x Global Lorentzian Geometry Beem
Short Description
Download x Global Lorentzian Geometry Beem...
Description
bout the first edition
...
'RE A N D
A P P L I E D
M A T H E M A T I C S
A Series o f Monographs and Textbooks
...a definitive source-book. " -General Relativity and Gravitation bout the second edition . ..
his fully revised and updated Second Edition of an incomparable referenceltext 5dges the gap between modem differential geometry and the mathematical ~ysicsof general relativity by providing an invariant treatment of Lorentzian 2ometry-reflecting the more complete understanding of Lorentzian geometry zhieved since the publication of the previous edition. arefully comparing and contrasting Lorentzian geometry with Riemannian geoietry throughout, Global Lorentzian Geometry, Second Edition offers a compreznsive treatment of the space-time distance function not available in other books... :cent results on the general instability in the space of Lorentzian metrics for a wen manifold of both geodesic completeness and geodesic incompleteness...new laterial on geodesic connectibility... a more in-depth explication of the behavior I the sectional curvature function in a neighborhood of a degenerate two-plane .and more. bout the authors
GLOBAL LORENTZ GEOMETRY Second Edition
. ..
K. BEEM is a Professor of Mathematics at the University of Missouri, olumbia. The coauthor or coeditor of three books and the author or coauthor of ver 60 professional papers, he is a member of the American Mathematical ociety, the Mathematical Association of America, and the International Society )r General Relativity and Gravitation. Dr. Beem received the Ph.D. degree !968) in mathematics from the University of Southern California, Los Angeles. 3HN
E. EHRLICHis a Professor of Mathematics at the University of Florida, iainesville. A member of the American Mathematical Society, the Mathematical .ssociation of America, and the International Society for General Relativity and iravitation, he is the author or coauthor of numerous professional papers that :flect his research interests in differential geometry and general relativity. Dr. .hrlich received the Ph.D. degree (1974) in mathematics from the State University f New York at Stony Brook. AUL
EASLEYis an Associate Professor of Mathematics at Truman State Jniversity, Kirksville, Missouri. He is a member of the American Mathematical ociety, the Mathematical Association of America, and the American Association 3r the Advancement of Science. Dr. Easley received the Ph.D. degree (1991) in lathematics from the University of Missouri, Columbia, where he studied ~rentziangeometry under Professors Beem and Ehrlich. ~ V I NL.
John K. Beem
Paul E. Ehrlich Kevin L. Faslev
GLOBAL LORENTZIAN GEOMETRY
PURE AND APPLIED MATHEMATICS
MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS Earl J . Taft Rutgers University New Brunswick, New Jersey
Zuhair Nashed University of Delaware Newark, Delaware
EDITORIAL BOARD M. S. Baouendi University of California, Sun Diego Jane Cronin Rugers University Jack K. Hale Georgia Institute of Technology
Anil Nerode Cornell University Donald Passman University of Wisconsin, Madison Fred S. Roberts Rugers University
S. Kobayashi University of California, Berkeley
Gian-CarloRota Massachusetts Institute of Technology
Marvin Marcus University of California, Santa Barbara
David L. Russell Virginia Polytechnic Institute and State University
W. S. Massey Yale University
Walter Schempp Universitiit Siegen
Mark Teply University of Wisconsin, Milwaukee
1. K. Yano, Integral Formulas in Riemannian Geometry (1970) 2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) 3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, ed.; A. Littlewood, trans.) (1970) 4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation ed.; K. Makowski, trans.) (1971) 5. L. Narici et a/., Functional Analysis and Valuation Theory (1971) 6. S. S. Passman, Infinite Group Rings (1971) 7. L. Dornhoff, Group Representation Theory. Part A: Ordinary Representation Theory. Part B: Modular RepresentationTheory (1 971, 1972) 8. W. Boothby and G. L. Weiss, eds., Symmetric Spaces (1972) 9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, trans.) (1972) 10. L. E. Ward, Jr., Topology (1972) 11. A. Babakhanian, Cohomological Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener lntegral (1973) 14. J. Barros-Nero, lntroduction to the Theory of Distributions (1973) 15. R. Larsen, Functional Analysis (1973) 16. K. Yano and S. lshihara, Tangent and Cotangent Bundles (1973) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonn&, lntroduction to the Theory of Formal Groups (1973) 21. 1. Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry of Submanifolds (1973) 23. M . Marcus, Finite Dimensional Multilinear Algebra (in t w o parts) (1973, 1975) 24. R. Larsen, Banach Algebras (1973) 25. R. 0.Kujala andA. 1. Vitrer, eds., Value Distribution Theory: Part A; Part 8 : Deficit and Bezout Estimates by Wilhelm St011 (1973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1 974) 29. J. Satake, Linear Algebra (S. Koh et al., trans.) (1975) 30. J. S. Golan, Localization of Noncommutative Rings (1975) 31. G. Klambauer, Mathematical Analysis (1975) 32. M . K. Agoston, Algebraic Topology (1976) 33. K. R. Goodearl, Ring Theory (1976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman, Matrix Theorv and Its ADDlications (1976) 36. B. R. ~ c ~ o n a l ~eometric d, Algebra over Local Rings (1976) 37. C. W. Groetsch, Generalized Inverses of Linear Operators (1977) J. E. Kuczkowskiand J. L. Gersting, Abstract Algebra (1977) C. 0.Christenson and W. L. Voxman, Aspects of Topology (1977) M . Nagata, Field Theory (1977) R. L. Long, Algebraic Number Theory (1977) W. F. Pfeffer, lntegrals and Measures (1977) R. L. Wheeden and A. Zygmund, Measure and Integral (1977) J. H. Curtiss, lntroduction to Functions of a Complex Variable (1978) K. Hrbacek and T. Jech, lntroduction to Set Theory (1978) W. S. Massey, Homology and Cohomology Theory (1978) M. Marcus, lntroduction to Modern Algebra (1978) E. C. Young, Vector and Tensor Analysis (1978) S. B. Nadler, Jr., Hyperspaces of Sets ( 1 978) S. K. Segal, Topics in Group Kings (1978) A. C. M. van Rooij, Non-ArchimedeanFunctional Analysis (1978) L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979) C.Sadosky, interpolation of Operators and Singular Integrals (1979)
54. 55. 56. 57. 58. 59. 60. 61. 62.
J. Cronin, Differential Equations (1980)
C. W. Groetsch, Elements of Applicable Functional Analysis (1980) 1. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980) H. I. Freedan, Deterministic Mathematical Models in Population Ecology (1980) S. B. Chae, Lebesgue Integration (1980) C. S. Rees etal., Theory and Applications of Fourier Analysis (1981) L. Nachbin, lntroduction to Functional Analysis (R. M . Aron, trans.) (1981) G. Onech and M. Orzech, Plane Algebraic Curves (1981) R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical Analysis (1981) 63. W. L. Voxman and R. H. Goetschel, Advanced Calculus (1981) 64. L. J. Corwin and R. H. Szczarba, Multivariable Calculus (1982) 65. V. I. Ist@tescu, lntroduction t o Linear Operator Theory (1981) 66. R. D.Jsirvinen, Finite and Infinite Dimensional Linear Spaces (1981) 67. J. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981) 68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981) 69. J. W. Brewer and M . K. Smith, eds., Emmy Noether: A Tribute (1981) 70. K. H. Kim, Boolean Matrix Theory and Applications (1982) 71 T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments (1982) 72. D. B.Gauld, Differential Topology (1982) 73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983) 74. M. Carmeli, Statistical Theory and Random Matrices (1983) 75. J. H. Carruth et a/., The Theory of Topological Semigroups (1983) 76. R. L. Faber, Differential Geometry and Relativity Theory (1983) 77. S. Bamett, Polynomials and Linear Control Systems (1983) 78. G. Karpilovsky, Commutative Group Algebras (1983) 79. F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1983) 80. 1. Vaisman, A First Course in Differential Geometry (1984) 81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984) 82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984) 83. K. Goebeland S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings (1984) 84. T. Albu and C. Ngstasescu, Relative Finiteness in Module Theory (1984) 85. K. Hrbacek and T. Jech, lntroduction t o Set Theory: Second Edition (1984) 86. F. Van Oystaeyen and A. Verschoren, Relative lnvariants of Rings (1984) 87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984) 88. M. Namba, Geometry of Projective Algebraic Curves 11984) 89. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics (1985) 90. M. R. Bremner et a/., Tables of Dominant Weight Multiplicities for Representations of Simple Lie Algebras (1985) 91. A. E. Fekete, Real Linear Algebra (1985) 92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985) 93. A. J. Jerri, lntroduction t o Integral Equations w i t h Applications (1985) 94. G. Karpilovsky, Projective Representations of Finite Groups (1985) 95. L. Nariciand E. Beckenstein, Topological Vector Spaces (1985) 96. J. Weeks, The Shape of Space (1985) 97. P. R. Gribik and K. 0. Kortanek, Extremal Methods of Operations Research (1985) 98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic Analysis (1986) 99. G. D. Crown etal., Abstract Algebra (1986) 100. J. H. Carruth etal., The Theory of Topological Semigroups, Volume 2 (1986) 101. R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras (1986) 102. M . W, Jeter, Mathematical Programming (1986) 103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution Equations with Applications (1986) 104. A. Verschoren, Relative lnvariants of Sheaves (1987) 105. R. A. Usmani, Applied Linear Algebra (1987) 106. P. Blass and J. Lang, Zariski Surfaces and Differential Equations in Characteristic p > 0 (1987) 107. J. A. Reneke eta/., Structured Hereditary Systems (1987) 108. H. Busemann and B. B. Phadke, Spaces w i t h Distinguished Geodesics (1987) 109. R. Harre, lnvertibility and Singularity for Bounded Linear Operators (1988)
.
110. G. S. Ladde e t a/., Oscillation Theory of Differential Equations w i t h Deviating Arguments (1987) 111. L. Dudkin et al., Iterative Aggregation Theory (1987) 112. T. Okubo, Differential Geometry (1987) 113. D. L. Stancland M. L. Stancl, Real Analysis with Point-Set Topology (1987) 114. T. C. Gard, lntroduction t o Stochastic Differential Equations (1988) 115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1 988) 116. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their Representations (1988) 117. J. A. Huckaba, Commutative Rings w i t h Zero Divisors (1988) 118. W. D. Wallis, Combinatorial Designs (1988) 119. W. Wipslaw, Topological Fields (1988) 120. G. Karpilovsky, Field Theory (1988) 121. S. Caenepeel and F. Van Oystaeyen, Brauer Groups and the Cohomology of Graded Rings (1989) 122. W. Kozlowski, Modular Function Spaces (1988) 123. E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps (1989) 124. M. Pavel, Fundamentals of Pattern Recognition (1989) 125. V. Lakshmikantham eta/., Stability Analysis of Nonlinear Systems (1989) 126. R. Sivaramakrishnan, The Classical Theory of Arithmetic Functions (1989) 127. N. A. Watson, Parabolic Equations on an lnfinite Strip (1989) 128. K. J. Hastings, lntroduction t o the Mathematics of Operations Research (1989) 129. B. fine, Algebraic Theory of the Bianchi Groups (1989) 130. D. N. Dikranjan et a/., Topological Groups (1989) 131. J. C. Morgan 11, Point Set Theory (1990) 132. P. Biler and A. Witkowski, Problems in Mathematical Analysis ( 1 990) 133. H. J. Sussmann, Nonlinear Controllability and Optimal Control (1990) 134. J.-P. Florens et a/., Elements of Bayesian Statistics (1 990) 135. N. Shell, Topological Fields and Near Valuations (1990) 136. B. F. Doolin and C. F. Martin, lntroduction t o Differential Geometry for Engineers (1990) 137. S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method (1990) 138. J. Oknirfski, Semigroup Algebras (1990) 139. K. Zhu, Operator Theory in Function Spaces (1990) 140. G. B. Price, An lntroduction t o Multicomplex Spaces and Functions (1991) 141. R. B. Darst, lntroduction t o Linear Programming (199 1) 142. P. L. Sachdev, Nonlinear Ordinary Differential Equations and Their Applications (1991) 143. T. Husain, Orthogonal Schauder Bases (1991) 144. J. Foran, Fundamentals of Real Analysis (1991) 145. W. C. Brown, Matrices and Vector Spaces (1991) 146. M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces (1991) 147. J. S. Golan and T. Head, Modules and the Structures of Rings (1991) 148. C. Small, Arithmetic of Finite Fields (1991) 149. K. Yang, Complex Algebraic Geometry (1991) 150. D. G. Hoffman etal., Coding Theory (1991) 151. M. 0. Gonzdlez, Classical Complex Analysis ( 1 992) 152. M. 0. Gonzdlez, Complex Analysis (1992) 153. L. W. Baggett, Functional Analysis (1992) 154. M. Sniedovich, Dynamic Programming (1992) 155. R. P. Agarwal, Difference Equations and Inequalities (1992) 156. C. Brezinski, Biorthogonality and Its Applications t o Numerical Analysis (1992) 157. C. Swarrz, An lntroduction to Functional Analysis (1992) 158. S. B. Nadler, Jr., Continuum Theory (1992) 159, M. A. Al-Gwaiz, Theory of Distributions (1992) 160. E. Perry, Geometry: Axiomatic Developments with Problem Solving (1992) 161. E. Castillo and M. R. Ruiz-Cobo, Functional Equations and Modelling in Science and Engineering (1992) 162. A. J. Jerri, Integral and Discrete Transforms with Applications and Error Analysis (1992) 163. A. Charlier e t a/., Tensors and the Clifford Algebra (1992) 164. P. Biler and T. Nadzieja, Problems and Examples in Differential Equations (1992) 165. E. Hansen, Global Optimization Using Interval Analysis (1992)
183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202.
S. Guerre-Delabridre, Classical Sequences i n Banach Spaces (1992) Y. C. Wong, Introductory Theory of Topological Vector Spaces (1992) S. H. Kulkarniand B. V. Limaye, Real Function Algebras (1992) W. C. Brown, Matrices Over Commutative Rings (1993) J. Loustau and M. Dillon, Linear Geometry with Computer Graphics (1993) W. V. Petryshyn, Approximation-Solvability of Nonlinear Functional and Differential Equations (1993) E. C. Young, Vector and Tensor Analysis: Second Edition (1993) T. A. Bick, Elementary Boundary Value Problems (1993) M. Pavel, Fundamentals of Pattern Recognition: Second Edition (1993) S. A. Albeverio eta/., Noncommutative Distributions (1993) W. Fulks, Complex Variables (1993) M. M. Rao, Conditional Measures and Applications (1993) A. Janicki and A. Weron, Simulation and Chaotic Behavior of a-Stable Stochastic Processes (1 994) P. Neittaanmakiand D. Tiba, Optimal Control of Nonlinear Parabolic Systems (1994) J. Cronin, Differential Equations: lntroduction and Qualitative Theory, Second Edition (1994) S. Heikkila and V. Lakshmikantham, Monotone Iterative Techniques for Discontinuous Nonlinear Differential Equations (1994) X. Mao. Exoonential Stabilitv of Stochastic Differential Eauations 11994) B. S. ~ h o m i o nSymmetric , Properties of Real Functions (i994) J. E. Rubio. Ootimization and Nonstandard Analvsis (1994) J. L. Bueso eral., Compatibility, Stability, and sheaves (1995) A. N. Micheland K. Wang, Qualitative Theory of Dynamical Systems (1995) M. R. Darnel, Theory of Lattice-Ordered Groups (1995) Z. Naniewicz and P. D. Panagiotopoulos, Mathematical Theory of Hemivariational Inequalities and Applications (1995) L. J. Convin and R. H. Szczarba, Calculus in Vector Spaces: Second Edition (1995) L. H. Erbe eta/., Oscillation Theory for Functional Differential Equations (1995) S. Aaaian et a/.. Binaw Polvnomial Transforms and Nonlinear Diaital Filters 11995) M. l.v~i/',~ o r m ' ~ s t i m a t i o nfor s Operation-Valued Functions and-~pplications(1995) P. A. Grillet, Seminroups: An lntroduction t o the Structure Theorv (1995) S. ~ichenassam~,- onl linear Wave Equations (1996) V. F. Krotov, Global Methods i n Optimal Control Theory (1996) K. I. Beidar eta/., Rings with Generalized Identities (1996) V. I. Arnautov et a/., lntroduction t o the Theory of Topological Rings and Modules (1996) G. Sierksma, Linear and Integer Programming (1996) R. Lasser, lntroduction t o Fourier Series (1996) V. Sima, Algorithms for Linear-Quadratic Optimization (1996) D. Redmond, Number Theory (1996) J. K. Beem eta/. Global Lorentzian Geometry: Second Edition (1996) Additional Volumes in Preparation
GLOBAL LORENTZIAN GEOMETRY Second Edition
John K. Beem Department of Mathematics University of Missouri- Columbia Columbia, Missouri
Paul E. Ehrlich Department of Mathematics University of Florida- Gainesville Gainesville, Florida
Kevin L. Easlev d
Department of Mathematics Truman State University Kirksville, Missouri
Marcel Dekker, Inc.
New YorkeBasel Hong Kong
PREFACE TO THE SECOND EDITION Library of Congress Cataloging-in-Publication Data Beem, John K. Global Lorentzian geometry. -2nd ed. /John K. Beern, Paul E. Ehrlich, Kevin L. Easley. p. cm. - (Monographs and textbooks in pure and applied mathematics ; 202) Includes bibliographical references and index. ISBN 0-8247-9324-2 (pbk. : alk. paper) 1. Geometry, Differential. 2. General relativity (Physics). I. Ehrlich, Paul E. 11. Easley, Kevin L. 111. Title. IV. Series. QA649.B42 1996 51 6 . 3 ' 7 6 ~ 2 0 96-957 CIP The publisher offers discounts on this book when ordered in bulk quantities. For more information, write to Special Sales/Professional Marketing at the address below.
The second edition of this book continues the study of Lorentzian geometry, the mathematical theory used in general relativity. Chapters 3 through 12 contain material, slightly revised in some cases, which was discussed in Chapters 2 through 11 of the first edition. Much new material has been added to Chapters 7 and 11, and new Chapters 13 and 14 have been written reflecting the more
complete and detailed understanding that has been gained in the intervening years on many of the topics treated in the first edition. Inspired by an example of P. Williams (1984), additional material on the instability of both geodesic completeness and geodesic incompleteness for general space-times has been provided in Section 7.1. Section 7.4 has been added giving sufficient conditions on a spacetime to guarantee the stability of geodesic completeness for metrics in a neighborhood of a given complete metric. New material has also
This book is printed on acid-free paper. Copyright O 1996 by MARCEL DEKKER, INC. All Rights Reserved. Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher.
been added to Section 11.3 on the topic of geodesic connectibility. Appendixes A, B, and C of the first edition have now been expanded into Chapter 2, which also contains new material on the generic condition as well as Section 2.3, which gives a proof that the null cone determines the metric up to a conformal factor in any semi-Riemannian manifold which is neither positive nor negative definite. Also, a deeper treatment of the behavior of the sectional curvature function in a neighborhood of a degenerate two-plane is given in Chapter 2.
MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016
In the concluding Chapter 11 of the first edition, which is now Chapter 12, we showed how the Lorentzian distance function and causally disconnecting
Current printing (last digit): 1 0 9 8 7 6 5 4 3 2 1
sets could be used to obtain the Hawking-Penrose Singularity Theorem concerning geodesic incompleteness of the space-time manifold. Around 1980, S. T. Yau suggested that the concept of "curvature rigidity," well known for
PRINTED IN THE UNITED STATES OF AMERICA
iv
PREFACE TO THE SECOND EDlTION
PREFACE T O THE SECOND EDITION
v
the differential geometry of complete Riemannian manifolds since the publica-
present, it has been interesting to observe the enormous expansion in the
tion of Cheeger and Ebin (1975), might be applied to the seemingly unrelated
journal literature on space-time differential geometry. This is reflected in the
topic of singularity theorems in space-time differential geometry. (Earlier. Ge-
substantial growth of the list of references for the second edition. However,
roch (1970b) had suggested that spatially closed space-times should fail to be
this wealth of new material has precluded our treating many interesting new
timelike geodesically incomplete only under special circumstances.) As a step
developments in space-time geometry since 1980 which are less closely tied in
toward this conjectured rigidity of geodesic incompleteness, the Lorentzian
with the overall viewpoint and selection of topics originally discussed in the
analogue of the Cheeger-Gromoll Splitting Theorem for complete Riemann-
first edition.
ian manifolds of nonnegative Ricci curvature needed to be obtained. This was accomplished in a series of research papers published between 1984 and
comments about the first edition and urged us to issue a second edition after
The authors would like to thank all those who have provided encouraging
in the new Chapter 14. Another new chapter in the second edition, Chapter
the first edition had gone out of print, especially Gregory Galloway, Steven Harris, Andrzej Kr6lak, Philip Parker, and Susan Scott. We thank Gerard
1990. Aspects of the proof of the Lorentzian Splitting Theorem are discussed 13, draws upon investigations of Ehrlich and Emch (1992a,b,c, 1993) and is
Emch for insisting that the second edition be undertaken, and the first two
devoted to a study of the geodesic behavior and causal structure of a class
authors thank Stephen Summers and Maria Allegra for independently sug-
of geodesically complete Ricci flat space-times, the gravitational plane waves,
gesting that a third author be added to the team to share the duties of the
which were introduced into general relativity as astrophysical models. These
completion of this revision. It is also a pleasure to thank Maria Allegra and
space-times provide interesting and nontrivial examples of astigmatic conju-
Christine McCafferty at Marcel Dekker, Inc., for working with us to see the
gacy [cf. Penrose (1965a)l and of the nonspacelike cut locus, a concept dis-
second edition into print. We are also indebted to Lia Petracovici for much helpful proofreading and to Todd Hammond for valuable technical advice con-
cussed in Chapter 8 of the first edition. As for the first edition, this book is written using the notations and meth-
cerning A M S W .
ods of modern differential geometry. Thus the basic prerequisites remain a
John K. Beem
working knowledge of general topology and differential geometry. This book
Paul E. Ehrlich
should be accessible to advanced graduate students in either mathematics or
Kevin L. Easley
mathematical physics. The list of works to which we are indebted for the two editions is quite extensive. In particular, we would like to single out the following five important sources: The Large Scale Stmcture of Space-time by S. W. Hawking and G. I?. R. Ellis; Techniques of Diflerential Topology in Relativity by R. Penrose; Riemannsche Geometric im Grossen by D. Gromoll, W. Klingenberg, and W. Meyer; the 1977 Diplomarbeit a t Bonn University, Exzstenz und Bedeutung von konjugierten Werten in der Raum-Zeit, by G. Bolts; and Semi-Riemannian Geometry by B. O'Neill. In the time from the late 1970's when we wrote the first edition to the
PREFACE TO THE FIRST EDITION
This book is about Lorentzian geometry, the mathematical theory used in general relativity, treated from the viewpoint of global differential geometry. Our goal is t o help bridge the gap between modern differential geometry and the mathematical physics of general relativity by giving an invariant treatment of global Lorentzian geometry. The growing importance in physics of this approach is clearly illustrated by the recent Hawking-Penrose singularity theorems described in the text of Hawking and Ellis (1973). The Lorentzian distance function is used as a unifying concept in our book. Furthermore, we frequently compare and contrast the results and techniques of Lorentzian geometry to those of Riemannian geometry to alert the reader to the basic differences between these two geometries. This book has been written especially for the mathematician who has a basic acquaintance with Riemannian geometry and wishes to learn Lorentzian geometry. Accordingly, this book is written using the notation and methods of modern differential geometry. For readers less familiar with this notation, we have included Appendix A which gives the local coordinate representations for the symbols used. The basic prerequisites for this book are a working knowledge of general topology and differential geometry. Thus this book should be accessible t o advanced graduate students in either mathematics or mathematical physics. In writing this monograph, both authors profited greatly from the opportunity to lecture on part of this material during the spring semester, 1978, a t the University of Missouri-Columbia. The second author also gave a series of lectures on this material in Ernst Ruh's seminar in differential geometry a t Bonn University during the summer semester, 1978, and would like to thank
...
vlll
PREFACE TO THE FIRST EDITION
Professor Ruh for giving him the opportunity to speak on this material. We would like to thank C. Ahlbrandt, D. Carlson, and M. Jacobs for several help ful conversations on Section 2.4 and the calculus of variations. We would like to thank M. Engman, S. Harris, K. Nomizu, T. Powell, D. Retzloff, and H. Wu for helpful comments on our preliminary version of this monograph. We also thank S. Harris for contributing Appendix D to this monograph and J.-H. Es-
CONTENTS
chenburg for calling our attention to the Diplomarbeit of Bolts (1977). To anyone who has read either of the excellent books of Gromoll, Klingenberg, and Meyer (1975) on Riemannian manifolds or of Hawking and Ellis (1973) on general relativity, our debt to these authors in writing this work will be
Preface to the Second Edition
iii
Preface to the First Edition
vii
List of Figures
xiii
obvious. It is also a pleasure for both authors to thank the Research Council of the University of Missouri-Columbia and for the second author to thank the Sonderforschungsbereich Theoretische Mathematik 40 of the Mathematics Department, Bonn University, and to acknowledge an NSF Grant MCS7718723(02) held at the Institute for Advanced Study, Princeton, New Jersey,
for partial financial support while we were working on this monograph. Finally it is a pleasure to thank Diane Coffman, DeAnna Williamson, and Debra Retzloff for the patient and cheerful typing of the manuscript.
1. Introduction: Riemannian Themes in Lorentzian Geometry
2. Connections and Curvature 2.1 Connections
1
15
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .16
John K. Beem
2.2 Semi-Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Paul E. Ehrlich
2.3 Null Cones and Semi-Riemannian Metrics . . . . . . . . . . . . . . . . . . . . 25 2.4 Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.5 The Generic Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.6 The Einstein Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 3. Lorentzian Manifolds and Causality
49
3.1 Lorentzian Manifolds and Convex Normal Neighborhoods . . . . . . 50 3.2 Causality Theory of Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.3 Limit Curves and the C" Topology on Curves . . . . . . . . . . . . . . . . . 72 3.4 Two-Dimensional Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.5 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
.......................................... 94 3.7 Semi-Riemannian Local Warped Product Splittings . . . . . . . . . . . 117 3.6 WarpedProducts
CONTENTS
x
CONTENTS
.
4 Lorentzian Distance
.
135
9 T h e Lorentzian C u t Locus 295 9.1 The Timelike Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 298 9.2 The Null Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9.3 The Nonspacelike Cut Locus . . . . . . . . . . . . . . . . . . . . . . . . . . . . .311 9.4 The Nonspacelike Cut Locus Revisited . . . . . . . . . . . . . . . . . . . . 318
............................. 135 Distance Preserving and Homothetic Maps .................. 151 The Lorentzian Distance Function and Causality ............. 157 Maximal Geodesic Segments and Local Causality ............. 166
4.1 Basic Concepts and Definitions 4.2 4.3 4.4
.
5 Examples of Space-times 173 5.1 Minkowski Space-time .................................... 174 5.2 Schwarzschild and Kerr Space-times ........................ 179
.............................. Space-times ............................
5.3 Spaces of Constant Curvature 5.4 Robertson-Walker
5.5 Bi-Invariant Lorentzian Metrics on Lie Groups
.
6 Completeness a n d Extendibility
6.1 Existence of Maximal Geodesic Segments
181 185
............... 190
I I
11. S o m e Results i n Global Lorentzian G e o m e t r y
399 11.1 The Timelike Diameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401 11.2 Lorentzian Comparison Theorems . . . . . . . . . . . . . . . . . . . . . . . .406 11.3 Lorentzian Hadamard-Cartan Theorems . . . . . . . . . . . . . . . . . . 411
.................... 198
......................................... 214 6.5 Local Extensions ......................................... 219 6.6 Singularities ............................................. 225 6.4 IdealBoundaries
.
7 Stability of Completeness and Incompleteness 239 7.1 Stable Properties of Lor(M) and Con(M) ................... 241 7.2 The C1 Topology and Geodesic Systems ..................... 247 7.3 Stability of Geodesic Incompleteness for Robertson-Walker Space-times . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 7.4 Sufficient Conditions for Stability ........................... 263 8. Maximal Geodesics and Causally Disconnected Space-times
.
10 Morse Index T h e o r y o n Lorentzian Manifolds 323 10.1 The Timelike Morse Index Theory ........................ 327 10.2 The Timelike Path Space of a Globally Hyperbolic Space-time ............................................ 354 10.3 The Null Morse Index Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
197
6.2 Geodesic Completeness .................................... 202 6.3 Metric Completeness ...................................... 209
271 8.1 Almost Maximal Curves and Maximal Geodesics ............. 273 8.2 Nonspacelike Geodesic Rays in Strongly Causal Space-times ... 279 8.3 Causally Disconnected Space-times and Nonspacelike GwdesicLines ........................................... 282
xi
.
12 Singularities 425 12.1 JacobiTensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 426 12.2 The Generic and Timelike Convergence Conditions . . . . . . . . . 433 12.3 Focal Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444 12.4 The Existence of Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467 12.5 SmoothBoundaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472
.
13 Gravitational P l a n e Wave Space-times 479 13.1 The Metric. Geodesics. and Curvature . . . . . . . . . . . . . . . . . . . . 480 13.2 Astigmatic Conjugacy and the Nonspacelike Cut Locus . . . . . 486 13.3 Astigmatic Conjugacy and the Achronal Boundary . . . . . . . . . 493
.
14 T h e 14.1 14.2 14.3 14.4 14.5
Splitting P r o b l e m i n Global Lorentzian G e o m e t r y 501 The Busemann Function of a Timelike Geodesic Ray . . . . . . . . 507 Co-rays and the Busemann Function . . . . . . . . . . . . . . . . . . . . . . 519 The Level Sets of the Busemann Function . . . . . . . . . . . . . . . . . 538 The Proof of the Lorentzian Splitting Theorem ............. 549 Rigidity of Geodesic Incompleteness ...................... 563
xii
CONTENTS
Appendixes 567 Jacobi Fields and Toponogov's Theorem for Lorentzian A. Manifolds by Steven G. Harris ........................... 567 From the Jacobi, to a Riccati, t o the Raychaudhuri B. Equation: Jacobi Tensor Fields and the Exponential Map Revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .573
!
1 LIST OF FIGURES
References List of Symbols
Figure
Page
Brief Description of Figure
Index chronological and causal future basis of Alexandrov topology, I f ( p ) n I - ( q ) reverse triangle inequality for p 0
< E < R, the metric ball B ( p , E) is geodesically convex. Thus for any E with 0 < E < R, the set B(p, E) is diffeomorphic to the such that for any
results for the Lorentzian distance function.
if X and Y are arbitrary smooth vector fields on N , the function N
for all p E N and
E
with 0
n-disk, n = dim(N), and the set {q E N : do(p,q) =
E)
is diffeomorphic to
sn-1 Removing the origin from R2 equipped with the usual Euclidean metric and setting p = (-1, O), q = (1, O), one calculates that do(p, q) = 2 but finds no curve c E R,,, with Lo(c) = &(p,q) and also no smooth geodesic from p to q. Thus the following questions arise naturally. Given a manifold N, find conditions on a Riemannian metric go for N such that (1) All geodesics in N may be extended to be defined on all of R.
as follows. Let Q , , denote the set of piecewise smooth curves in N from p to q. Given a curve c E flp,, with c : [O,1] + N , there is a finite partition 0 = tl < tz < . . . < t k = 1 such that c 1 It,, ti+l] is smooth for each i. The Riemannian arc length of c with respect to go is defined as
(2) The pair (N, do) is a complete metric space in the sense that all Cauchy sequences converge. (3) Given any two points p,q E N, there is a smooth geodesic segment c E Q,,, with Lo(c) = do(p, q).
A distance realizing geodesic segment as in (3) is called a minimal geodesic segment. The word minimal is used here since the definition of Riemannian
4
1
INTRODUCTION
distance implies that Lo(y) 2 do(p, q) for all y E R,,,.
RIEMANNIAN THEMES IN LORENTZIAN GEOMETRY
More generally, one
may define an arbitrary piecewise smooth curve y E R,,, to be minimal if Lo(y) = &(p, q). Using the variation theory of the arc length functional, it is minimal, then y may be reparametrized to a may be shown that if y f a,,, smooth geodesic segment.
5
tition of unity argument, it follows that N also admits a complete Riemannian metric. We now turn our attention to the Lorentzian manifold (M, g). A Lorentzian metric g for the smooth paracompact manifold M is the assignment of a nondegenerate bilinear form glp : T,M x TpM -+ R with diagonal form (-,
+, . . . , +)
The question of finding criteria on go such that (I), (2), or (3) holds was
to each tangent space. It is well known that if M is compact and x ( M ) # 0,
resolved by H. Hopf and W. Rinow in their famous paper (1931). In modern
then M admits no Lorentzian metric. On the other hand, any noncompact
terminology the Hopf-Rinow Theorem asserts the following.
manifold admits a Lorentzian metric. Geroch (1968a) and Marathe (1972)
T h e o r e m (Hopf-Rinow).
For any Riemannian manifold (N, go) the fol-
metric is paracompact.
lowing are equivalent: (1) Metric completeness: (N, &) is a complete metric space. (2) Geodesic completeness: For any v E T N , the geodesic c(t) in N with c'(0) = v is defined for all positive and negative real numbers t E R. (3) For some p E N , the exponential map exp, is defined on the entire tangent space TpN t o N at p. (4) Finite compactness: Every subset K of N that is do bounded (i.e., sup{&(p, q) : p, q E K)
have also shown that a smooth Hausdorff manifold which admits a Lorentzian
< m) has compact closure.
Furthermore, if any one of (1) through (4) holds, then
Nonzero tangent vectors are classified as timelike, spacelike, nonspacelike,
< 0, > 0, 5 0, or = 0, respectively. [Some authors use the convention (+, -, . . . ,-) for the Lorentzian metric,
or null according to whether g(v,v)
and hence all of the inequality signs in the above definition are reversed for them.] A Lorentzian manifold (M, g) is said to be time oriented if M admits a continuous, nowhere vanishing, timelike vector field X. This vector field is used to separate the nonspacelike vectors a t each point into two classes called future directed and past directed. A space-time is then a Lorentzian manifold (M,g) together with a choice of time orientation. We will usually work with
(5) Minimal geodesic connectibility: Given any p, q E N , there exists a smooth geodesic segment c from p to q with Lo(c) = do(p, q).
A Riemannian manifold (N,go) is said to be complete provided any one (and
space-times below. In order to define the Lorentzian distance function and discuss its properties, we need to introduce some concepts from elementary causality theory. It is
0.
(spacelike plane)
T h e sectional curvature o f the nondegenerate plane section E with basis
30
2
24
CONNECTIONS AND CURVATURE
{v, w) where v = C v i d / d x i and w = C wid/dxi is then given by
SECTIONAL CURVATURE
31
constructed which have all timelike sectional curvatures bounded in one direction [cf. Harris (1979)j. However, if dim(M)
> 3 and if the sectional curvatures
of all nondegenerate planes are bounded either from above or from below, then the sectional curvature is constant [cf. Kulkarni (1979)l. Sectional curvature -
R w , v, w, V) g(v, v)g(w1w) - [g(v,w)I2
(sectional curvature)
has been further investigated by Dajczer and Nomizu (1980b), Nomizu (1983), Beem and Parker (1984), and Cordero and Parker (1995a,c).
A semi-Riemannian manifold (M, g) which has the same sectional curvature on all (nondegenerate) sections is said to have constant curvature. If (M,g) has For positive definite manifolds the Ricci curvature evaluated at a unit vector
constant curvature c, then R(X, Y ) Z = c [g(Y, Z ) X - g(X, Z)Y] [cf. Graves
is sometimes thought of as more or less an average sectional curvature weighted
and Nomizu (1978)l. Thorpe (1969) showed that the sectional curvature can
by a factor of (n- I). More precisely, let w be a unit vector at p in the Riemann-
only be continuously extended to degenerate planes in the case of constant
ian manifold (M, g), extend to an orthonormal basis {el, ez, . ..,en-l, en = w),
curvature. Spaces of constant sectional curvature have been investigated in
and let Ei = span{ei, w) for 1 5 i 5 n - 1. Equations (2.22) and (2.37) yield
connection with the space form problem [cf. Wolf (1961, 1974)l.
.
Ric(w, w) =
-
C g ( ~ ( e iW)W, , ei) = C K(p, E,).
The curvature tensor, Ricci curvature, scalar curvature, and sectional curvature may all be calculated in local coordinates using the metric tensor components and the first two partial derivatives of these con~por~ents. Thus, the
It is instructive to contrast the above with the Ricci curvature evaluated a t
metric tensor determines the curvatures. In contrast, curvature does not nec-
a unit timelike vector in a Lorentzian manifold. In this case the interpretation
essarily determine the metric. Nevertheless, for most Lorentzian manifolds
is in terms of the negative of the timelike sectional curvature. For let u be a
the metric will be determined either completely or up to a constant conformal
unit timelike vector at a point p of the Lorentzian manifold (M,g). Extend to
factor given sufficient information concerning curvature in local coordinates
a n orthonormal basis {el, e2,. . . ,en-1, en = u) and let Ei = span{ei, u) for
[cf. Hall (1983, 1984), Hall and Kay (1988), Ihrig (1975), Quevedo (1992)j.
I
5 i 0, then in some sense the
"average" sectional curvature for planes in the pencil of u is negative. For Riemannian manifolds one has a number important "pinching" theorems [cf. Cheeger and Ebin (1975)l. However, similar results fail for Lorentzian
-+
-+
M is a
T M along c is a Jacobi field
if it satisfies the Jacobi equation, (2.38)
J" + R(J,cJ)cJ= 0,
where J" = V,,(V,lJ).
