because M is Hausdorff), we can apply theorem 3.19 and lemma 3.20 to deduce the Chapter 3. Convergent Sequences of Metric Tensors 36 existence of a subsequence [jnk] that converges in the Vietoris topology to some compact connected subset S of J(p, q). We simply need to verify that S is indeed a causal path. Choose any points r and s in S. Given any neighborhoods Nr and Ns of r and s respec-tively all but a finite number of the [ynk] intersect Nr and Ns, because [ynk] converges to S in the Vietoris topology. So, we may construct a sequence of pairs (r^, s^) converging to (r, s) such that for each k, r^ and Sfc lie on [ynk]. In other words, for each k, (i) <g Sk or (ii) sk <s rk- Since (i) or (ii) must hold for an infinite number of k, and since < g is closed when (M, g) is globally hyperbolic, we must have either r <g s or s <g r. So S is compact, connected and linearly ordered by < g , and by theorem 3.18, S must be a causal path. • 3.4.2 Length and distance in a Lorentz manifold We have formulated the statements of last section for continuous Lorentz manifolds. Now in this section, we discuss several statements for which there are no known proofs if the metric tensor is assumed to be continuous only. In fact, the standard proofs of many of these results depend on the existence of convex normal neighborhoods. We roughly follow the same approach found in Lerner [16]. For a C2 Lorentz manifold (M, g), we define a causal trip from p to q to be a causal path [7] for which 7 is a piecewise geodesic, future causal curve whose pieces are either timelike or null. Let us denote the set of all causal trips from p to q by Tg(p, q). Lemma 3.22 Let (M, g) be a C2 Lorentz manifold. Then Tg(p, q) is dense in Cg(p, q) with respect to the Vietoris topology. Proof. Let [7] be any causal path in Cg(p,q). Choose any Vietoris neighborhood of [7], B(AQ; Ai,A2...A{). Cover [7] with a finite collection A = {0\... Of] of convex normal neighborhoods so that each such neighborhood is contained in Ao and each Ai (for i € {1,2... /}) contains at least one such neighborhood. For the map 7 : [a, b] -> M there is a partition P : a = SQ < s i < . . . sf = b such that each jj := J\[SJ-I,SJ] has its image contained in some convex normal neighborhood in the cover A and every convex normal neighborhood in A contains one such TJ. S O , we relabel the sets Oj so that jj is contained in Oj for all j € {1 . . . / } . Chapter 3. Convergent Sequences of Metric Tensors 37 Now, for each j, there is a unique future timelike or null geodesic from j ( s j - i ) to 7(SJ) with image in Oj. Joining these geodesies together, we construct a causal trip from p to q that intersects every Oj (and thus intersects every Ai) and that is contained in UjOj C Aq. • Definition 3.23 The length of any [7] G T9(p,q) is Clearly Lg ([7]) < 00 for any [7] G Tg (p, q). In contrast with the Riemannian length function, we see it is possible for Lg([y]) = 0 even though 7, being a future causal curve, is not degenerate (i.e. not a trivial curve in M). The obvious example is where 7 is a piecewise future null curve. In fact, for any continuous Lorentz manifold, inf Lg([j]) = 0, where the infimum is taken over all [7] G Tg(p, q). From these considerations, we also expect that Lg is not continuous on Tg(p, q). More specifically, take any causal trip [7] in Tg(p, q) that is the image of a smooth future timelike geodesic. [7] has strictly positive length with respect to g. We can find a sequence of causal trips [7„] converging to [7] such that each pyn] is the image of piecewise geodesic, future null curve. Since each [yn] has zero length, this clearly shows that Lg cannot be continuous. On the other hand, we have the following. Theorem 3.24 Let (M, g) be a C2 strongly causal Lorentz manifold and p, q be any points in M for which p <g q. Then the functional Lg is upper semicontinuous on Tg(p, q) in the Vietoris topology. Proof. In Lerner [16], it is shown that for a C2 strongly causal (M, g), the functional Lg is upper senucontinuous