(Jacobz equatzon)
In an intuitive sense, one thinks of a Jacobi field
manifolds. In particular, if the sectional curvatures of timelike planes are
as representing the relative displacement of "nearby" geodesics [cf. Hawking
> 3, then (M,g) has constant
and Ellis (1973), Hicks (1965), or Misner, Thorne and Wheeler (1973)l. In
curvature [cf. Harris (1982a), Dajczer and Nomizu (l980a)l. Nevertheless, fam-
particular, let c be a unit speed timelike geodesic. Then c represents the path
ilies of Lorentzian manifolds conformal to ones of constant curvature may be
of a "freely falling" particle moving a t less than the speed of light. Taking J
bounded both above and below, and if dim(M)
32
2
2.5
CONNECTIONS AND CURVATURE
as a Jacobi field which is orthogonal to the tangent vector c', one interprets J
2.5
THE GENERIC CONDITION
The Generic Condition
as a vector from the original particle to another particle moving on a nearby
The sectional curvature can be used to study the generic condition, which
timelike geodesic, and one interprets J" as the relative (or tidal) acceleration of the second particle as measured by the first. If g(J, J )
# 0, then the definition
of sectional curvature together with g(cl, c') = -1 and g(J, c') = 0 yield
will be of importance in the singularity theorems t o be considered in C h a p ter 12. If W =
Wad/dxais a tangent vector, then the values W a are the contravariant components. Using the metric g one has values Wb = Cr=lgabWawhich are the covariant components. Thus, Wbdxb is the
xr=l
cotangent vector corresponding to the original W . The generic condition is
At points where J does not vanish, the vector J / d m is a unit vector in the direction of J. Using J" = -R(J, cl)c', one finds that the radial component
said to be satisfied for a vector W at p E M if
of the tidal acceleration is given by
n
wCwdwfa R
(2.40)
~wfl# ~ 0. ~ (generic ~ [ condition) ~
c.d=l
~f condition (2.40) fails to hold (i.e., if C:,,=,WCWdWI,Rblcdl,Wfl = o), then
W will be called nongeneric. The generic condition is said to hold for a geodesic c : (a, b)
-+
M if at some
point c(t0) the tangent vector to the geodesic is generic, i.e., one has (2.40) This equation shows that for "close" particles the radial component of the tidal acceleration varies directly with the separation distance IJI and with the sectional curvature K(cl, J ) of the plane containing both c' and J. Thus, using
satisfied with W = cl(to). Notice from continuity that if (2.40) holds for some W = cl(to), then it will hold for W = cl(t) whenever t is sufficiently close to to.
our sign conventions, a timelike plane with positive sectional curvature cor-
We will show in Proposition 2.6 that for a vector W which is nonnull (i.e.,
responds to freely falling particles accelerating away from each other, and
g(W, W ) # O), the nongeneric condition is equivalent to requiring that all
negative sectional curvature of a timelike plane corresponds to particles accel-
plane sections containing W have zero sectional curvature. For null vectors,
erating toward each other. Since Ric(cl, c')
> 0 corresponds to the timelike
the nongeneric condition is slightly more complicated.
planes containing c' having negative average sectional curvature, it follows that Ric(cl, c') > 0 corresponds to average attractive (i.e., focusing) tidal forces. It should be kept in mind that some authors use different sign conventions and may have sectional curvature equal to the negative of ours. For a constant value of I JI the maximum tidal acceleration will be radial
Lemma 2.5. Let Rabcdrepresent the components of the curvature tensor with respect to an oi.thonormal basis { v l ,vz, . . . ,v,) of T,M.
The vector W = vn satisfies the generic condition (2.40) iff there exist b and e with
1 < b, e
< n - 1 such that Rbnne# 0.
[cf. Beem and Parker (1990, p. 820)j. Thus at any fied value to, a n observer traversing the timelike geodesic c will have zero tidal accelerations if and only if all planes E containing cl(to) have zero sectional curvature.
Proof. The components of W are given by W 1 =
. .. = Wn-1 = 0, W" = 1, and W, = +l or
. . . = Wn-'
-1. Consequently,
= Wl =
34
2
CONNECTIONS AND CURVATURE
2.5
THE GENERIC CONDITION
35
Proposition 2.6 implies that the only way for a timelike geodesic c to fail to satisfy the generic condition is for the corresponding observer to fail to ever experience any tidal accelerations. -
wa Rbnn j We + Wb Rann j W e )
- bna Rbnnj bne
+ bnb Rannj S n e ) .
In Chapter 12 we will make use o f an alternative formulation o f the generic condition using the curvature tensor. Let c be a unit speed timelike geodesic, p = c(to), and W = cl(to). T h e set of vectors orthogonal t o W is an ( n - 1)-dimensional linear subspace W' = V L ( c ( t o ) lying ) in T p M , and the metric induced on this linear subspace is positive definite. Thus v L ( c ( t o ) )is
It is easily seen that this expression is nonzero i f and only i f Rbnne # 0 for
a spacelike hyperplane in T p M . I f y E T p M ,then g ( W ,R(y, W ) W ) = 0 which
someb,ewithl~b,e~n-1.
implies that R ( y , W ) W lies in V L ( c ( t o ) ) .It follows that the curvature tensor
Proposition 2.6. If W E T p M is a nonnull vector, then W fails to be
R induces a linear map from V L ( c ( t o ) )to V L ( c ( t o ) ) :
generic (i.e., is nongeneric) i f ffor each nondegenerate plane section E containing W the sectional curvature K ( p ,E ) vanishes. Set W = vn and extend to an orthonormal basis { v l ,v2, . . . ,v,) o f T p M . Let
Since g I V L ( c ( t o ) )is positive definite, this curvature map is nontrivial i f and ) g ( y n , R ( y l ,W ) W )# 0. T h e only i f there arc vectors yl, y2 f V L ( c ( t o ) with
Rabcdrepresent the components o f the curvature tensor with respect t o this
next result shows this map is nontrivial i f and only i f W is generic.
Proof. W e may assume without loss o f generality that W is a unit vector.
basis. From Lemma 2.5, i f W fails t o satisfy the generic condition, then Rbnne = 0 for all 1 5 b, e 5 n - 1. This and the skew symmetry of Rabcdin both the first pair o f indices and the second pair o f indices yield Rbnne = 0 for all 1 5 b, e 5 n. Hence Rnbne = -Rbnne = 0 for all such b and e. I t follows that i f U is another tangent vector at p, one has z:,b,c,e=lRabceWaUbWCUe =
Proposition 2.7. If W = cl(to) is a timelike vector in the Lorentzian
manifold ( M ,g ) , then the following three conditions are equivalent: ( 1 ) The timelike vector W is generic. ( 2 ) At least one plane containing W has nonzero sectional curvature. (3) R( . , W ) W is not the trivial map.
~ ~ , eR,~,,U~U" = l = 0. Fkom (2.37), we conclude that i f E is the plane spanned by { W ,U),and i f E is nondegenerate, then K ( p ,E ) = 0. Conversely, assume all nondegenerate planes containing W have zero sec-
lent. Let Rabcdrepresent the components o f the covariant curvature tensor with
tional curvature. Then (2.37) easily shows that all terms o f the form Rbnnb must be zero. Assume b # e. Notice that E = span{vb,vb + 2ve) cannot
isfies the generic condition, then Lemma 2.5 implies that -Rbnen = Rbnne # 0
-
-
be degenerate. Using (2.37) we obtain R(vn,vb, v,, vb) = R(vn,v,, v,, v,) =
-
-
R(vn, vb + 2ve,vn, vb + 2ve) = 0. T h e multilinearity of R and standard curva-
-
+ 2ve,v,, vb + 2ve) = 4 @v,,
Proof. Clearly, Proposition 2.6 shows the first two conditions are equivarespect to an orthonormal basis { v l , v 2 , .. .vn-l,vn = W ) o f T p M . I f W satfor some 1 5 b, e 5 n - 1. Consequently, g(vb,R ( v e ,v n ) v n ) # 0 which shows R(v,, vn)vn is a nontrivial vector. Thus, when W is generic the map R( . ,W ) W is not the trivial map.
vb, v,, we).
Conversely, assume that R ( . , W ) W is not trivial. Let yl, yz be vectors in
This shows f i b n e = 0 and hence Rbnne= 0. Using Lemma 2.5, it follows that W fails t o satisfy the generic condition as desired. U
W L = spanjvl, v2, . . . ,vn-l) with g(y2,R ( y l ,vn)vn) # 0. This last inequality Z , X , Y ) = g(W, R ( X , Y ) Z ) yield b together with the multilinearity o f 2(~,
ture identities then yield 0 = R(vn,vb
2
36
2.5
CONNECTIONS AND CURVATURE
<
THE GENERIC CONDITION
37
W and finds an orthonormal basis {en-', en) of E with W = (en-l +en)/&,
and e with 1 5 b, e n - 1 such that Rbnne# 0. Thus, Lemma 2.5 shows that W is generic as desired.
en-1 spacelike, and en timelike. Then one extends to an orthonormal basis {el, e2,. . . , e n ) and sets vi = ei for 1
The next proposition shows that a sufficient condition for the nonnull vector
vn-1
W to be generic is that Ric(W, W) # 0.
=N =
2m) x
S2.
Then M is topologically lR2 x S2. Let A = 0 and T = 0, and set dR2 = do2
+ ~ i n ~ ( e ) d $Then ~ . M with this A and T admits both the flat metric
ds2 = -dt2
+ dr2 + r2dR2 as well as the Schwarzschild metric
(= R) to the energy-momentum tensor
T. The tensor T is to be determined from physical considerations dealing with
(2.59)
ds2 = -
dr2
+ r2dR2
(Schwarzschild)
the distribution of matter and energy [cf. Hawking and Ellis (1973, Chapter 3), Misner, Thorne, and Wheeler (1973, Chapter 5)]. The Einstein equations
as solutions to the Einstein equations. Each of these rnetrics is asymptotically
may be written invariantly as
flat, and each is Ricci JEat (i.e., Ric = 0). However, the Schwarzschild metric
(2.57)
1 Ric - - R g + A g = 87rT 2
(Einstein equations)
has a nonzero curvature tensor, and hence the two metrics cannot be isometric. Nevertheless, a counting argument shows that, in general, one expects the Ein-
where A is a constant known as the cosmological constant. The constant factor
stein equations to determine the metric up to diffeomorphism [cf. Hawking and
of 87r is present for scaling purposes. In local coordinates, one has
Ellis (1973, p. 74)j. First, notice that the metric tensor g has 16 components
1
(2.58) Rij - -Rgij 2
+ h g i j = 87rTij
which, by symmetry, reduce to ten independent components. Furthermore, four of these ten components can be accounted for by the dimension of M (Einstein equations in coordinates)
which allows four degrees of freedom. Thus the metric tensor is thought of
46
2
2.6
CONNECTIONS AND CURVATURE
T H E EINSTEIN EQUATIONS
47
This last equation shows that the condition Ric(v,v) 2 0 is equivalent to
as having six independent components after symmetry and diffeomorphism freedom are taken into account. Consequently, the Einstein equations yield
the inequality T(v, v) 2 [tr(T)/2 - A/8n] g(v, v). If follows that when A = 0
six independent equations to determine six essential components of the metric
and dim(M) = 4, the condition
tensor. More rigorous approaches to the problem of existence and uniqueness of solutions to the Einstein equations using Cauchy surfaces with initial data may
is equivalent to the condition
be found in a number of articles and books such as Chrusciel (1991), Hawking and Ellis (1973, Chapter 7), Marsden, Ebin, and Fischer (1972, pp. 233-264), and Choquet-Bruhat and Geroch (1969). The Einstein equations may be used to relate the timelike convergence condition (Ric(v, v) 2 0 for all timelike, and hence also all null vectors v) t o the
[cf. Hawking and Ellis (1973, p. 95)]. Note that (2.60) and (2.61) show that if A = 0, then T = 0 (i.e., vacuum) is equivalent to Ric = 0 (i.e., Ricci flat).
energy-momentum tensor. In order to evaluate the scalar curvature R in terms
The Einstein equations are fundamental in the construction of cosmological
of T at p E M , let {el, e 2 , . . . , e n ) be a n orthonormal basis of T,M, and use
models. Consider a fluid which moves through space. This motion generates
equation (2.57) to obtain
timelike flow lines in space-time. Let v be the unit speed timelike vector field which is everywhere tangent to the flow lines of the fluid. The fluid is said to be a perfect fluid if it has an energy density p , pressure p, and energy-momentum tensor T such that
Using the fact that the scalar curvature R is the trace of the Ricci curvature,
(2.62)
this last equation becomes
T = (p
+ p) w 8 u + p g,
(perfect fiuid)
which is
Tij = (p+p)vivj + p g i j in local coordinates. Here w = z v i d x i is the one-form corresponding to the
Hence,
vector field v =
R = - 8 tr(T) ~ + 411.
(2.60)
The Einstein equations become 1
Ric - -2 ( - 8 ~tr(T)
+ 4A)g + Ag = 8x2".
4
(2.63) Thus,
via/dxi. It follows from the above form of T that a perfect
fluid is an isotropic fluid which is free of shear and viscosity. Let (M,g) be a manifold for which T has the above perfect fluid form. If the vectors {el, e:!, e3, e4) form an orthonormal basis for T,M, then the trace of T may be calculated as follows: tr(T) =
g(ei, el) T(e,, e,) i=l = -(P+P) $ 4 ~ = 3p - p.
2
48
CONNECTIONS AND CURVATURE
Using equation (2.61), it follows that the timelike convergence condition for a perfect fluid is equivalent to
CHAPTER 3
for all timelike (and null) w. For Lorentzian manifolds, it is easy to verify
LORENTZIAN MANIFOLDS AND CAUSALITY
that the inner product of a timelike vector and another timelike (or nontrivial null) vector is nonzero. Thus, we may assume without loss of generality that g(v, w)
# 0.
Using equation (2.62) we obtain
Sections 3.1 and 3.2 give a brief review of elementary causality theory basic to this monograph as well as to general relativity. Then Section 3.3 describes an important relationship between the limit curve topology and the C0 topology for sequences of nonspacelike curves in strongly causal space-times. Namely,
which simplifies to
if y : [a,b ]
--+
M is a future directed nonspacelike limit curve of a sequence
{y,) of future directed nonspacelike curves, then a subsequence converges to y in the C0topology. This result is useful for constructing maximal geodesics in Since g(w, w) 5 0 and g(v, w)
# 0, equation (2.64) shows that a negative
cosmological constant has the effect of making the timelike convergence condition more plausible and that a positive cosmological constant has the opposite effect. Einstein originally introduced the cosmological constant because the Einstein equations with A = 0 predict a universe which is either expanding or contracting, and in the early part of this century it was believed that the universe was essentially static. After the discovery that the universe was expanding, the original motivation for the cosmological constant was removed;
strongly causal spacetimes using the Lorentzian distance function (cf. Chapter
8 and Chapter 12, Section 4). In Section 3.4 we study the causal structure of two-dimensional Lorentzian manifolds. In particular, we show that if (M, g) is a space-time homeomorphic to R2, then (M,g) is stably causal. Section 3.5 gives a brief discussion of the theory of Lorentzian submanifolds and the second fundamental form needed for our discussion of singularity theory in Chapter 12.
however, removing A from the theory has been more difficult. While astro-
An important splitting theorem of Geroch (1970a) guarantees that a glob-
nomical experiments have failed to detect a A different from zero, one may
ally hyperbolic space-time may be written as a topological (although not nec-
always argue that A is so small that the experiments have not been sufficiently sensitive. Discussions of the experimental evidence for general relativity may be found in a number of books such as Misner, Thorne, and Wheeler (1973) and Will (1981).
essarily metric) product R x S where S is a Cauchy hypersurface. This result suggests that product space-times of the form (R x M, -dt2 @ g) with (M, g) a Riemannian manifold should be studied. While this class of space-times includes Minkowski space and the Einstein static universe, it fails to include the physically important exterior Schwarzschild and Robertson-Walker solutions to Einstein's equations. In Sections 3.6 and 3.7 we study a more general class of product space-times, the so-called warped products, which are spacetimes MI x f M2 with metrics
50
3
of the form gl
LORENTZIAN MANIFOLDS AND CAUSALITY
@I fg2.
This class of metrics, studied for Riemannian manifolds
3.1
LORENTZIAN MANIFOLDS, CONVEX NORMAL NEIGHBORHOODS
51
More precisely, we have the following definition.
by Bishop and O'Neill (1969) and later for semi-Riemannian manifolds by Robertson-Walker space-times. The following result, which may be regarded
Definition 3.1. (Space-time) A space-time (M,g) is a connected Cm Hausdorff manifold of dimension two or greater which has a countable basis,
as a "metric converse" to Geroch's splitting theorem, is typical of the results
a Lorentzian metric g of signature (-,
O'Neill (1983), includes products, the exterior Schwarzschild space-times, and
of Section 3.6. Let (R x M, -dt2 @ g) be a Lorentzian product manifold with (M, g) an arbitrary Riemannian manifold. Then the following are equivalent: (1) (M, g) is a complete Riemannian manifold. @I g)
We now show how to construct a time oriented two-sheeted Lorentzian covering manifold n : ( E , i j )
-+
(M,g) for any Lorentzian manifold ( M , g)
which is not time orientable.
(2) (R x M , -dt2 @ g) is globally hyperbolic.
(3) (R x M , -dt2
. . ,+), and a time orientation.
+?.
To this end, first let (M,g) be an arbitrary Lorentzian manifold. Fix a base point PO E M . Give a time orientation to TpoM by choosing a timelike
is geodesically complete.
tangent vector vo E TpoM and defining a nonspacelike w E TpoM to be future
3.1
Lorentzian Manifolds a n d Convex N o r m a l Neighborhoods
Let M be a smooth connected paracompact Hausdorff manifold, and let
TM denote the tangent bundle of M with n : T M
-+
M the usual bundle map
[respectively, past] directed if g(v0, w) < 0 [respectively, g(v0, w)
> 01. Now let
q be any point of M . Piecewise smooth curves y : [O, 11 -+ M with y(0) = po and $1) = q may be divided into two equivalence classes as follows. Given
for M is a smooth symmetric tensor field of type (0,2) on M such that for each
M with ~ l ( 0 )= ~ ( 0 = ) PO and yl(1) = y2(1) = 91 let Vl (respectively, T/z) be the unique parallel field along y l (respectively, yz) with
p f M, the tensor glp : TpM x TpM
Vl(0) = V2(0) = vo. Wesay that yl and 7 2 areequivalent ifg(Vl(l),V2(1)) < 0.
taking each tangent vector to its base point. Recall that a Lorentzzan metric g
of signature (1, n
- 1) [i.e., (-,
-+
R is a nondegenerate inner product
+, . . . , +)I.
All noncompact manifolds admit
Lorentzian metrics. However, a compact manifold admits a Lorentzian metric if and only if its Euler characteristic vanishes [cf. Steenrod (1951, p. 207)]. The space of all Lorentzian metrics for M will be denoted by Lor(M).
A continuous vector field X on M is timelike if g(X(p),X(p)) < 0 for all
~ 1 ~ :7 [0,11 2
If yl and
72
-+
are homotopic curves from po to q, then yl and 7 2 are equivalent.
But equivalent curves are not necessarily homotopic. Given y : [O, 11 -+ M with y(0) = po, let [y] denote the equivalence class of y. Let
2 consist of all such equivalence classes of piecewise smooth curves
y : [O, 11 -+ M with $0) = PO. Define n :
points p E M . In general, a Lorentzian manifold does not necessarily have
is time orientable, then M = M .
globally defined timelike vector fields. If (M,g) does admit a timelike vector
covering [cf. Markus (1955, p. 412)].
field X E X(M), then (M,g) is said to be time oriented by X . The timelike
z
-+
Otherwise,
M by ~ ( [ y ]= ) y(1). If (M, g) T
:
%
-+
M is a two-sheeted
Suppose now that the Lorentzian manifold (M, g) is not time orientable. It
ii;j a
topology and dif-
vector field X divides all nonspacelike tangent vectors into two separate classes,
is standard from covering space theory to give the set
called future and past directed. A nonspacelike tangent vector v E T,M is
ferentiable structure such that n : M + M is a two-sheeted covering manifold.
said to be future [respectively, past] directed if g(X(p), v)
< 0 [respectively,
g(X(p),v) > 01. A Lorentzian manifold (M,g) is said to be time onentable if (M,g) admits a time orientation by some timelike vector field X. In this
Define a Lorentzian metric Then the map
7r
:
z
-+
- by ij = n*g, i.e., ;(v,
g for M
w) = g ( ~ , vn,w). ,
M is a local isometry.
In order to show that (%,g) is time orientable, it is useful to establish a
Fo E T - ' ( ~ ~for) M. Let Go E T~,Mbe
case, (M,g) admits two distinct time orientations defined by X and -X,
preliminary lemma. Fix a base point
respectively. A time oriented Lorentzian manifold is called a space-time.
the unique timelike tangent vector in TFoM with r,Go = VO.
-
3
52
3.1
LORENTZIAN MANIFOLDS AND CAUSALITY
LORENTZIAN MANIFOLDS, CONVEX NORhlAL NEIGHBORHOODS
53
two
if, fixing any base point po E M and timelike tangent vector vo E Tp,M, the
piecewise smooth curves with Yl(0) = 72(O) = Po and ;j;l(l) = 72(1) = g. If
following condition is satisfied for all q f M. Let yl,yz : [0,1] --+ M be any
fi, - 1/2 are the parallel vector fields along y1 and y2, respectively, with
two smooth curves from p to q. If V , is the unique parallel vector field along
Lemma 3.2. Let
qE
be arbitrary and let yl,;j;2 : [0, I]
V2(0) = 6, then g(v1(1), %(I)) Proof. Let y l = .n o yl and
< 0.
7 2 = 7r o
-
72,
(0) =
yi with V,(O) = vo for i = 1,2, then g(V1(l),V2(1)) < 0. This condition means that parallel translation of the future cone determined by vo at po to any
72. Since .n : ( M , $ )
a local isometry, the vector fields Vl = .n,(vl) and fields along 71 and
% be
-+
---t
(M, g) is
6 = 7riT,(v2)are parallel
-
respectively, with Vl(0) = h(O) = n,vo = vo. Also,
g ( K ( l ) , Wl))= g(x*G(l),x*%(l)) = g ( G ( l ) >%(I)). Suppose now that $(Ql(l), V2(1)) y! 0. Since ?1(1) and ?2(1) are timelike tangent vectors, it follows that 5(?1(1), V2(1)) > 0. Thus g(Vl(l), h ( 1 ) )
>0
at q = .n(@). By definition of the equivalence relation on piecewise smooth curves from po to q, we have [yl] # [y2]. F'rom the construction of M , we know that T1(l) = [yl] and T2(l) = Im]. Thus %,(I) # 72(1), in contradiction.
other point q of M is independent of the choice of path from p to q. Hence a consistent choice of future timelike vectors for each tangent space may be made by parallel translation from po. Recall that a smooth curve in (M,g) is said to be timelike (respectively, nonspacelike, null, spacelike) if its tangent vector is always timelike (respectively, nonspacelike, null, spacelike). As in the Riemannian case, a geodesic c : ( a ,b)
-+
M is a smooth curve whose tangent vector moves by parallel dis-
placement, i.e., V,jcl(t) = 0 for all t E (a,b). The tangent vector field cl(t) of a geodesic c satisfies g(cl(t), cl(t)) = constant for all t 6 (a,b) since
Theorem 3.3. Suppose that (M,g) is not time orientable. Then the twosheeted Lorentzian covering manifold
(%,$)
of (M,g) constructed above is
time orientable and hence is a space-time.
Consequently, a geodesic which is timelike (respectively, null, spacelike) for
T E %, let o : [O, 11 % be a smooth curve with a(0) = Po,a(1) = $ Let V be the unique parallel vector
some value of its parameter is timelike (respectively, null, spacelike) for all
<
The exponential map expp : TpM + M is defined for Lorentzian manifolds
0). By Lemma 3.2, the definition of F + ( a ) is independent of the choice of
just as for Riemannian manifolds. Given v E TpM, let cu(t) denote the unique
Proof. Let
& and Go be as above.
Given any
-+
field along a with ?(o) = Go. Set F + ( q ) = (timelike w E T& : ~ ( v ( l )w) , u. Hence if+
T ~ % ,q E
F+(T) consistently assigns a future cone t o each tangent space
G.
Now let h be an auxiliary positive definite Riemannian metric for %. We by choosing may define a continuous nowhere zero timelike vector field X on
values of its parameter.
geodesic in M with cv(0) = p and cul(0) = v. Then the exponential exp,(v) of v is given by expp(v) = c v ( l ) provided c,(l) is defined. Let vl, v2, . . . ,vn be any basis for the tangent space TpM. For sufficiently small (21, 22,. . . , x,) E Rn, the map
X ( 5 ) to be the vector in F + ( q ) which is the unique h-unit vector in F + ( q ) having a negative eigenvalue of continuous function X : M
g with respect
to h. That is, we may find a
xlvl
+ 22212 + . . . + X n V n --' ~ X P ~ (+X2 ~2 ~U+ 2 ~. . . + xnvn)
(-a, 0) and a continuous timelike vector field
is a diffeomorphism of a neighborhood of the origin of T,M onto a neigh-
X ( q ) E F+(q"), h(X(T), X ( c ) ) = 1, and $(X(g), v) = X(y) h(X(q), v) for all v f T& and 5 E %. U
borhood U(p) of p in M . Thus, assigning coordinates (xl,x2,. . . , x n ) to the
Implicit in the proof of Theorem 3.3 is an alternative definition for the time orientability of a Lorentzian manifold (M, g). Namely, ( M g) is time orientable
called normal coordinates based at p for U(p). The set U(p) is said to be a (simple) convex neighborhood of p if any two points in U(p) can be joined by
X on
+
% satisfying
point expp(xlvl
+ 22712 + . . . + xnvn) in U(p) defines a coordinate chart for M
3
54
LORENTZIAN MANIFOLDS AND CAUSALITY
3.2
a unique geodesic segment of (M,g) lying entirely in U(p). Whitehead (1932) has shown that any semi-Riemannian (hence Lorentzian) manifold has convex
CAUSALITY THEORY OF SPACE-TIMES
55
For a given p E M , the chronological future Ii(p), chronological past I-(p), causal future Ji(p), and causal past J-(p) of p are defined as follows:
neighborhoods about each point [cf. Hicks (1965: pp. 133-136)]. In fact, it may even be assumed that for each q E U(p), there are normal coordinates based
I f (p) = { q f M
a t q containing U(p). We call such a neighborhood U(p) a convex normal
I-(p) = { q E M : q 0
that (3.25)
-[4'(t)I2dim H < 2d1'(t)< -[4'(t)12
provided that
4" < min{G(O),G ( P ) ) ,which yields inequality
(3.26).
for all t E ( a ,b). Thus i f Ric2(v,v) 2 0 for all v E T H and condition (3.25)
W e now consider the scalar curvature o f warped product manifolds o f the
holds, the space-time (v,ij) will have everywhere positive Ricci curvatures. A globally hyperbolic family o f such space-times is provided by warped prod-
form = R x f H , 3 = -dt2 @ f h . W e will let n = dim H below. Given ( t , p ) E M ,choose el E T,H for 1 < j < n such that i f Zl = (0,e,) E T(t,p)%,
ucts
=
= (0,co) x f H , where ( H ,h ) is a complete Riemannian manifold of
-
-
then { d l & = ( d l d t ,O,),El,. . . ,En) forms a 3-orthonormal basis for T ( t , p ) M .
116
LORENTZIAN MANIFOLDS AND CAUSALITY
3
SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS
3.7
117
Hence {mel,. . . , m e , } forms an h-orthonormal basis for T p H . Thus
Lorentzian metric ijx = -dt2 @ eXth,i.e., f (t) = ext. By Theorem 3.70, for all
if r : 2 -+ W and
X
TH : H
-+
W denote the scalar curvature functions of ( 2 , 3 )
and (H,h) respectively, we have
> 0 the space-time (Rn+',?jx) is future null geodesically complete but past null geodesically incomplete, and for all X < 0, the space-time (Rn+1,3x) is past null geodesically complete but future null geodesically incomplete. Using formulas (3.28), (3.29), and (3.30), we obtain
and and Now formulas (3.23) and (3.24) above simplify to
Thus if X
# 0, (zA,?jx)
is an Einstein space-time with constant positive scalar
curvature.
and
E x a m p l e 3.74. Let li?x = (0,co) x f x3, where gA = -dt2 @ f h with
> 0, and h the usual Euclidean metric on LR3. It is then immediate from formula (3.31) that r(gX)= 0 for all X > 0. Since d ( t ) = ln(Xt), it may
f (t) = At, X
for 1 5 j 5 n. Consequently, we obtain the formula 1 r ( t , p) = - r ~ (p)
f (4
+ n+I1(t) +
1 (n2
be checked using formulas (3.28) and (3.29) that
+ n) [+'(t)12.
nor Einstein for any X
(z~, 3,) is neither Ricci flat
> 0. Also we have for any X > 0 that
Recalling that $(t) = In f (t), this may be rewritten as are "inextendible across" for all t > 0. It follows that the space-times ( z A ,?jA) where dim H = n as above. In particular, in the case that n = 3 as in general relativity, we obtain the simpler formula
(0) x R3 (cf. Section 6.5). Also, ( R x , g x )is future null geodesically complete
by Theorem 3.70.
3.7
Semi-Riemannian Local W a r p e d P r o d u c t Splittings
In each of the exact solutions to Einstein's equations which are presented E x a m p l e 3.73. With the formulas of this section in hand, we are now
as warped product manifolds, the warped product decomposition emerges as a
ready to give an example of a 1-parameter family gAof nonisometric Einstein
natural mathematical expression of assumed physical symmetries. Moreover,
metrics for Wn+' such that for X = 0, (Wn+l,go) is Minkowski space-time of
formulas for warped product curvatures (cf. Proposition 3.76) indicate that any
dimension n + 1. Let (Wn, h) be Euclidean n-space with the usual Euclidean
semi-Riemannian manifold (M, g) must possess certain measures of symmetry
+
metric h = dx12 dx22+ - . . +dzn2, and put MA = lipn+' = W x f Wn with the
and flatness in order to be (locally or globally) isometric to a warped product
118
3
LORENTZIAN MANIFOLDS A N D CAUSALITY
3.7
SEMI-RIEMANNIAN LOCAL WARPED P R O D U C T SPLITTINGS
119
B x F. In this section we identify geometric conditions on a semi-Riemannian
The geometry of a warped product B x f F is expressed through the ge-
manifold (M,g) which are necessary and sufficient to ensure that (M,g) is
ometries of the base (B,gB) and fiber ( F , g F ) and various derivatives and
locally isometric to a warped product B x f F. We shall call such a local
integrals of the warping function f E 5(B). We will consider warped products
isometry a 'Llocalwarped product splitting."
M = B x f F where both (B, g s ) and (F,gF) may be semi-Riemannian man-
There are a number of different "splitting" or decomposition theorems
ifolds (thus generalizing the Lorentzzan warped products of Definition 3.51). :M
B and a
: M -+
F denote the standard projections.
throughout differential geometry. For example, the splitting theorem of Geroch
The symbols
(1970a), Theorem 3.17 in this chapter, demonstrates that a globally hyperbolic
As in portions of Section 3.6, we will find it convenient to consider the square
space-time may be written as a particular type of topological product but not
root S of the warping function f . Throughout this sectzon, we wzll adhere t o
necessarily as a metric product. By contrast, a well-known result of de Rham
the conventzon that S ( b ) =
[cf. Kobayashi-Nomizu (1963, p. 187)] asserts that a complete simply con-
warped product wzth metric tensor
T
-+
for b E B where M = B x f F denotes a
nected Riemannian manifold which has a reducible holonomy representation is isometric to a Riemannian product. Along very different lines, Chapter 14 provides the following Lorentzian analogue of the Cheeger-Gromoll Splitting Theorem: if (M, g) is a space-time of dimension n 2 3 which (1) is globally hyperbolic or timelike geodesically complete, (2) satisfies the timelike convergence condition, and (3) contains a complete timelike line, then (M, g) is isometric to a product (R x V, -dt2
$ h ) , where
(V,h ) is a complete Riemann-
ian manifold. Since a product manifold is trivially a warped product, either
The function S will be called the root warpzng functzon. As defined in Section 3.6, the lzft of f E 5 ( B ) to a function defined by the formula
jE
S ( M ) is
7= f o T. The lifted function will simply be denoted
by f as well, when no ambiguity results. A vector field V E X(B) is lifted to
M by defining
E X(M) in such a manner that at each (p, q) E M , T/(p, q)
of these last two results clearly provides sufficient conditions to ensure that a
is the unique vector in the tangent space T(,,,]M such that both d ~ ( ? )= V
manifold is globally isometric to a warped product. However, the examples
and do(?) = 0. Similar definitions apply for lifts from F to M. We shall
(S1 x H, 3) of non-globally hyperbolic warped product space-times discussed
also denote lifted vector fields without the tildes, and we write V E C ( F ) to
in Section 3.6 indicate that causal assumptions such as global hyperbolicity
denote a vector field on M lifted from F. More generally, vectors tangent to
are not necessary for the existence of a global warped product splitting.
leaves B x q are called honzontal while those tangent t o fibers p x F are called
The global splitting question typically involves rather delicate topological considerations; our focus in this section will be on the simpler local warped
product splitting question: given a semi-Riemannian manifold (M,g) and a point p E M, what conditions are necessary and suficient for the existence of a n open neighborhood U of p such that the submanzfold (U,g lu) is isometric t o a warped product B x F ? The following assumption will be needed.
vertzcal. Lifts of covariant tensors on B and F are now defined in the obvious way through the use of the pullbacks
T*
and a*.
Let D denote the Levi-Civita connection on M, and use V to denote the Levi-Civita connections on both B and F. The curvature tensors on B x f F = M may be characterized through their actions on lifted horizontal and vertical vector fields. In the following proposition, the symbols B R and
F~
denote the
lifts t o M of the Riemannian curvature tensors on B and F , respectively. The
Convention 3.75. It will be assumed throughout the remainder of this section that the neighborhood U mentioned above is a connected, simply connected, open set.
-
symbol H S will be used to denote
5,the lift to M of the Hessian of S. Note
that in general H S ( X , Y) = H'(x, Y) only for horizontal vector fields X , Y. For notational simplicity, the bracket notation ( , ) will be occasionally used
3
120
LORENTZIAN MANIFOLDS AND CAUSALITY
3.7
SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS
121
to denote the metric g on M; the metrics on B and F will always be denoted
for all vector fields X , Y, and 2. The unit vector field i ( d / d y 2 ) does not have
by g~ and g ~ . Some basic curvature formulas for warped product manifolds are now given
vanishing curl, in general. These observations lead to the following result.
in Proposition 3.76 for ease of reference. The standard reference for this material is O'Neill (1983).
have Riemannian curvature tensor R, Ricci curvature Ric, and root warping
a.Assume X, Y, Z
E C ( B ) and U, V, W E C(F).
-
(1) If h f 5(B), then the gradient of the lift h o 7r of h to M = B x f F is the lift to M of the gradient of h on B, i.e., grad (71) = grad h. (2) DxY f L(B) is the lift of VxY on B, i.e., (3) DxV = D v X =
I V
D ~ =F(VxY).
(9) V.
l?.ic(X,Y) -
(4) Ric(X, Y) =
($) H'(x,
Y) where d = dim F.
(5) Ric(V, X ) = 0.
an open neighborhood of p.
Proof. The local existence of a nonvanishing Killing field about any point of a two-dimensional semi-Riemannian warped product was noted above. Conversely, suppose there exists a neighborhood U about p E M having a nonvanishing Killing field V. If the Killing field is null, then it is well known that (U, g lu) is isometric to (a portion of) Minkowski two-space RT and hence is (trivially) a warped product. If the Killing field V is not null, then we may complete the classical construction of local geodesic (or Fermi) coordinates (xl, x 2 ) [cf. do Carmo (1976)]
(6) Ric(V, W) = Ric(V, W) - (V,W)Sd where Sn =
and a point p E M, p has a neighborhood U such that (U, g lu) is isometric to
a warped product if and only if there exists a non-vanishing Killing field on
Proposition 3.76. Let the semi-Riemannian warped product M = B x f F function S =
L e m m a 3.77. Given a two-dimensional semi-Riemannian manifold (M, g)
9 + (d - ,1-)
d = dimF, and AS is the
Laplacian of the root warping function S on B.
such that V = d/dx2 on U and x1 measures g-arc length along the geodesics orthogonal to the integral curves of V. In these coordinates the metric assumes the form ds2 = €1 dzl @ dxl
Consider first the local warped product splitting question in dimension two. Assume a semi-Riemannian surface (M,g) is a warped product so that in the appropriate local coordinates (yl, y2) adapted to B and F , the metric is given by (3.34)
+ c2[s(y1)I2dy28 dy2
the elementary fact that a coordinate vector field d/dxk is Killing if and only
& = 0 for all i,j). Each (d/dyl)-integral
curve is a geodesic of M (more
generally, each leaf B x q of a warped product is a totally geodesic submanifold). Thus d/dy2 restricted to y is a Jacobi field for each such coordinate geodesic y, and of course, {dl&', d/dy2) = 0 along y. Further, direct calculation shows that d/dyl is ir-rotational-that
is, curl(d/dyl) = 0, where the curl of a vector
field is the skew-symmetric (0,2) tensor defined through the formula (3.35)
where ci = & I , i = 1,2. Since V = d/dx2 is Killing, we must have
[curlX](Y, 2 ) = (DUX,2 ) - (Y, D z X )
= 0,
yielding ds2 = €1 dxl 8 dxl
g = cldyl 8 dyl
where ci = r t l , i = 1,2. It is immediate that d/dy2 is a local Killing field (recall if
+ €2 F'(xl, x2)dx2 8 dx2 + €2 F'(x1)dx2 8 dx2
and leading to the desired local warped product representation.
0
In generalizing the preceding result to higher dimensions, the existence of a Killing field V must be supplemented by an integrability condition which allows the construction of leaves B x q orthogonal to V. This integrability condition holds trivially in dimension two but need not hold in higher dimensions where, in general, a Killing field need not be irrotational. It is also necessary to include the additional assumption that the Killing field be nonnull. Recall that a space-time (M, g) is called static if there exists on M a nowhere zero timelike Killing field X such that the distribution of (n- 1)-planes orthogonal to X is integrable. The following result parallels the formal construction
124
3
3.7
LORENTZIAN MANIFOLDS AND CAUSALITY
and a : (x1,x2,. . . , x n )
--t
(0,0,. . . ,O,xn). For arbitrary X , Y f X(U) with
coordinate basis expansions X =
Xi(d/dxi), Y =
Yi(d/dxi), we have
SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS
125
of dimension two. The following computational lemma is a necessary preliminary step. In the following result, the symbols "K and B K will be used to denote the sectional curvatures of F and B , respectively. Since these objects are bnctions on the surfaces F and B, they may be unambiguously lifted to M as well. Proposition 3.79. Let M = B x f F be a 4-dimensional semi-Riemannian warped product with dim B = dim F = 2 and root warping function S = fl. For M to be Ricci fiat, it is necessary and sufficient that (1) F have constant sectional curvature F K , (2)
B x f F with dim B = 1. One answer to this question involves a construction which bears strong similarities to the formal development of the RobertsonWalker cosmological models [cf. O'Neill (1983)l. Recall that a signal feature of Robertson-Walker space-time is the presence of a proper time synchronizable geodesic observer field U . Since the observer field U is irrotational, the infinitesimal rest spaces of U may be integrated to provide local rest spaces. Through a construction quite similar to that of the preceding lemma, it is possible to verify the following [cf. Easley (1991)]: given an n-dimensional (n 2 3)
+ gB(grad S, grad S )
= F K on B , where F~ is the
constant value from (1) and A denotes the Laplacian on B, and
The "dimensional dual" of the preceding result asks for conditions necessary and sufficient to ensure that (M, g) is locally isometric to a warped product
= sAs
(3)
D x(grad S ) =
(p ) X for all X E L(B).
Further, if M is Ricci flat, then
A s-BK (4) -3-
onB,
and the sectional curvatures and root warping function are related as follows:
(5) K . S3= 2 C,w on B, where CM is a const ant, and (6) FK = B K . S2 g~ (grad S, grad S ) on B.
+
Proof. Using Proposition 3.76, we see that M is Ricci flat if and only if
semi-Riemannian manifold and point p E M I the point p has a neighborhood
(3.36)
Ric(X, Y) - ~H'(X, Y) = 0
M such that (U,glu) is isometric to a warped product B x j F with dim B = 1 and dim F = (n - 1) if and only if (1) there exists an irrotational
(3.37)
Ric(V, W) - (V, W)S$ = 0
unit vector field U (either spacelike or timelike) on a neighborhood of p such
for all X , Y E C ( B ) and V,W E C(F), where SQ=
that (2) the flow J[r induced by U acts as a positive homothety on the local
denotes the Laplacian on B.
U
( n - 1)-dimensional submanifolds which are everywhere orthogonal to U near
and
*
and A
On the semi-Ftiemannian surface B we have
p. Condition (2) is of course equivalent to a number of different geometric
conditions expressible in terms of curvature and the connection. The local splitting problem in the context of Ricci flat (M,g) has an exsion four which will not be subsumed under more general results, namely, the
with the analogous formula holding on the surface F (here the symbol Ric denotes the Ricci tensor on B and not its lift to M). Projecting equations
case when M is locally isometric to a warped product with base and fiber each
(3.36) and (3.37) onto B and F respectively, we see that M is Ricci flat if and
tremely restricted class of solutions. We consider first the only case in dimen-
3
126
3.7
LORENTZIAN MANIFOLDS AND CAUSALITY
where
only if the following two conditions hold:
F~
SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS
127
denotes the constant value of the sectional curvature of F. Differ-
entiating both sides of equation (3.41) produces . g B ( X , Y ) = $H'(x,Y), FK . g ~ ( W) v = s ~ ~ F ( vW)S' ,
(3.38)
B~
(3.39)
and
+B = S 2 ( x B K )+ B = s 2 ( x B ~+) 3 . B
for all X , Y E X(B) and V,W E X(F). Equation (3.39) must hold as we project from each fiber p x F into F. Since S is constant on fibers we see that equation (3.39) will hold if and only if F~ =
+ 2gB(vx(grad 2 ~S ) , grad~S ) ~ + S . B2 ~ ( ~ ~ ~ ~ )
0 =s2(xBK)
Thus, X(BK) = -
constant, and
f 2 ~ g = S A S + g ~ ( g r a d S , g r a d S ) = F ~OnB.
(3 .
~
~
*K(xS)
~
)
(
~
~
for all
)
~ ~
.
x t X(B)
yielding
Rewriting the Hessian as H S ( x , Y) = gB(Vx(grad S), Y), equation (3.38) is equivalent to and this holds for all lifts X E C(B). It follows that S3. B K is constant on B (and hence its lift is constant on M ) . which in turn holds if and only if Vx(grad S ) =
It is now possible to characterize all (2 x 2) Ricci flat warped products
$) X
( B ~ .
M = B x j F with base B of constant curvature.
for all X E X(B).
Corollary 3.80. If M = B x f F is a four-dimensional Ricci flat warped
That conditions (I), (2) and (3) are necessary and sufficient for M to be Ricci
product with base B a surface of constant curvature, then M is simply a
flat now follows. Proposition 3.76-(2) is used to obtain the lifted form of
product manifold B x G where B and G are flat two-manifolds. Thus the
condition (3).
metric on M is semi-Euclidean.
From condition (3) it follows that for a local orthonormal frame field Eo,El on B with
~i = gp,(Ei,
Ei),
Proof. Assume B has constant curvature. Proposition 3.79-(5) shows that the root warping function S, and hence also the warping function f , must be constant on B , and thus M may be v~ewedas a product manifold. Equations (2) and (4) of Proposition 3.79 now imply that F and B are flat.
Thus the Laplacian of S is given by
The following result deals with (2 x 2) warped products B x f F where the curvature
1 6
By combining (3.40) with condition (21, we see that (3.41)
F~
= Ks2
+ g~ (grad S, grad S )
O
B $
F
B~
is not constant.
Proposition 3.81. Let M = B x j F be a R i c c ~flat semi-Riemannian warped product with dim B = dim F = 2, and assume that the sectional curvatureBK of the base B isnot constant. IfgradS isnonnuil and (grad S)I, # 0
128
3
3.7
LORENTZIAN MANIFOLDS AND CAUSALITY
a t a given point p E B, then there exist local coordinates (t, r ) on a neighborhood U
C B of p such
SEMI-RIEMANNIAN LOCAL W A R P E D P R O D U C T SPLITTINGS
129
Furthermore, since
that the following conditions hold on U.
(1) The metric on B has coordinate expression ds2 = E(r)dt2 where G(r) = ( F K- % ) - I ,
+ G(r)dr2,
E(T) = i G - ' , CM is a constant, and
the gradient of S is given in (T,t ) coordinates by
F K is the constant value of the sectional curvature on F. (2) The root warping function has the form S(t, T) = T.
9.
(3) The sectional curvature on B is given by B ~ (t) ~= ,
Equation (5) of Proposition 3.79 now shows that the curvature on the surface
The sign of E in condition (1) depends upon the signature of B. Proof. Assume grad S is nonnull and (grad S)I,
# 0.
B is a function solely of r and is given by
By continuity, we may
find a neighborhood U of p on which the gradient of S is non-vanishing. Let co = S-'(k) n U ,k E W+,denote a single level curve of S in U. Consider geodesics intersecting co orthogonally; these geodesics are also integral curves
Now gB(grad S,grad S ) = &, so formula (6) of Proposition 3.79 yields
of grad S, and the orthogonal trajectories of these geodesics are level sets of
S. Applying the classical geodesic coordinate construction, we introduce local coordinates (u, u) such that the geodesics are the u = constant curves and the orthogonal trajectories are the u = constant curves. In these coordinates the metric assumes the form We have determined the metric coefficient G: Rescale coordinates, introducing r = T(U)and t = t(v) such that r(u) = S(u)
and
Da,at(a/at) = 0.
It remains to determine the form of the metric coefficient E ( t , r ) . We first
Thus, r merely traces the values of the warping function S, and t is an affine coordinate along the geodesics. It should be noted that this will imply In ( t ,r ) coordinates we have metric tensor components
T
> 0.
show that E is a function only of the variable
r
and then derive the function
E(r). Note that
so that in particular
= 0. However, since
(t,r ) are orthogonal coordinates,
we have with the root warping function given simply by
1 E . G . ~ ; ,= l ~ t . ~ .
It follows that Et = 0 and E is a function solely of r .
130
3
LORENTZIAN MANIFOLDS AND CAUSALITY
Now using Proposition 3.79 with grad S = Dalat
1
(&g)
=
(BK .
S d Z ) z,
a
CM Dalat ( a l d r ) = 7-2 at' t1
a
&&,
3.7
we obtain
SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS
131
absolute values since the end result will show such considerations to be unnecessary.
or equivalently
giving
G.CM d
=
7 z.
However, in the orthogonal ( t ,T ) coordinate system, we have Since B K =
9,we see that a = 1, and the sign in this final term is deter-
mined by the signature of B and the sign of G ( r ) . We have therefore shown that It follows that
Observe that dt is a local nonvanishing Killing field on U in the above lemma, ~ C M Er = G(T)-
E
'
T-2
and thus
which is valid for spaces of arbitrary signature. Let us now fix the signature of all spaces under consideration to the Lorentzian signature (-,
+, +, +). Corol-
lary 3.80 and Proposition 3.81 now have the following import: a Ricci flat (2 x 2) warped product M is completely specified (locally) by the choice of
base ( B , g B ) and the constants With the substitution y = F~ In I El = I
l n ] E l = fIn lyl
.=fa.
working locally, we may assume the fiber F of constant sectional curvature F K
%,this becomes
J 3, ?f
is a subset of
giving
+ c,
or finally
(1) the sphere S 2 ( p )if " K =
$,
(2) Euclidean space Ift2if
= 0, or
Theorem 3.82. Assume ( M ,g) is a Ricci Aat four-manifold of Lorentzian signature (-,
g=
m,the curvature is given by the classical formula
F~
( 3 ) hyperbolic space H 2 ( p )if F K = - 7 .
(FK_+)
where a = eC is positive. In orthogonal coordinates with e =
and CMin Proposition 3.79. Since we are
F~
+, +,+), and let p E M be any point of M.
The point p has a
neighborhoodU such that (U, g lu) is isometric to a (2 x 2) warped product with
and
both parameters (CM, F
~ positive )
and nonconstant sectional curvature B K
r
on B if and only if M is locally isometric to an open subset of Schwarzschild CM "space-time with mass Mo = k
T
i Since E and G are functions of r only, this reduces to the following simple formula. We are ignoring the case-by-case analysis of the signs involving the
b
a
r
( F K ) ~'
Proof. Assume M admits such a local warped product structure. Since we are working locally, we may assume the fiber F is the 2-sphere of radius 1 JF7;;'
132
3
3.7
LORENTZIAN MANIFOLDS AND CAUSALITY
Let du2 denote the canonical metric on the unit 2-sphere. Proposition 3.81 implies the metric may be expressed on U in the form
SEMI-RIEMANNIAN LOCAL WARPED PRODUCT SPLITTINGS
(2) (F, g ~ is) an Einstein manifold with
133
Ric = 2(grad S,grad S)gF =
CO. gF. This result indicates that a Ricci flat manifold ( M , g ) can admit only one of a restricted class of local warped product splitting at p f M if we require
with E(r) = (
%).
F ~ -
Upon making the change of variables u = --$-- and v = tm,the metric F K
becomes
dim B = 1. If dim F = 2, for example, it follows that F is a surface of constant
~ in gthis ~case. If we also assume F K = 0,
curvature F K since FRic = F we have S =
Jf = constant, and the warped
product is merely a standard
product manifold with a semi-Euclidean metric. If dim F = 3, it follows that the Einstein manifold ( F , ~ F )has constant thus demonstrating the first claim. The converse is now clear.
13
This result should be contrasted with Birkhoff's Theorem [cf. Hawking and
curvature [cf. Petrov (1969, p. 77) 1. If (F, g ~ is)Riemannian, we may consider F a subset of (1) the sphere S3(r) if
F~
=
5'
Ellis (1973, p. 372)j on spherically symmetric solutions to Einstein's vacuum
(2) Euclidean space IR3 if FK = 0, or
field equations, which also offers an alternate proof of Theorem 3.82.
(3) hyperbolic space ~ ~ (if rF~) = - 7
1
Theorem (Birkhoff). Any C2 solution of Einstein's empty space equa-
with analogous restrictions holding if g~ is indefinite. An exact solution to
tions which is spherically symmetric in an open set V is locally equivalent to
Einstein's equations which uses F = S3 is the spatially homogeneous Taub-
part of the maximally extended Schwarzschild solution in V.
NUT model [cf. Hawking and Ellis (1973, p. 170)].
The typical construction of Schwarzschild space-time [cf. O'Neill (1983)l follows the proof of Birkhoff's theorem. A number of strong physical assump tions, not the least of which is spherical symmetry, lead one inevitably to Schwarzschild space-time as the unique model. What we have shown is that a Ricci flat space-time which possesses enough symmetry t o be expressed locally as a (2 x 2) warped product must have spherical, planar, or hyperbolic symmetry. In the first case we obtain the conclusion t o Birkhoff's Theorem:
M is locally isometric to a portion of Schwarzschild space-time. The following result may be established by an argument similar to the one employed in the proof of Proposition 3.79 [cf. Easley (1991)].
Theorem 3.83. Let M = B x f F be a semi-Riemannian warped product with dim B = 1 and dim F = n 2 2. For M to be Ricci flat, i t is necessary and sufficient that the following two conditions hold. (1) The root warping function S satisfies (grad S,grad S ) = Co,a constant.
CHAPTER 4
LORENTZIAN DISTANCE
With the basic properties of Riemannian metrics in mind (cf. Chapter I), it is the aim of this chapter to study the corresponding properties of Lorentzian distance and to show how the Lorentzian distance is related to the causal structure of the given space-time.
We also show that Lorentzian distance
preserving maps of a strongly causal space-time onto itself are diffeomorphisms which preserve the metric tensor. While there are many similarities between the Riemannian and Lorentzian distance functions, many basic differences will also be apparent from this chapter. Nonetheless, a duality between "minimal" for Riemannian manifolds and "maximal" for Lorentzian manifolds will be noticed in this and subsequent chapters. 4.1
Basic Concepts and Definitions
Let (M,g ) be a Lorentzian manifold of dimension n with p
< q, let fl,,,
> 2.
Given p, q E M
denote the path space of all future directed nonspacelike
curves y : [0, I] + M with $0) = p and $1) = q. The Lorentzian arc length functional L
= L, : Clp,q -' R
is then defined as follows [cf. Hawking and Ellis
(1973, p. 105)]. Given a piecewise smooth curve y E 0 = to < tl < ... < t,-l
< tn = 1 such that
Qp,,,
choose a partition
y 1 (t,.t,+l) is smooth for each
i = 0 , 1 , . . . ,n - 1. Then define
It may be checked as in elementary differential geometry [cf. O'Neill (1966, pp. 51-52)] that this definition of Lorentzian arc length is independent of
136
LORENTZIAN DISTANCE
4
BASIC CONCEPTS AND DEFINITIONS
4.1
are curves y E
Q,,q
137
with arbitrarily small Lorentzian arc length. Hence, the
infimum of Lorentzian arc lengths of all piecewise smooth curves joining any two chronologically related points p 0
if and only if
q E I'(p).
Thus the Lorentzian distance function determines the chronological past and future of any point. However, the Lorentzian distance function in general fails FIGURE4.1.
The timelike curve y from p to q is approximated
by a sequence of curves {y,) with yn
+y
in the C0 topology but
L(yn) --t 0.
imply g E J+(p) - I+(p). But at least if q E J+(p) - Ii(p), then d(p, q) = 0. We emphasize that the Lorentzian distance d(p, q) need not be finite. One
the parametrization of y. Since an arbitrary nonspacelike curve satisfies a local Lipschitz condition, it is differentiable almost everywhere. Hence the Lorentzian arc length L(7) of 7 may still be defined using (4.1). Alternative but equivalent definitions of L(y) for arbitrary nonnull nonspacelike curves may be given by approximating y by C' timelike curves [cf. Hawking and Ellis (1973, p. 214)] or by approximating y by sequences of broken nonspacelike geodesics {cf.Penrose (1972, p. 53)j. The Lorentzian arc length of an arbitrary null curve may be set equal to zero. Now fix p, q E M with p
0. On the other hand, y may be approximated by a sequence (7,)
of piecewise smooth "almost null" curves y, : [O,1] yn(l) = q such that yn
to determine the causal past and future sets of p since d(p,q) = 0 does not
+
+
M with yn(0) = p and
y in the Co topology, but L(yn) -+ 0 (cf. Figure 4.1).
This construction shows, moreover, that given any p, q E M with p 0. If
u, E R , , is the timelike curve obtained by traversing y exactly n times, then L(u,) = nL(y)
-+ ca
as n
--t
ca. Thus d(p,p) = m .
(2) Suppose (M,g) is totally vicious. Fix p, q E M, and let n > 0 be any positive integer. Since p E I+(p), we may find yl E R,,,
with L(yl) 2 n by
part (1). Since q E I+(p), there is a timelike curve
from p to q. Then
y = yl
* yz E R,,,
72
is a timelike curve with length L(y) = L(y1)
+ L(yz) > n.
Hence d(p, q) = m. Conversely, suppose d(p, q) = m for all p,q E M . Fixing r E M , we have d(r,p) > 0 and d(p,r)
> 0 for all p
E M . Thus by (4.2), it follows that
I + ( r ) n I-(r) = M . (3) Inspired by Lemma 4.2-(2), T. Ikawa and H. Nakagawa (1988) proved (3) using the Lorentzian distance function. Subsequently, B. Wegner (1989) noted that the following elementary argument yields the desired result. Let q E M = I+(p)nI-(p). Then q E I+(p) so I-(p)
0 is possible. But if (M, g) is chronological, then d(p,p)
A Reissner-Nordstrom space-time with e2
= m2 is
shown. By taking timelike curves y from p to q close to Jf and 3-. we can make L(y) arbitrarily large. Thus d(p, q) = m , which
= 0 for all
means an accelerated observer may take arbitrarily large amounts
p E M. Also, the Lorentzian distance function tends to be nonsymmetric.
of time in going from p to q.
More precisely, the following may be shown for arbitrary space-times.
Remark 4.3. If p # q and d(p, q) and d(q,p) are both finite, then either d(p,q) = 0 or d(q,p) = 0. Equivalently, if d(p,q) > 0 and d(q,p) > 0, then d(p, 9) = 4 9 , P) = a. Proof. If d(p, q) > 0 and d(q,p)
A further consequence of Definition 4.1 is that if y : [O,ca) -+ (M, g) is any future directed, future complete, timelike geodesic in a n arbitrary space-time (M, g ) , then limt,c,
d(r(O), y(t)) 2 limt,m L(y I 10,t ] ) = m . By contrast,
complete Riemannian manifolds (N,go) may contain (nonclosed) geodesies
> 0, we may find future directed timelike
a : [0,m )
(NIgo) for which sup{do(u(0),u(t)) : t 2 0) is finite. Fur-
curves yl from p to q and yz from q to p, respectively. Define a sequence (7,)
ther assumptions are needed for Riemannian manifolds to guarantee that
by yn = yl * (72 * y ~ E )Rp,q. ~ AS n infinite. Similarly, d(q, p) = m .
and Ebin (1975, pp. 53 and 151)].
--+
m , L(yn)
-+
m , whence d(p,q) is
limt,,
d(u(O), u(t)) = co for all geodesics a : [O,m)
--+
(N,go) [cf. Cheeger
140
4
4.1
LORENTZIAN DISTANCE
BASIC CONCEPTS AND DEFINITIONS
While the Lorentzian distance function fails to be symmetric and nondegenerate, at least a reverse triangle inequality holds (cf. Figure 1.3). Explicitly, If P < r
(4.3)
< q, then d(p, q) 2 d(p, r ) + d(r, q).
We now discuss some properties of the Lorentzian distance that make it a useful tool in general relativity and Lorentzian geometry. First, the Lorentzian distance function is lower semicontinuous where it is finite [cf. Hawking and Ellis (1973, p. 215)]. Lemma 4.4. For Lorentzian distance d, if d(p,q) < m, p,
qn
-+
q,
-+
q, then d(p,q) 5 liminf d(p,,q,). q, then limn,,
-+
Also, if d(p,q) = m, p,
-+
p, and p, and
d(p,, q,) = m.
Proof. First consider the case d(p,q) < m. If d(p, q) = 0, there is nothing to prove. If d(p,q)
> 0, then q E
I+(p) and the lower semicontinuity follows
from the following fact. Given any
6
> 0, a timelike curve y of length L
FIGURE^.^. L e t M b e { ( x , y ) : O < y < 2 ) - { ( x , l ) : - 1 5 x 5 1 )
>
with the identification (x, 0)
d(p, q) - €12 from p to q and sufficiently small neighborhoods Ul of p and U2 of q may be found such that y may be deformed to give a timelike curve of = R
<
m.
-+
p as shown.
Then p, E I+(pn) and hence d(pn,pn)= m for all n. For large n we have q E I+(p,) and thus d(p,,q) = m. On the other hand,
Since
d(p, q) = m there exists a timelike curve y from p to q of length L(y)
d(p, q) = 112 which yields d(p, q)
>
> +
L' R 1 from any point r of Ul to any point s of U2. This contradicts lim inf d(p,, q,) = R. 0 In general, the Lorentzian distance function fails to be upper semicontinuous. We give an example of a space-time (M, g) containing an infinite sequence {p,) with p, -+ p and a point q E I+(p) such that d(p,,q) = m for all large n but d(p, q) < ca (cf. Figure 4.3). For globally hyperbolic space-times, on the other hand, the Lorentzian
be upper semicontinuous in causal space-times (cf. Figure 4.6).
Proof. To prove the finiteness of d, cover the compact set J+(p) n J-(q) with a finite number of convex normal neighborhoods B1, B2,. .. ,B, such that no nonspacelike curve which leaves any Biever returns and such that every nonspacelike curve in each Bi has length at most one. Since any nonspacelike curve y from p to q can enter each Bi no more than once, L(y) 5 m. Hence d(p1q) 5 m. If d failed to be upper semicontinuous at (p,q) E M x M, we could find
distance function is finite and continuous just like the Riemannian distance
a 6
function.
such that d(p,, q,) 2 d(p, q)
L e m m a 4.5. For a globally hyperbolic space-time (M,g), the Lorentzian
< lim inf d(p,, q). This space-
time is not causal. However, the distance function may also fail to
R + 2. This implies that there exist neighborhoods Ul and Uz of p and q, respectively, such that y can be deformed to give a timelike curve of length
distance function d is finite and continuous on M x M.
(x, 2) and the flat Lorentzian metric
ds2 = dx2 - dy2. Let p = (O,O), q = (0,1/2), and p,
length L' 2 d(p, q) - c from any point r of Ul to any point s of U2. Suppose now that d(p,q) = oo but liminfd(p,,q,)
-
> 0 and sequences ipn) and {q,) converging to
+ 26
p and q respectively,
for all n. By definition of d(p,, q,), we
may then find a future directed nonspacelike curve y, from p, to q, with L(y,) 2 d(p, q) + 6 for each n. By Corollary 3.32, the sequence {y,) has a
142
4
LORENTZIAN DISTANCE
BASIC CONCEPTS AND DEFINITIONS
4.1
nonspacelike limit curve y from p to q. By Proposition 3.34, a subsequence {y,) of {y,) converges to y in the C0 topology. Hence L(y) 2 d(p, q) + 6 by Remark 3.35. But this contradicts the definition of Lorentzian distance. Thus d is upper semicontinuous a t (p, q). U
We now define the following distance condition [cf. Beem and Ehrlich (1977, Condition 4)].
Definition 4.6. (Finite Distance Condition) isfies the finite distance condition if d(g)(p, q)
The space-time (M, g) sat-
< m for all p, q f M.
Lemma 4.5 then has the following corollary.
Corollary 4.7. If (M, g) is globally hyperbolic, then (M,g) satisfies the finite distance condition and d(g) : M x M
+R
is continuous.
If (MIg) is globally hyperbolic, all metrics in the conformal class C ( M , g) are globally hyperbolic. Hence all metrics in C(M,g) satisfy the finite distance condition. We will examine the converse of this statement in Section 4.3, Theorem 4.30.
FIGURE 4.4.
The set B + ( p , ~ )= {q E I+(p) : d(p,q)
< E)
in Minkowski space-time does not have compact closure, is not geodesically convex, and does not contain p. Furthermore, sets of the form B+(p, E) do not form a basis for the manifold topol-
Since the given topology of a smooth manifold coincides with the metric
ogy. But in general, if (M,g) is a distinguishing space-time with
topology induced by any Riemannian metric, it is natural to consider the sets
a continuous Lorentzian distance function, then a subbasis for the
{m f I+(p) : d(p, m) < E) for a Lorentzian manifold. However, as Minkowski
manifold topology is given by sets of the form B+(p, E)and B-(p, E)
space shows, these sets do not form a basis for the given manifold topology
[cf. Proposition 4.311. Hence these sets do form a subbasis for the
(cf. Figure 4.4). Indeed, this same example shows that no matter how small
given topology of Minkowski space-time.
E
> 0 is chosen, the sets {m E J f (p) : d(p, m) 5
E)
may fail to be compact
and fail to be geodesically convex as well as failing to be diffeomorphic to the closed n-disk. The sphere of radius
E
for the point p f M is given by K ( ~ , E= ) {q E M :
d(p,q) = E). This set need not be compact. However, the reverse triangle inequality and (4.2) imply that K(p, E) is achronal for all finite E > 0 and all p~ M .
In arbitrary space-times, neither the future inner ball
nor the past inner ball B-(P, €1 = {q f I-(P) : d(q,P) < €1 need be open. On the other hand, when the distance function d : M x M
-+
R U {m) is continuous, these inner balls must be open. In Section 4.3 we will show that for distinguishing space-times with continuous distance functions, the past and future inner balls form a subbasis for the manifold topology.
144
4
LORENTZIAN DISTANCE
4.1
BASIC CONCEPTS AND DEFINITIONS
145
A different subbasis for the topology of any strongly causal space-time (M,g) with a possibly discontinuous distance function d = d(g) : M x M -+ WU{co) may be obtained by using the outer balls O+(p, E)and 0-(p, E)rather
than the inner balls.
Definition 4.8. (Outer Balls O+(p,E), 0-(p, e))
The outer ball O+fp,E) [respectively, 0 - ( p , ~ ) ]of I+(p) [respectively, I-(p)] is given by
respectively, O-(P,E) = {q E M : d(q,p) > E) (cf. Figure 4.5) Since the Lorentzian distance function is lower semicontinuous where it is finite, the outer balls O+(p, E)and 0 - ( p , E) are open in arbitrary space-times. The reverse triangle inequality implies that these sets also have the property that if m, n f O+(p, E ) [respectively, m, n E 0-(p, E)]and m ,< n , then any future directed nonspacelike curve from m to n lies in Ot(p,
E)
[respectively,
0-(p, E)]. Moreover, we have
FIGURE4.5.
The outer balls Of (p, E) = {q E M
:
d(p, q)
> E)
and 0-(p, E) = {q f M : d(q,p) > E) are open in arbitrary spacetimes. Furthermore, O+(p, 0. Choose constants el, t2
implies that if the geodesic y is not maximal, there exist variations of y which
with 0 < €1 < d(p1,m) and 0 < €2 < d(m,pz). Then m E O + ( p l , ~ l )n 0-(pz, €2). Since 0' (PI, €1) C I+(pl) and 0 - ( p z , ~ z )G I-(pz), we also have
yield curves from p to q "close" to y having longer Lorentzian arc length than
(< p2
O+(PI,€1) n 0-(pz, €2) E I+(pl) n I-fpz)
c UI c u as required.
y. If y is maximal in this sense, however, no small variation of y keeping p
and q fixed will produce timelike curves u from p to q with L(u) > L(y).
146
4
4.1
LORENTZIAN DISTANCE
BASIC CONCEPTS AND DEFINITIONS
147
in M from p to q
that y(t) $ Int(@). Let yl be the unique null geodesic in U from p to y(t),
("far" from y) with d(p,q) = L(o1) > L(y). Thus maximality as defined by Hawking and Ellis does not imply "maximality in the large." To study
and let yz be the unique null geodesic in U from y(t) t o q. By Proposition 2.19 of Penrose (1972, p. 15), y l * is~either ~ a smooth null geodesic or p 0.
Suppose now that d(p, q) = 0, and let y be any nonspacelike curve in U from p to q. Then L(y) 5 d(p, q) = 0. Thus y : [O,1]
-+
M is a null curve. Suppose
null geodesic.
t
Ik-
Note in Corollary 4.14 that since the null geodesic is maximal, it cannot contain any null conjugate points to p prior to q [cf. O'Neill (1983, p. 404)j. A sometime useful special case of Corollary 4.14 occurs when p = q is assumed.
i;
In this case, one may deduce that if the space-time ( M , g ) is chronological
t
but not causal, then there exists a smooth null geodesic
I
/
0
: [O, 1 1 -i (M, g)
with P(0) = @(I) and p ( 1 ) = Ap(0) for some A > 0. (If p ( 1 ) and p ( 0 )
4
148
LORENTZIAN DISTANCE
4.1
were not proportional, then ,O could be deformed near P(0) to be future time-
BASIC CONCEPTS AND DEFINITIONS
hyperbolic. Because 5 = ~ ' g it , follows that c = ?r oE : [0, I]
149
like, contradicting the condition d(m, m) = 0 for all m in M , since (M, g) is
M is a timelike geodesic. Since Z(0) = r l and E(1) = +(rl), we also have c(0) = c(1). If c
chronological [cf. Proposition 2.19 of Penrose (1972, p. 15)]. Even more dra-
were not smooth at c(O), we could deform c to a timelike curve u : [O, 11 + M
matically, a closed timelike curve could be produced to violate chronology [cf.
with L,(o) > L(c), a(0) = u(1) E ~ ( S I )which , lifts to a curve 3 : [O,11 -+ M
Proposition 10.46 of O'Neill (1983, pp. 294-295)].)
with a(0) E S1and Z(1) = q(Z(0)) E
As a second application of the elementary properties of the distance function, we give a proof of the existence of a smooth closed timelike geodesic
L a (Z) = A, in contradiction. 0
-+
-
S2.
But then L z ( a ) = L,(u)
> L,(c)
=
on any compact space-time having a regular cover with a compact Cauchy
More recently, G. Galloway (1984b) has obtained, by elementary geometric
surface. Using infinite-dimensional Morse theory, it may be shown [cf. Klin-
arguments, the existence of a closed timelike geodesic from any stable nontrivial
genberg (1978)] that any compact Riemannian manifold admits a t least one
free homotopy class of closed timelike curves on an arbitrary compact space-
smooth closed geodesic. However, the method of proof relies crucially on the
time. Galloway's approach is based on geodesic convexity methods which were
positive definiteness of the metric and thus is not applicable to Lorentzian
earlier used to obtain a result, basic to the development of global Riemannian
manifolds. Nonetheless, one may obtain the following theorem of Tipler by
geometry during the early part of this century, given first by J. Hadamard for
direct methods [cf. Tipler (1979) for a stronger result].
surfaces and then by E. Cartan for general Riemannian manifolds of higher
Theorem 4.15. Let (M,g) be a compact space-time with a regular covering space which is globally hyperbolic and has a compact Cauchy surface. Then (M, g) contains a closed timelike geodesic.
dimension. Their result concerns the existence, within any nontrivial free homotopy class of curves on a compact Riemannian manifold, of a shortest curve in the given homotopy class, which must then be a nontrivial smooth closed geodesic. From a directly geometric viewpoint, certain basic ideas involved in
Proof. Since M is compact, there exists a closed, future directed, timelike
the existence of this geodesic are the following.
curve y : [0,1] + M. Set p = y(0) = y(1). Let T : G -+ M denote the given covering manifold, and let 7 : [0, I] + M be a lift of y, i.e., ~ o y ( t = ) y(t) for all
> 0 denote the infimum of all lengths of curves in this homotopy class. Choose a minimizing sequence {ck) with lim L(ck) = LOin the given free
t E [0, 11. Then 7 is a future directed timelike curve in
homotopy class. By compactness and convexity radius arguments, one covers the given manifold by a finite number of geodesically convex neighborhoods
z.Put pl
= y(0) and
implies pl and p2 are distinct p2 = r(1). Then the global hyperbolicity of points which cannot lie on any common Cauchy surface. Since T : G -+ M is regular, there must be a deck transformation 11, : G -t
% taking pl
Let Lo
(each having compact closure in a larger geodesically convex neighborhood).
to pz
Using these neighborhoods in succession, each ck may be approximated by
G
a piecewise smooth geodesic, or equivalently, may be viewed as an N-tuple
containing pl, and define S 2 = $I(&). Since (%,7j) is globally hyperbolic, the - distance function d = d(Z) : M x M + WU (m) is finite-valued and continuous.
of successive points pl(k),p2(k),... ,pN(k), where a uniform bound needs to be given for the number N of points required. Since a geodesic segment in
Thus we may define a continuous function f : S1 -t W by f (s) = d(s, $(s)).
a convex neighborhood is the shortest curve between any two of its points,
Since f 071) > 0, we have A = sup{d(s, +(s))
E S1) > 0. Moreover, since
this approximation procedure produces a shorter curve than ck, but one which
S1 is compact, A < w and there exists an rl E S1 with d(rl111,(rl)) = A. Let E : [0, I] be a timelike geodesic segment with Z(0) = TI, Z(1) = $(rl), and L(E) = d(rl,$(rl)) = A. This geodesic exists since ( z , ~ is) globally
By compactness, a diagonalizing argument gives points pi, pz, . . . , p which ~ are limits of a subsequence of all of the above sequences. Joining p, to p,+l
[cf. Wolf (1974, pp. 35-38, 60)j. Choose a compact Cauchy surface S1 of
-+
M
:s
is also still homotopic to ck, and hence is in the given free homotopy class.
150
4
4.2
LORENTZIAN DISTANCE
DISTANCE PRESERVING AND HOMOTHETIC MAPS
151
successively produces a piecewise smooth geodesic c in the given free homotopy
class of closed timelike curves to be stable. Finally, we note that in Galloway
class which realizes the minimal length Lo. Now the geodesic c must in fact be
(198613) it is shown by covering space arguments that every compact two-
smooth a t the pi's, or an even shorter closed curve in the free homotopy class
dimensional Lorentzian manifold contains a closed timelike or null geodesic.
could be produced by the usual "rounding off the corners" procedure. Since
Here one uses dimension two in the essential way that a closed timelike curve for
Lo > 0, this closed geodesic c must be nontrivial, i.e., not a "point curve." A
(M, g) corresponds to a closed spacelike curve (hence a spacelike hypersurface)
detailed discussion of the steps involved in the above argument may be found
for (M, -g), which is also a Lorentzian manifold since dim(M) = 2. Thus,
in Spivak (1979, p. 358).
certain techniques in general relativity involving spacelike hypersurfaces may
Now in carrying these ideas over to the space-time setting, it is clear that
be applied to the existence problem.
"minimal geodesic segment" should be replaced by "maximal timelike geodesic segment." Technical difficulties arise, however, since the set of unit timelike tangent vectors based at a given point is not a compact set. Hence, problems can arise with timelike tangent vectors tending toward a null direction when trying to d o subsequence arguments. Equivalently, in the above context, it is necessary t o prevent timelike geodesic segments joining pi(k) t o pi+l(k) from converging t o a null geodesic segment from pi to pi+l when the diagonalization procedure is carried out. Indeed, Galloway (198613) gives a n example of a compact space-time which contains no closed timelike geodesics but which
4.2
Distance P r e s e r v i n g a n d H o m o t h e t i c M a p s
Myers and Steenrod (1939) and Palais (1957) have shown that i f f is a distance preserving map of a Riemannian manifold (Nl, gl) onto a Riemannian manifold (N2,g2), then f is a diffeomorphism which preserves the metric tensors, i.e., f *gz = 91. In particular, every distance preserving map of (Nl,gl) onto itself is a smooth isometry. In this section we give similar results for Lorentzian manifolds following Beem (1978a). Recall that a diffeomorphism f : (M1,gl)
contains a closed null geodesic. In view of these difficulties, Galloway (1984b) considers timelike free homotopy classes which are "stable" for a given compact space-time (M,go). Here a given free timelike homotopy class C for (M, go) is said to be stable if there exists a "wider" Lorentzian metric g for M , i.e., go < g in the sense of the discussion following Remark 3.14, such that if L, denotes the Lorentzian arc length for (M,g), then the given timelike homotopy class C satisfies the
+
(M2,gz) of the Lorentzian
manifold (M1,gl) onto the Lorentzian manifold (M2,gz) is said to be homothetic if there exists a constant c > 0 such that gz(f,v, f,w) = cgl(v, w) for all v, w E TpMl and all p E MI. In particular, if c = 1, then f is a (smooth) isometry. The group of homothetic transformations is important in general relativity since it has been shown to be the group of transformations which preserves the causal structure for a large class of space-times [cf. Zeeman (1964, 1967), Gobel (1976)j.
condition sup L,(c)
< +m.