on Tip, q) in the Hausdorff metric topology (proposition 6.2). One may verify that every open neighborhood of [7] in Hausdorff metric topology contains some open Vietoris neighborhood of [7]. It follows immediately that Lg must also be upper semicontinuous on Tip, q) in the Vietoris topology. • where \\i{s)\\g = [-g{1'{S)n'{s))^2. Chapter 3. Convergent Sequences of Metric Tensors 38 Since Tg(p, q) is dense in Cg(p, q), we may extend Lg to a upper semicontinuous functional on all of Cg(p, q). Now the Lorentz distance function of a continuous Lorentz manifold ( M , g), sometimes called the time separation function, r 9 : M x M ->• R is defined as follows: rg(p,q) = supLs([7]), where the supremum is taken over all [7] e Cg(p, q). Definition 3.25 A causal path [7] from p to q is a maximizing path of a continuous Lorentz manifold (M,g) ijfLg{[~/]) = rg(p,q). For Lorentz manifolds ( M , g) of class C2, one may use the existence of convex normal neighborhoods to show that any maximizing path from p to q with p <^g q is actually the image of a future timelike geodesic 7 : [0,1] -> M from p to q (for example, see [2]). We call this a maximizing geodesic path. In general, the time separation between two points need not be finite. To see this intu-itively, consider the case where there exists a closed future timelike curve 7 : [0,1] ->• M in the Lorentz manifold ( M , g). Then, we may define a curve jn : [0, n] —> M that is obtained by simply going around 7 a total of n times. So we obtain a sequence of future timelike curves jn so that each [7n] is a causal path, and l i m ^ o o Lg([yn]) = 00. Furthermore, even if the time separation is finite, it is not clear whether there exists a maximizing path. However, for a globally hyperbolic Lorentz manifold we have the follow nice result. Theorem 3.26 Let ( M , g) be a C2 globally hyperbolic Lorentz manifold. Then for any p <9 q, there is a maximizing geodesic path from p to q. Proof. Since (M, g) is globally hyperbolic, we know that Cg{p, q) is compact in the Vietoris topology (theorem 3.21) and the length functional Lg is upper sermcontinuous on Cg(p, q) with respect to the Vietoris topology. Since an upper semicontinuous function on a com-pact domain attains its maximum, we see that there exists a maximizing path from p to q. • Since a geodesic is a smooth map, it also follows that the time separation between any p and q in a C2 globally hyperbolic Lorentz manifold is finite. Chapter 3. Convergent Sequences of Metric Tensors 39 Finally we prove one more lemma that will be useful in the proof of our main result. Our proof utilizes the following theorem. Theorem 3.27 In a C2 globally hyperbolic Lorentz manifold ( M , g), there is a smooth function t : M -> R such that the vector field Vt, defined by dt = g(Vt, •), is a continuous nonvanishing timelike vector field on (M, g). A more general theorem is proved in Hawking and Ellis [9] and Seifert [23] under the assumption of stable causality. In Beem and Erhlich [2] , it is proved that a C2 globally hyperbolic hyperbolic Lorentz manifold is also stably causal, and so we have the theorem stated above. Lemma 3.28 Let (M, g) be a C2 globally hyperbolic Lorentz manifold. Let h be any continuous Riemannian metric on M. Given p <g q, there is a constant p such that L^ij) < p, where 7 is any future causal curve from p to q. Proof. Consider the time function t that is guaranteed to exist by the above theorem. We may assume without loss of generality that Vt agrees with the time direction of (M,g). Now define the C2 function / = \/[g(Vt,Vt)]2. One may easily verify that the tensor h = 2fdt <g> dt + g is a continuous Riemannian metric on M. Now for any causal v E TM, we have K(v)(v,v) = 2[f o ir{v)dtK(v)(v)]2 + g*(v)(v,v) < 2[f o ir(v)dt^v)(v)]2. Denote K := J(p,q) for any points p <g q. Now let p = supp^xfip)- So for any future causal curve 7 : [a, b) —> M from p to q, we have ^( s)(V(*),7 '(s)) < 2p[dt^s)(7'(s))}2, for all s E [a, b] where 7'(5) is defined. Since Vt agrees with the time