C€E
Galloway (1984b) shows that this concept of stability gives the control needed
We will let dl denote the Lorentzian distance function of (Ml, gl) and d2 denote the Lorentzian distance function of (M2, 92) below. The distance analogue of a smooth homothetic map is defined as follows.
to force convergence arguments to be successful and hence obtains the theorem that for any compact Lorentzian manifold, each stable free timelike homotopy class contains a longest closed timelike curve which is of necessity a closed timelike geodesic. Galloway also shows how his result may be used to recover Tipler's Theorem 4.15 given above and gives a criterion for a free homotopy
Definition 4.16. (Distance Homothety)
A map f : ( M I ,gl) --+ (M2,gz) is said to be distance homothetic if there exists a constant c > 0 such that d2(f(p), f(q)) = cdl(p,q) for all p,q E 1M. If c = 1, then f is said to be distance preserving.
152
4
4.2
LORENTZIAN DISTANCE
It is important to note that for arbitrary Lorentzian manifolds, distance preserving maps are not necessarily continuous.
For if ( M , g ) is a totally
vicious space-time, we have seen that d@,q) = 03 for all p, q E M [cf. Lemma
4.2-(2)]. Hence any set theoretic bijection f : M
+M
is distance preserving
but need not be continuous.
that a map f : Ml
+ M2
153
is said to be open if f maps each open set in M I t o
an open set in M2. L e m m a 4.20. Let ( M I ,g l ) be strongly causal and let ( M 2 , g 2 )be an arbitrary space-time. If f is a (not necessarily continuous) distance homothetic map of ( M I ,g l ) onto (M2,g2), then f is open and one-to-one.
T h e o r e m 4.17. Let ( M I ,g l ) be a strongly causal space-time, and let
(M2,g2) be an arbitrary space-time. If f : ( M l , g l )
DISTANCE PRESERVING AND HOMOTHETIC MAPS
-+
( M 2 ,g 2 ) is a dis-
tance homothetic map (not assumed to be continuous) of Ml onto M 2 , then f
is a smooth homothetic map. That is, f is a diffeomorphism, and there exists
Proof. The openness of f is immediate from part ( 2 ) of Lemma 4.19. It remains to show that f is one-to-one. Assume there are distinct points p and q of M I with f ( p ) = f(q). Let U ( p ) be an open neighborhood of p with q $! U ( p ) and such that no nonspacelike curve intersects U ( p ) more than once.
a constant c > 0 such that f *g2= cg1. In particular, every map of a strongly
Choose r l , 7-2 E U ( p )with rl 0 such that if m
a and d coincide.
= y(t), then m E V and z(p, m) = Z(y 1 [0, t ] ) is
finite. We then obtain
the conformal class of space-times to be globally hyperbolic. If a space-time
with the global Lorentzian distance function on U x U. Hence the Lorentzian distance function is forced to be finite-valued on U x U. For the purposes of this section, it will be convenient to reformulate Defi-
d ( ~ , mL) L(T / 10, t ] ) = Z(y 1 10,t ] ) = a(p, m). But since m E V we have d(p,m) = z(p,m), whence L(y I [O,t]) = d(p,m). Hence y I [O, t ] is a geodesic in (M,g) by Theorem 4.13. Thus we have shown that if u E TpN is any future directed tangent vector, the geodesic in (M,g) with initial direction u is also a geodesic in (N,?j) near p. Therefore Sn(u, u) =
nition 4.10 slightly as follows. Definition 4.35. (Mmimal Segment) curve c : la, b]
-,
A future directed nonspacelike ( M , g) is said to be a maximal segment provided that
L(c) = d(c(a), c(b)) and hence L(c 1 [ s , t ] ) = d(c(s), c(t)) for all s , t with a s s l t l b .
0 for all future directed nonspacelike tangent vectors. Since the sum of two
Evidently, the Lorentzian distance function restricted t o the image of a
nonparallel future directed nonspacelike tangent vectors is future timelike, it
maximal segment must be finite-valued and continuous. In the terminology
follows by polarization that S,(u, w) = 0 for all future directed nonspacelike tangent vectors u, w E TpN, as required.
0
Combining Propositions 4.32 and 4.33 yields the following characterization of totally geodesic timelike submanifolds of strongly causal space-times in terms of the Lorentzian distance function.
of Definition 4.35, the previous Proposition 4.12 may be rephrased as follows. Suppose that (M, g) is strongly causal. Given p in M , let U b e a local causality neighborhood of p, i.e., a causally convex neighborhood about p which is also a geodesically convex normal neighborhood. Then any nonspacelike geodesic segment lying in U is a maximal segment in the space-time (M,g). In Chapter 14 the Lorentzian Splitting Theorem for timelike geodesically
Theorem 4.34. Let (M, g) be a strongly causal space-time of dimension
complete (but not necessarily globally hyperbolic) space-times is studied.
n 2 2, and suppose that (N, i'g) is a smooth timelike submanifold of (M,g),
Since global hyperbolicity is not assumed and hence the global continuity
i.e., ?j = i*g is a Lorentzian metric for N . Then (N,?j) is totally geodesic
and finite-valuedness of the space-time distance function may not be taken
iff given any p E N, there exists a neighborhood V of p in N such that the
for granted, it is necessary to make a careful study of the space-time distance
Lorentzian distance functions 2 of ( N , 3 ) and d of ( M , g ) agree on V x V. 4.4
Maximal Geodesic Segments and Local Causality
function and asymptotic geodesics in a neighborhood of a given timelike line. What emerged in a series of papers, as summarized in the introduction to Chapter 14 which will not be repeated here, is that the existence of a global
We have seen in this chapter that certain causality conditions placed on a
maximal timelike geodesic has important implications for the distance function
space-time are related to pleasant local or global behavior of the Lorentzian
and the Busemann function of the timelike line in a neighborhood of the given
distance function. For instance, we saw that if (M,g) is globally hyperbolic,
line [cf. Eschenburg (1988), Galloway (1989a), Newman (1990), Galloway and
then all metrics conformal to g for M are not only continuous but also satisfy
Horta (1995)l. Especially, Newman (1990) made a thorough study of the geo-
the finite distance condition as well. Further, the finite distance condition for
desic geometry in the case that timelike geodesic completeness, but not global
168
4
LORENTZIAN DISTANCE
hyperbolicity, is assumed. In this section we give certain elementary preliminaries which will be germane to Chapter 14 but which fit into the spirit of this chapter. It is interesting that the reverse triangle inequality plays an important role, hence this material is decidedly non-Riemannian. Also the lower
4.4
MAXIMAL GEODESIC SEGMENTS, LOCAL CAUSALITY
169
(2) For any p, q in J+(c(s)) n J-(c(t)) with p ,< q, we have d(p, q) = 0;
and (3) Chronology holds at all points of J+(c(O)) n J- ( ~ ( 1 ) ) . Proof. (1) By assumption, c(s)
< r < c(t) and d(c(s), c(t)) = 0 , so that the
semicontinuity of the Lorentzian distance function for an arbitrary space-time
reverse triangle inequality d(c(s), c(t)) 2 d(c(s),r ) +d(r, c(t)) implies equation
(cf. Lemma 4.4) is useful here.
(4.5). Further, since c(s)
Note as an immediate first example that if a space-time (M,g) contains
< r < c(t) is assumed, there exist future causal curves
cl from c(s) to r and cz from r to c(t) which by (4.5) must both be maximal
a single maximal segment, then (M,g) cannot be totally vicious, since the
null geodesic segments. Since c(s) and c(t) are not chronologically related,
Lorentzian distance function is finite-valued on the particular segment while
the concatenation of cl and cz must constitute a single null geodesic by basic
totally vicious space-times satisfy d(p, q) = +m for all p, q in M. Newman
causality theory [cf. Penrose (1972, Proposition 2.19)], whence r E c(I(c)).
(1990) noted the more interesting consequence that the existence of a maximal
(2) This is immediate from the reverse triangle inequality applied to
timelike segment c : [a, b] -4 (M,g) implies that strong causality holds a t all points of c((a, b)). Since strong causality is an open condition [cf. Penrose (1972, p. 30)], this thus yields an open neighborhood U of c((a, b)) for which strong causality at q is valid for all q in U . Hence, not only does the existence of a local causality neighborhood in a strongly causal space-time guarantee the local existence of maximal segments, but a kind of converse holds: the
(3) Condition (3) now follows since if chronology fails to hold, then d(p,p) is infinite which contradicts (2) applied with q = p. We should caution that I(c) is not necessarily equal to [O, +m) (cf. Lemma
existence of a maximal timelike segment implies that some local region of
7.4). In the case of a maximal timelike segment, the reverse triangle inequality
(M,g) containing all interior points of the given maximal timelike segment
yields the finiteness of Lorentzian distance from points in some neighborhood
must be strongly causal.
of the segment despite the possible general lack of finiteness of distance for
For our use in Chapter 14, these basic consequences of the existence of maximal segments will be treated in the present section. We begin with a maximal null segment.
chronologically related points (cf. Figure 4.2). Similar arguments to those used in Lemma 4.36 yield the following finiteness of the distance function in the causal hull of any causal maximal segment.
Lemma 4.37. Let c : [O,1] -, (M,g) be a causal maximal segment. Then Lemma 4.36. Let c : [O, 11 -4 (M,g) be a maximal null geodesic segment,
and let I(c) denote the domain of c extended to be a future inextendible null geodesic emanating from c(0). Then (1) For any s, t with 0 5 s < t 5 1 and r E J+(c(s)) fl J-(c(t)), we have
(1) Given any p, q in J+(c(O)) n J - ( ~ ( 1 ) )with p 6 q , the distance d(p, q) is finite. (2) Chronology holds at all points of J+(c(O)) n J - ( ~ ( 1 ) ) .
Having dealt with chronology, let us now consider the stronger requirement of causality. The cylinder M = S' x W with metric ds2 = dBdt contains closed null geodesics c : [0, +m)
-t
M which satisfy d(c(O), c(t)) = 0 for all t 2 0.
Hence, the existence of a maximal null geodesic does not imply more than hence T lies on c(I(c));
local chronology.
170
4
4.4
LORENTZIAN DISTANCE
Lemma 4.38. Let c : [0, I] + (M,g) be a maximal timelike segment. Then causality holds a t all points of c([O, 11).
Proof. Suppose that cl is a closed nonspacelike curve beginning and ending at c(t1) with tl > 0. Since chronology holds at c(t1) by Lemma 4.37, cl must be a maximal null segment (cf. Corollary 4.14). Consider the composite causal curve y = (c I [0, tl]) * cl from c(O) to c(tl). Since 7 is a causal curve from c(0) to c(tl) which is a timelike followed by a null geodesic, first variation "rounding the corner" arguments produce a causal curve yl from c(0) t o c(tl) which is longer than y,hence longer than ~110,tl] [cf. O'Neill (1983, p. 294)). But this contradicts the maximality of c 1 [O,tl].
If t l = 0, apply the samc type of argument to the composition of the closed null geodesic cl followed by c 1 [O,l]. Now we turn to a result with a somewhat more difficult proof first given
MAXIMAL GEODESIC SEGMENTS, LOCAL CAUSALITY
171
lower semicontinuity of distance at (q,p), we have d(y,, x,) > 0 for some n sufficiently large. Hence y,
2 and .rrl(S:)
= Z.
(2) Sf. is globally hyperbolic and geodesically complete.
FIGURE 5.6.
(3) H f . is nonchronological since y(t) = ( T cos t ,r sin t , 0 , . . . ,0) is a closed
The Kruskal diagram for the maximal analytic ex-
tension of the exterior Schwarzschild space-time is shown. The ex-
timelike curve. Also
tended space-time is the connected nonconvex region I U I1 U I' U 11'
bolic.
bounded by the hyperbola corresponding t o r = 0. The points of this hyperbola are the true singularities of this space-time.
The
lines a t f45' separate the space-time into four regions. Region I corresponds to the exterior Schwarzschild solution. Region I1 is the "interior" of a nonrotating black hole. Region I' is isometric t o region I and corresponds to a n alternative universe on the "other side" of the black hole. There is no nonspacelike curve from region I to region 1'.
@, while strongly causal, is not
globally hyper-
The de Sitter space-time represented in Figure 5.7 may be covered by global
< cm,0 < x < 0 < B 5 T , and < $ 5 27r. Here t is the coordinate on IR and (x,B,$) represent coordinates
coordinates (t,x,B,$) with -m < t
0
T,
on S 3 Icf. Hawking and Ellis (1973, pp. 125, 136)]. In these coordinates, the metric for de Sitter space-time of constant positive sectional curvature l / r 2 is given by
184
5
EXAMPLES OF SPACGTIMES
5.4
ROBERTSON-WALKER SPACGTIMES
185
Riemannian metric h of constant negative sectional curvature -1 on the hyperbolic three-space H = W3, this space-time may be represented as a warped product of the form (Rx f H , -f dt2@h),where the warping function is defined on the Riemannian factor H (cf. Remark 3.53). 5.4
Robertson-Walker Space-times
In this section we discuss Robertson-Walker space-times in the framework of Lorentzian warped products. These space-times include the Einstein static universe and the big bang cosmological models of general relativity. In order to give a precise definition of a Robertson-Walker space-time, it is necessary to first recall some concepts from the theory of two-point homogeneous Riemannian manifolds and isotropic Riemannian manifolds. Let (H, h) be a Riemannian manifold. Denote by I ( H ) the isometry group of (H, h) and by & : H x H
-+ R
the Riemannian distance function of (H, h).
Definition 5.3. (Homogeneous and Two-Point Homogeneous Manifolds) FIGURE5.7. The n-dimensional de Sitter space-time with positive constant sectional curvature r-2 is the set -x12 +x22+-. .+x:+~ = r 2 in Minkowski space-time R;'~. The geodesics of S,"lie on the intersection of S," with the planes through the origin of R;".
The Riemannian manifold (H, h) is said to be homogeneous if I ( H ) acts transitively on H , i.e., given any p,q f H , there is an isometry
4
E I ( H ) with
4(p) = q. Further, (H, h) is said to be two-point homogeneous if given any PI, ql, p2,92 E H with do(~1,ql)= do(p2, 4, there is a n isometry
4
E I(H)
with +(PI) = p2 and 4(ql) = 92. This may be reinterpreted as a Lorentzian warped product metric (cf. Section
(0,m) be given by f (t) = r2cosh2(t/r), and
Since it is possible to choose p, = q, for z = 1 , 2 , a two-point homogeneous
let S3 be given the usual complete Riemannian metric of constant sectional
Riemannian manifold is also homogeneous. Two-point homogeneous spaces
3.6) as follows. Let f : W
+
curvature one. Then the de Sitter space-time described in local coordinates
were first studied by Busemann (1942) in the more general setting of locally
as above is the warped product (W x S3, -dt2 @ fh).
compact metric spaces. Wang (1951, 1952) and Tits (1955) classified two-point
Universal anti-de Sitter space-time of curvature K = -1 may be given coordinates (t', r, 8,4) for which the metric has the form
homogeneous Riemannian manifolds. Notice that in Definition 5.3 it is not required that (H, h) be a complete Riemannian manifold. Nonetheless, homogeneous Riemannian manifolds have the important basic property of always being complete.
[cf. Hawking and Ellis (1973, pp. 131, 136)). Regarding -(dt')2 as a negaas the complete tive definite metric on W and dr2 sinh2(r)(a2 sin2 B
+
+
L e m m a 5.4. If (H, h) is a homogeneous Riemannian manifold, then (H, hf is complete.
186
5
EXAMPLES OF SPACE-TIMES
5.4
ROBERTSON-WALKER SPACE-TIMES
187
Proof. By the Hopf-Rinow Theorem, it suffices to show that ( H ,h ) is geodesically complete. Thus suppose that c : [a,1) -+ H is a unit speed
with $,cl(0) = -cJ(0). Hence by geodesic uniqueness, g!~(c(t)) = c ( - t ) for
geodesic which is not extendible t o t = 1. Choosing any p E H , we may find a constant a > 0 such that any unit speed geodesic starting at p has length
c 1 [0,6). Since p may be taken t o be any point o f the geodesic c, it follows that a = -m and 6 = f m . Thus ( H ,h ) is geodesically complete. Hence by
> 0. Since isometries preserve geodesics, it
the Hopf-Rinow Theorem, given any two points pl, p2 E H , there is a geodesic
1 2 a. Set 6 = min{a/2, ( 1- a ) / 2 )
all t E ( a ,6). This implies that the length of c 1 ( a ,0] equals the length o f
follows from the homogeneity o f ( H ,h ) that any unit speed geodesic starting
segment
at c(1 - 6 ) may be extended t o a geodesic o f length 1 2 26. In particular,
o f Q. As ( H ,h ) is isotropic, there is an isometry # E I,(H) which reverses Q. I t follows that # ( P I ) = p2. Hence ( H ,h ) is homogeneous. It remains to show
c may be extended to a geodesic c : [a,1
+6)
-4
H , in contradiction t o the
inextendibility o f c to t = 1. 0
Q
o f minimal length &(pl,pz) from pl to p2. Let p be the midpoint
false in general for homogeneous Lorentzian manifolds [cf.Wolf (1974, p. 95),
that i f P I , ql, p2, q2 E H with & ( P I , ql) = do(p2,q2) > 0 are given, we may find an isometry 4 E I ( H ) with # ( P I ) = p2 and # ( q l ) = q2. Choose minimal unit speed geodesics cl from pl t o ql and c2 from p2 to 92. Since ( H ,h ) is
Marsden (1973)l.
homogeneous, we may first find an isometry 1I, E I ( H ) with lI,(pl)= p2. Then
R e m a r k 5.5. It is important to note that the conclusion o f Lemma 5.4 is
W e now recall the concept o f an isotropic Riemannian manifold. Given p 6 ( H ,h ) , the isotropy group I p ( H ) o f ( H ,h ) at p is the closed subgroup
as ( H ,h ) is isotropic, we may find 77 E I,,(H) with
v*((+o c1)'(0))= ~ ' ( 0 ) .
I t follows that 4 = 71 0 $ is the required isometry. Now suppose that ( H ,h ) is two-point homogeneous. Fix any p E M , and
I p ( H )= { 4 E I ( H ) : $(p) = p ) o f I ( H ) consisting of all isometries of ( H ,h ) which fix p. Given any 4 E I p ( H ) ,the differential #, maps T p H onto T p H
let U be a convex normal neighborhood based at p. Choose a > 0 such that
since 4 ( p ) = p. As h(4,v, & v ) = h ( v ,v ) for any v E T p H , the differential q5,, also maps the unit sphere SpH = {v E T,M : h ( v ,v ) = 1 ) in T,H onto itself.
o f nonzero tangent vectors with h ( v ,v ) = h ( w ,w )< a/2. Set ql = exppv and
Definition 5.6. (Isotropic Riemannian Manifold) A Riemannian manifold ( H ,h) is said t o be isotropic at p i f I p ( H ) acts transitively on the unit
expp(v)E U for all v E T,H with h ( v , v ) 5 a . Now let v?w E T p H be any pair = q2 = exp, w.Then ql, q2 E U and d(p,q l ) = = d(p,q2). Since ( H ,h ) is two-point homogeneous, there is thus an isometry $ E I ( H )
sphere SpH o f T p H ,i.e., given any v , w E S,H, there is an isometry 4 E I,(H)
with 4 ( p ) = p and $(ql) = q2. I t follows that $,v = w. The linearity o f qap: T p H --t T p H for any 11 E I p ( H )then implies that I,(H) acts transitively
with 4,v = w.The Riemannian manifold ( H , h ) is said t o be isotropic i f it is
on S p H . Thus ( H ,h ) is isotropic at p. As the same argument clearly holds for
isotropic at every point.
all p E H , it follows that ( H ,h ) is isotropic as required.
W e now show that the class o f isotropic Riemannian manifolds coincides with the class of two-point homogeneous Riemannian manifolds [cf.Wolf (1974,
Corollary 5.8. Any isotropic Riemannian manifold is homogeneous and complete.
p 289)]. Proposition 5.7. A Riemannian manifold ( H ,h ) is isotropic iff it is twopoint homogeneous.
R e m a r k 5.9. ( 1 ) T h e two-point homogeneous Riemannian manifolds are well known [cf.Wolf (1974, pp. 290-296)]. In particular, the odd-dimensional two-point homogeneous (hence isotropic) Riemannian manifolds are just the
Proof. Recall that & denotes the Riemannian distance function o f ( H ,h).
odd-dimensional Euclidean, hyperbolic, spherical, and elliptic spaces [cf.Wang
First suppose that ( H ,h ) is isotropic. Then for each p E H and each inextendible geodesic c : ( a ,b) -+ H with c(0) = p, there is an isometry 4 E I,(H)
(1951, p. 47311. ( 2 ) Astronomical observations indicate that the spatial universe is approxi-
188
EXAMPLES OF SPACE-TIMES
5
5.4
mately spherically symmetric about the earth. This suggests that the spatial
ROBERTSON-WALKER SPACE-TIMES
189
universe should be modeled as a three-dimensional isotropic Riemannian mani-
Theorem 3.69 we also know that every level surface n.-'(c) = {c) x H is a Cauchy surface for Mo x f H.
fold. Hence the possibilities are limited to the Euclidean, hyperbolic, spherical,
Next to Minkowski space Rn = R x Rn-' itself, the Einstein static universe
and elliptic spaces. However, if one only assumes local isotropy, there are more possibilities [cf. Misner, Thorne, and Wheeler (1973, pp. 713-725)]. (3) Any three-dimensional isotropic Riemannian manifold (H, h) has constant sectional curvature, and also dim I ( H ) = 6 [cf. Walker (1944)l.
is the simplest example of a Robertson-Walker space-time.
Example 5.11. (Einstein Static Universe)
ative definite metric -dt2, and let H = Sn-' with the standard spherical Riemannian metric. I f f : R
We are now ready to define Robertson-Walker spacetimes using Lorentzian
Let Mo = R with the neg-
-+
(0, m ) is the trivial warping function f = 1,
then the product Lorentzian manifold M = Mo x H = Mo x f H is the n-
warped products and isotropic Riemannian manifolds.
dimensional Einstein static universe. If n = 2, then M is the cylinder R x S'
Definition 5.10. (Robertson-Walker Space-time) A Robertson-Walker space-time (M, g) is any Lorentzian manifold which can be written in the form
not flat since Sn-' has constant positive sectional curvature K = 1.
with Aat metric -dt2
+ do2. If n 2 3, then this metric for M = R x Sn-' is
<
For the rest of this section we restrict our attention to four-dimensional
b 5 -too,given the negative definite metric -dt2, with (HI h) an isotropic
Robertson-Walker spacetimes. By Remark 5.9, these are warped products
Riemannian manifold, and with warping function f : Mo -+ (0, m).
Mo x f H , where (H, h) is Euclidean, hyperbolic, spherical, or elliptic of di-
of a Lorentzian warped product (Mo x f H,g) with Mo = ( a , b ) , -co 5 a
In the notation of Section 3.6, we thus have g = -dt2 @ f h and Mo x f H
mension three. In the first two cases, H is topologically R3. In the third case
is also topologically the product Mo x H. Letting du2 denote the Riemannian
H = S3, and in the last case H is the real projective three-space RP3.We thus have the following
metric h for H and defining S(t) =
m,the Lorentzian metric g for Mo x f H
may be rewritten in the more familiar form
Corollary 5.12. All four-dimensional Robertson-Walker spacetimes are topologically either R4, R x S3, or R x RP3.
Also by Remark 5.9, the sectional curvature K of (H, h) is constant. If K is The map n. : Mo x f H
--+
R given by x(t,z) = t is a smooth time function
nonzero, the metric may be rescaled to be of the form ds2 = -dt2
+ S2(t)du2
on Mo x f H so that the Lorentzian manifold Mo x f H , of Definition 5.10
on M so that K is either identically $1 or -1. This is the form of the metric
actually is a (stably causal) spacetime. Also each level surface n.-l(c) of the
usually studied in general relativity.
c R is an isotropic Riemannian manifold which is
In physics, cosmological models are built from four-dimensional Robertson-
homothetic to (HIh). Furthermore, the isometry group I ( H ) of (HIh) may be
Walker space-times assumed to be filled with a perfect fluid. The Einstein
map
T
: Mo
xfH
--+
Mo
T(H)of I(Mo x f H ) as follows.
Given 4 E I(H),
equations (cf. Chapter 2) are then used to find the form of the above warping
h) = (r,$(h)) for all (r, h) E Mo x H. With this
definition, F(H) restricted to the level surfaces s-'(c) of s acts transitively on
function S2(t). Among the models this technique yields are the big bang cosmological models [cf. Hawking and Ellis (1973, pp. 134-138)]. These models
each level surface.
depend on the energy density p and pressure p of the perfect fluid a s well as
Since all isotropic Riemannian manifolds are complete, Theorem 3.66 implies that all Robertson-Walker spacetimes are globally hyperbolic. From
the value of the cosmological constant A in the Einstein equations. In the big bang cosmological models, the inextendible nonspacelike geodesics are all past
identified with a subgroup defbe
E F(H) by
$(T,
190
5
5.5
EXAMPLES O F SPACE-TIMES
LIE GROUPS: BI-INVARIANT LORENTZIAN METRICS
191
incomplete. The stability of this incompleteness under metric perturbations
Thus any compact Lie group is furnished with a large supply of bi-invariant
will be considered in Section 7.3. Astronomical observations of clusters of
Riemannian metrics.
galaxies indicate that distant clusters of galaxies are receding from us. This
On the other hand, while (5.1) equips any Lie group with left-invariant
expansion of the universe suggests the existence of a "big bang" in the past and
Lorentzian metrics, the standard Haar integral averaging procedure used for
also suggests that the universe is a warped product with a nontrivial warping
Riemannian metrics fails to preserve signature (-,
function rather than simply a Lorentzian product. Observations of blackbody
used to convert left-invariant Lorentzian metrics into bi-invariant Lorentzian
radiation support these ideas [cf. Hawking and Ellis (1973, Chapter
lo)].
+, . - . ,+), so it cannot be
metrics. But we will see shortly that a large class of bi-invariant Lorentzian metrics
5.5
may be constructed for noncompact Lie groups of the form
Bi-Invariant Lorentzian Metrics o n Lie G r o u p s
R x G, where G is
any Lie group admitting a bi-invariant Riemannian metric. The purpose of this section is to show how Theorems 3.67 and 3.68 of
Before giving the construction, we need to discuss product Lie groups briefly.
Section 3.6 may be used to construct a large class of Lie groups admitting
Let G and H be two Lie groups. The product manifold G x H is then turned
globally hyperbolic, bi-invariant Lorentzian metrics.
into a Lie group by defining the multiplication by
We first summarize some basic facts from the elementary theory of Lie groups. Details may be found in a lucid exposition by Milnor (1963, Part IV) or at a more advanced level in Helgason (1978, Chapter 2). A Lie group is a
It is immediate from (5.2) that if u = (g, h) E G XH , then the translation maps
group G which is also an analytic manifold such that the mapping (g, h)
L,, R, : G x H
gh-I from G x G
-+
translation maps L,,
-+
G is analytic. This multiplication induces left and right
R, for
, ) for G
It is straighforward to check that for any u E G x H and any tangent vector
t = (v, w) f T,(G
is then said
to be left invariant (respectively, right invariant) if (L,*v, L,,w)
G x H are given by L, = (L,, Lh), and R, = (R,, Rh),
i.e., L,(gl, hl) = (L,gl, Lhhl)>etc. Recall that T,(G x H) 2 T,G x ThH.
each g E G I given respectively by L,(h) = gh
and Rg(h) = hg. A Riemannian or Lorentzian metric (
-+
x H ) 2 TgG x ThH, one has
= (v, w)
(respectively, (Rgtv,R,,w) = (v, w)) for all g E G and v, w E TG. A metric and
which is both left and right invariant is said to be bi-invariant. By an averaging procedure involving the Haar integral, any compact Lie group may be given a bi-invariant metric [cf. Milnor (1963, p. 112)]. In fact, the Haar integral
Now if ( ,
may be used to produce a bi-invariant Riemannian metric for G from any left left invariant Riemannian (or Lorentzian) metric by starting with a positive
(
,
)I,:T,GxT,G+
W by
is a Lorentzian metric for G and ( , j2 is a Riemannian
metric for H , the product metric (( , )) = ( , )1 @ ( , )2 is a Lorentzian metric for G x H . Explicitly, recalling Definition 3.51, we have for tangent
invariant Riemannian metric for G. Any Lie group may be equipped with a definite inner product (respectively, inner product of signature n - 2) ( , )I, on the tangent space T,G to G a t the identity element e E G, then defining
)l
vectors X
i I
i P
El
= (vl, wl) and
t2= (vz, w2) in T,(G
x H ) the formula
((Ei,E2)) = (v1,~2)1+ (~1,w2)2. It is then immediate from (5.3) and (5.4) that if ( ,
is a bi-invariant
Lorentzian metric for G and ( , )2 is a bi-invariant Riemannian metric for H , then (( , )) is a bi-invariant Lorentzian metric for G x H. To summarize,
5
192
EXAMPLES OF SPACE-TIMES
Proposition 5.13. Let (G, (
,
5.5
be a Lie group equipped with a bi-
LIE GROUPS: BI-INVARIANT LORENTZIAN METRICS
193
for any [ E T,,(G x H) we have
invariant Lorentzian metric, and let (H, ( , )2) be a Lie group equipped with a bi-invariant Riemannian metric. Then the product metric ((
~ u . 0, the set {x E M : p 0 for (XI,.. . ,x,) E U. While U does not have compact closure in Minkowski space-time, f ( U ) does have
:
[O, I]
--t
+
(M1,g') with o(0)
E
and a(1) = p. Since F ( M ) is open in M' and the curve 7 = F-'
such that U is contained in some open set {(XI,. . . ,x,) E Rn : 0 < XI < a ) for
F(M)
B(F(M)), and choose a geodesic a to E (0, I] such that u(t) E
Then f o y : [I, ca) -+ M' is a spiral which is asymptotic to the circle t = B3 = . . = 8, = 0 in M'. Let U be an open tubular neighborhood about y in M
[0, to)
0 a][o,to) :
-+
M is b-incomplete, inextendible to
:U
-t
compact closure in M ' since f (U) is contained in the compact set [O, a]x Tn-l.
6
COMPLETENESS AND EXTENDIBILITY
6.6
SINGULARITIES
225
Thus f : (U, glU) --, (M', g') is an analytic local extension of Minkowski spacetime. Notice that if y~ : [0,a)
--+
U is any curve with noncompact closure in
M , then f o yl cannot be extended to t = a in MI.
Let d M denote an ideal boundary of M (i.e., d M represents either dbM or d,M).
A point q E d M is said to be a regular boundary point of M if
there exists a global extension (MI, g') of (M, g) such that q may be naturally identified with a point of MI. A regular boundary point may thus be regarded as being a removable singularity of M. Let y : 10, a)
-+
M be an inextendible curve such that y(a) corresponds
to an ideal point of M. The curve y is said to define a curvature singularity [cf. Ellis and Schmidt (1977, p. 916)] if some component of Ra6cd;el,,,,,ek is not C0on [0,a] when measured in a parallelly propagated orthonormal basis along
y. A curvature singularity is an obstruction to a local b-boundary extension about y because if there is a local b-boundary extension about y, then the curvature tensor and all of its derivatives measured in a parallelly propagated orthonormal basis must be continuous and hence converge to well-defined limits as t
FIGURE 6.7. Minkowski space-time has analytic local extensions. Let M = Wn be given the usual Minkowskian metric g, and let Tn-I be the (n - 1)-dimensional torus with the usual positive definite flat metric h. Let M' = W x Tn-' be given the Lorentzian product
-+
a-. A related but somewhat different notion is that of strung curvature
singularity which may be defined using expansion 8 along null geodesics. The notion of strong curvature singularity has been useful in connection with cosmic censorship [cf. Kr6lak (1992), Kr6lak and Rudnicki (1993)l.
A b-boundary point q
E
dbM which is neither a regular boundary point
metric g' = -dt2 8 h. Then (M,g) is the universal covering space
nor a curvature singularity is called a quasi-regular singularity. Clarke (1973,
of (MI, g'), and the quotient map f : M
p. 208) has proven that if y : [0, a)
-+
M' is locally isometric.
-+
M is an inextendible b-complete
Choose U to be an open set in M about y(s) = (s-P, s, .. . ,0),
curve which correspo,lds to a quasi-regular singularity, then there is a local
M , such that f ] U is one-to-one and f ( U ) has compact
b-boundary extension about y. This shows that curvature singularities are
7 : [I, m)
-+
closure in Mi. Then f 1~ : (U,gu)
-+
(M1,g') is an analytic local
extension of Minkowski space, but this extension is not across points of dcM and not across points of dbM.
the only real obstructions to local b-boundary extensions. In general, it can be quite difficult to decide if a given space-time has local extensions of some type. However, for analytical local b-boundary extensions of analytic space-times, the situation is somewhat simpler (cf. Theorem 6.23).
226
6
COMPLETENESS AND EXTENDIBILITY
6.6
For the proof of Theorem 6.23, it is useful to prove the following proposition about real analytic space-times and local isometries. Recall that a local isometry F : M
-+ M'
is a map such that for each p E M , there exists an open
neighborhood U(p) of p on which F is an isometry. Thus local isometries are
227
71 : [O,to) -+ M. But since F is smooth, F,x = F,X(to) = lim,,5
Because F, o X is a parallel vector field for all t with 0 that lim,-,o
F,X(t)
# 0.
F,X(t).
< t < to, it follows
Hence Fax # 0. Thus F is nonsingular a t the point
r, whence r E V in contradiction.
local diffeomorphisms but need not be globally one-to-one.
0
We are now ready to turn to the proof of Theorem 6.23 on local b-boundary
Proposition 6.22. Let (M, g) and (Ml,gl) be real analytic space-times of the same dimension and suppose that F : M
SINGULARITIES
MI is a real analytic map. If M contains an open set U such that Flu : U + Ml is an isometry, then F
extensions of real analytic space-times.
-+
is a local isometry.
Theorem 6.23. Suppose (M, g) is an analytic space-time with no imprisoned nonspacelike curves, which has an analytic local b-boundary extension about y : [O,a) + M . Then there are timelike, null, and spacelike geodesics
Proof. Let W = {m E M
:
F,v # 0 for all v # 0 in T,M),
which is an
open subset of M by the inverse function theorem. Since Flu is an isometry,
of finite &ne parameter which are inextendible in one direction and which do not correspond to curvature singularities.
U is contained in W. Fix any p E U, and let V be the path connected
The proof of Theorem 6.23 will involve two lemmas.
component of W containing p. We will establish the proposition by showing Lemma 6.24. Suppose (M,g) is an analytic space-time with no impris-
first that Ffvis a local isometry and second that V = M . Let q be any point of V. Choose a curve y : [0,1] -+ V with $0) = p and $1) = q. By the usual compactness arguments, we may cover y[O, I] with a finite chain of coordinate charts (Ul,
U, is simply connected, Flu, : U, Ul C u n v , q E UI;, and U,nU,+l
-+ MI
oned nonspacelike curves, which has an analytic local b-boundary extension about y : [O, a )
(U2,+2), . . . ,(Uk, &) such that each is an analytic diffeomorphism, p E
# 0 for each i with 1 I i I k-1. Since UI C
UnV, we have g = (Flul)*gl on Ul. Thus g = (FIUl)*glon UlnU2. Since Uln
-+
M . Then (M, g) has an incomplete geodesic.
Proof. Let f : (U, glU)
-+
(U1,g') be an analytic extension about y. We
may assume U contains the image of y. Also, f o y is extendible in U'. Thus
f
o
y(t)
-+
p E U' as t
-+
a-.
Let W' be a neighborhood of p such that
M . Choose any point
W' is a convex normal neighborhood of each of its points. Then exp;l : W' -+ T,U' is a diffeomorphism for each fixed x E TV'. Assume to is chosen with f 0 y(t) E W' for all to < t < a. Set q = y(t0) and T = f(q). Then H = exp, of;' o exP;' : W' -+ M is analytic and is a t least defined near r. The map H takes geodesics starting a t r to geodesics starting a t q, and H preserves lengths along these geodesics. In fact, H agrees with f-' near r .
rl E M - V. Let yl : [O, 11 -+ M be a smooth curve with y(0) = p and
The map H need not be one-to-one since expq is not necessarily one-to-one.
U2 is an open subset of U2 and F is a real analytic diffeomorphism of U2 onto its image, it follows that g = ( FIU2)*glon U2. Continuing inductively, we obtain g = (FIUk)*glon Uk, whence F is an isometry in the open neighborhood Uk of q. Thus FIv : V
MI is a local isometry. It remains to show that V = M. Suppose V -+
#
= y(to) E M - V. Then
Because the domain of exp, is a union of line segments starting at the origin
F restricted to the neighborhood V of yl{O,to) is a local isometry. It suffices to show that r E V to obtain the desired contradiction. Since r E M - V,
of T,M, the domain V' of H must be some subset of W' which is a union of
y(1) = rl. There is a smallest to E [O, 1) such that
there exists a tangent vector x
T
# 0 in T,M with F,x
= 0. Let X be the
geodesic segments starting a t r. Hence the set V' fails to be all of W' only when expq : T,M
-t
M is defined on a proper subset of TqM which does not
f;' o exp;l(W1). Thus if we show V' # W', there is
unique parallel field along 7 with X(to) = x. Then F, o X is a parallel field
include all of the image
along F o yl : [O,to) -+ MI since F is a local isometry in a neighborhood of
some incomplete, inextendible geodesic starting at q. But the analytic maps
6
228
H and f
6.6
COMPLETENESS AND EXTENDIBILITY
SINGULARITIES
229
-'must agree on the component o f f (U) nV' which contains r. This
implies V'
# W'. Otherwise, H and f -'would agree on f (U) nW' and hence < a and
on a neighborhood of f o ?[to, a). This yields H o f o y = y for to 5 t implies y is extendible in M across the point H(p), in contradiction.
-
We will continue with the same notation in the next lemma. L e m m a 6.25. The map H : V'
+
f
M is a local isometry.
Proof.The space-time (M, g) is analytic, (U', g') is analytic, and H is analytic. Furthermore, H agrees with the isometry f -l near r , and H is defined on an arcwise connected set V'. Thus Proposition 6.22 implies H is a local isometry. O We are now ready to complete the Proof of Theorem 6.23. There are three cases to consider corresponding to incomplete timelike, null, and spacelike geodesics. We only give the proof for the timelike case. Let U, U', f , etc., be as in Lemmas 6.24 and 6.25. Assume without loss of generality that there is some point x E W' such that in a chronological ordering on W', we have x to 5 t If x
0.
an example MI x M2 which shows that geodesic completeness (respectively,
In this chapter the manifold M is always fixed. However, one-parameter families ( M A gx) , of manifolds and Lorentzian metrics have been considered in
incompleteness) may fail to be stable for signature ( s , ~ whenever ) neither s
general relativity [cf. Geroch (1969)l.
nor T is zero. The situation for positive definite (i.e., s = 0) and negative definite (i.e.,
7.2
T
= 0) signatures is quite different. For these signatures, both geodesic com-
pleteness and geodesic incompleteness are C0stable. For example, if (M, g) is positive definite, then the set U(g) consisting of all positive definite metrics h on M with
T h e C' Topology a n d Geodesic Systems
If (M,g) is an arbitrary Lorentzian manifold, then metrics in Lor(M) which are close to g in the fine C' topology have geodesic systems which are close to the geodesic system of g. The purpose of this section is to give a more analytic formulation of this concept, which is needed for our investigation of the C1 stability of null geodesic incompleteness for Robertson-Walker space-tlmes in
for all nontrivial vectors v is a Co-open subset of the collection Kem(M) of all Riemannian metrics on M . Clearly, U(g) contains the metric g. If h is
We begin by recalling a well-known estimate from the theory of ordinary
fixed metric in U ( g ) , then any given curve has an h-length LI, and a g-lengt
fferential equations [cf. Birkhoff and Rota (1969, p. 155)]. We will al-
L, with
ays use IlzIlz to denote the Euclidean norm
1 -L, 5 Lh 1 2Lg. 2 This inequality shows that either g and h are both complete or else bot
[C:,x Z 2 ]
roposition 7.9. Suppose that f = ( f l , . . . ,f,)
of the point x =
and h = ( h l , . . . , h,)
are
incomplete. It follows that geodesic completeness and geodesic incomple are CO stable for positive definite negative definite case, and consequently one obtains the following result Proposition 7.7. Let (M, g)
Ilf ( s >x) - f (.,z)llz 5 Lllx - z112
inite ,cjemi-fi~annianmanifold.
& r 2 0 thexe i s a CY neighborhood .\M. 3) & *xm*e-
k,GT,i
*edLz?-s'-
-J
1 1 1 -
. ,y,(s))
be solutions for
248
7
7.2
STABILITY O F COMPLETENESS AND INCOMPLETENESS
Now let M be a smooth manifold, and let (U, x) be any coordinate chart for M . We may obtain an associated coordinate chart
THE C1TOPOLOGY AND GEODESIC SYSTEMS
249
We will use the notation expq[g,] for the exponential map at q E (M,g,), a = 1,2. If v E TMIU, then s --+ expq[ga](sv)is the solution of (7.1) in U with initial conditions (q,v) for (M,g,). In order to apply Proposition 7.9
z = ( ~ 1 , ~ 2. - ,,xn,xn+lr. . - .1 ~ 2 n ) for TMlu as follows. Let d/dxl, . . . ,d/dxn be the basis vector fields defined on U bv the local coordinates x = (xi,. . .,xn). Given v E TqM for q E n Then E(v) is defined to be Z(v) we may write v = ai
to these exponential maps, we identify TMlu with a subset of W2" using the coordinate chart (TMIu ,Z ) and define f (s,X ) = f (X) and h(s, X ) = h(X)
zEl$1".
(xl(q), x2(q),. . . ,xn(q),al, a2,. . . ,an). These coordinate charts may then be
fi+n(X) = -rjk(gl)~j+n~k+n,
used to define Euclidean coordinate distances on U and TMIU. Explicitly, and
given p, q E U and v, w f TMIU,set
= -rjk(92)xj+nxk+n
for 1 5 i , j , k 5 n a n d X = (xl ,..., x2,) E WZn. Lemma 7.10. Let (U, x) be a local coordinate chart for the n-manifold
M. Let (p,v) E TMIu, and assume that cl(s) = exp,[gl](sv) lies in U for all
< s 5 b.
> 0, there exists a constant 6 > 0 such that JJv- wl12 < 6 and llgl-g2111,u < 6implythatca(s) = expq[g2](sw)liesinU ford10 5 s < b,
and
0
Given E
and moreover,
0 to mean that calculating wi
respectively. Also if T 2 0 is given, we will use the notation IJgl-g2\lr,v gl,g2 E Lor(M) and a positive constant 6
the local coordinates (U,x), all the corresponding entries of the two metri tensors and all their corresponding partial derivatives up to order r are 6clos at each point of U. We will denote the Christoffel symbols of the second kind for gl,g2 E
Ibj 0 CI)'(S)- (53
O
CZ)'(S)I< E
for all 1 5 j 5 n a n d 0 5 s 5 b.
Proof. Let f (X) and h(X) be defined as above. Then X(s) = (cl (s), cll(s)) and Y(s) = (cz(s),czl(s)) are solutions to the differential equations dX/ds =
Lor(M) by rJk(gl) and rJk(g2):respectively. Then for a = 1,2, the geodesic
f(Xf and dY/ds = h(Y), respectively. Choose Do to be an open set in TMlu
equations in the coordinate chart (U, x) for (M, g,) are given by
about the image of the curve X(s) such that DOis compact. Then there exists a constant L such that f satisfies a Lipschitz condition 11 f ( X ) - f ( x ) ] / 25 L1IX - Xllz on Do. We may make the term jlX(0) - Y(0)IJ2 in the estimate of Proposition
for 1 5 i, j, k 5 n, where we employ the Einstein summation convention throughout this chapter.
7.9 as small as required by making Ilu - wll2 small. Furthermore, since the Chriitoffel symbols depend only on the coefficients of the metric tensor and
on their first partial derivatives, we may make 11 f (X) - h(X)ll2 as small as we
250
7
7.3
STABILITY OF COMPLETENESS AND INCOMPLETENESS
GEODESIC INCOMPLETENESS
251
wish on Do by requiring that IJgl - g2jll,u be small. Hence for a sufficiently
we will formulate the results in the first portion of this section for the larger
> 0, Proposition 7.9 may be applied to guarantee that c2(s) E U for all 0 5 s 5 b and also to yield the estimate IIX(s) - Y(s)l12 < E for all 0 5 s 5 b. Consequently, Ixi(s) - yi(s)I < E for all 1 5 i 5 2 n and 0 5 s < b. In view of
class of Lorentzian warped products M = (a, b) x f H with
small 6
(7.1), this establishes the desired inequalities. 0
-00
5 a 0 may be chosen
such that
Definition 7.12. (Adapted Normal Neighborhood) vex normal neighborhood
An arbitrary con-
U of (M,g) with compact closure 7 7 is said to
be an adapted normal neighborhood if
U
is covered by adapted coordinates
(xl,x2,. . . ,x,) which are adapted at some point of U such that the following
7.3
Stability of Geodesic Incompleteness for Robertson-Walker
Space-times
hold: (1) At every point of U , the components gij of the metric tensor g ex-
pressed in the given coordinates (XI,xz, . . . ,x,) differ from the matrix In this section we investigate the stability in the space of Lorentzian rnetrics of the nonspacelike geodesic incompleteness of Robertson-Walker space-times
M = (a,b) x f H (cf. Definition 5.10). It turns out, however, that the proof of the C0 stability of timelike geodesic incompleteness uses only the homogeneity of the Riemannian factor (H,h) and not the isotropy of (H, h). Accordingly,
diag(-1, +I,. ..,+1) by at most 112. (2) The metric g satisfies g < U 71, where 71 is the Minkowskian metric
ds2 = -2dx12 +dxz2
+ ... + dxn2 for U (see the definition of stably
causal in Section 3.2 for the notation g
where the functions kiJ : U -+ IR satisfy Ikajl I 1/2 for all 1 5 i, j I n. For use in the sequel, we need t o establish the existence o f countable chains {Uk) o f adapted normal neighborhoods covering future directed, past inextendible, timelike geodesics o f the form c(t) = ( t ,yo). Since in Definition a
Proof. W e will say that {Uk), { t k ) is an admissible chain for c I (9,wo],9 a, i f {Uk), { t k ) satisfy the properties o f Definition 7.14 except that tk -+Bt as k + oo instead o f t k --+ a+ as k -t co. Set T
= inf{8 E [a,wo] : there is an admissible chain
{Uk), I t k ) for c 1
wo] for all 0'
(el,
7.14 and in Lemma 7.15 it is possible that a = -co, we adopt the following W e must show that
convention throughout this section. Convention 7.13. Let wo denote any fixed interior point o f the interval
( a ,b).
T
= a.
B y taking an adapted normal neighborhood centered at c(wo),it is easily seen that T < W O . Suppose that T > a. Let U be any adapted normal neighborhood adapted at the point
Now we make the following definition. Definition 7.14. (Admissible Chain)
( T , yo)
E
M . Choose T > T such that all future
directed nonspacelike curves u ( t ) = (t,u l ( t ) ) originating at (r,yo) lie in U for Let M = ( a ,b) x f H denote a Lor-
entzian warped product with metric g = -dt2
@ fh.
Fix any yo E H and let
c : ( a ,wo]-+ ( M ,3 ) be the future directed, past inextendible, timelike geodesic
given by c ( t ) = ( t ,yo). A countable covering {Uk)p=l o f c by open sets and a strictly monotone decreasing sequence {tk)& with tl = wo and tk
as k -+ co is said t o be an admissible chain for c : (a,wo] following two conditions hold:
-+
-+
a+
( M , g ) i f the
(1) Each Uk is an adapted normal neighborhood containing c(tk) = (tk,yo) which is adapted at some point of c. ( 2 ) For each k , every future directed and past inextendible nonspaceliie curve u ( t ) = ( t , a z ( t ) )with u ( t k )= ( t k ,yo) remains in Uk for all t with tk+l I t I
9).
tk.
Any future directed nonspacelike curve u in ( M , g ) may be given a parametrization of the form u ( t ) = (t,u l ( t ) ) . Thus condition ( 2 ) applies t o all future directed nonspacelike curves issuing from (tk,y o ) W e now show that admissible chains exist. Lemma 7.15. Let M = (a,b) x f H with a 2 -m and 3 = -dt2 $ fh be a Lorentzian warped product. For any yo E H , the timelike geodesic
where E > 0. There exists an admissible chain {U,), {t,) for C ( ( T + 7 ) / 2 ,W O ] with tm < r for some m. Define U, = = U . Extending the finite chain {U1,U2,. . . ,Urn-1,Urn,Um+l),{ t l , t 2 , . . . ,t m - l l t m l ~ 6) to an infinite admissible chain yields the required contradiction. 0 all
T
-E I t I
T,
W e now show that the subset o f Uk for which property (2) o f Definition 7.14 holds may be extended from the point ( t k , yo) to a neighborhood I t k ) x V k ( y o ) in { t k ) x H . T h e notation Ilg - glllo,pk 7.2.
< 6 has been introduced in Section
Lemma 7.16. Let {Uk), { t k ) be an admissible chain for the timelike ge-
odesic c ( t ) = (t,yo), c : (a,wo] -+ ( M , g ) . For each k , there is a neighborhood Vk(y0) of yo in H such that any future directed nonspacelike curve a ( t ) = ( t , a l ( t ) ) with ~ ( t k E) { t k ) x &(yo) remains in Uk for d l t with tk+l 5 t 5 t k . Furthermore, Vk(yo)and 6 > 0 may be chosen such that i f gl E Lor(M) and I(g - glllo,uk < 6, then the following two conditions are satisfied. If y ( t ) = (t,-yl(t))is any nonspacelike curve of ( M , g l ) with
), ~ ( t kE) { t k ) x V ~ ( Y O then ( 1 ) y remains in Uk for tk+l 5 t 5 t k ; and ( 2 ) The gl-length o f yl [ t k + ~tk] , is at most &n(tk - tk+l).
254
7
7.3
STABILITY OF COMPLETENESS AND INCOMPLETENESS
GEODESIC INCOMPLETENESS
255
as a global time function for M . In particular, the vector field V z satisfies
Assuming now that the Riemannian factor ( H ,h ) o f the Lorentzian warped product is homogeneous, we may extend Lemma 7.16 from Uk t o [tk+l,t k ]x H .
g ( V z ,V z )< 0 at all points of M . Define g E Lor(M) by
W e will use the notation lgl - glo < 6, as defined in Section 3.2.
Proof. First recall that z : M = (a,b) x H
--+
W given by ~ ( ht ),= t serves
Lemma 7.17. Let ( M , g ) be a Lorentzian warped product with ( H ,h )
I t follows that g < $ on M so that U2 = { i 2 E C o n ( M ) : g2 < ~ ( 9 )is) an open neighborhood o f C ( M , g ) in Con(M). Let Ul = r-l(U2). Then Ul is a Co-open neighborhood o f g in Lor(M) such that i f gl E U l , the projection map z : M --+ W is a global time function for ( M ,gl). Hence the hypersurfaces { t ) x H , t E ( a ,b), remain spacelike in ( M l g l ) . Thus any nonspacelike curve y of ( M ,g l ) , gl E U l , may be parametrized as y ( t ) = ( t ,y l ( t ) ) . Thus the lemma will apply to any nonspacelike curve o f ( M ,g l ) originating at any point o f { t k ) x Vk(yO)provided g1 E Ul is sufficientlyclose t o g on Uk. Let ( x l , . . . ,x,) denote the given adapted coordinate system for the adapted normal neighborhood Uk. In view of condition ( 2 ) o f Definition 7.12, we may find 61 > 0 such that llg - g1110,~,< 61 implies gl < 72 on Uk, where 772 is the Lorentzian metric on Uk given in the adapted local coordinates by 772 = -3 dx12+ dx22 + - . . + dxn2. Secondly, since Co-close metrics have close light cones, it follows by a compactness argument that there exist a neighborhood Vk(yo)o f yo in H and a constant 62 > 0 such that i f gl E Lor(M) satisfies llg-glllo,vk < 62 and y ( t ) = ( t ,y l ( t ) ) is any future directed nonspacelike curve o f ( M , g l ) with y(tk) E { t k ) x V ~ ( Y Othen ) , y ( t ) E Uk for tk+l i t I tk. I t remains t o establish the length estimate (2). Set 6 = min{61,62,1/21. Suppose that g' E Lor(M) satisfies jig'-gllo,uk < 6, and let y ( t ) = (t,y l ( t ) ) be any nonspacelike curve o f ( M ,g') with y (tk) E { t k ) x Vk (yo),y : [ t k + l , tk] M . Let L ( y ) denote the length o f y in ( M ,g'). Thus -+
homogeneous, and let {Uk), { t k ) be an admissible chain for c ( t ) = ( t ,yo), c : ( a ,wo] M . For each k , there is a continuous function bk : [tk+l,tk]x H -+ -+
(0,co)such that i f gl € Lor(M) and lg - gllo < 6k on
[&+I,
tk] x H , then any
nonspacelike curve y ( t ) = ( t ,y l ( t ) ) , y : [tk+l,tk]-' ( M , g l ) ,joining any point of {tk+l) x H to any point o f { t k ) x H , has length at most &n(tk - t k i l ) . Proof. Fix any k > 0. Let 6 > 0 be the constant given by Lemma 7.16 such that i f gl E Lor(M) satisfies Ilgl - gllo,pk < 6, then any nonspacelike curve y ( t ) = ( t ,y l ( t ) ) in ( M , g l ) with y(tk) E { t k ) x &(yo) remains in Uk for tk+l
I t i t k and has length at most &n (tk - & + I ) . Also let ( x l , . . . , x,)
denote the given adapted coordinates for Uk. W e may find isometries {$,)El in I ( H ) such that i f yi = $i(yo) and V k ( ~ i= ) $i(Vk(yo)), then the sets {l/k(yi))& together with Vk(y0) form a locally finite covering of H . Let @, : M -+ M be the isometry given by @i(t,h ) = ( t ,+i(h)),and set Ui = @i(Uk)for each i. Then the sets {Gi) cover [tk+l,t k ] X H and ( X I , x2 0 ah1,. . . , x, 0 @, l ) form adapted local coordinates for Ui for each i. Since everything is constructed with isometries, the constant 6 > 0 that works in Lemma 7.16 for Uk and c ( t ) = ( t ,yo) works equally well for each
6i and
@i 0
c, provided that the adapted coordinates
( ~ 1 ~0x@ 2 i l l . .. ,x , 0 a;') are used for Ui. ~f we let bk : [tk+l,tk] x H -+ M be any continuous function such that ilgl - gllo < hk on [tk+1,t k ] x H implies that llgl - g110,5i < 6 for each i, then the lemma is immediate from Lemma 7.16 Cl W e are now ready t o prove the C0 stability of timelike geodesic incomplete-
Fkom Definition 7.12 and the choice o f the 6's, we have lgi31 5 (1+ 1/2)+1/2 = 2 and Iyil(t)l 5 & for all 1 2 i,j 5 n. Thus, as required,
ness for Lorentzian warped products M = ( a ,b) x f H with a > -m and ( H .h ) homogeneous. Theorem 7.18. Let ( M , g ) be a warped product space-time o f the form M = (a,b) x f H with a > -03, g = -dt2 @ fh, and ( H ,h ) a homogeneous
256
7
STABILITY O F COMPLETENESS AND INCOMPLETENESS
Riemannian manifold. Then there exists a fine
CO
7.3
neighborhood U ( g ) o f g
GEODESIC INCOMPLETENESS
257
'I'heorem 7-19. Let ( M ,g) be a Robertson-Walker space-time o f the form
in Lor(M) o f globally hyperbolic metrics such that all timelike geodesics o f
M = ( a ,b) x f H with a > -W. Then there exists a fine C0neighborhood U ( g )
( M ,g l ) are past incomplete for each gl E U ( g ) .
of g in Lor(M) o f globally hyperbolic metrics such that all timelike geodesics of ( M ,g l ) are past incomplete for each gl E U(g).
Proof. Fix any yo E M and let c : (a,wo] 4 M be the past inextendible future directed geodesic given by c(t) = (t,yo). Let {Uk), { t k ) be an admissible chain for c, guaranteed by Lemma 7.15. Also choose 6k : [&+I,t k ]x H 4 (0,a) for each tk according t o Lemma 7.17. Let 6 : M -+ (0,CO) b e a continuous function such that 6(q) I &(q) for all q E [tk+i,tk]X H and each k > 0. Set v ~ = (~ {gl )E Lor(M) : lgl - glo < 6). Since global hyperbolicity is a Co-open condition, we may also assume that all metrics in q ( g ) are globally hyperbolic. By the first paragraph o f the proof of Lemma 7.16, we may choose a
CO
) , hyperV2(g)o f g in Lor(M) such that for all gl E V Z ( ~each
surface { t ) x H , t E (a,b), is spacelike in ( M , g l ) . Then every nons~acelike curve y
p) -+ ( M ,g
: (a,
~ may ) be parametrized in the form y ( t ) = ( t ,yl(t)).
Hence Lemma 7.16 may b e applied to all inextendible nons~acelikegeodesics in ( M , g l ) with gl E Vz(g). Now U ( g )= V l ( g ) n V z ( g )is a fine
I f we change the time function on ( M ,9 ) to T I : M -+ R defined by
( t ,h) =
-t, and apply Lemmas 7.16 and 7.17 t o the resulting space-time, we obtain the exact analogue o f these lemmas for the future directed timelike geodesic c : ["o, b) ( M ,9 ) given by c(t) = (t,yo) in the given space-time. Hence i f ( M ,g ) is a Lorentzian warped product M = (a,b) x f H with ( H ,h ) homogeneous and +
b < oo, the same proof as for Theorem 7.18 yields the C0 stability o f the future timelike geodesic incompleteness. Combining this remark with Theorem 7.18 then yields the following result. T h o r e m 7-20. Let ( M , g ) be a Lorentzian warped product o f the form
M = ( a ,b)
Xf
H , g = -dt2 @ f h , with both a and b finite and ( H ,h ) ho-
mogeneous. Then there is a fine C0 neighborhood U ( g ) o f g in Lor(M) of globally hyperbolic metrics such that all timelike geodesics o f ( M ,g l ) for each 91 f U ( g ) are both past incomplete and future incomplete.
neighborhood o f in the CO topology.
Let gl E U ( g ] ,and let y : ( a ,p) -t ( M ,gl) be any future directed inextendible timelike geodesic. W e may assume that { t l ) x H is a Cauchy surface
( M ,g l ) by the arguments o f Geroch (1970a, p. 448), and hence there exists an so E ( a ,p) such that y(s0) E { t ~x}H . In passing f k m {tk+i) X H t o { t k ) the gl length of y is at most &n ( t k - &+I), applying Lemma 7.17 for each k. Summing up these estimates, it follows that the gl length o f ( a ,so] is at most f i n (tl - a ) . Since yl (a,so] is a past inextendible timelike geodesic finite gl length, it follows that y is past incomplete in ( M , g l ) . 0 Lerner raised the following question (1973, p. 35) about the Roberts Walker big bang models ( M ,g): under small C 2 perturbations o f the me does each nonspacelike geodesic remain incomplete? Since the Ftiemannl factor ( H ,h) o f a Robertson-Walker spacetime is homogeneous, we obt
It is interesting t o note that while the finiteness o f a and b is essential t o the proof o f Theorem 7.20, the proof is independent of the particular choice o f warping function f : (a,b) -t (0,CO). While the homogeneity o f the Riemannian factor ( H ,h ) is also used in the proof o f Theorem 7.20, no other geometric r topological property o f ( H ,h ) is needed.
In general relativity and cosmology, closed big bang models for the unise are considered [cf.Hawking and Ellis (1973, Section 5.3)]. These models Robertson-Walker space-times for which b - a < oo and H is compact. ce Theorem 7.20 implies, in particular, the 60 stability o f timelike geodesic mpleteness for these models. now turn t o the proof o f the C 1 stability o f null geodesic incompleteness Robertson-Walker space-times. Taking M = ( 0 , l ) x f W with f ( t ) = and 3 = -dt2 @ fdx2, it may be checked using the results o f Section
)-2
the following corollary t o Theorem 7.18 which settles affirmatively for time
that the curve 7 : (-m,O) -t ( M , g ) given by ~ ( t=) (et, e 2 t ) is a past
geodesics the question raised by Lerner (1973, P. 35).
plete null geodesic. Thus by choosing the warping function suitably, it
258
7
STABILITY OF COMPLETENESS AND INCOMPLETENESS
GEODESIC INCOMPLETENESS
7.3
is possible to construct Robertson-Walker space-times with a > -m which are past null geodesically complete. Thus unlike the proof o f stability for timelike geodesic incompleteness, it is necessary to assume that ( M ,g) contains a past incomplete (respectively, past and future incomplete) null geodesic to obtain the null analogue o f Theorem 7.19 (respectively, Theorem 7.20). Not
259
Lemma 7.21. Let V be an adapted normal neighborhood adapted at p E ( M , g ) . Given a > 0 and
> 0, there exists 6 > 0 such that Ilp - ql/2 < 6, g1 E Lor(M) with Ilg - g1110,v < 6, and la1 - a / < 6 together imply that dist(S(91cul,gl),S(P,a , 9 ) ) < c. E
surprisingly, the proof of the C 1 stability of null geodesic incompleteness is
Now let ( M ,g) be a Robertson-Walker space-time ( a ,b) x f H which is past null incomplete. Thus some past directed, past inextendible null geodesic
more complicated than for the timelike case since affine parameters must be
c : [O,A)-+ ( M , g ) is past incomplete (i.e., A < ca). Since ( H ,h ) is isotropic
used instead of Lorentzian arc length to establish null incompleteness. Also for the proof o f Lemma 7.22, we need the isotropy as well as the homogeneity
it follows that all null geodesics are past incomplete. W e now fix through the
o f ( H ,h). Thus we will assume that M = (a,b) x f H is a Robertson-Walker
proof o f Theorem 7.23 this past inextendible, past incomplete, null geodesic
space-time in the rest o f this section. Let (V,x l , . . . ,x,) denote an adapted normal neighborhood of ( M ,g) with adapted coordinates ( x l , .. . ,x,).
and spatially homogeneous, and since isometries map geodesics t o geodesics,
For the proof o f Lemma 7.22, it is neces-
c : [0,A )
--+
( M ,g ) with the given parametrization.
Let (wo,yo) = c(0) E M = (a,b) x f H. W i t h this choice o f wo, apply Lemma 7.15 t o the future directed timelike geodesic t -+ ( t ,yo), t 5 wo, to
sary t o define a distance between compact subsets o f vectors that are null for
get an admissible chain {Uk), { t k ) for this timelike geodesic. Using this choice
different Lorentzian metrics for M and are attached at different points o f V .
of { t k ) , we may find sk with 0 = sl < s:, < . . . < sk < ... < A such that
Recall from Section 7.2 that local coordinates ( x l ,... ,x,) for V give rise to
c ( s ~f) { t k ) x H for each k. Set Ask = sk+l - sk. AS above, let xl
local coordinates Z = ( x l , . . . ,x,, x,+l,.
denote the projection map x l ( t , h ) = t on the first factor o f M = ( a ,b) x f H.
. .,x2,) for T V = T M I V . Thus given
any q E V , gl E Lor(M), and cu > 0 , we may define
:M
-+
W
Notice that i f (V,X I , x2,.. . ,x,) is any adapted coordinate chart, then the
R coincides with this projection map. I f y is any smooth curve o f M which intersects each hypersurface { t ) x H o f M exactly once and y ( s ) E { t ) x H , we will say that l ( x l o y)'(s)l is the $1 speed o f y at { t ) x H . In particular, we will denote by ak = / ( X I o c)'(.qk)l the x l speed o f the fixed null geodesic c : [0,A ) ( M ,g) at { t k ) x H for each k . coordinate function xl : V
S ( q , a , g l ) = { v E T,M : gl(v,v) = 0 and z,+l(v) = -a). Then S(q, a,g l ) is a compact subset o f T,M for any cr > 0 and gl E Lor(M). Given p,q f V , gl,g2 E Lor(M), and cul,az > 0, define the Hawdorff distance
-r
-+
between S ( p ,~
1gl) ,
and S(q,a2,g2) by
Lemma 7.22. Let
E
> 0 be given. Then for each k > 0 , there exists
continuous function 6k : [tk+l,tk] X H
-'
a
( 0 , m ) with the following properties.
Let gl E Lor(M) with lg - g1]1 < 6k on [tk+l,tk]x H , and let y : [O,B) M be any past directed, past inextendible, null geodesic with $0) E i t k ) x H --+
T h e continuity o f the components o f the metric tensor g as functions g,?
:
V x V -+ W and the closeness o f light cones for Lorentzian metrics close in the C0 topology imply the continuity o f this distance in p, a , and g [cf.Busemann (1955, pp. 11-12)].
and with x1 speed o f cuk at { t k ) x H . Then y reaches { & + I )x H with an ncrease in f i n e parameter o f at most 2 A s k , and moreover, the x1 speed 9 of y at {tk+l) x H satisfies the estimate
260
7
7.3
STABILITY OF COMPLETENESS AND INCOMPLETENESS
Proof. Let c : [0,A) (M,g) be the given past incomplete null geodesic as above. Fix k > 0. By the spatial homogeneity of Robertson-Walker spacetimes, we may find an isometry $ € I ( H ) such that $ = id x 4 € I ( M , g ) satisfies +(c(sk)) = (tk,YO)with yo as above. Since k is fixed during the
GEODESIC INCOMPLETENESS
261
such that for any geodesic c2(s) = exp[gl](sw) with gl and w both 61-close to
-+
g and v, we have
as 61
21 0 c ~ ( s ' )= tk+l
for some st f [Ask - bo, Ask
+ bo]. Hence
< 60, we may apply estimate (2) above with so = s' t o obtain
course of this proof, we may set p = (tk,yo) without danger of confusion. Put cl(s) = $ o c ( s + sk). Then cl is a past inextendible, past incomplete, null
We now need to extend these estimates from a neighborhood of v E T,M
geodesic of (M,g) with cl(0) f {tk) x H, cl(Ask) E {tk+l) x H , and cl(s) =
to a neighborhood of S(p, a k , g). TO this end, note that since (a, b) x f H =
exp,[g](sv) for v = $,(cl(sk)). Choose b > 0 with Ask < b < 2Ask such that
M is a warped product of H and the one-dimensional factor (a, b) with f :
cl(s) E Uk for all s with 0 5 s
< b.
Since b > Ask, we have cl(b) € {t) x H
< tk+l. Hence (xi o cl) (Ask) - (xi 0 cl)(b) = tk+i - t > 0. Set €1 = min{c, tk+l - (xi 0 cl)(b)) > 0.
for some t
Now let gl E Lor(M), and let q E Ukn ({tk) x H). Suppose that y : [0, B) --+
(a, b) any
-,
t €
R,it follows that I ( H ) acts transitively on S(p, ak,g). Thus given S(p, ak,g), we may apply the previous arguments using the same
admissible chain {Uk), {tk) to find a constant 61(t) > 0 such that if w E T M
(M, gl) is any past directed, past inextendible, null gl geodesic with y(0) = q
satisfies Ilw - 4 2 < 61(z), .rr(w) E Uk n ({tk) x HI, llgl - 9lll,uk < bl(z), and 4 s ) = exp[gl](sw) has xl speed a k at {tk) x H , then c2 has an increase
a t q. Then w = yl(0) f TqM satisfies gl(w, w) = 0 and
in affine parameter of at most 2Ask in passing from {tk) x H to {tk+l) x H
and with xl speed xn+,(w)
a k
< 0. Moreover, y(s)
= expq[gl](sw). Applying Lemmas 7.10 and
and satisfies estimate (7.4). Using the compactness of S(p, ak,g), we may
7.11 to cl and c2 = y with the constant €1 as above, we may find a constant
choose null vectors vl, v2, . . . ,v, E S(p, ak, g) such that S(p, a k , g) is covered
60 > 0 with 0 < 60 < Ask such that Ilv - wll2 Iso - Ask\ < 60 imply that
by the sets {w f S(p,ak,g) : IIw - vrnI12 < 61(vrn)) for m = 1,2,. . . ,j. Set
(I)
) (21 0 ca)(~)I< €1 I(XI 0 c ~ ) ( s -
for all s with 0 5 s 5 b and (2)
1 - €1 <
< 60, 119 - gllll,uk < 60, and
b2 = mini61 (v,)
< tk+l - (XI
0
("I O C2)'(s0) (21 O cl)'(Ask)
1
m 5 j}. By Lemma 7.21 we may find a constant b3 with
0
such that i f llg - gllll,w < E and i f c is a geodesic o f gl with c l ( t l )E V , then the domain o f c includes the value t 2 and c ( t ) E W for all tl
< t 5 t2.
Let M be given an auxiliary, positive definite, and complete Riemannian metric. This metric has a complete distance function do : M x M -+ W,and the Hopf-Rinow Theorem [cf.Hicks (1965)l guarantees that the compact subsets o f M are exactly the subsets which are closed and bounded with respect to do. The proof of Theorem 7.30 [cf.Beem (1994)l will use a sequence o f n-
dimensional annuli {W,) [i.e., sets bounded by spherical shells with respect
Lemma 7.28. Let ( M , g ) be a given semi-Riemannian manifold, fix a geodesic y : (a,b) -+ M o f ( M , g ) ,and let W = W ( yI [tl,t2]) be a neighborhood
to the distance do which are centered at a fixed point y(to)]. T h e approach involves requiring the perturbed metric gl t o be sufficientlyclose to the original
o f y 1 [ t l , t 2 ] .Given any neighborhood Vl o f y l ( t l ) in T M , there is a constant
metric g on the sequence {W,). For a fixed gl, one constructs a sequence o f
> 0 and a neighborhood fi o f y1(t2)in T M such that i f llg - glll1,w < 6 and
geodesics cj o f gl such that a given c, is either tangent or almost tangent to
i f c is a geodesic o f gl with c1(t2)f V2, then cl(tl) E Vl and c(t) E W for all tl < t < t2.Furthermore, i f y is timelike (respectively, spacelike), then V2 and E > 0 may be chosen such that each v E 7 2 o f each such metric gl is timelike (respectively, spacelike). If y is null, then E > 0 may be chosen such that each such metric gl has some null vectors in V2.
the original y at a certain point y ( t j ) o f W j . A limit geodesic o f the sequence
E
W e now define partial imprisonment and imprisonment. Definition 7.29. (Imprisonment and Partial Imprisonment)
will be the desired geodesic c o f gl. Theorem 7.30. Let ( M ,g) be a semi-Riemannian manifold. Assume that ( M , g ) has an endless geodesic y : (a,b)
( a ,b) -+ M be an endless (i.e., inextendible) geodesic. ( 1 ) T h e geodesic y is partially imprisoned as t -+ b i f there is a compact b- such that y ( t j ) E K for set K M and a sequence { t j ) with t , -+
all j . (2) T h e geodesic y is imprisoned i f there is a compact set K such that the
entire image of y is contained in K .
In other words, y is partially imprisoned in K as t 4 b i f either y ( t ) E K for all t sufficiently near b or i f y leaves and returns t o K an infinite number of times as t + b-. An imprisoned geodesic is clearly partially imprisoned, but for general space-times one may have some partially imprisoned geodesics which fail t o be imprisoned. T h e stability o f incompleteness result (Theorem 7.30) will hold for geodesics which fail t o be partially imprisoned in the direction
M such that y is incomplete in
the forward direction (i.e., b # oo). I f y is not partiaI1y imprisoned in any compact set as t
Let y :
-+
-+
b, then there is a C 1 neighborhood U ( g ) o f g such that
each gl in U ( g ) has at least one incomplete geodesic c. Furthermore, i f y is timelike (respectively, null, spacelike) then c may also be taken as timelike (respectively, null, spacelike). Proof. Choose some to in the interval (a,b). We will construct sequences t,, D,, and L,.
Let tl and Dl be chosen with Dl > 1 and t l > to such
that d o ( y ( t o ) , y ( t l ) )= Dl and do(y(to),y ( t ) ) > Dl for all tl < t < b. In other words, y 1 [to,b) leaves the closed ball o f radius Dl for the last time at t l . T h e existence o f t l follows using the fact that y is not partially imprisoned as t
-+
b. Set Lo = 0 and L1 = 1
+ sup{do(y(to),y ( t ) )1 to < t < t l ) . Notice
that on the interval [to,t l ] the geodesic y remains within the closed ball o f radius L1 - 1 about y(t0). I f t l , . . . ,t,-1, D l , . . . ,D,-1, and Lo, L1, . . . ,L,-1 have been defined, then t, and D, are defined by letting D, > L, - 1 i-1 and requiring both do(y(to),y(t,)) = D, and &($to), ? ( t ) )> D, for all t, < t < b.
266
STABILITY O F COMPLETENESS AND INCOMPLETENESS
7
7.4
Existence again follows using the nonimprisonment hypotheses. Define L j = 1
+ sup{do(y(to),r(t)) It0 < t
j
-+
co,and hence t j
-+
< tj). Note that limy(tj) does not exist as w . Set Wl = {x E M I &(y(to), x) < L1), < L2), and Wj = {X E M I Lj-2 < do(y(to), X) <
b as j
-+
SUFFICIENT CONDITIONS FOR STABILITY
267
the sequence {cj) yield that v is timelike (respectively, null or spacelike) for gl if the original geodesic y is timelike (respectively, null or spacelike) for g. Let c be the endless (i.e., inextendible) geodesic of gl with cl(to) = v. Note
W2 = {X E M I &(y(to), X) Lj) for j > 2. The sets Wj are n-dimensional annuli with respect to the
that each geodesic c, satisfies h ( t j ) E .rr(V,) for all 0 5 j 5 m. Also, for each t in the domain of c, cm(t) converges to c(t) and cml(t) converges to cl(t)
distance do. We now define a sequence of positive numbers {cj) and a sequence
6 implies c(tj+l) exists. Furthermore, ckt(tj+1) f V,+I for all k > j + 1 yields cl(tj+l) 6 vj+l.Thus, c is defined for
of open sets {V,) of T M . Let Vo be an open neighborhood of yl(to) in TM with
as m
-+
w . By construction, cl(tj) f
compact closure, and assume that the closure of Vo does not contain any trivial
all values of tj, and &(y(to), c(tj))
vectors. If g(yl(to),yl(to)) > 0 [respectively, g(yl(to),yl(to)) < 01, we assume
limit c is not partially imprisoned in any compact set as t
70satisfy
>
-+
oo as j
-+
w . It is easily shown that the +
b. It only remains
0 [respectively,
to show that c cannot have any domain values above b (i.e., c(b) does not exist).