direction of ( M , g) This implies Lh{l) < {2p)1'2 f dt(7'(s))ds = {2pfl2[t{q)-t{p)]. J a So every causal curve from p to q must have /i-length no greater than (2p)1/2[t(q) — t(p)]. Chapter 3. Convergent Sequences of Metric Tensors 40 Now note that K := J(p,q) is compact, since (M, g) is globally hyperbolic. For any other continuous Riemannian metric h on M , there always a constant v > 0 for which hn(v) (u> v) < ^n-fv) ( w i v) f ° r every u G TK. So the h-length of any causal curve from p to q must not exceed (2pv) ll 2[t{q) — t(p)] • 3.4.3 Comparing neighboring Lorentz geometries Before discussing the main proposition, we present two simple results. Lemma 3.29 Let g be any continuous Lorentz metric on M, and let {gn} be any sequence of smooth Lorentz metrics converging to g in the compact-open topology. Suppose that 7 : [0,1] M is a piecewise C 1 timelike curve of (M, g). Then there exists an integer N such that a is a timelike curve of (M, gn)for all n > N. Proof. Suppose that 7 : [0,1] —> M is a C 1 timelike curve of {M,g) and suppose the sequence {gn} (with each gn smooth) converges to g in the compact-open topology. Let h be any fixed Riemannian metric on M. Now a is contained in some compact set K c M. Since {gn} converges to g in the compact-open topology, {gn} converges to g uniformly on K. So there is sequence of positive real numbers {Sn} converging to 0 such that for all n, \(9n)ir(v){v,v)-g v^){v,v)\ < 6n h^{v)(v,v), for every v € TK. So there is an integer N > 0 such that for all n > N, 0 < 5n < —p/2u where where p, and v are defined by H = supc77(s)[7'(s),7;(s)] < 0, v = sup/i7(s)[7'(s),7'(s)], where the supremum is taken over all s e [0,1] at which j'(s) is defined. Since 7 is piece-wise C 1 , p. and v are finite constants. This means that for all n > N, (0n)7(.)[V(a),7'(s)] < 37(s)[7'(s),7'(s)] + *n hl(s)[^(s),j'(s)] < + (-n/2u)u = fi/2 < 0, for all s E [0,1] at which j'(s) is defined. So 7 is timelike with respect to gn for all n > N. • Chapter 3. Convergent Sequences of Metric Tensors 41 Lemma 3.30 Let g be a C2 Lorentz metric on M and let {gn} be a sequence of smooth Lorentz metrics converging to g in the compact-open topology such that each (M, gn) is globally hyperbolic. Finally, let p and q be arbitrarypoints for which p <C9 q. Then rg(p,q) < lim^oo rgn(p, q). Proof. Let (M, g) be of class C2 and p <CS q. Since any maximizing path from p to q must be the image of a future timelike geodesic curve from p to q, we see this implies Tg{P-, q) = supL s([a]), where the supremum is taken over all [a] E Cg{p, q) for which [a] is a smooth future timelike curve. Since {gn} converges to g in the compact-open topology and K :— Jg(p, q) is compact (because (M, g) is globally hyperbolic), lemma 3.1 tells us that for any continuous Rieman-nian metric h on M, we have a sequence of real number Sn converging to 0 such that for every v E TK. If v is a causal vector of both (M, gn) and (M, g) then the above inequality implies yj-9n(v)(v,v) - ^-(gn)ir(v)(v,v) < ^ &nh^v)(v,v). Now consider an arbitrary smooth future timelike curve a in the Lorentz manifold (M, g) from p to q. Lemma 3.29 tells us that there exists some N > 0 such that for all n > N, a is a future timelike curve of (M, gn). Applying the above inequality, we have Lg([a\) < Lgn([a]) + y/SnLh{a) < rgn(p,q) + y/SnLh{a), for all n> N. Therefore, L9([a]) < linin-yoo rgn (p, q). Taking the supremum over all [a] for which a is a smooth future timelike curve, we have rg(p, q) < l i m n _ > 0 0 r S n (p, q). • Note that we have not shown limn-xx) r 9 n (p, q) < oo in general. As we have mentioned earlier, the choice of convergence of a sequence of Lorentz met-rics {gn} in the compact-open topology does not constrain the causal structure of each (M, gn). In fact, in every neighborhood of a Lorentz metric g in the compact-open topology, there exists a metric g for which the Lorentz manifold (M, g) has closed timelike curves [9]. So to obtain reasonable results, we have to put a constraint on the metrics