< 01. Assume that &, Vl, .. . ,T/;-l have been defined. Use Lemma 7.28
Assume the domain of c contains b. Since c is continuous, the set c([to,b])
V, = T/;(yl(tj)) of yl(tj) in T M and a positive number
would be compact in contradiction to the condition do(yo(to),c(tj)) -+ co. 0
< cj where P = Wj, and if c is a geodesic of gl with Vj, then cl(tj-1) E y-l and c(t) E Wj for all tj-l 5 t 5 tj. Lemma
An interesting aspect of the construction used in the above proof is that the
7.28 implies that if the original geodesic y is timelike (respectively, spacelike),
final geodesic c of the metric gl has the same value b for the least upper bound
then we may assume each v in Vj is timelike (respectively, spacelike) for gl. If
of its domain as the original geodesic y of the metric g. In essence, this is
without loss of generality that all v E g(v, v)
to obtain an open set
g(v, v)
cj such that if Ilg-gljll,p cl(tj) E
y is null, then we may assume that gl has some null vectors in V,. Lemma 7.27
due to cjl(tj) E T/, implying cjl(to) E Vo and to the fact that do(cj(tj),cj(to))
vj,
diverges to infinity. Notice that the methods used in the proof of Theorem
then the domain of c contains tj+l. We may also assume without loss of
7.30 show that if gl is another semi-Riemannian metric on M and if gl is close
implies we may assume that if c is a geodesic of such a gl with cl(tj) E
generality that the diameter with respect to & of K(V,) is less than 112 and
to g in the C' sense on a neighborhood of y , then gl has a corresponding
< cj for all j . Notice that points of M are in a t most two consecutive
geodesic which is incomplete. In other words, the proof of Theorem 7.30 only
sets of the sequence Wj. It follows that there is a continuous positive-valued
really requires that g and gl be metrics which are C1 close near y. Thus,
( 0 , w ) with ~ ( x < ) ej for all j with x 6 Wj. Let gl satisfy
if (M,g) has a finite number N of incomplete geodesics and none of these
< ~ ( x ) We . will construct a sequence of geodesics of gl. Assume
are partially imprisoned in any compact set, then one may construct a C1
first that the original geodesic y is either timelike or spacelike. For each j,
neighborhood U(g) of g such that each gl in U(g) has at least N incomplete
let cj(t) be the geodesic of gl which satisfies cjl(tj) = yl(tj). Notice that if
geodesics. On the other hand, one may construct space-times which have all
cj+l
function
E
:M
119 - gllll,M
-+
y is timelike (respectively, spacelike), then each cj is timelike (respectively,
geodesics complete except for a single geodesic which is incomplete and which
spacelike). If the original geodesic y is null, then we choose c j such that
is not partially imprisoned [cf. Beem (1976c)j. Thus, the existence of a single
cjl(tj) E Vj and cjl(tj) is null. The above construction yields that cjt(tk) E Vk
incomplete geodesic does not imply that there is more than one such geodesic.
for all 0 5 k 5 j. Since Vo has compact closure and this closure does not contain any trivial vectors, one obtains a nontrivial vector v in T o and a subsequence { m ) of {j) such that cml(to)
-+
v. The constructions of Vo and
Corollary 7.31. If (M,g) is a strongly causal space-time which is causally geodesically incomplete, then there is a C1 neighborhood U(g) of g such that each gl in U(g) is nonspacelike geodesically incomplete.
268
7
STABILITY OF COMPLETENESS AND INCOMPLETENESS
Proof. This follows from Theorem 7.30 using the fact that for strongly causal space-times, no causal geodesic is future or past partially imprisoned in any compact set. 0
7.4
SUFFICIENT CONDITIONS FOR STABILITY
269
can, of course, consider this property for other classes such as the class of all null geodesic segments, all spacelike segments, etc. Furthermore, pseudoconvexity and disprisonment have also been applied to sprays [cf. Del Riego and
It is known that no null geodesic in a twc-dimensional Lorentzian manifold has conjugate points. However, a chronological space-time in which all null
Parker (1995)l.
geodesics have conjugate points is strongly causal and has dimension at least
Definition 7.34. (Nonspacelike Pseudoconuexity) We say (M, g) has a pseudoconuex nonspacelike geodesic system if for each compact subset K of M ,
three. Thus, we obtain the following corollary.
there is a second compact set H such that each nonspacelike geodesic segment
Corollary 7.32. Let (M, g) be a chronological space-time of dimension at least three which is causally geodesically incomplete. If (M,g) has conjugate points along all null geodesics, then there is a C1 neighborhood U(g) of g such
with both endpoints in K lies in H. The following [cf. Beem and Ehrlich (1987, p. 324)] gives sufficient conditions for the stability of nonspacelike completeness.
that each gl in U(g) is nonspacelike geodesically incomplete. Let (a, b) x f H be given the metric g = -dt2
+ f (t)dcr2. If either a or b is
finite, then (a,b) x f H is timelike geodesically incomplete. Note that (a, b) x H is necessarily stably causal since f (t,y) = t is a time function. Using Corollary 7.31 and the fact that stably causal space-times are strongly causal, we obtain
Corollary 7.33. Assume (H, h) is a positive definite Riemannian manifold, and let the warped product (a, b) x f H be given the Lorentzian metric g = -dt2
@fh.
If either a # -ca o r b # +ca, then there is a fine C1 neighborhood
U(g) of g such that each metric gl in this neighborhood is timelike geodesically incomplete. This last corollary applies to a much more general class of spacetimes than the Robertson-Walker space-times because we have not made symmetry assumptions on (H, h). Of course, for the special case of Robertson-Walker space-times we have already established the stronger conclusion that there is a C0 neighborhood U(g) with each gl in U(g) having all timelike geodesics incomplete (cf. Theorem 7.18).
Theorem 7.35. Let (M,g) be a Lorentzian manifold which has no imprisoned nonspacelike geodesics and which has a pseudoconvex nonspacelike geodesic system. If (M, g) is nonspacelike geodesically complete, then there is a C1 neighborhood U(g) ofg in Lor(M) such that each gl f U(g) is nonspacelike complete. Pseudoconvexity is a type of "internal" completeness condition in somewhat the same sense that global hyperbolicity is such a condition. The next proposition shows that nonspacelike pseudoconvexity is a generalization of global hyperbolicity. An example of a nonspacelike pseudoconvex space-time which fails to be globally hyperbolic is given by the open strip { ( t ,x) : 0 < x
< 1) in
the Minkowski plane.
Proposition 7.36. If (M,g) is a globally hyperbolic space-time, then (M,g) has a pseudoconvex nonspacelike geodesic system. Proof. Let K be a compact subset of M . For each p E K choose points q and r with q in the chronological past I-(p) of p and r in the chronological
If K is a subset of Wn, then the conuez hull K H of K is the union of all
future If (p) of p. Then U(p) = I+(q) n I-(T) is an open set containing p.
Euclidean line segments with both endpoints in K. It is well known that the
Since (M, g) is globally hyperbolic, the set U(p) has compact closure given by
n J-(T).
convex hull of a compact set is again a compact set. Pseudoconvexity is a gen-
J+(q)
eralization of this property to manifolds. In Definition 7.34 below, we assume the class of geodesic segments under consideration to be nonspacelike. One
U(p,) = I+(q,) n I-(r,) for i = 1 , 2 , .. . ,k, and let H be the union of the k2 compact sets of the form J+(q,) nJ-(r,) for 1 5 i,j 5 k. It is easily seen that
Cover the compact set K with a finite number of open set
270
7
STABILITY OF COMPLETENESS AND INCOMPLETENESS
all nonspacelike geodesic segments with both endpoints in K must lie in the compact set H. Since globally hyperbolic space-times are strongly causal, they have no imprisoned nonspacelike geodesics. Consequently, Theorem 7.35 and Proposition
CHAPTER 8
7.36 yield the following corollary.
M A X I M A L GEODESICS A N D CAUSALLY DISCONNECTED S P A C S T I M E S
Corollary 7.37. Let (M,g) be a globally hyperbolic space-time. If (M,g) is nonspacelike geodesically complete, then there is a C1 neighborhood U(g) of g in Lor(M) such that each gl E U(g) is nonspacelike complete.
Many basic properties of complete, noncompact Riemannian manifolds stem
Minkowski space-time is globally hyperbolic and geodesically complete. Furthermore, its (entire) geodesic system is pseudoconvex. In other words,
from the principle that a limit curve of a sequence of minimal geodesics is
given any compact set K , there is a larger compact set H such that any ge-
been given by Hopf and Rinow (1931), Rinow (1932) and Myers (1935) were
odesic segment (timelike, null, or spacelike) with endpoints in K must lie in H. In fact, one may take H to be the usual convex hull of K . The next
able t o establish the existence of a geodesic ray issuing from every point of
itself a minimal geodesic. After the correct formulation of completeness had
proposition gives the C1 stability of global hyperbolicity, nonimprisonment,
a complete noncompact Riemannian manifold using this principle. Here a geodesic y : [0,co) --t (N,go) is said to be a ray if y realizes the Riemannian
completeness, and inextendibility for Minkowski space-time. The C1 stability
distance between every pair of its points. Rinow and Myers constructed the
of global hyperbolicity follows from the stronger C0 stability result of Geroch
desired geodesic ray as follows. Since (N, go) is complete and noncompact, there exists an infinite sequence (p,) of points in N such that for every point
(1970a). The stability of nonimprisonment and of geodesic completeness ( all geodesics) follows from Beem and Ehrlich (1987, pp. 324-325). See a Beem and Parker (1985, p. 18). The stability of inextendibility follows fro
p E N, do(p,p,) -+ CXI as n -+m. Let y, be a minimal (i.e., distance realizing) unit speed geodesic segment from p = y,(O) to p,. This segment exists by
the stability of completeness using Proposition 6.16 which guarantees th
the completeness of (N,go). If v E T,N is any accumulation point of the
geodesically complete space-times are inextendible.
sequence {ynl(0)) of unit tangent vectors in TpN, then y(t) = exp, tv is the
Proposition 7.38. There is a C1 neighborhood U(7) of n-dimension Minkowski space-time (M, 7 ) such that for each metric gl in this neighborhood: (1) (M, gl) is globally hyperbolic; (2) No geodesic of (M,gl) is imprisoned; (3) (M, gl) is geodesically complete; and (4) (M, gl) is an inextendible space-time.
required geodesic ray. Intuitively, y is a ray since it is a limit curve of some subsequence of the minimal geodesic segments (7,). The existence of geodesic rays through every point has been an essential tool in the structure theory of both positively curved [cf. Cheeger and Gromoll (1971, 1972)] and negatively curved [cf. Eberlein and O'Neill (1973)l complete noncompact Riemannian manifolds.
A second application of this basic principle of constructing geodesics as limits of minimal geodesic segments is a concrete geometric realization for complete Riemannian manifolds of the theory of ends for noncompact Hausdorff topological spaces [cf. Cohn-Vossen (1936)l. An infinite sequence (p,) of points in a manifold is said to diverge to infinity if, given any compact sub-
272
8
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
8.1
ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS
273
complete Riemannian manifold (N,go) has more than one end, there exists a
disconnecting two divergent sequences is said to be causally disconnected. It is evident from the definition that causal disconnection is a global conformal
compact subset K of N and sequences {p,) and {q,) which diverge to infinity such that 0 < &(p,, q,) --+ oo and every curve from p, to q, meets K for each
strongly causal space-time (M,g) is causally disconnected by the compact set
set K , only finitely many members of the sequence are contained in K. If a
n. Let -y, be a minimal (i.e., distance realizing) geodesic segment from p, to
q,. Since each y, meets K , a limit geodesic y : R -+ M may be constructed. Moreover, y is minimal as a limit of a sequence of minimal curves. Then "y(-m)" corresponds to the end of N represented by { p , ) and "y(+m)" to
the end represented by (9,).
In particular, a complete Riemannian manifold N with more than one end contains a line, i.e., a geodesic y : (-co,+m) that is distance realizing between any two of its points. Motivated by these Riemannian constructions, we study similar existence theorems for geodesic rays and lines in strongly causal space-times. From the viewpoint of general relativity, it is desirable to have constructions that
invariant of C(M, g). Applying the principle of Section 8.1, we show that if the
K, then (M, g) contains a nonspacelike geodesic line y : (a, b) -4 M which intersects K. That is, d(y(s), y(t)) = L ( y I [s,t]) for all s , t with a < s t < b. This result is essential to the proof of the singularity theorem 6.3 in Beem and Ehrlich (1979a), as will be seen in Chapter 12. We conclude this chapter by studying conditions on the global geodesic structure of a given space-time (M,g) which imply that (M,g) is causally disconnected. In particular, we show that all two-dimensional globally hyperbolic space-times are causally disconnected. Also, it follows from one of these conditions and the existence of nonspacelike geodesic lines in strongly causal, causally disconnected spacetimes that a strongly causal space-time containing no future directed null geodesic rays contains a timelike geodesic line.
are valid not only for globally hyperbolic subsets of space-times, but also for strongly causal space-times. However, if we only assume strong causality, it not true in general that causally related points may be joined by maximal g e odesic segments. Thus a slightly weaker principle for construction of maximal geodesics is needed for Lorentzian manifolds than for complete Riemannian manifolds. Namely, in strongly causal space-times, limit curves of sequences o "almost maximal" curves are maximal and hence are also geodesics. In Secti 8.1 we give two methods for constructing families of almost maximal curv whose limit curves in strongly causal space-times are maximal geodesics. T strong causality is needed to ensure the upper semicontinuity of arc length the C0 topology on curves and also so that Proposition 3.34 may be appli In Section 8.2 we apply this construction to prove the existence of past and ture directed nonspacelike geodesic rays issuing from every point of a stron causal space-time. In Section 8.3 we study the class of causally disconnec space-times. Here a space-time is said to be causally disconnected by a pact set K if there are two infinite sequences {pn) and { q n ) , both divergi infinity, such that p, < q,, p, # qn, and all nonspacelike curves from Pn meet K for each n. A space-time (M, g) admitting such a compact K caus
8.1
Almost Maximal Curves and Maximal Geodesics
The purpose of this section is to show how geodesics may be constructed
as limits of "almost maximal" curves in strongly causal spacetimes. In both constructions, the upper semicontinuity of Lorentzian arc length in the C0 topology on curves for strongly causal space-times and the lower semicontinuity of Lorentzian distance play important roles. The strong causality of (M,g ) is used in our approach here so that convergence in the limit curve sense and in the CO topology on curves are closely related (cf. Proposition 3.34). The first nstruction may be applied to pairs of chronologically related points p, q with (p, q) < a.While this approach is therefore sufficient to show the existence of nonspacelike geodesic rays in globally hyperbolic space-times [cf. Beem and Ehrlich ( 1 9 7 9 ~Theorem ~ 4.2)], it is not valid for points at infinite distance. cordingly, for use in Sections 8.2 and 8.3, we give a second construction ich may be used in arbitrary strongly causal space-times. In Section 14.1 a ghtly different approach to constructing maximal segments from sequences of onspacelike almost maximal curves is presented [cf. Galloway (1986a)J. Here,
274
8
8.1
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS
275
We are now ready to give an example of the principle that for strongly
the use of uniform convergence in a unit speed reparametrization with respect t o an auxiliary complete Riemannian metric is employed to dispense with the
causal space-times, limits of almost maximal curves are maximal geodesics.
global requirement of strong causality which we assume in this chapter.
Strong causality is used here (with all curves given a "Lorentzian parametrization") since convergence in the limit curve sense and in the C0 topology are
Let (M, g) be an arbitrary space-time and suppose that p and q are distinct points of M with p
< q.
closely related for strongly causal space-times but not for arbitrary space-
If d(p,q) = 0, then letting y be any future directed
nonspacelike curve from p to q, we have L(y) 5 d(p, q) = 0. Hence L(y) =
times (cf. Section 14.1).
d(p, q) and 7 may be reparametrized to a maximal null geodesic segment from p to q by Theorem 4.13. Thus suppose that p 0. If d(p, q) < co as well, then by Definition 4.1 there exists a future
y, : [a, b ]
-+
M be a future directed nonspacelike curve from p , to q, with
where
--, 0 as
directed nonspacelike curve y from p to q with
for any E E
> 0. Of course, inequality (8.1) is only a restriction on L(y) provided
6,
n
-+
m.
If y
:
[a,b ]
--+
M is a limit curve of the sequence
{m)with y(a) = p and y(b) = q, then L(y) = d(p,q). Thus y may be reparametrized to be a smooth maximal geodesic from p to q.
< d(p, q). In this case, we will call y an almost maximal curve. We note the following elementary consequence of the reverse triangle in-
equality.
Proof. First, y is nonspacelike by Lemma 3.29. Second, by Proposition 3.34, a subsequence {y,) of {y,) converges to y in the C0 topology on curves. By the upper semicontinuity of arc length in this topology for strongly causal
Remark 8.1. Let y : [O, 11 -+ M be a future directed nonspacelike curve from p to q, p # q, with
space-times (cf. Remark 3.35), we then have Then for any s < t in [O,l], we have
Proof. Assume that L ( y I [s, t]) Then
< d(y(s),y(t)) - c for some s < t in [O,l].
k
using the lower semicontinuity of Lorentzian distance (Lemma 4.4). But by definition of distance, d(p,q) 2 L(y). Thus L(y) = d(p, q) and the last state-
[ ment of the proposition follows from Theorem 4.13. El We now consider a second method for constructing maximal geodesics in strongly causal space-times (M,g) which may be applied to points at infinite Lorentzian distance. For this purpose, we fix throughout the rest of Chapter 8 an arbitrary point po E M and a complete (positive definite) Riemannian
in contradiction. 0
metric h for the paracompact manifold M. Let do : M x M
-+
R denote the
276
8
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
Riemannian distance function induced on M by h. For all positive integers n , the sets
ALMOST MAXIMAL CURVES AND MAXIMAL GEODESICS
8.1
277
For strongly causal space-times, the Lorentzian distance function d and the d [Bn]'s are related by the following lower semicontinuity.
-
B n = {m E M : do(po,m) F n)
L e m m a 8.5. Let (M,g) be strongly causal. If p,
are compact by the Hopf-Rinow Theorem. Thus the sets
{B,
:n
> 0) form a
d ( ~9),
compact exhaustion of M by connected sets. For each n , let
-+
p and q,
-+
q, then
< lim inf d [Bn](pn qn).
Proof. If d(p, q) = 0, there is nothing to prove. Thus we first assume that 0 < d(p, q) denote the Lorentzian distance function induced on B, by the inclusion B, (M, g). That is, given p f B,, set d [Bn](p,q) = 0 if q $! J + ( ~R), , and for 4 E J + ( ~ , B , ) , let d [Bn](p,q) be the supremum of lengths of future directed nonspacelike curves from p to q which are contained in B,. It is then immediate that d
q) 5 d(p, q) for all p, q E B,. However, d [B,] is not in general
the restriction of the given Lorentzian distance function d of (MIg) to the set
-
B, x
B,. Nonetheless, for strongly causal space-times
these two distances
coincide "in the limit."
d(p, 9) = lim d [Bn](p,9).
16)], a timelike curve y from p to q may be found with d(p, q) - 6 < L ( y ) 5 d(p, q). Since y is timelike and L(y) > d(p,q) - E , we may find TI,rz E y with d(p, q) - 6
< L(y[rl,rz]) and p k for each k, we have for each
for all n sufficiently large as above. Hence limd [B,] (pn,qn) = co as required. 13 6
Since we are assuming that (M,g) is strongly causal but not necessarily globally hyperbolic, it is possible that the Lorentzian distance function d : M x M --t W U {m) assumes the value + w . Nonetheless, for any given Bn the distance function d [B,] : Bn x
B,
R U {m) is finite-valued. This is a consequence of the compactness of the Bn and the compactness of certain subspaces of nonspacelike curves in the C0 topology on curves [if. Penrose (1972, p. 50, Theorem 6.5)]. Moreover, this compactness also implies the existence of curves realizing the d [B,] distance for points p, q E Bn with -+
E J+(P, Bn).
L e m m a 8.6. Let (M,g) be a strongly causal space-time and let n arbitrary. If q E J + ( ~ , B , ) , then d [Bn](p,q)< directed nonspacelike curve y in d[B,l(p,q).
B, joining
XI
> 0 be
and there exists a future
p to q which satisfies L(y) =
278
8
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
8.2
Proof. By definition o f the distance d [B,], i f d [ B n ] ( p , q )= 0 and q E
J + ( ~ , B , )then ,
there exists a future directed nonspacelike curve y in
GEODESIC RAYS IN CAUSAL SPACE-TIMES
d(p,q) L lim inf d [Bn] ( P , q)
B,
= lim inf L ( y n )
from p t o q with L ( y ) 5 d [Bn](p,q) = 0. Hence L ( y ) = d [Bn](p,q) as required. Thus we may suppose that d q) > 0. Again by definition
L lim sup L ( y n ) L L ( y )
o f d [&I,
L d ( ~q)!
we may find a sequence { y k ) o f future directed nonspacelike curves
from p t o q with L(yk) -t d [ R ] ( p l q ) (. I f d [B,](p,q) = oo,choose yk with L(yk)
> k for each k.)
Since
3, is compact
and ( M , g ) is strongly causal,
there exists a future directed nonspacelike curve y in B, joining p t o q with the property that a subsequence {y,} o f { y k )converges t o y in the C Otopology on curves by Theorem 6.5 o f Penrose (1972, pp. 50-51). But then using the upper semicontinuity o f arc length in the C 0 topology on curves, we have d [B,](p, q) = lim L(y,)
5 L ( y ) which implies the finiteness o f d [B,](p,q).
279
as required. C3 Now let p, q be distinct points o f an arbitrary strongly causal space-time with p
< q and
let a sequence {y,} of nonspacelike curves from p to q be
chosen as in Proposition 8.7. While a limit curve y for the sequence {y,) with y(0) = p may always be extracted by Proposition 3.31, we have no guarantee that y reaches q unless ( M , g ) is globally hyperbolic. Indeed, i f d(p,q) = co, then there is no maximal geodesic from p to q, and hence no limit curve y o f
Since L ( y ) 0 such that
( M ,g ) is not globally hyperbolic. Examples may easily be constructed by
yo is contained in
BN.
Lemma 8.6 we may find a future directed nonspacelike curve y, from p to q with L(y,) = d [B,](p, q) for each n 2 N . For C 0 limit curves o f the sequence {y,),
deleting points from Minkowski space-time.
Hence q E J + ( p , B n ) for all n 2 N . Thus using
we then have the following analogue o f Proposition 8.2.
8.2
Nonspacelike Geodesic R a y s i n Strongly Causal Space-times
T h e purpose o f this section is t o establish the existence o f past and future directed nonspacelike geodesic rays issuing from every point o f a strongly causal space-time ( M ,g).
Proposition 8.7. Let ( M Ig) be strongly causal and let p, q f M be distinct points with p
< q.
For all n > 0 sufficiently large, let yn be a future
q). If -y directed nonspacelike curve from p t o q in B, with L(y,) = d [Bn](p, is a nonspacelike curve from p to q such that {y,} converges t o y in the C0 topology on curves, then L ( y ) = d(p,q) and hence y may be reparametrized to a maximal geodesic segment from p t o q.
Proof. Using Lemma 8.5 and the upper semicontinuity of arc length in the C 0 topology on curves in strongly causal space-times, we have
Definition 8.8. (Future and Past Directed Nonspacelike Geodesic Rays) A future directed (respectively, past directed) nonspacelike geodesic ray is a future (respectively, past) directed, future (respectively, past) inextendible, nonspacelike geodesic y : [O,a)-t ( M , g ) for which d(y(O),y ( t ) ) = L ( y 1 [0,t ] ) (respectively, d ( y ( t ) y, ( 0 ) ) = L ( y I [0,t ] ) )for all t with 0 5 t < a. T h e reverse triangle inequality then implies that a nonspacelike geodesic ray is maximal between any pair o f its points. Using Lemmas 8.5 and 8.6 we first prove a proposition that will be needed not only for the proof o f the existence o f nonspacelike geodesic rays, but also
280
8
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
8.2
for the proof of the existence of nonspacelike geodesic lines in strongly causal, causally disconnected spacetimes in Section 8.3. Let
-
B,
-+
and d [B,] :
x
R be constructed as in Section 8.1.
Proposition 8.9. Let (M, g) be a strongly causal space-time and let K be any compact subset of M. Suppose that p and q are distinct points of M such that p < q and every future directed nonspacelike curve from p to q meets K. Then at least one of the following holds: (1) There exists a future directed maximal nonspacelike geodesic segment from p to q which intersects K. (2) There exists a future directed maximal nonspacelike geodesic which
GEODESIC RAYS IN CAUSAL SPACE-TIMES
x, E y, with x, x as m w. Passing to a subsequence {yk) of {y,) if necessary, we may assume by Proposition 3.34 that yk[p,xk]converges to -+
-+
y[p, x] in the C0 topology on curves (recall Notational Convention 8.4). Since y[p, x] is closed in M and q # y, there exists an open set V containing y[p, x]
#
V . Since yk[p,xk] -+ y[p,x] in the C0 topology on curves, there exists an Nl > O such that yk[p,xk]2 V for all k 2 N l . Hence q @ yk[p,xk] for all k 2 N I . Thus yk[p,xk] E yk[p, q] for all k 2 N1 which implies that ) all k > N1. By Lemma 8.5 and the upper L(yk[p,xk]) = d [ B k ] ( P , ~ kfor semicontinuity of arc length in the C0 topology on curves for strongly causal spacetimes, we have
with q
starts at p, intersects K, and is future inextendible.
d(p, x)
(3) There exists a future directed maximal nonspacelike geodesic which
< lim inf d [Bk](p,xk) = lim inf L(yk[p,~ k ] )
ends at q, intersects K , and is past inextendible.
5 f i m s u L~ ( y k \ ~xk]) , 5 L(y[p, XI).
(4) There exists a maximal nonspacelike geodesic which intersects K and is both past and future inextendible.
281
Since L(y[p,x])
< d(p,x) by definition of Lorentzian distance, we thus have
d(p, x) = L(y[p, XI) as required.
Proof. Let yo be any future directed nonspacelike curve in M from p to q. Since K U yo is compact, there exists an N > 0 such that K U yo is contained in
Bn for all n 2 N.
Hence q E J+(P,B,) for all n 2 N. Thus by Lemma
8.6, for each n >_ N there exists a future directed nonspacelike curve y, in Bn joining p to q with L(yn) = d [B,](p, q). By hypothesis, each yn intersects K in some point r,. Since K is compact, there exists a point T E K and a subsequence {r,) of {r,) such that r, r as m + m. Extend each curve
For globally hyperbolic space-times, case (1) of Proposition 8.9 always applies because J+(p) n J-(q) is compact and no inextendible nonspacelike curve is past or future imprisoned in a compact set. However, space-tirnes which are strongly causal but not globally hyperbolic and which have chronologically related points p
0 such that y[x,y] C Int(BN). By definition of the C0 topology on curves, there is then an Nl 2 N such that yk[zk, yk] C Int(BN) for all k 2 Nl. Since {pk) diverges to infinity and BN is compact, there is an Nz > Nl such that pk $ BNfor all k 2 Nz. Consequently, xk comes after pk on yk for all k 2 N2 so that yk[xk,yk]is maximal for all
286
k
8
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
> N2. We thus have
8.3
DISCONNECTED SPACSTIMES, NONSPACELIKE LINES
287
hyperbolic, causally disconnected spacetimes which do not have null geodesic lines. Thus the existence of a null geodesic line is not a necessary condition
d(x, y) 5 lim inf d(xk, yk) = lim inf L(yk[zk, yk])
5 l i m s u ~ L ( ~ k [ ~ k5, ~L(Y[x, k ] ) Y])< ~ ( x , Y ) .
for a globally hyperbolic space-time to be causally disconnected. Evidently, the strong causality in Proposition 8.14 may b e replaced by any other nonimprisonment condition which guarantees that both ends of c diverge
Hence d(x, y) = L(y[x,y]) as required. 0
to infinity, together with the requirement that (M,g) be causal.
We now give several criteria, expressed in terms of the global geodesic struc-
In the next proposition we will give a sufficient condition for a strongly
ture, for globally hyperbolic space-times and for strongly causal space-times
causal space-time (M,g) to be causally disconnected. For the proof of this
to be causally disconnected. In particular, we are able to show that all two-
result (Proposition 8.18), it is necessary to recall some additional concepts
dimensional globally hyperbolic spacetimes are causally disconnected. Also
from elementary causality theory. A subset S of (M, g) is said to be achronal
one of our criteria (Proposition 8.18) together with Theorem 8.13 implies that
if no two polnts of S are chronologically related. Given a closed subset S of (M,g), the future Cauchy development or domazn of dependence D f (S)
if a strongly causal space-time (M,g) has no null geodesic rays, then (M, g) contains a timelike geodesic line. Recall that an inextendible null geodesic y : (a, b) -+ (M, g) is said to be a null geodesic line if d(y(s),y(t)) = 0 for all s, t with a
< s 5 t < b.
Proposition 8.14. Let (M,g) be strongly causal. If (M, g) contains a null geodesic line, then (M, g) is causally disconnected.
(M, g) be the given null geodesic line. Choose d with a < d < band put K = {c(d)). Choose sequences {s,), {t,) with s, < d < t, and s, -+ a, t, + b. Put p, = c(s,) and q, = c(t,). Since c is both future and past inextendible and (M, g) is strongly causal, both {p,) and {q,) diverge to infinity by Proposition 3.13. Now because c is a maximal null geodesic, any future directed nonspacelike curve a from p, to q, is a reparametrization of cl [s,, t,], whence a meets K as required. (First, a may be reparametrized to be a smooth future directed null geodesic segment a : [O, 11 --+ M with a(0) = p,, a(1) = q, by Theorem 4.13. If a l ( l ) # Xcl(tn) for some X > 0, then q,+~ E I+(pn). But this contradicts d(p,, qn+l) = 0 since c is maximal. Hence, a'(1) = Xc1(tn)for some X > 0. But then since (M, g) is causal, a must simply be a reparametrization of c 1 [s,, t,].) D Proof. Let c : (a,b)
-+
Proposition 8.14 implies that Minkowski space-time, de Sitter space-time, and the Friedmann cosmological models are all causally disconnected. Ako, the Einstein static universe (cf. Example 5.11) shows that there are globally
of S is defined as the set of all points q such that every past inextendible nonspacelike curve from q intersects S. The future Cauchy honzon H+(S) is given by Hf (S)= D+(S) - I-(D+(S)). The future honsmos E+(S) of S is defined to be E+(S) = J + ( S ) - I+(S). An achronal set S is said to be future trapped if E + ( S ) is compact. Details about these concepts may be found in Hawking and Ellis (1973, pp. 201, 202, 184, and 267, respectively). For the proof of Proposition 8.18 we also need to use a result first obtained in Hawking and Penrose (1970, p. 537, Lemma 2.12). This result is presented somewhat differently during the course of the proof of Theorem 2 in the text of Hawking and Ellis (1973, p. 266). In the proof of this theorem, it is assumed that dim M
> 3 and that
(M,g) has everywhere nonnegative nonspacelike
Ricci curvatures and satisfies the generic condition [conditions (1) and (2) of Theorem 21. However, it may be seen that in the proof of Lemma 8.2.1 and the following corollary in Hawking and Ellis (1973, pp. 267-269), it is only necessary to assume that (M, g) is strongly causal to obtain our Lemma 8.15 and Corollary 8.16. We now state these two results for completeness. Lemma 8.15. Let A be a closed subset of the strongly causal space-time
(M,g). Then H + ( E f (A)) is noncompact or empty. From this lemma, one obtains as in Hawking and Penrose (1970, p. 537) or Hawking and Ellis (1973, pp. 268-269) the following corollary.
288
8.3
MAXIMAL GEODESICS, DISCONNECTED SPACE-TIMES
8
Corollary 8.16. Let (M,g) be strongly causd. If S Ss future trapped in (M,g), i.e., E+(S) is compact, then there is a future inextendible timelike
DISCONNECTED SPACGTIMES, NONSPACELIKE LINES
289
With these preliminaries completed, we may now obtain a sufficient condition for strongly causal space-times to be causally disconnected. Minkowski space-time shows that this condition is not a necessary one. Recall that a future directed, future inextendible, null geodesic y : [0, a ) --+ M is said to be
curve y contained in D+(E+(S)). It will also be convenient to prove the following lemma for the proof of Proposition 8.18.
L e m m a 8.17. Let (M, g) be strongly causal. If E+(p) is noncompact, then E+(p) contains an infinite sequence {q,) which diverges to infinity.
a null geodesic my if d(y(O),y(t)) = 0 for all t with 0 _< t
< a.
Proposition 8.18. Let (M,g) be strongly causal. If p E M is not the origin of any future [respectively,past] directed null geodesic ray, then (M,g) is causally disconnected by the future [respectively, past] horismos E+(p) = J+(p) - I+(p) [respectively, E- (p) = J- (p) - I- (P)] of p.
xn
Proof.We first show that the assumption that p is not the origin of any future directed null geodesic ray implies that E+(p) is compact. For suppose that E+(p) is noncompact. Then there exists an infinite sequence {q,) C E+(p) which diverges to infinity by Lemma 8.17. Since q, E E+(p), we have
yn from p to x, for each n. Extend each ^/, beyond x, to a future inextendible
d(p, q,) = 0 for all n. As qn E Jf (p), there exists a future directed null geodesic yn from p to q, by Corollary 4.14. Extend each yn beyond qn to a
nonspacelike curve still denoted by y,. By Proposition 3.31, the sequence {y,)
future inextendible nonspacelike curve, still denoted by 7,.Let y be a future
has a future inextendible, future directed, nonspacelike limit curve y : [O,a) +
inextendible nonspacelike limit curve of the sequence (7,) with y(0) = p,
M with y(0) = p. We may assume that the sequence {y,) itself distinguishes
guaranteed by Proposition 3.31. Using Proposition 3.34 and the fact that the qn's diverge to infinity, it may be shown along the lines of the proof of Theorem 8.10 that if q is any point on y with q # p, q 2 p, then L(y [p, q]) =
Proof. If E+(p) is closed, this is immediate since a closed and noncompact subset of M must be unbounded with respect to do. Thus assume that E+(p) is not closed. Then there exists an infinite sequence {x,)
C E+(p) such that
-t x 4 E+(p) as n -4 co.Since x, E E+(p), we have d(p, x,) = 0 and hence as x, E J+(p), there exists a maximal future directed null geodesic segment
y. If x E y, then x E J+(p). Since d(p,x) 5 liminfd(p,z,) = 0, we then have x E J+(p) - I+(p) = E+(p), in contradiction to the assumption that x 4 E+(p). Thus x f y. We now show that y[O, a) is contained in E+(p). To this end, let z E y be arbitrary. Since {y,) distinguishes y,we may find z, E y, such that z,
-t
z as n
-+ a. By
d(p, q). Thus y may be reparametrized to a null geodesic ray issuing from p, in contradiction. Hence Ef (p) is compact.
Proposition 3.34, there is a subsequence {yk)
We now show that E+(p) causally disconnects (M,g). Since E+(p) is com-
c0topology on curves.
pact, the set {p) is future trapped in M. Thus by Corollary 8.16, there is a
of {?,)
such that 7k[P,zk] converges to yip, z] in the
Since x
4 7, we may find an open set U containing 7[p, z] such that
n.
future inextendible timelike curve y contained in D+(Ef (p)). Extend y to a
x, it follows that zk comes before xk on yk for all k sufficiently
large. Thus y[p, zk] is maximal and d(p, zk) = 0 for all k sufficiently large.
past as well as future inextendible timelike curve, still denoted by y. From the definition of D+(Ef (p)), the curve y must meet E+(p) at some point r .
Hence d(p, z) 5 liminf d(p, zk) = 0. Since z was arbitrary, we have thus shown that d(p, z) = 0 for all z E y. Since y is a nonspacelike curve, 7 is then a
Since E+(p) is achronal and y is timelike, y meets E+(p) at no other point than r. Now let {p,) and (9,) be two sequences on y both of which diverge
maximal, future directed, future inextendible geodesic ray. Letting {t,) be any infinite sequence with t, -4 a- and setting q, = y(t,) gives the required
to infinity and which satisfy p, L ( y 1 [O,t]); and (iv) the future cut point y(to) comes at or before the first future conjugate point of y(0) along y. Since many of the theorems for Riemannian cut points are true only for complete Riemannian manifolds, it is not surprising that the more "global"
there exist a t least two maximal geodesic segments from p to q. In Section 9.2 we study null cut points. Even though null geodesics have zero arc length, null cut points may still be defined using the Lorentzian distance function. Let y : [O,a) -+ M be a future directed, future inextendible, null geodesic with p = y(0). Set to = supit E [O,a) : d(p, y(t)) = 0). If 0 < to < a, then $to) is called the future null cut point of y(0) along y. The null cut point, if it exists, has the following properties: (i) y is maximizing up to and including the null cut point; (ii) there is no timelike curve joining p to y(t) for any t 5 to; (iii) if to < t
< a, there is a timelike curve from y(0) to y(t); and
(iv) the future null cut point comes at or before the first future conjugate point of y(0) along y. For globally hyperbolic space-times, it is true for null as well as timelike cut points that the analogue of Poincark's Theorem for complete Riemannian manifolds is valid. Thus if (M,g) is globally hyperbolic and q is the future null cut point of p = y(0) along the null geodesic y, then either one or possibly both of the following hold: (i) q is the first future conjugate point of p along y; or (ii) there exist at least two maximal null geodesic segments from
p to q. We conclude Section 9.2 by using null cut points t o prove singularity theorems for null geodesics following Beem and Ehrlich (1979a, Section 5).