in the sequence Chapter 3. Convergent Sequences of Metric Tensors 42 {gn} 'by hand'. We write g -< g if and only if for all v € TM, 9TV(V) (V, V) < 0 whenever g^v) (v, v) < 0 In other words, every causal vector of g is also a causal vector of g. Pictorially, this says that the null cones of g are more 'opened-out' than the null cones of g. We use relation -< between metrics to constrain the behaviour of the converging sequence of metrics {gn}-Proposition 3.31 Let gbeaC2 Lorentz metric on M such that (Af, g) is globally hyperbolic. Let {gn} be a sequence of smooth Lorentz metrics converging to g in the compact-open topology such that (i) each (M, gn) is globally hyperbolic, and(ii) gn -< gfor all n. Finally, let p and q be arbitrary points for which p <C9 q. Then there is a sequence of causal paths (with respect to g) [jn] such that (i) for each n, [7„] is a maximizing geodesic path of (M, gn) from p to q, and (ii) there is a subsequence [jnk] that converges in the Vietoris topology to a maximizing path [7] of (M,g) from ptoq. Proof. Suppose that {gn} converges to g in the compact-open topology and that for each nt 9n -< 9- Since p <§C9 q, there is a future timelike curve a from p to q. By lemma 3.29, there exists an integer N > 0 such that for all n > N, a is a future timelike curve of gn. So for all n > N,p <Cfln q. And since each (M,gn) (n > N) is a smooth globally hyperbolic Lorentz manifold, by theorem 3.26 there exists a maximizing path [yn] of (Af, gn) from p to q; furthermore yn : [0,1] -> M is a future timelike geodesic of (Af, gn). So we have proved (i). Since gn -< g, we see that C9n (p, q) C Cg (p, q) for every n; so we have the sequence [jn] is a sequence of causal paths in (Af, g) from p to q. By theorem 3.21, the set Cg (p, q) is compact in the Vietoris topology. So there exists a subsequence [jnk] of [7n] that converges to some causal path [7] e Cg(p, q). Now we need to verify that [7] is indeed a maximizing path of {M,g). By hypothesis {gn} converges to g in the compact-open topology, and K := Jg(p, q) is compact (because (Af, g) is globally hyperbolic), lemma 3.1 tells us that for any continuous Riemannian metric h on Af, we have a sequence of real number 8n converging to 0 such Chapter 3. Convergent Sequences of Metric Tensors 43 that \9n(v)(v,v) - {gn)n(v)(v,v)\ < 6nhn(v){v,v) for every v e T K . If v is a timelike vector of (M, gn), it is also a timelike vector of (M, g) (since gn -< g for all n). In such a case, the above inequality implies \J-9-K(V)(V,V) - y/-(9n)n(v)(v,v) < yJSnh^{v)(v,v). Now each j n is a timelike geodesic of (M, <?„). By applying the above inequality we have Tgn(P>0) = L9n(hn]) < Lg{{ln]) + V ^ - M l n ) -By theorem 3.28, there is a constant /x that bounds the lengths of all future causal curve from p to q. So The functional L g is lower semicontinuous on Cg (p, q) (by lemma 3.24) so we have lim inf r 9 n (p, q) < lim inf L g ( [ 7 n ] ) < L 9 ([7]). By lemma 3.30, rg(p,q) < l im^oo rg(p, q). Therefore, Tg(p,q) < Lg([y]). So [7] must be a maximizing path of (M, g) from p to q. • There are several unsatisfactory features of this proposition. First, the assumption of C 2 differentiability of the metric, while certainly undesirable, was necessary in our formula-tion and proof of the proposition. Also, this proposition suffers the same deficiency as the analogous result in the Riemannian case. Roughly speaking, the statement only tells us that some maximizing path of (M,g) appears the same as a maximizing path of (M, gn). Indeed, if we are given a particular maximizing path [7] of (M, g), the above result is able to tell us nothing; we do not know if there is a sequence [jn] of maximizing paths of the (M, g n ) that converge to [7]. Perhaps the most unsatisfactory aspect of proposition 3.31 is that we had to introduce a constraint on the causal structure of the sequence of metric tensors. Chapter 4 Conclusions and Future Research 4.1 Conclusions We have presented three sets of propositions on the comparison of the semi-Riemannian geometries corresponding to a convergent sequence of metric tensors. The original moti-vation for this work was to gain some understanding of whether certain geometric prop-erties of a semi-Riemannian manifold (M, g) are shared by a sequence of approximating manifolds (M,gn). The formulation of proposition 3.6 for semi-Riemannian manifolds is perhaps the nearest to the type of result originally sought after. This proposition gives us the existence of a sequence of geodesies a n of the (M, gn) that converges to a given geodesic of (M, g). It allows us to understand some features of the geodesies of (M, g n ) if we know something about the geodesies of (M, g). Propositions 3.14 and 3.15 for Riemannian manifolds tell us the manner in which dis-tances are changed if the Riemannian metric is perturbed (that is when the 'surface' is 'deformed'). They also tell us that there is at least one path of least distance of (M, g) for which there is some path of least distance of (M, g n ) in its neighborhood. Compared with proposition 3.6 for semi-Riemannian manifolds, these results for Riemannian manifolds are conceptually more elegant, but perhaps of less utility. For Lorentz manifolds, proposition 3.31 was modelled after the results for the case of Riemannian manifolds. This proposition the least satisfactory of all the results we have presented. There are several unwanted hypotheses in the statement of the proposition, and there does not yet appear to be any obvious ways of removing these unwanted assump-tions. One difficulty seems to be more purely technical (C2 differentiability was needed 44 Chapter 4. Conclusions and Future Research 45 so that we could apply some standard known results), whereas the other is more directly concerned with the nature of Lorentz geometry. Indeed, though we proved some results for the case of Riemannian manifolds to help us formulate and prove our result for the case of Lorentz manifolds, our discussion in fact served to underline the significant dif-ferences between Riemannian and Lorentz geometry that often make it difficult to obtain Lorentzian analogues of theorems in Riemannian geometry. 4.2 Future research 4.2.1 Further elaborations There are several directions in which one might want to take this research. One general direction is to follow the same type of approach outlined here, but vary some of the details involved. For example, one might want to examine not only compare the 'straight line geometry' of various semi-Riemannian manifolds, but also the properties concerning the curvature. Perhaps one might want follow the fruitful approach of modern comparison geometry [8] in imposing certain bounds on the curvatures of the converging sequence of metrics. One might also consider other notions of convergence. For example, it might be in-teresting to consider the much more difficult case of a true sequence of semi-Riemannian geometries (Mn,gn), where both the underlying manifold and the metric are converging. One such notion of convergence has been developed by Geroch [5] and is particularly well adapted to problems in Lorentz geometry and general relativity. For Riemannian mani-folds, it may be worthwhile to consider Hausdorff-Gromov convergence or Lipschitz con-vergence [8]. It may also be worthwhile to remove some of the conditions in proposition 3.31 (for Lorentz manifolds) on the differentiability of the metric. One would have to find proofs of similar statements found in section 3.4.2 concerning the interaction of the causal struc-ture of a Lorentz manifold with the behaviour the length functional. Such an extension of the standard tools of global Lorentz geometry would perhaps be of some interest to the mathematical relativist. Chapter 4. Conclusions and Future Research 46 4.2.2 Some examples and applications The above mentioned possibilities are elaborations of the same theme. It may be more interesting to apply the results that we already have to some concrete examples. Propo-sition 3.6 is applicable to many