298
9
THE LORENTZIAN CUT LOCUS
9.1
T H E TIMELIKE CUT LOCUS
299
The nonspacelike cut locus, the union of the null and timelike cut loci of
Definition 9.2. Let T-IM = {v E T M : v is future directed and g(v,v) =
a given point, is studied in Section 9.3. For complete Riemannian manifolds,
-1). Given p E M , let T-lMlp denote the fiber of T-l M a t p. Also given v E T-1M, let c, denote the unique timelike geodesic with cV1(0)= v.
if q is a closest cut point to p, then either q is conjugate t o p or there is a geodesic loop based a t p passing through q. The globally hyperbolic analogue
It is immediate from the reverse triangle inequality that if y : [0,a)
-+
M is
a future directed nonspacelike geodesic and d(y(O), y(s)) = L ( y 1 [O,s]), then
(Theorem 9.24) of this result has a slightly different flavor, however. If (M, g) is a globally hyperbolic space-time and q E M is a closest (nonspacelike) cut
d(y(O),y(t)) = L ( y 1 [O,t]) for all t with 0 I t 5 s. Also, the reverse triangle
point to p, then q is either conjugate to p or else q is a null cut point to p.
inequality implies that if d(y(O),y(s)) > L ( y 1 [0, s]), then d(y(O), y(t))
Thus there is no closest nonconjugate timelike cut point to p. We also show
L ( y 1 [O,t]) for all t with s
< a. Hence the following definition makes
that for globally hyperbolic space-times, the nonspacelike and null cut loci are closed (Proposition 9.29). It can be seen (Example 9.28) that the hypothesis of global hyperbolicity is necessary here. In Section 9.4 we treat null and timelike cut points simultaneously, but nonintrinsically, using a different tool than the unit timelike sphere bundle of Section 9.1. This approach, developed in collaboration with G. Gallow while non-intrinsic, enables the hypothesis of timelike geodesic completenes to be deleted from Proposition 9.30.
Definition 9.3. Define the function s : T-lM
-,
Ru
{ m ) by s(v) =
supit 2 0 : d(r(v), c,(t)) = t). We may first note that if d(p,p) = m , then s(v) = 0 for all v E T-lM with r(v) = p. Also S(V)> 0 for all v E T-1M if (M,g) is strongly causal. The number s(v) may be interpreted as the "largest" parameter value t such that c, is a maximal geodesic between c,(O) and c,(t). Indeed from Lemma 9.1 we
know that the following result holds.
Recall that a future directed nonspacelike curve y from p to q is said t
Corollary 9.4. For 0 < t < s(v), the geodesic c, : [0,t ] -+ M is the unique maximal timelike curve (up to repararnetrization) from c,(O) to c,(t).
be maximal if d ( p ,q) = L(y). We saw above (Theorem 4.13) that a maxim
The function s fails to be upper semicontinuous for arbitrary spacetimes as
9.1
The Timelike Cut Locus
future directed nonspacelike curve may be reparametrized to be a geode
may easily be seen by deleting a point from Minkowski space. But for timelike
We also recall the following analogue of a classical result from Rieman
geodesically complete space-times we have the following proposition.
geometry. The proof may be given along the lines of Kobayashi (1967, p. using in place of the minimal geodesic segment from pl to p2 in the Rieman proof the fact that if p
0. Suppose either that (v) = +m, or ~ ( v is) finite and c,(t) = exp(tv) extends to [0, s(v)]. Then s is pper semicontinuous at v f T-1M. Especially, if (M, g) is timelike geodesially complete, then s : T-1M
-+
W U { ~ ) is everywhere upper semicontinuous.
Proof. It suffices to show the following. Let v,
-+
v in T-lM with (s(vn))
onverging in R U {a). Then s(v) 2 lim s(v,). If s(v) = m, there is nothing prove. Hence we assume that s(v) < lims(vn) = A and s(v) < co and rive a contradiction. We may choose 6 > 0 such that s(v)
d also assume that s(v,) 2 s(v)
+6 =
+6
< A is in the domain of c, b for all n. Let c, = tun. Since
300
9
T H E LORENTZIAN C U T LOCUS
9.1
<
b s(v,), we have d(n (v,), c,(b)) = b for all n. Since v, 4 v, we have by lower semicontinuity of distance that d(r(v),~ ( b )5) liminf d(r(v,), ~ ( b )=) b.
<
Thus d(x(v), c,(b)) b = L(c, 1 [0,b]), this last equality by definition of arc length. On the other hand, d(n(v), ~ ( b ) 2 ) L(q, 1 [0, b]) so that d(r(v),q,(b)) = L(c, diction. 0
I
[0, b]) = b. Hence s(v) 2 b = s(v)
+ 6, in contra-
In order to prove the lower semicontinuity of s for globally hyperbolic space-
past b,
+ 6). Since v, v and c, is defined for some parameter values + 6, the geodesics c, must be defined for some parameter values whenever n is larger than some N 2 No. Let q, = c,(b,) for n > N.
Now c,
I
and q = c,(A
4
beyond A
[O,b,] cannot be maximal since b,
> s(v,).
Because M is globally
hyperbolic and c,(O) 0. Hence Theorem 4.13 implies that the curve c may be reparametrized to be a maximal timelike geodesic segment from p to q. Finally, w = c'(0) / [-g(cl(0), c'(o))]'/~ is the required tangent vector.
bolic space-times. Proposition 9.7. If (M,g) is globally hyperbolic, then the function -+
W U {co) is lower semicontinuous. --+
+ 6) exists even if G, does not extend to s(v).)
and let No be such that b,
with the
induced metric (cf. Figure 9.1). Let p = (O,O),p, = (l/n,O), v = bldyl,, and v, = b/dylpn for all n 2 1. Then v, and yn(t) = (p,, t) for all t
--+
v as n
-+
m. Also let y(t) = (0,t)
2 0. Conformally changing g on the compact set
C shown in Figure 9.1 which is blocked from I+(p) by the slit {(I, y) E It2 : 1 < y 5 21, we obtain a metric Zj for M with the following properties. First there exists a timelike curve on passing through the set C and joining p, to a point q, = ( l l n , y,) on yn with y, 5 4 such that L(a,) > L(ynIpn,q,]). Hence y,[p,, q,] fails to be maximal for all n so that s(v,) 5 4 for all n. Thus s is not lower semicontinuous. Note that (M,g ) is strongly causal but (M, Zj) is not globally hyperbolic since J + ( ( l , 0)) n J-((1,3)) is not compact. The
Proof.It suffices to prove that if v, -+ v in T h l M and s(v,) A W U {m}, then s(v) 5 A. If A = oo, there is nothing to prove. Thus supp A < m. Assuming s(v) > A, we will derive a contradiction. Choose 6 > 0 such that A + 6 < s(v). (Since A + 6 < s(v), the c,(A
< y < 2)
y is still a maximal geodesic in (M,Zj) so that s(v) = +m. But for each n
We are now ready to prove the lower semicontinuity of s for globally hype
T-lM
time formed by equipping M = W2 - {(l,y) € R2 : 1
Define b, = s(v,
< s(v) for all n 2 No. Put
c, = cvn,pn = ~n
analogous construction may be applied to n-dimensional Minkowski space to roduce a strongly causal n-dimensional space-time for which the function s ils to be lower semicontinuous.
Globally hyperbolic examples also may be constructed for which the function s is not upper semicontinuous. However, this discontinuity may occur at
a tangent vector v E T-lM in a globally hyperbolic space-time only if s(v) is
302
9
T H E LORENTZIAN C U T LOCUS
9.1
T H E TIMELIKE CUT LOCUS
finite and ~ ( tdoes ) not extend to t = s(v). Combining Propositions 9.5 and 9.7, we obtain the following result. Theorem 9.8. Let ( M , g ) be globally hyperbolic, and suppose for v E
T - l M that either s ( v ) = +co, or s ( v ) is finite and c,(t) = exp(tv) extends to [0,s ( v ) ] . Then s is continuous at v E T - l M . Especially, i f ( M , g ) is globally hyperbolic and timelike geodesically complete, then s : T-1M
4
BU
{CQ}
is
everywhere continuous. W e are now ready t o define the timelike cut locus. Definition 9.9. (Future and Past Timelike Cut Loci)
T h e future timelike
in T,M is defined t o be r+(p) = { s ( v ) v : v E T-lMIp and 0 < < co). T h e future timelike cut locus C , f ( p ) of p i n M is defined t o be C t ( p ) = exp,(r+(p)). I f 0 < s ( v ) < co and c,(s(v)) exists, then the point
cut locus
s(v)
c,(s(v)) is called the future cut point o f p = c,(O) along c,. T h e past timelike cut locus C;(p) and past cut points may be defined dually. W e may then interpret s ( v ) as measuring the distance from p up t o the ture cut point along c,. Thus Theorem 9.8 implies that for globally hyperboli timelike geodesically complete spacetimes, the distance from a fixed p
E
t o its future cut point in the direction v E T - l M l p is a continuous function v. W e now give Lorentzian analogues of two well-known results relating cut a conjugate points on complete Riemannian manifolds. T h e following proper o f conjugate points is well known [Hawking and Ellis (1973, pp. 111-116 Lerner (1972, Theorem 4(6))]. Theorem 9.10. A timelike geodesic is not maximal beyond the first co
jugate point.
FIGURE 9.1. A strongly causal space-time ( M ,5) is shown with unit timelike tangent vectors v, which converge t o v, but with s ( v ) = +m and s(v,) 5 4 for alln 2 1. Hence s : T - I M -, R ~ { c o } is not lower semicontinuous.
In the language o f Definition 9.9 this may be restated as follows. Corollary 9.11. The future cut point o f p = c,(O) along c, comes no la
than the first future conjugate point o f p along c,. Utilizing this fact, we may now prove the second basic result on cut an conjugate points in globally hyperbolic space-times.
Theorem 9.12. Let ( M ,g) be globally hyperbolic. I f q = c ( t ) is the future cut point o f p = c(0) along the timelike geodesic c from p t o q, then either one
or possibly both o f the following hold: ( 1 ) The point q is the first future conjugate point o f p along c.
304
9
9.2
T H E LORENTZIAN CUT LOCUS
(2) There exist a t least two future directed maximal timelike geodesic segments from p to q. Proof. Without loss of generality we may suppose that c = c, for some v E T-lM and thus that t = d(p, q) = s(v). Let {t,) be a monotone decreasing sequence of real numbers converging to t. Since c(t) E M , the points c(tn) exist for n sufficiently large. By global hyperbolicity, we may join c(0) to c(tn) by
T H E NULL CUT LOCUS
305
Riemannian manifolds [cf. Wolter (1979, p. 93)]. The dual result also holds for the past timelike cut locus C c (p). 9.2
The Null Cut Locus
The definition of null cut point has been given in Beem and Ehrlich (1979a, Section 5) where this concept was used to prove null geodesic incompleteness
= s(v),
for certain classes of space-times. Let y : [0,a ) -+ (MIg) be a future directed
we have v # v, for all n. Let w E T-lM be the timelike limiting vector for {v,) given by Lemma 9.6. If v # w, then c and c, are two maximal timelike
If 0 < to < a , we will say y(t0) is the future null cut point of p on y. Past null
a maximal timelike geodesic c, = c,,, with v, E T-1 MIp. Since t,
>t
null geodesic with endpoint p = $0). Set to = supit E [0,a ) : d(p,y(t)) = 0).
It remains to show that if v = w, then q is the first future conjugate point
cut points are defined dually. Let C$(p) [respectively, CG (p)] denote the future [respectively, past] null cut locus of p consisting of all future [respectively,
v.
past] null cut points of p. The definition of Cz(p) together with the lower
If v were not a conjugate point, there would be a neighborhood U of v in
semicontinuity of distance yields d(p, q) = 0 for all q E C2(p). We then define
T-lMI, such that expp : U -+ M is injective. On the other hand, since c, and
the future nonspacelike cut locus to be C+(p) = C$(p) U Cz(p). The past
c 1 [0,t,] join c(0) to c(t,) and v,
Thus q is a future conjugate point of p along c. By Corollary 9.11, q must be
nonspacelike cut locus is defined dually. For a subclass of globally hyperbolic space-times, Budic and Sachs (1976) have given a different but equivalent
the first future conjugate point of p along c.
definition of null cut point using null generators for the boundary of I+(p).
geodesic segments from p to q. of p along c. If v = w, then there is a subsequence {v,)
-+
of {v,) with v,
-+
v, no such neighborhood U can exist.
Theorem 9.12 has the immediate implication that for globally hyperbolic spacetimes, q E C:(p)
if and only if p E Cc(q).
The timelike cut locus of a timelike geodesically complete, globally hyperbolic space-time has the following structural property which refines Theorem 9.12. We know from this theorem that if q f C$tp) and q is not conjugate to p, then there exist at least two maximal geodesic segments from p to q. Accordingly, it makes sense to consider the set Seg(p) = { q E C$(p) : there exist at least two future directed maximal geodesic segments from p to q).
The geometric significance of null cut points is similar to that of timelike ut points. The geodesic y is maximizing from p up to and including the cut oint ?(to). That is, L ( y 1 [O,t]) = d(p,y(t)) = 0 for all t 5 to. Thus there no timelike curve joining p to y(t) for any t with t 5 to. In contrast, the eodesic y is no longer maximizing beyond the cut point y(t0). In fact, each oint y(t) for to
< t < a may be joined to p by a timelike curve.
Utilizing Proposition 2.19 of Penrose (1972, p. 15) and the definition of aximality, the following lemma is easily established.
Lemma 9.13. Let (M, g) be a causal space-time. If there are two future irected null geodesic segments from p to q, then q comes on or after the null cut point of p on each of the two segments.
Since
s :
TP1M
-+
WU
(m)
is continuous by Theorem 9.8 and maximal
geodesics joining any pair of causally related points exist in globally hyperbolic space-times, it may be shown that Seg(p) is dense in C,f(p) for all p E M. The proof may be given along the lines of Wolter's proof of Lemma 2 for complete
The cylinder S1 x W with Lorentzian metric ds2 = dedt shows that the assumption that (M,g) is causal is needed in Lemma 9.13. We next prove the null analogue of Lemma 9.6.
306
9
T H E LORENTZIAN C U T LOCUS
9.2
-
Lemma 9.14. Let (M, g) be globally hyperbolic, and let p, q be distinct points in M with p 6 q and d(p,q) = 0. Assume that p, p,
< q,.
-+
p, q,
T H E NULL CUT LOCUS
307
models in the Einstein static universe is given in Penrose (1968) [cf. Hawking
q and
and Ellis (1973, pp. 132, 141)].
Let c, be a maximal geodesic joining p, to q, with initial direction
We now digress briefly to give an explicit computational discussion of the
v,. Then the set of direction {v,) has a limit direction w, and c, is a maximal
well-known fact that null pregeodesics are invariant under global conformal
null geodesic from p to q.
changes. An indirect proof of the conformal invariance of null cut points may also be given using the Lorentzian distance function.
Proof. Using Corollary 3.32 we obtain a future directed nonspacelike limit
Recall that a smooth curve y : J
curve X from p to q. Since d(p, q) = 0, the curve X must satisfy L(X) = 0. It
-+
M is said to be a pregeodesic if y can
be reparametrized to a smooth curve c which satisfies the geodesic differential
follows that X may be reparametrized to a maximal null geodesic.
= 0. Also recall the following definition. equation VC~cr(t)
We may now obtain the null analogue of Theorem 9.12.
Definition 9.16. (Global Conformal Dzffeomorphism)
A diffeomorphism f : (MI, gl) --+ (M2, g2) of MI onto M2 is said to be a global conformal diffeo-
Theorem 9.15. Let (M,g) be globally hyperbolic and let q = c(t) be the
morphism if there exists a smooth function R : MI
future null cut point of p = c(0) dong the null geodesic c. Then either one or
-+
R such that
possibly both of the following hold: (1) The point q is the first future conjugate point of p along c.
(2) There exist a t least two future directed maximal null geodesic segments
The space-time (M1,gl) is said to be globally conformally diffeomorphic to an
from p to q.
open subset U of (M2,gz) if there exists a diffeomorphism f : M1 -+ U and a smooth function R : Ml
-r
R such that
Proof. Let v = c'(O), and let t, be a monotone decreasing sequence of real numbers with t, -+ t. Since q E M , we know that c(t,) exists for all sufficiently large n. Since (M, g) is globally hyperbolic, we may find maximal nonspacelike The purpose of using the factor e20 rather than just a positive-valued
geodesics c, with initial directions v, joining p to c(t,). By Lemma 9.14 the set of directions {v,) has a limit direction w. If v
# w, then the geodesic c,
smooth function in Definition 9.16 is to give the simplest possible formula
9 for (M2,92). Explicitly, it may
is a second maximal null geodesic joining p to q as d(p, q) = 0. If v = w, then
relating the connections V for (M1,gl) and
q is conjugate to p along c. Since d(p,q) = 0, q must be the first conjugate point of c (cf. Theorem 10.72). D
be shown that if f*g2 = e20gl where j : (M1,gl)
-+
(Mz,g2) is any smooth
function, then
We now show how the null cut locus may be applied to prove theorems on the stability of null geodesic incompleteness. The key ideas needed for this application are first, that many physically interesting space-times may be conformally embedded in a portion of the Einstein static universe (cf. Exa
:
ple 5.11) that is free of null cut points and second, that null cut points
j
irivariant under conformal changes of metric. A discussion of global con
:
ma1 embeddings of anti-de Sitter space-time and the Friedmann cosmologi
b h
for any pair X, Y of vector fields on MI. Here "gradRn denotes the gradient vector field of C2 with respect to the metric g1 for MI. Using formula (9.1), it is now possible to show that if y : J --+ MI is a null geodesic in (MI,gl), then a = f 0 y : J M2 is a null pregeodesic in (M2,g2). -+
308
9
T H E LORENTZIAN C U T LOCUS
The crux of the matter is that because yl(t) is null and V,q'
9.2
= 0, formula
(9.1) simplifies to
309
THE NULL CUT LOCUS
geodesic y, then f (q) is a null cut point of f (p) along the null pregeodesic f o r : I0,a)
?bjal(t) = 2y1(t)(R)a'(t)
-+
(M2,92).
Proof. It suffices to assume that q is the future null cut point of p along
for all t E J. Note, however, that if y were a timelike geodesic, then the factor
y and that (M2,g2) is time oriented so that f o y is a future directed null
gl (y', y') f*(grad R) in formula (9.1) would prevent f o y from being a timelike
curve in M2. Let a : [0, b)
pregeodesic in (M2,g2).
future directed null geodesic guaranteed by Lemma 9.17 with f (p) = a(0).
Lemma 9.17. Let f : (MI, gl) -+ (M2,g2) be a global conformal diffeomorphism of MI onto M2. Then y : J -+ (MI, gl) is a null pregeodesic for (MI, gl) iff f o y is a null pregeodesic for (M2,gz). Proof. Since f
-': (M2,g2) + (MI,gl) is also a conformal diffeomorphism,
+
(M2,g2)be a reparametrization of f
oy
to a
Then f (q) = u(t1) for some tl E (0, b). Let d, denote the Lorentzian distance function of (M,, g,) for i = 1,2. We show first that d2(a(0),o(t)) = 0 for any t with 0
< t 5 tl.
For suppose
<
t 5 tl. Then we may find a future d2(a(O),u(t)) # 0 for some t with 0 directed nonspacelike curve P in M2 from a(0) to a(t) with Lg,(P) > 0. Thus
-'
pregeodesic of (M2,92). That is, we must show that a can be reparametrized
o p is a future directed nonspacelike curve in MI from p to f -l (a(t)) with L,, (f o p ) > 0. Since f (q) = cr(tl), we have f-l(u(t)) = y(t2) for some
to be a null geodesic of (M2,g2). But it has already been shown that
t2 5 to. Hence dl(p, y(t2)) 2 L,, (f
it is enough to show that if y : J + MI is a null geodesic, then a = f oy is a null
f
-'
-'op) > 0, which contradicts the fact that
dl(p, y(t2)) = 0 since t2 5 to and y(t0) = q was the future null cut point to p along y.
J --t W. Just as in the classical theory of projectively equivalent connections in Riemannian geometry, however, formula
for some smooth function f
:
(9.2) implies that a is a pregeodesic [cf. Spivak (1970, pp. G37 ff. )].
We show now that d2(u(O),a(t)) # 0 for any t > tl. This then makes f (q) = u(t1) the future null cut point of f (p) along a as required. To this end, fix t > tl. There is then a t2 > to so that f-'(a(t)) = y(t2). Since y(t0) = q is the future null cut of p along y, we have dl (p, y(t2)) > 0. Hence
We are now ready to prove the conformal invariance of null cut points
there is a future directed nonspacelike curve a from p to y(t2) with L,, ( a ) > 0.
under global conformal diffeomorphisms f : (M1,gl) -r (M2, g2). Notice that if (MI, gl) is time oriented by the vector field XI, then (M2,g2) is time oriented
Then f o a is a future directed nonspacelike curve from f (p) to u(t) Hence d2(f(p),a(t))2Lg,(f oa)>Oasrequired. !J
either by X2 = f*X1 or -X2. If M2 is time oriented by X2, then f maps future directed curves to future directed curves and thus future (respectively, past) null cut points to future (respectively, past) null cut points in Proposition 9.18. On the other hand, if M2 is time oriented by -X2, then f maps future (respectively, past) null cut points to past (respectively, future) null cut points since f maps future directed curves to past directed curves. Proposition 9.18. Let f : (MI, gl)
With Proposition 9.18 in hand, we may now apply the null cut locus to study null geodesic incompleteness. Recall that a geodesic is said to be incomplete if it cannot be extended to all values of an affine parameter (cf. Definition 6.2). Let R and Ric denote the curvature tensor and Ricci curvature tensor of (M,g), respectively. Recall that an inextendible null geodesic y will satisfy the generic condition if for some parameter value t, there exists a nonzero
(M2, g2) be a global conformal
tangent vector v E T,(t)M with g(v, yl(t)) = 0 such that R(v, yl(t))y'(t) is
diffeomorphism of (Ml,gl) onto (M2,gz). Let y : [O,a) --t (MI, gl) be a null geodesic of MI. If q = $to) is a null cut point of p = y(0) along the
nonzero and is not proportional to yl(t) (cf. Proposition 2.11, Section 2.5). In particular, if Ric(v, v) > 0 for all null vectors v E T M , then every inextendible
--t
310
9
T H E LORENTZIAN CUT LOCUS
9.3
null geodesic of (M,g) satisfies the generic condition. In Section 2.5, it was noted that dim M 2 3 is necessary for the null generic condition to be satisfied. Thus we will assume that dim M 2 3 in the following proposition. Proposition 9.19. Let (M,g) be a space-time of dimension n
> 3 such
that all inextendible null geodesics satisfy the generic condition and such that
T H E NONSPACELIKE CUT LOCUS
311
(M,g) is globally conformally diffeomorphic to some portion of the subset M' of the Einstein static universe, then d l null geodesics of (M,g) are incomplete. Since Minkowski space-time is free of null cut points, the above result remains valid if we replace M' with Minkowski space-time.
Ric(v, v) 2 0 for all null vectors. If (M, g) is globally conformally diffeomorphic
Friedmann space-times are used in cosmology as models of the universe. In these spaces it is assumed that the universe is filled with a perfect fluid having
to an open subset of a space-time (MI, g') which has no null cut points, then
zero pressure. We will also assume that the cosmological constant A is zero for
all null geodesics of (M, g) are incomplete.
these models. These space-times are then Robertson-Walker spaces (cf. Sec-
Proof. Proposition 4.4.5 of Hawking and Ellis (1973, p. 101) shows that if
tion 5.4) with p = A = 0. These space-times may be conformally embedded in
the Ricci curvature is nonnegative on all null vectors, then each complete null
the subset M' of the Einstein static universe defined above [cf. Hawking and
geodesic which satisfies the generic condition has conjugate points (cf. Propo-
Ellis (1973, pp. 139-141)). Furthermore, Ric(g)(v, v)
sition 12.17). Since maximal geodesics do not contain conjugate points, we
vectors in a F'riedmann cosmological model (M, g). By Proposition 7.3, there
> 0 for all nonspacelike > 0 for all
need only show that all null geodesics of (M,g) are maximal to establish their
is a C 2 neighborhood U(g) of g in C(M,g) such that Ric(gl)(v, v)
incompleteness. Assume that y is a future directed null geodesic from p to q
gl E U(g) and all nonspacelike vectors v in (M, gl). Since R ~ C ( ~ I )v) ( V>, 0
which is not maximal. Then there is a timelike curve from p to q. Since con-
implies that the generic condition is satisfied by all null geodesics in (M, gl),
formal diffeomorphisms take null geodesics to null pregeodesics and timelike
Corollary 9.20 yields the following result.
curves to timelike curves, the image of y must be a nonmaximal null geodesic in (MI, g'). This implies that (M', g') has a null cut point and hence yields a contradiction.
Corollary 9.21. Let (M, g) be a Friedmann cosmological model. There is a
c2neighborhood U(g) of g in C(M, g) such that
every null geodesic in
(M, gl) is incomplete for all gl E U(g).
Recall that the four-dimensional Einstein static universe (Example 5.11) is the cylinder x 2 + y 2 + ~ 2 + w 2= 1 embedded in R5with the metric induced from the Minkowski metric ds2 = -dt2 +dx2 +dy2 + dz2 +dw2. The Einstein static universe is thus
9.3
The Nonspacelike C u t Locus
Recall the following definitions.
B x S3 with a Lorentzian product metric. The geodesics and
null cut points are easy to determine in this space-time. The null cut locus of the point (t, x, y, t,w) merely consists of the two points ( t i n , -x, -y, -2, -w)
Definition 9.22. (Future and Past Nonspacelike Cut Loci) nonspacelike cut locus C+(P) of p is defined as C+(p) = C:(P)
U C$(P).
The
past nonspacelike cut locus is C-(p) = CL(p) U Cg(p), and the nonspacelike
Consequently, the subset
cut locus is C(p) = C-(p) M'=((t,s,y,z,w):O T}
to obtain two maximal null geodesics from p to q. In this case q is a cut point
n). The future timelike cut locus Cz(p) consists of {(n, t) : t point (T,
and is not closed. On the other hand, C+(p) = C$(p) U C$(p) is closed. n/4,277) from M , we obtain a strongly causal If we delete the two points (f space-time (M', glM,) which is not globally hyperbolic. In (MI, glM,) the future nonspacelike cut locus C+(p) is not closed. Proposition 9.29. Let (M,g) be a globally hyperbolic space-time. For any p E M, the sets C$(p), C i ( p ) , C+(p), and C-(p) are all closed subsets of M. In particular, the null cut locus and the nonspacelike cut locus o f p are
of p. On the other hand, v and w may determine the same direction. If this is so, we first note that there are constants a
> 0 and b > 0
such that av = bw
and expp(av) = expp(bw) = q. Applying Lemma 9.25, it follows that for some subsequence ( m ) of ( n ) we have urn -+ av and w,
+ av.
However v,
# w,
and expp(vrn) = expp(wrn) then yield that (exp,), must be singular at av. Thus q is conjugate to p along y(t) = exp,(tav), and since d(p, q) = 0 we find that q is the cut point of p along y(t). ! I
each closed. Proof. Since the four cases are similar, we will show only that C$(p) is closed in M . To this end, let q, E C$(p) and q,
-+
q. Since (M,g) is
globally hyperbolic, q # p and q E Jt(p). From the definition of C$(p) we see that d(p, q,) = 0. Using the continuity of Lorentzian distance for globally hyperbolic space-times (Lemma 4.5),it follows that d(p,q) = limd(p, q,) = 0.
Surprisingly, the above result holds without any assumptions about the timelike or null geodesic completeness of (M,g). In particular, the null cut locus and the nonspacelike cut locus in globally hyperbolic space-times are closed by Proposition 9.29, even though the function s : T T I M -+ R U {co) may fail to be upper semicontinuous.
Let 7 be any nonspacelike geodesic from p to q. Then d(p, q) = 0 implies that
It is well known [cf. Kobayashi (1967, pp. 100-101)] that using the cut locus,
y is a maximal null geodesic and that the cut point to p along y cannot come
it may, be shown that a compact Riemannian manifold is the disjoint union
before q. There are two cases to consider. Either infinitely many points q, are future
of an open cell and a closed subset (the cut locus of a fixed p E M) which is a continuous image (under expp) of an (n - 1)-sphere. Thus the cut loci
nonspacelike conjugate to p or else no q, is future nonspacelike conjugate to p
inherit many of the topological properties of the compact manifold itself. For
for large n. In the first case we apply Lemma 9.27 and find that if q is future
Lorentzian manifolds, cut points may be defined (using the Lorentzian distance
nonspacelike conjugate to p, then if this conjugacy is along y, q is the cut point
at least) only for nonspacelike geodesics. Hence a corresponding result for
along y because the cut point along y comes before or a t the &st conjugate point along y. If q is a future nonspacelike conjugate point to p along some
space-times must describe the topology not of all of M itself, but only of J+(p) for an arbitrary p E M. To obtain this decomposition with the methodology of
other nonspacelike geodesic y', then y' must be null, and Lemma 9.13 shows that q is the cut point along both y and y'.
this section, we need to assume that (M,g) is timelike geodesically complete as well as globally hyperbolic, so that the function s : T - I M --t R U {co)
Assume now that q, is not future nonspacelike conjugate to p for all sufficiently large n. Theorem 9.15 implies that for large n there exist a t least two
defined in Definition 9.3 will be continuous and not just lower semicontinuous (cf. Propositions 9.5 and 9.7).
maximal null geodesics from p to q,.
Thus for large n there are two future
Recall also that the future horismos E+(p) of any point p E M is given by
directed null vectors v, and w, with expp(vn) = exp,(w,) = q,. The future directed nonspacelike directions a t p form a compact set of directions. Thus
E+(p) = J+(p) - I+(p) and that C+(p) denotes the future nonspacelike cut locus of p.
318
9
THE LORENTZIAN C U T LOCUS
9.4
T H E NONSPACELIKE C U T LOCUS REVISITED
319
Proposition 9.30. Let (M, g) be globally hyperbolic and timelike geodesi-
cut loci for globally hyperbolic spacetimes could not draw on the continuity
cdly complete. For each p E M the set J+(p) - [C+(P) U E+(P)] is an open
properties of the s-function established in Proposition 9.7. Rather, we relied
cell.
PTOO~. Let B = J+(p)- [C+(p)uE+(p)]. Then q E B if and only if there is a
on a more indirect proof method which called upon the characterization of nonspacelike cut points given in Theorems 9.12 and 9.15. In this section wc
maximal future directed timelike geodesic which starts at p and extends beyond
will give a more elementary method to derive the closure of the nonspacelike
q. Thus B = I+(p) - C+(p) which shows that B is open. Now T-'MI,
TpM : v is future directed and g(v,v) = -1) is homeomorphic to lRn-l. Let
cut loci for globally hyperbolic spacetimes which treats both null and timelike cut points simultaneously but is non-intrinsic. The treatment given in this
H : T-'MIP + IRn-' be a homeomorphism, and define 3 : Itn-' -+ R U {m) by 3 = s o H-'. There is an induced homeomorphism of B with {(x,t) E
section has been worked out in discussions with G. Galloway. We have noted in Section 9.3 just prior to Example 9.28 that Magerin (1993)
x R : 0 < t < ~ ( x ) defined ) by q + (H(v), t), where v is the vector in T-IM such that exp,(sv) is the unique (up to reparametrization) maximal
has given examples for Riemannian spaces that show that, contrary to folk lore, the first conjugate locus of a point in a compact Riemannian manifold need not
geodesic from p to q, and expp(tv) = q with v f T-IM. Let f : [0,m ] + [o, 11
be closed. Thus for completeness, we present an elementary proof that the cut
be a homeomorphism with f (0) = 0 and f ( m ) = 1. Then the map (x, t ) -4
locus of any point in a complete Riemannian manifold is closed, nonetheless.
(x, f ( t )/ f ( ~ ( 2 ) )shows ) B is homeomorphic to Etn-' x (0,I), which establishes
T h e o r e m 9.31. Let (N, go) be a complete Riemannian manifold, and let
= {v E
wn-1
the proposition. Since I+(p)
p in N be arbitrary. Then the cut locus C(p) of p in N is a closed subset of J + ( p ) , Proposition 9.30 yields some indirect topological in-
formation about I+(p). The Einstein static universe shows that I+(P)need not be an open cell even if (M,g) is globally hyperbolic. However, (I+(p), glI+(p))
Proof. We have the basic fact that the distance to the cut point in unit direction v, which we will denote as is customary by s : SN -, W u {rn)
is globally hyperbolic whenever (M, g) is globally hyperbolic. Thus I+(p) may
corresponding to Definition 9.3 for unit timelike vectors, is continuous pro-
be expressed topologically as a product I+(p) = s x W,where S is an (n - 1)dimensional manifold (cf. Theorem 3.17).
9.4
T h e Nonspacelike C u t Locus Revisited
In Sections 9.2 and 9.3 we treated the null and timelike cut loci separate1 because the intrinsic unit observer bundle T-1M was used as the prim technical tool for investigating the timelike cut locus. This approach s the drawback, however, that if {q,) is a sequence of future timelike cut poi to p and q, = exp(s(vn)v,) converges to a null cut point q to p, then
vided (N, go) is complete. Now let {q,) G C(p) be given with qn -+ q in M . Represent q, = expp(s(vn)vn)with llvnll = 1. Since the set of unit vectors in TpM is compact, we may assume by passing to a subsequence if necessary that un -,v f TpM. By continuity of the s-function, lims(vn)vn = s(v)v. Since (N,go) is geodesically complete, exp,(s(v)v) is defined. Hence q = lim qn = lim exp,(s(vn)vn) = exp,(s(v)v). Therefore, q is a cut point to p. Now Galloway suggested a non-intrinsic way to deal with the nonspacelike
(9.3)
lim s(vn) = 0.
As a result of the technical differences in handling the null and timelike loci, the proof in Proposition 9.29 of the closure of the nonspacelike and n
cut locus in such a manner as to bring an analogue of (9.4) to bear. The wo facts that (i) future inextendible nonspacelike geodesics are not trapped m compact sets in strongly causal spacetimes, and (ii) a modified s-function
I
320
9
THE LORENTZIAN C U T LOCUS
9.4
T H E NONSPACELIKE CUT LOCUS REVISITED
321
defined for all future causal directions t o be constructed below is lower semicontinuous for globally hyperbolic space-times suffice t o ensure that the basic
L e m m a 9.34. Let ( M , g ) be a globally hyperbolic space-time. Suppose that c, : [0,b,] -+ ( M ,g) is a sequence o f future nonspacelike maximal geodesic
ideas o f the above proof are valid in the space-time setting. Fix throughout the rest o f this section an auxiliary complete Riemannian
segments with v, = cnl(0) in U M and c,(O) = p. Put q, = cn(bn), and suppose that q, --t q in M , b, -+ b > 0 in W U { m ) , and v, -+ v in U M . Let
metric h for the space-time ( M ,g). Let
c : 10, a ) + ( M ,g ) denote the maximally extended nonspacelike limit geodesic
U M = { v E T M : h(v,v ) = 1 and v is future directed ). Then we define the modified s-function on U M , which will be denoted by s l , as would be expected. Definition 9.32. Define the function sl : U M
+W U
{ m ) by
given by c(t) = exp,(tv). Then ( 1 ) b < a, so that b is finite and positive, ( 2 ) c(b) = q, and ( 3 ) s1(v) 2 b. Proof. ( 1 ) Suppose a 5 b. Fix any q' with q 0 for any v in U M .
by uniform convergence.
= c(b)
( 3 ) Since the geodesic segments c, 1 10, b,] are assumed t o b e maximal and b =
Proposition 9.33. ( 1 ) Let v E U M with s l ( v ) > 0. Suppose either that s l ( v ) = +m, or s l ( v ) is finite and cv(t) extends to [O,sl(v)].Then sl is upper semicontinuous at v in U M . Especially, i f ( M , g ) is future nonspacelike geodesically complete, then sl : U M semicontinuous.
( 2 ) Since b < a, we find that c(b) is defined and q = lim q, = lim c,(b,)
-+
JR U { m ) is everywhere upper
( 2 ) I f ( M ,g) is globally hyperbolic, then sl : U M semicontinuous.
limb,, the limit segment
c
I [0,b ] must also be maximal, whence s l ( v ) 2 b.
O
T h e o r e m 9.35. Let ( M , g ) be globally hyperbolic. Then the future nonspacelike cut locus and the future null cut locus o f any point p in M are closed subsets o f M . Proof. T h e proof may be given in a uniform fashion either for (q,) a se-
--t
R U { m ) is lower
Before giving the elementary proof o f the closure o f the nonspacelike cut locus for globally hyperbolic space-times, it is helpful t o note the following technical result. I t is not required a priori in this result that either o f a or b is finite. Similar issues are addressed in a more general context and on a more systematic basis in Lemma 11.20 and the proof o f Theorem 11.25.
quence o f null cut points t o p or a sequence o f nonspacelike cut points to p, so we will not specify. Suppose qn -+ q in M . Choose v, E U M with
and put b, = sl(v,). Because {w E U M : w E T p M ) is compact, by taking successive subsequences we may suppose that v, --t v in U M and that b, -+ b in W U { m ) . Put c ( t ) = expp(tv).By Proposition 9.33-(2), we have b 2 s l ( v ) ,
322
THE LORENTZIAN CUT LOCUS
9
so that b > 0. By Lemma 9.34, b is finite? c(b) = q, and s l ( v ) 2 b. Hence s l ( v ) = b, and thus q is a cut point of the required type.
G. Galloway has also remarked that with non-intrinsic methods, the hyCHAPTER 10
pothesis of "timelike geodesic completeness" may be removed from Proposition
9.30.
MORSE INDEX THEORY ON EORENTZIAN MANIFOLDS
Proposition 9.38 (Galloway). Let (M, g) be globally hyperbohc. Then for each p in M the set
B = Ji(p) - [C+(p)U E+(p)]is an open cell.