examples of Lorentz manifolds in General Relativity. For example, in any spacetime manifolds with a boundary surface that separates a matter-filled region from a vacuum region, the curvature of the metric will be discontinuous at the boundary surface. These spacetimes, and the ones for thin shells of matter and grav-itational radiation mentioned in the introduction, may be investigated using some of the results we have proved. Proposition 3.6 could be used to obtain a more rigorous version of certain comments indicated in section 8.4 in Hawking and Ellis [9] on singularities in C 2 ~ Lorentz manifolds. The lemmas that we have proved on the way to obtaining the main results also have some interesting applications. For another example in general relativity, Burnett [4] has proposed a counterexample to passive topological censureship of K (n, 1) prime factors. The construction of his counterexample is by the 'cut and paste' method we described in the introduction. It seems that lemma 3.29 can be used to easily refine his counterexample so that all the technical details concerning the differentiability of the metric can be resolved. Finally, while our results on Riemannian geometry can not be directly applied to prob-lems concerning the Lorentz manifolds used in general relativity, they can be applied to problems involving the initial value formulation in general relativity (i.e. the initial data for the Einstein equation consists of specifying the matter fields, a Riemannian metric h and an auxiliarly symmetric tensor field p on a three dimensional manifold E so that the resulting solution of Einstein's equation has S as an embedded submanifold with induced first and second fundamental form given by h and p respectively). Most notably, the posi-tive mass theorem can be discussed in the context of the initial value formulation, and one might examine whether some of the differentiability conditions of the theorem might be weakened using our results. Bibliography [1] Thierry Aubin. Nonlinear Analysis on Manifolds. Monge-Ampere Equations. Springer-Verlag, 1982. [2] John K. Beem and Paul E. Ehrlich. Global Lorentzian Geometry. Marcel Dekker, New York, NY, 1981. [3] Garrett Birkhoff and Gian-Carlo Rota. Ordinary Differential Equations. Ginn and Com-pany, 1962. [4] Gregory A. Burnett. Counterexample to the passive topological censorship of k(ir, 1) prime factors. Physical Review D, 52(12):6856-6859,1995. [5] Robert Geroch. Limits of spacetimes. Communications in mathematical Physics, 13:180-193,1969. [6] Robert Geroch. Domain of dependence. Journal of Mathematical Physics, ll(2)A37-M9, 1970. [7] Robert Geroch and Jennie Traschen. Strings and other distributional sources in gen-eral relativity. Physical Review D, 36(4):1017-1031, 1987. [8] Karsten Grove and Peter Peterson. Comparision Geometry. Mathematical Sciences Re-search Institute Publications, Cambridge University Press, 1997. [9] S.W. Hawking and G.F.R. Ellis. The Large Scale Structure of Space-Time. Cambridge University Press, 1973. [10] Noel J. Hicks. Notes on Differential Geometry. D. Van Nostrand Company, Princeton, NJ, 1965. [11] Morris W. Hirsch. Differential Topology. Springer-Verlag, 1976. [12] Werner Israel. Singular hypersurfaces and thin shells in general relativity. II Nuovo Cimento B, 44:1-14,1966. [13] A . N . Kolmogorov and S.V. Fomin. Elements of the Theory of Functions and Functional Analysis. Dover, 1990. [14] Marcus Kriele. Spacetime. Springer-Verlag, 2000. [15] Serge Lang. Differential Manifolds. Springer-Verlag, 1985. [16] David Lerner. Techniques of topology and differential geometry in general relativity. In J. Farnsworth, D. Fink and J. Porter, editors, Methods of Local and Global Differential Geometry in General Relativity, pages 1-44. Springer Lecture Notes in Physics, 1972. 47 Bibliography 48 [17] David Lerner. The space of lorentz metrics. Communications in mathematical Physics, 32:19-38,1973. [18] J. Morrow. The denseness of complete riemannian metrics. Journal of Differential Ge-ometry, 4:225-226,1970. [19] Barrett O'Neill. Semi-Riemannian Geometry With Applications to Relativity. Academic Press, 1983. [20] R. Penrose. The geometry of impulsive gravitational waves. In L. O'Raifeartaigh, editor, General Relativity: Papers in Honour of J. L. Synge, pages 101-118, Ely House, London, 1972. Oxford University Press. [21] H.L. Royden. Real Analysis. Prentice-Hall, Englewood Cliffs, NJ, third edition, 1988. [22] Walter Rudin. Principles of Mathematical Analysis. McGraw-Hill, second edition, 1964. [23] H.-J. Seifert. Smoothing and extending cosmic time functions. General Relativity and Gravitation, 8:815-831,1977. [24] R. D. Sorkin and E. Woolgar. A causal order for spacetimes with c° lorentzian metrics: Proof of compactness of the space of causal curves. Classical and Quantum Gravity, 13:1971-1994,1996. [25] M . Spivak. A Comprehensive Introduction to Differential Geometry, volume one. Publish or Perish, Berkeley, CA, second edition, 1979. 6. Appendix A Notations and Conventions for Calculus on Manifolds The purpose of this section is mainly to fix our notations and conventions. We follow the basic approach to calculus on manifolds found in Spivak [25], though our notations will differ slightly. For our purposes, an m-dimensional manifold M is a connected, Hausdorff, paracom-pact topological space that is locally homeomorphic to Rm, and that is furnished with a smooth differential structure (i.e. all transition functions between overlapping charts are smooth). Essentially, we choose to say 'manifold' in place of the phrase 'connected para-compact Hausdorff smooth differentiable manifold'. The term 'smooth' will always mean ' C 0 0 ' . Charts of M are typically denoted by (x, U), where x = (x1, x2 ... xm) : U -> K. A chart at pis a chart (x, U) with p e U. If (x, U) and (y, V) are overlapping charts of M (that is, if U fl V 0, then the smooth function y o x'1 : x(U n y ) -> y(U fl V) is the transition function from the x coordinates to the y coordinates. A map / : U C M m -» Rl satisfies a Lipschitz condition on K C U iff there is a constant k > 0 such that ||/(u) — f{u)\\ < k\\u — u|| for all u,u € K. In this case, we also say that / is rC-Lipschitzian on K; the constant k is called the Lipschitz constant of / on K. The map / is locally Lipschitz, if for any u G U, there is a neighborhood Nu such that / satisfies a Lipschitz condition on Nu. A function on a manifold / : M —> R is of class C f e ~ iff it is of class Ck~l and for every chart (x, U), all (k — l)th partial derivatives of / o x~l : x(U) -> K are locally Lipschitz; a Cl~ function on a manifold is a locally Lipschitz function. The tangent space at p, TPM is the vector space of all derivations of functions that are differentiable at p; elements of TpM are called tangent vectors at p. Tangent vectors at a 49 Appendix A. Notations and Conventions for Calculus on Manifolds 50 point may also be equivalently defined according to the contravariant transformation rule of classical tensor analysis, or by the concept of an equivalence class of curves tangent at p. The cotangent space T*M is the vector space dual of TPM. If {x, U) is a chart at p, the natural basis of TPM is {(d/dx1 )p... {d/dxm)p} and the dual basis (of covectors in T*M) is {{dxl)p... (dxm)p}. The vector space of type (r, 5) tensors at p is denoted T J p M ; a vector is a type (1,0) tensor and a covector is a type (0,1) tensor. In a chart (x, U) at p any type (r, s) tensor at p can be written as a linear combination of suitable tensor products of the basis vectors (d/dxl)p and covectors {dxl)p. In particular a type (0,2) tensor batp can be expanded as b = 52ij=i bij (dx*)p ® (dxj)p for some collection of real numbers bij. TM — UptitfTpM is the tangent bundle; n is the projection map that maps v to p when-ever v e TPM. We also write TU = \JpeUTpM. To any chart (x, U) of M, an associated chart of TM is defined as follows. Define x = (x1... xm) : TM -» Rm by x(v) = £ whenever v = 2~2i Cid/dx1)^. Then (z, TU) with z := {XOTV, x):TU - 4 - x{U) x Rm is a chart of TM. T*M = UpeMT*M is the cotangent bundle; the bundle of type (r, s) tensors is T J M = UpeitfTspM. A type (r, s) tensor field on M is a section of T J M ; a vector field is a section of TM and a one-form is a section of T * M . If A is a tensor field on M, A(p) is the tensor at p that is the image of p under A. Given any local chart (x, U) the coordinate vector fields {d/dx1... d/dxm} are defined on U so that (d/dxi)(p) = (<9/cV) p. Similarly, there is also a set of coordinate one-forms {dx1... dxm} so that [dx^){p) = {dx^)p. With these conventions, we can consistently use the notation Ap = A(p) for any tensor field A on M. In the chart (x, U) any type (r, s) tensor field A may be written in terms of combinations of suitable tensor products of the d/dx1' and dxi with real functions as coefficients. For a type (0,2) tensor field b, we have b = J2Tj=i hj{x)dxl ® dxi for some collection of real functions b^ : x(U) ->• M. We always consider the coefficients by to be functions on some real domain, so that b^ (x) is a function on U that is the composition of fry with x. A tensor field is of class Ck~ iff in every chart, its coefficients are of class Ck~. A vector field X can be written in the chart {x, U) as X = YT=\ Cix)d/dxi and Xp = YT=i C(x(p))(d/dx%- The m a P £ = (f1 • • • £ w ) : x(U) -> Rm can also be expressed as Appendix A. Notations and Conventions for Calculus on Manifolds 51 £ = x o X o x~l; for convenience, we will sometimes call £ the representation of X in (x, U). So the representation of a vector field in some chart is simply an arrangement of its coefficients into a vector-valued function. A Ck (k > 0) curve in M is a Ck map a : E -4- M where E is some interval of R (this interval can be open, closed, or neither). If a : E —> M is a C1 curve, then a' : E -4- TM is the field of tangent vectors defined by a'(t) = YHL\(xl ° a)'(t)(d/dxl)a(t), where (x, U) is any chart at a(t). Here we interpret (xl o a)'(t) as a right-hand or left-hand derivative if necessary (i.e. when t is an endpoint of the interval I). If a is a C f e (A; > 1) curve in M , then a' is a C f c - 1 curve in TM. The map a : [a, 6] —• M is a piecewise (A; > 1) curve if a is continuous and there exists a partition a = to < t\ < • • • < tf = b such that (i) a|[i 0,*i) and a\(tf-i,tf] are Ck curves, and (ii) a|(ij_i, U) is a C f c curve for each i G { 2 , 3 . . . ( / - 1)}. So the tangent vector to a is defined at all but a finite possible number of points in (a, b).
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Comparison of neighboring semiriemannian geometries Ling, Henry Ho-Kong 2001
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Title | Comparison of neighboring semiriemannian geometries |
Creator |
Ling, Henry Ho-Kong |
Date Issued | 2001 |
Description | Given a convergence sequence of metric tensors gn → g on a manifold M, in what sense does the corresponding geometries (M, gn) converge to (M, g)? This question is examined by comparing the nature of the straight lines in each (M, gn) with the nature of the straight lines in (M, g). The cases of semi-Riemannian geometry Riemannian geometry, and Lorentz geometry are considered in succession. The relevance of this question to problems in general relativity is emphasized, and some potential applications of our results to questions concerning, for example, topological censureship, and singularity theorems are indicated. |
Extent | 2960970 bytes |
Genre |
Thesis/Dissertation |
Type |
Text |
FileFormat | application/pdf |
Language | eng |
Date Available | 2009-08-06 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0080061 |
URI | http://hdl.handle.net/2429/11804 |
Degree |
Master of Science - MSc |
Program |
Mathematics |
Affiliation |
Science, Faculty of Mathematics, Department of |
Degree Grantor | University of British Columbia |
GraduationDate | 2001-11 |
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Scholarly Level | Graduate |
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