Proof. -4s before, B = I + ( p ) - C+(p).Fix c we set V = ( v E T M
,
>O
Given a nonspacelike geodesic y sufficiently small that if
h ( v , v ) = c, v future timelike), then C
contained in B. Also, fix a diffeomorphism f : U
-
expp(V) is
Rn-l. Define a projection map r, : B i U by letting ?(m) denote the point of intersection of U with the unique future timelike gcodcsic in M from p to m. Now choose a second +
auxiliary complete Riemannian metric hl just for the open subset B of M. considered as a manifoid in its own right. For each q in U ,let "14 denote the g-geodesic cexp;~(q)reparametrized as an hl-unit speed curve in B with efq(0) = q and parametrized with the same orientation a s the corresponding g-geodesic. By Lemma 3.65. y, : W -+ 3 for each q in U. Now define t : B -+ R by the requirement that
Then we may define a homeomorphism F :
f?
[O.a )
-+
M In a causal space-tlme, Xe
have seen in Chapter 9 that if ?(to) is the future cut point to 7 ( 0 )along 2 , then for any t
< t o the geodesic segment y I [O, t ]is the longest
from ~ ( 0 to ) y(t)
In
nonspacclike curve
all of M . We could ask a much less stringent quest~on:
among all nonspacelike curves u from $0) to "((to) sufficiently "closc" to 7, is L ( y ) 2 L(u)? If so, y(to) comes at or before the first future conjugate point of y(0) along y. The crucial difference here is between "in all of M" for cut points and "close to y" for conjugate points. The importance of t h ~ s distinction is illustrated by the fact that while no two-dimensional space-time has any null conjugate points, all null geodesics in the two-dimensional Einstein static universe have future and past null cut points. Since only the behavior of "nearby" curves is considered in studying con-
(t(m)) = m
.
-
jugate poirits, it is natural to apply similar techniques from the calculus of
B
-+
R
x Rn-' by F ( m ) =
variations to geodesics in arbitrary Rlernannian manifolds and to nnnspacelike geodesics in arbitrary Lorentzian manifolds. To ~ndicatethe flavor of the Lorentzian index theory. we sketch the Ivlorse Index theory of an arbitrary (not, necessarily complete) Riemannian manifold (iV, go). Let c : [a,b j
-+
N
be a fixed geodesic segment and consider a one-parameter family of curves a, starting s t c(a) and ending at c(b). More precisely, let a : 'a, bl x ( - c , be a continuous, piecewise smooth map with and &(a,s ) = c(a), a(b, s) = c(b) for all s f clirve a, . t
E)
--,AT
0) = ~ ( tfor ) all t E [a,b ] ,
CY(~.
(-t, t)
Thus each neighbor~ng
cr,(t) = a(t,s ) is piecewlse smooth. The vanatzon vector field (or s-parameter denvatzve) V of this deformation is given by 4
324
10
10
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
MORSE INDEX THEORY ON LORENTZIAN RlANIFOLDS
325
Also, c has only finitely many points in (a, bj conjugate to c(a).
Since c is a smooth geodesic, one calculates that
With the Morse Index Theorem in hand, the homotopy type of the loop space for complete Riemannian manifolds may now be calculated geornetrically [cf. Milnor (1963, p. 95)]. One obtains the result that if (N,go) is a complete Riemannian manifold and p, q E N are any pair of points which are
and the second variation works out to be
not conjugate along any geodesic, then the loop space fl(p,,)of all contiriuous
d2
-(b(ffs))1 ds2 S=O
paths from p to q equipped with the compact-open topology has the homotopy
b
It
bo(V1, V ' ) - go(R(V, c1)c',V) - c'(go(V, cl))l dt =a
b
+ 90(VvV,c1) I,.
type of a countable CW-complex which contains a cell of dimension X for each geodesic from p to q of index A. The purpose of Sections 10.1 and 10.3 is to prove the analogues of (1) through (3) for nonspacelike geodesics in arbitrary space-times Let c : [a,b]
This second variation formula naturally suggests defining an zndez form
-,
&I [respectively, P : [a.b] M ] denote an arbitrary timelike [respectively, null] geodeslc segment in ( M , g ) . Let V , ( c ) [respectively, V$(P)] denote the -+
infinite-dimensional vector space of piecewise smooth vector fields Y along c [respectively, P] perpendicular to c [respectively, P] with Y ( n ) = Y(b) = 0. on the infinite-dimensional vector space V$ (c) of plecewise smooth vector fields X along e orthogonal to e with X(a) = X(b) = 0 It IS then shown that
The timelike index form I : q ( c ) x Vo-(c)
-+
R may be defined by
necessary and sufficient cond~tlonfor e to be free of conjugate points is tha
I ( X ,X ) > 0 for all nontr~vialX E V&(c). T h ~ ssuggests that for a geodeslc segment c : [a,b] -+ N wlth conjugat points to t = a, the index Ind(c) of c with respect to I : Vg'(c) x V$(c) -+
in analogy with the Riemannian index form. It may also be shown that c
-
.
dimensional vector space, the Morse Index Theorem for arbitrary Riemannia
[a.b] A4 has no conjugate points in (a. b) if and only if I . *(c) x V$(c) i R is negative definite. Similarly, an index form I : V&(P) x @(P) + R may be defined by r(X, Y) = - J , ~ [ ~ Y '()x - 'g,( R ( X .P1)P1,Y)]dt. But since P is a null geo-
manifolds asserts that.
desic, g(P1,@') = 0. Consequently, vector fields of thc form V(t) = f (t)P1(t)
should be defined as the supremum of dimensions of all vector subspaces o V&(c) on which I is negative definite. Even though e ( c ) is an infinit
R piecewise smooth and f ( a ) = f (b) = O are always : V;(f?) x e(P) -- R yet never give rise to null conjugate points. One way around this difficulty is to consider the quotient bundle Xo(P) of V$(P) formed by identifying YI and Y2 in V$(/3) if
(1) Ind(c) is finlte; and
with f : [a. b]
(2) Ind(ef equals the geodesic index of c, l.e., the number of conjugat
in the null space of I
points along c counted wlth multlphattes. More precisely, if we let Jt(c) denote the vector space of smooth vector fie1 -0wi Y along c sat~sfyingthe Jacob1 differential equation Y" R(Y, c') c' -
+
boundary conditions Y ( a ) = Y(t) = 0, then (2) is equivalent to the formula
(3) Ind(c) =
dlm Jt (c).
-+
Yl- Y2= fP/ for some piecewise smooth function f . [a.b ] --+ W. The index form I : Vo-(@) x Vo-(6) -+ R may also be projected to a quotient index form 7 : Xo(Pj x Xo($) 4 R. It may then be shown that the null geodesic segment ,B : [a, b ]
-+
M has no conjugate points in [a,b ] if and only if
326
10
10.1
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
-
THE TIMELIKE MORSE INDEX THEORY
327
geodesics, the procedure of forming the quotient bundle has also been utilized
M are any two points with p 0
10.6, this restriction is unnecessary by the following lemma.
such that for all s with Is/ 5 E,, the tangent vectors as1(t:)
-+
-+
M be a piecewise smooth variation of the
lif. There is by Defin~tion10 6 a finite partition
a = to < t l
timelike and a, I [t,, t,,~]
1s a
and a,'(t;+,)
are
timellke curve. Taking b = min(6~.b l , . . . ,6k-1)
shen yields the required 6.
Lemma 10.7. Let a : [a, b] x ( - 6 , ~ )-, A4 be a piecewise smooth variation of the tirnelike geodesic segment c : [a,b ] -+ Al. Then there exists a constant
O > 0 depending on a such that the neighboring curves a, are timelike for all s with Is] 5 6
Remark 10.8. There is no result corresponding to Lemma 10.7 for variations of null geodesics (cf. Definition 10.58 ff.). The index form may now be related to variat~onsof timelike geodesic segments c : [a,b]
-, ht as
variation a : [a,b] x Proof We first suppose that a is a smooth variation. Choose any el with
0<
€1
< E. Then a is differentiable on the compact set [a,b] x [-el,el] by
Definition 10.6. Hence by definition of differentiability, a extends to a smooth mapping of a larger open set containing [a,b] x [-el, el]. Since c is a timelike geodesic, the vectors cl(a+) and cl(b-) are timelike. It follows from this and the extension of
N
to an open set containing [a,h ] x [-el, el] that there exists
> 0 such that the tangent vectors a,'(ai) and cuS1(b-) to the xeighboring curves a, are timelike for all s with ( s j < bl. a constant El
Suppose now that no 6
> 0 can be found such that all the curves a, are
timelike for Is] < 6. Then we could find a sequence s, --+ 0 such that the curves a,,, failed to be timelike. Thus there would be t, E [a,b] so that g (orin (t,), a:, (t,))
2 0 for each n. Since [a, b ] x [-el. el] is compact, the
sequence ((t,,
has a poirit of accumulation ( t ,s). Since s,
5,))
point must be of the for111 (t,O). and also the existence of
-
:
[a. b ] x (-E,E)
4
(-6,
(10.5)
E) -+ M of c by setting 4 t . s) = exp,(z)(sy(t)).
It should first be noted that slnce c([a,G]) is a compact subset of M , and the differential of the exponential map, exp,-. is nonsingular at the origin of T,M for all p E M, it is possible given c([a, b]) to find an
E
> 0 such that
exp,(,)(sY(t)) is defined for all s with Is/ 5 c and for each t E [a,b]. Secondly, fro111 Definition 10.1 it follows that a ( t , s) defined as in (10.5) 1s a piecewise sn~oothvariation of c. Nence given Y t VoL(c),we know from Lemma 10.7 that there exists some constant 5 > 0 such that all the neighborlng curves a, : t
-+
a(t, s) are timelike for all s with -6 < s < 6.
Given an arbitrary smooth variation a . [a,bj x (-e,
E ) -,
M of c : [a,b ] -+
M . the van'atzon vector field V of a is defined 50 be the vector field V(t) along c
given by the formula
0 , this
61 above shous
that t # a, b. But then as g (a:,, (t,), a,'n(t,)) 2 0 for each n, it follows that g(cJ(t),cl(t)) >_ 0, in contradic~ionto the fact that c was a timelike gecdesic segment. Thus we have seen that ~f a
follows. Given Y E V ~ ( C )define , the canonzcal proper
M is a smooth
variation of the timelike geodesic c : [a,b] 4M , then there is a constant 5 > 0
More precisely, lettlng a l a s be the coordinate vector fieid on [a, b ] x (-e, e) corresponding to the s parameter, the variation vector field is given 'ny
332
10
MORSE INDEX THEORY ON LORENTZIAN MAXIFOLDS
For a piecewise smooth variation a : [a. b ] x (-E, E ) --+ M , one obtains a continuous piecewise smooth varlatlon vector field as follows. Let a = to <
< tk = b be a partit~onof [a, b ] such that cu / It,. t,+l] x (-E,E) IS smooth for z = 0,1,. . . ,k - 1. Given t E [a, b], choose an index z such that t, < t < t,+l, and set t2 <
<
~ ( t =) [ aI it,, t,+ll x
(-E,
~11,
10.1
(Y', c"} = (Y", c')
THE TIMELIKE MORSE INDEX THEORY
333
= -(R(Y, cl)c', c') = 0 by the skew symmetry of the
Riemann-Ghristoffei tensor Thus (Y: c') is an affine function. Corollary 10.10. If Y is any Jacob1 field along the timelike geodesic
IM and Y ( t ~ j= Y(tz)
c : [a,b; Y E VL(c).
--+
= 0 for distinct t ~ . t nE [a,bj, then
Corollary 10.11. If Y E @ ( c ) is a Jacobi field; then Yf E VL(c).
The canonical variation (10.5) has the property that each curve s -+ a ( t , s) 1s
Using the canonical variation, we are now ready to derive the following
just the geodesic s -exp,(,)(sY(t)) i which has initial direct~onY(t) at s = 0.
geometric consequence of the existence of a conjugate point to E (a, b) to t = a
Hence the variation vector field of the canonical variation (10.5) is just the
along c.
glven vector field Y E V$ (c). If we put L(s) = L(a,) = L(t c
IS
a ( t , s)), then L'(0) = y \ 8 = o= 0 since
-+
a smooth tlmelike geodesic, and
(10.8) Thus if I(Y, Y)
L"(0)=2L(s) dZ
Lo
> 0 for some Y E Vg'(c),
=I(Y,Y).
then the canonical proper variation
a ( t , s) defined by (10.5) using Y will produce timelike neighboring curves a, joining c(a) to c(b) with L(a,) > L(c) for s sufficiently small. Thus if the
M is maximal [i.e., L(c) = d(p,q)], then I . V$(c) x V&(c) --+ W must be negative semidefinite. Before proving the Morse Index Theorem for tirnelike geodesics, we must establish the following more precise relationship between corljugate points, Jacobi fields, and the index form. First, the null space of the index form on V$ (c) consists of the smooth Jacobi fields in VO-(c), and second, c has no conjugate points on [a, b] if and only if the index form is negative definite on V&(c). We first derive an elementary but important consequence of the Jwobi differential equation.
timdike geodesic c : ;a, b]
-i
Proposition 10.12. Suppose that the timel~kegeodes~cc : [a. b ]
-+
M
contains a conjugate point to E (a,b) to t = a along c Then there exists a piecewise smooth proper variation a , ofc such that L(a,) > L(c) for all s # 0. Thus c : [a,b]
--+
M is not maximal.
Proof. In view of (10.5), (10.8), and Lemma 10.7, it is enough to construct a piecewise smooth vector field
Y
E V$(c) with I(Y, Y)
> 0 and let a be
the canonical variation associated with Y. To this end, let Yl be a nontrivial Jacobi field along c with YI (a) = Yl (to) = 0. By Corollary 10.10, Yl E VL(c). Hence as Yl(a) = &(to) = 0, we have YI' E VL(c) by Corollary 10.11. Since Yl(to) = 0 and Yl is a nontrivial Jacobi field. it follows that YI1(to) is a (nonzero) spacelike tangent vector. Let I( , ): denote the restriction of the index form to cj [a,s ] , that
IS.
Then since Yl is a Jacobi field, we have from (10.2) of Definition 10.4 that
Lemma 10.9. Let c : [a, b ] --+ M be a timelike geodesic segment an Y be any Jacob] field along c. Then (Y(t), cl(t)) 1s an affine function of (Y ( t ) ,cl(t)) = a t + /3 for some constants a,p E R.
Proof. First (Y, c')'
= (Y', c'}
+ (Y, c")
= (Y',c'), as c" = V,lcr =
along the geodesic c. Differentiating again, we o b t a ~ n(Y, c')" = (Y", d )
r any Z E Vi(c). We are now ready to construct a piecewise smooth vector iield Y 6 V$(c) ith IfY,Y)
> 0. Let $J: [a,b] --,IR be a smooth function with @(a)= @(b)=
0 and + ( t o ) = 1. Also let Z1 be the uniqae smooth parallel vector Seld along
334
10
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
10.1
T H E TIMELIKE MORSE INDEX THEORY
c with &(to) = -fil(to). Then Z = ~ Z E1 V ~ ( C )Define . a oneparamete1
If Y is a Jacobi field. I(Y, Z) = - (Y', 2) . :1
family Y, E @(c) by
z E VoL(c).
K(t) =
Y I ( ~+ ) €Z(t)
ifastto, if to t b.
< <
t:Z(t)
Then using (10.9) we obtain
+ I(Y,, yE)fo = I(Y1 + € 2 ,Yl + E Z ) + ~ I(&, EZ):~ = I(Y1, fi)? + 2tI(Yl, Z)? + C~I(Z,Z)? + t21(2, z ) i 0 = - (Yll, Y I ) I :- 2 6 z)(: + E~I(Z,2).
I ( K , K ) = I(Y,, Y,)2
(~1'.
= 2E 11 Yjl(fJo)/I2
+ €21(2.
and Y E v ( c ) , we have (Y', c') = 0 and (Y"
E'I(Z, 2).
As Yll(to) is a (nonzero) spacelike tangent vector and I ( Z , Z) is finite, it follows that there is some t: > 0 such that I ( K ,K ) Y
= YE for
this value of
E.
+ R(Y,c')cl, cl) = 0 at all points
where Y is differentiable. Taklng left-hand limits. we have (Y1(t,),cl(t,)) = 0 also at the finitely many polnts of d~scontinuitya = to < tl < , . < tk = b of Y. By contmuity, the right-hand limit lim,+,;(Y1(t),c'(t)) = 0 as well. Hence the vectors Ot*(Y1)defined in Remark 10.5 also satisfy (At,(Y').cl(t,)) = 0
By (10.4) of Remark 10.5, the index form may be calculated as k
(10.10)
0 and #(t)
b
Z(t,)) I(Y. Z) = X ( O t , (Y'), ,=o
Let p : [a, b]
2)
Thus l(Y, Z)vanishes for all
To show (2) =+ (1). we first note that since c is a tirnelike geodesic segment
Since Yl(a) = Yl(to) = 0, this simplifies to
I ( X . Y,) = -2€(Y11(to),Z(t0))
335
-+
+
(Y" + R(Y, cl)c', 2 ) dt
[0, 11 be a smooth function with d(to) = +(tl) = . . . = Q(tk) =
> 0 elsewhere. Then the vector field Z1 = d(Y" + R(Y, c')cl) is In
V$(c) and Zl(t,) = 0 for all
2.
Since it
1s
assumed that I(Y. 2)= 0 for all
Z E V&(c),we obtain from (10.10) that
> 0. Put the required vector field
@
We now turn to an important characterization of Jacobi fields in terms of
As
4
2 1
is a spacelike vector field, smooth except at the t,'s, and ~ ( f >) 0 if
+ R(Y(t). cl(t))c'(t) = 0 if t d
the index form. The same characterization holds for Riemannian spaces with
t
an identical proof. It is important to note that the index form characterizes
piecewise Jacobi field and formula (10.10) reduces to
ita). we obtain Y1'(t)
it,). Thus Y is a
smooth Jacob1 fields among all pzecewzse smooth vector fields in VoL(c) and not just among all smooth vector fields in Vg'(c). Proposition 18.13. Let c : [a, b ] -+ ( M ,9) be a timclike geodesic segment.
Then for Y E Vo-(c), the following me equivalent: (1) Y is a (srnootl-i) Jacobi field along c.
+ (2)
2,
a vector field Z2 E
V$(c) may be constructed with Z2(t,) = At, (Y') for 1. = 1,2.. . , k - 1. Then we have
(2) I (Y, Z) = 0 for all Z E V$ (c) .
Proof. First (1)
Recalling from above that (At,(Y').cl(t.)) = 0 for each
k-1
/lAl%(Y1) l2 2=1 Since all of the tangent ~ e c t o r sin this sum are spacelike. it follows that 0 = I(Y, 22) =
is immediate, since for smooth vector fields Y and
arbitrary Z the index form may be written as I(Y.Z)=-(Y1,z)/:+
i"
At,(Y1) = 0 for z = 1,2,. . . ,k (Y1l+R(Y,e')e',Z)dt.
-
1. This :hen implles that Y1 has no breaks
at the t,'s. Since for any t E [a,bj there
IS
a .lniqu3 Jacobi field along c
336
10
MORSE INDEX THEORY Oh' LORENTZIAN MANIFOLDS
10.1
with Y ( t )= v and Y 1 ( t )= tour? it follows that the Jacobi fields Y / [ti,ti+,] fit
THE TIMELIKE MORSE INDEX THEORY
Definition 10.15. (Canonical Isomorphzsm)
337
The canonzcal isomo7;iihssm
r, : TpM -+ T, (T,M) is given by
together to form a smooth Jacobi field. In view of Propositions 10.12 and 10.13, it should come as no surprise that the negative definiteness of the Lorentzian index form should be relzted to the absence of conjugate points just as the positive definiteness of the Riemannian index form is guaranteed by the nonexistence of conjugate points [Gromoll,
where, as in Definition 10.14,
Klingenberg, and Meyer (1975, p. 145)j. The negative semidefiniteness of the index form in the absence of conjugate points has been given in Hawking and Ellis (1973, Lemma 4.5.8). It has been noted in Bolts (1977. Satz 4.4.5) and Beem and Ehrlich (1979c, p. 376) that the negative definiteness of the
In particular, let v = Q,, the zero vector in the tangent space T,M. Then
index form in the absence of conjugate points follows "algebraically" from the
4,
semidefiniteness just a s in the proof of positive definiteness for the Eemannian
ro,(w) = 4,.
index form. In order to give a proof of the negative semidefiniteness of the
by identifying the vector w E TpM and the map 4,.
Lorentzian index form in the absence of conjugate points, we need to obtain
then since exp, : T'M -t
:
R
-+
TpM is the curve & ( t ) = tu: in Top(TpiZif). and we find that Thtis To,(T,M) is often canonically identified with T,M itself
If p E M and v E T p M , M . the definition of the differentlal gives
the Lorentzian analogues of several important results in Riemannian geometry [cf. Gromoll, Klingenberg, and Meyer (1975, pp. 132, 136-137, 140)]. For the purpose of constructing Jacobi fields, it is useful to introduce some notation for the identification of the tangent space T,(TpM) with TpM its
In particular, for v = Op we have
by "parallel translation in T,M." Definition 10.14. (Tangent Space T,(T,M))
exp,,
Given any p E M an
v E TpM, the tangent space T,(T,M) to the tangent space
T'M at v is give
by
: To,
(TpM) --t T,IM
since expp(O,) = p. If E = ~ o (v) p E Top(TpM) and we define
q(t) = tu, then exp,,(b) = exp,.(q5, d/dzlo), where T,(T,M) = (4,
:R
-+
T,M)
bector field for
a/&
Q :R
- T, bf hy
is the usual bask
TR defined above. Thus we obtain
where
&(t) = v i r t u . Then T,(TpM) may intuitively be identified with T,M by identifying th image of 9, in TpM with the vector w. More formally, let 1be given th usual manifold coordinate chart x ( r ) = r for all r E W ,i.e., s = id. The
,+,
4,
-. TpM and &(0) = v, we hav : TOR-+ T, (T,hf). We may then make the following definition.
B/dz is a vector field on R. Since
:R
hus exp,
o r v = I ~ T , M .This fact 1s commonly stated a s "the differential .%' the exponential map at the origin of TpJVi 1s the identity."
The following proposition shows how the differential of the exponeritial map ay be used to construct Jacobi fields.
338
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
10
Proposition 10.16. Let c : [O, b]
-, M
10.1
THE TIMELIKE MORSE INDEX T H E O W
339
be a geodesic with c(0) = p. Let
Proof. If d(t) = tz, then a = dl(l). If c is the geodesic c(t) = exp,(tv) =
w E T,M be arbitrary. Then the unique Jacobi field J aiong c with J(O) = 0
exppc+(t), we then have expp.a = cl(l). Also set w = rV-l(b) E T,M. Let
and J1(0) = w is given by
Y be the unique Jacobi field along c with Y(0) = 0 and Y1(0) = w. By Proposition 10.16, we know that Y ( t )= exppw( t ~ ~ , w )1x1 . particular, Y ( l j = J(t) = ex?,- ( t ~ i ~ ~ ( ~ ) ~ ) .
Prooj. Set v = cl(0). We may find an a : [a,b ] x (-E, E )
+M
of c : [a,b]
-+
E
expp, (rvw) = expp. (b).
> 0 so that the smooth variati
M given by a(t,s) = exp,(t(v
is defined. Since this is a variation of c whose s-parameter curves t
+ sw
+ a(t,s
are geodesics, it follows that the variation vector field of this deformation i a Jacobi field. Since a , D/dsl(,,,) = expp*(rt(v+sw)(tw)), the variation vecto
From Definition 10.17, we have ((a,b)) = (.r;'(a),r;'(b)) the Gauss Lemma
IS
= ( v , ~ )Hence .
proved if we show that (z., w) = (cl(l),Y(1)). But by
Lemma 10.9, the function f (t) = (cl(t),Y(t)) =
cyt
+ /3 for some constants
a , P E R. Since Y(0) = 0, we have O = O and j ( t ) = t f l ( 0 )= t(cf(0),Y1(0)) = t(v, w).In particular, ( c l ( l ) ,Y(1)) = f (1) = ( L , w)
%
required.
D
. a(0,s ) = c(O) for a1 field is just J ( t ) = ex?,. (rt,tw) = exp,. (t ~ ~ , w )Since
The Gauss Lemma has the following geometric consequences. The proofs of
s, we have J(0) = 0, and a calculation also gives J1(0) = w Icf. Gromoll
these corollaries, which are given along the lines of Gromoll. Klingenberg, and
Klingenberg, and Meyer (1975, p. 132)].
Meyer (1975, pp. 137-138) in Bolts (1977, pp 75-77), will be omitted. The
As in the Riemannian theory, it is now possible to prove the Gauss Eemm using Proposition 10.16. We first need to put a natural inner product ( ( on T,(T'M) using the given Lorentzian metric (
, )
, ) on TpM and the canonic
isomorphism.
((
The inner product
Corollary 10.19. Let U be a convex normal neighborhood in M, and let
+
, ) for M
q = c(1) in
U . Then if P : [0, 11
is given by
is
The basic idea of the proof is that since lJ is convex, B and c may be lifted
OF
for any a, b E T,(TpM).
and hence the lengths of
T'M
be a t a n g e ~ ~vect t
in the domain of definition of the exponential mapping and let a = T,(%) Then for any b E T,(TpM), we have ((a, b:: = (exp,. a , exp,. b),
Thus the exponential map is a "addial isometry."
< L(c) unless
to rays : [0,1] + TpM and F : [O, 11 -,T,M. with F(t) = tcl(0) Then the Gauss Lemma may be appl~edto compare P I = expp* and d = exp,. o Z
((a, b ) ) = (7,'(a),.r,-l(b))
T h e o r e m 10.18 (Gauss Lemma). Let v E
- U is any future directed timelike piecewise
srnooth curve from p to q, we have L(P) 5 L(c), and L(O) j u s t a reparametrization of c.
, )) : Tv(Xp~v)x Tv(TpAd) R
associated with the Lorentzian metric (
(10.12 )
system in Penrose (1972, p. 53).
c : [O,lj -, U be a future directed timel~kegeodesjc segment from p = c(O) to
Definition 10.17. (Inner Product on T,(T,M))
T,(T,M),
use of the Gauss Lemma here replaces the use of a synchronous coord~nate
P and the geodesic c.
An alternative formulation of Corollary 10.19 is also given in Bolts (1977. pp. 75-77) as follows. Corollary 10.20. Let v E T,M be a timelike tangent vector In the domain of definition of exp,.
Let
+
T,M be the curve +(t)= tv. Let : ;0, lj + TpM be a piecewise smooth curve with d(0) = q(0) and +(1) = d(1)
such that exp, 3w : j0,11
--$
: [0,1] +
M is a future directed nonspaceljke curve Then
340
10
10.1
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
THE TIMELIKE MORSE INDEX THEORY
341
Corollary 10.20 then implies that L(a,) = L(exppoQ) 5 L(exp, od) = L(c)
L(exp o+) 5 L(exp 04); and moreover,
for each s with Is1 5 6, and equality holds only if a, is a reparametrization of c. C provided that there is a to E (O,1] such that the component b of +'(to) perpendicular to ~+(~,)($(to)/ll$(t~)/l) satisfies expp.b
With Proposition 10.21 in hand, we may now show that the negative definiteness of the index form on V$(c) is equivalent to the assumption that c has
# 0.
no points conjugate to t = a along c in [a,b]. We are now ready to show that if the timelike geodesic segment c has no conjugate points (recall Definition 10.3), then the length of c is a local
c : la, b ]
maximum. Proposition 10.21. Let c : [a,b]
M be a future directed timelike geo-
4
desic segment with no conjugate points to t = a and let a : [a, b] x (-E,
6)
-M
be any proper piecewise smooth variation of c. Then there exists a constant
6 > 0 such that the neighboring curves a, : [a, b] -+ M given by cr,(t) = a(t, s) satisfy L(cr,) 5 L(c) for ail s with Is/ < 6. Also L(a,) < L(c) ~f0 < 1st < 6 unless the curve a, is a reparmetrizat~onof c. Prooj. Reparametrize cr to a variation n : Lemma 10.7, there is an
€1
[O.P] x
(-c,~)
M . By
> 0 such that all the neigllboring curves a,
of a 1 [0,/3] x (-tl,el) are tirnelike. We may then restrict our attention to
I
[O,P] x ( - ~ 1 , ~ 1 ) . Set p = c(0) and let
4 : [O,$1
TpM be the ray +(t) = tcl(0). Since
4
c has no conjugate points, expp has maximal rank at p(t) E TpM for each
t
E [O, p ] . Thus by the inverse function theorem, there is a neighborhood of
+(t) in TpM which is mapped by exp, diffeomorphically onto a neighborhood of c(t) in M. Since +([O,
T h e o r e m 10.22. For a given future directed tirnelike geodesic segment
PI) is compact in TpM, we can find a finite partition = P and a neighborhood U, > 6([t3,t,+l]) in T,M for
0 = to < tl < - - . < t k each J = 0,1,. . . ,k - 1 such that h, = exp
p
I
: U,4
M is a diffeomorphism
M , the following are equiva!ent:
(1) The geodesic segment c has no conjugate points to t = a in (a,b ] . (2) The index form I : V/(c) x V$(c) -,R is negative definite.
Proof. ( 1 ) 3 (2) Suppose Y # 0 in V$(c) and I(Y. Y) > 0. Let cr(t. s) = exp,(,)(sY(t)) be the canonical variation associated to Y . Then we have
> 0 , so that for all
# 0 sufficiently small, L(cY,)> L(c). Bu&this then contradicts Proposition 10.21. Hence if Y # 0, Y) 5 0, so that the index form 1s negative semidefinlte. then I (Y. L1(0) = 0, L1'(0) = I ( Y . Y )
s
It remains to show that if Y E V&(c) and I(Y, Y) = 0. then Y = 0. To this end, let Z E V$(c) be arbitrary. By Remark 10.2, we have Y - t Z E V$(c) for all t E
R. Hence I(Y - tZ, Y - t Z ) 5 0 for all t
E
W by the negative
semidefiniteness of the index form just established. Since I(Y - tZ, Y - t Z ) = -2tI(Y, Z)
+ t21(Z,Z), it follows that I(Y,Z ) = 0.
As Z was arbitrary. this
then implies by Proposition 10.13-(2) that Y is a Jacobi field. Since c has no conjugate points, Y = 0. (2) + (1) Suppose Y is a Jacobi field with Y ( a ) = Y(tl) = 0, a < tl 5 b. By Corollary 10.10, Y E VL(c). Extend V to a nontrivial vector field Z E V$(c) by setting
u3
of U3 onto its image. By continuity, we may then find a constant 6, > 0 such that a([t,,t,+l]x (-6,, 6,))
exp,(U,) for each j = 0,1, . . . :li - 1. Set
6 = min(61,62,. . . ,6k). Then we may define a pierewise smooth map Q : [0,/3] x (-6.6)
Then I ( Z , 2 ) = I(Z. 2): -+
T,M
+ I(Z, Z)& =
-
(Z', z)/: + 0 = 0.
A consequence of Theorem 10.22 that is crucial to the proof of the Timelike
with exgp o@ = N and @(t.0) = d(t) for all t E [O.41 a s follows. Given (t, s) E
Morse Index Theorem is the following maximality pro?erty of Jacobi fields with
IO,@]x [-6,6]. choose j with t E [t,,t,+l], and set @(t,s) = (h,)-"a(t,
respect to the index form for timelike geodesic segments without conjugate
s)).
342
10
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
10.1
THE TIMELIKE MORSE INDEX THEORY
points. This result is dual to the minimality of Jacobi fieids with respect to the index form for geodesics without conjugate points in Riemannian manifolds.
Theorem 10.23 (Maximality of Jacobi Fields). Let c : [a, b] + M be a timelike geodesic segment with no conjugate points to t = a, and let J E VL(c) be any Jacobi field. Then for any vector field Y E VL(c) with Y # J , and (10.13)
Y(u) = J ( a )
and
Indo(c) = iuWdii11A . A is a vector subspace of v$(c) and I / .A x A is posltlve sem~definite). Also let Jt(c) denote the W-vector space of sclooth Jacobi fields Y along c with Y(a) = Y ( t )= 0. a < t 5 b. Mre now relate Ind(c) to Indo(c) and establish their finiteness in Proposition
Y(b) = J(b),
10.25. The maximality of Jacobi fields with respect to the index form for we have
timelike geodesics without conjugate points plays a key role in the proof of this proposition. The basic idcas involved in the proof of Proposition 10.25 and Theorem 10.27 are due to Marston Morse (1934)
Proof. The vector field W = J - Y E V&(c) by (10.13) and W Y
# J by hypothesis.
#
0 as
By Theorem 10.22. we thus have I(W, W) < 0. Now
Proposition 10.25. Let c : [a,b ] -+ M be a future directed timellke geodesic segment. Then Ind(c) and Indo(c) are finite and
calculating i(W, W) we obtain I(W, W) = I(Y, Y) = I(Y,Y)
-
+ 2 (J',Y)/! - (J', J)J;.
Since Y(a) = J ( a ) and Y(b) = J(b), we have (J',Y)I: = (J'.J)/:. Thus
+ = I(Y,Y) + (J', J)I:
Indo(c) = Ind(c)
(10.15)
2I(J, Y) + I ( J , J )
+ dim Jb(c).
Proof. The proof follows the usual method of dpproxirnating V$(c) by finite-dimensional ~ e c t o rspaces of piecewire srriooth Jacob! fields. To this end, dioose a finite partition a = to
< tl < . . . < t k
= b so that c 1
[t,,t,+l]
has no conjugate poirlts to t = t , for each z with 0 5 z 5 k - 1. Let J ( t , ) denote
I(W,W) = i ( Y , Y ) 2 { J 1 ,~ ) l b-, (J',J)/: = I(Y,Y) - I ( J , J ) .
the subspace of Vj'(c) consisting of all Y E V&(c) suck1 that Y / [t,,t,+l] is a Jacobi field for each z with 0 5
As i(W, W) < 0, this establishes (10.14).
points for each
2,
z
< k - 1. Since c / [t,, t,+l] has no conjugate
it follows that dim J{t,) = (n - l ) ( k - 1).
Now that Theorem 10.23 is obtained, a Morse Index Theorem may be estab-
Mre now define the approximation 4 : ~ & ( c 4 ) J{tz) of V,'-(c) by J { t , ) as
lished. First we must define the ~ndexof any timelike geodeslc c : [a, b] --+ M .
follows. For X E V&(c) let (pX) I [t,,t,+l] be the unique Jacobi field along
The definition given makm sense because Vo-(c) is a vector space. Definition 10.24. (Indez and Extended Indez)
The index Ind(cf and M are defined the extended zndez Indo(c) of the timelike geodesic c : [a,b ] -+
by
c / [tl,tB+l]with (q5X)(tE)= X(tb) and (4X)(t,+l) = X(t,,l) for each z with 0 ( i 5 k - 1. Thus X is approximated by a piecewise smooth Jacob] field OX such that (dX)(tt) = X ( t t ) at each t,, 0 < 5 k This approximat~onis useful in t h ~ scontext a s 9 is iqdex nondecr~asing KIore explicitly,
Q
1 .J{t,)
is just the identity map, so that I ( X , X) = I(4X. q X ) ~f X E J{t,). On the
Ind(c) = lub(dim A : A is a vector subspace of ~ & ( c ) and Il A x A is positive definite),
other hand, if X $ J j t , ) then the inequality
344
10
10.1 THE TIMELIKE MORSE INDEX THEORY
MORSE INDEX THEORY ON LORENTZIAX MANIFOLDS
may be obtained by applying Theorem 10.23 to each subinterval It,, t,+l] and
a = so < s l
summing.
and c /
We estabiish the following sublemma which shows the finiteness of Inda(c)
[s,,
.
= b so that {sl,sa, .. .,sm-3}n (t1,t2,. . . t r - l ) = Q
s,+lj has no conjugate points for each i with 0 5 z
I ( X , X ) by (10.16).
Thus $X = 0 implies that I ( X , X) < 0 which contradicts the assumption that
I ] A x A is positive semidefinite Thus if +X = 0: then X = 0. Now that we know that
c$
is injective on subspaces of V$(C) on which I is
positive semidefinite, we may prove the sublemma. For if A is a subspace of V/(c) on which 11 A x A is positive semidefinite, inequality (10.16) implies that the index form of J ( t , ) is positive semidefinite on the subspace d(A) of J{t,). Since q5 is injective on such a subspace from above, dim A = dim $ ( A ) . Hence Indb(c) 2 Indo(c). However, since J(t,) is a vector subspace of q ( c ) , we have Ind&(c) 5 Indo(c). Thus Indo(c) = Indb(c). The same argument shows Ind(c) = Ind(cl). 3 To conclude the proof of Proposition 10.25, we must establish the equality Indo (c) = Ind(c) + dim Jb(c). To this end, we choose a second partition
Applying the proof of Sublemma 10.26 to the partition {s,) of [a!b ] , we may choose a vector subspace
BA
of J{s,) with Il
Bh x
BA positive semidefinite
and with Indo(c) = dim B;. Since dim BA = Indo(c) < m, it follows that Jb(c) is a vector subspace of
BA. By
(10.17) and the proof of Sublemma 10.26, the
map
bjBA: BA
- J{t,}
is injective. Thus if we set Bo = &(B&),we have d!m Bo = dimBA = Indo(c). Since Bo is a finite-dimensional vector space and Jb(c) is a vector subspace, we may find a vector subspace B of Bo such that Bo = B 9Jb(c), where 9 denotes the direct sum of vector spaces We claim now that TI B x B is positive definite
By construction. we
Bh x B; is positive semidefinite. Also if 0 # Z E B and we X E Bb,then X 4 .ib(c). (For if X E J b ( c ) . we ) which is impossible since have b(X) = X so that Z = 4(X) = X E J ~ ( c also, B r Jb(c) = {O) by construction.) Hence I ( Z . Z ) = I($X, 4X) > I ( X , X ) , the last inequality by (10.17). Thus as I Bh x B& is positive semidefinite, we obtain I(Z, 2 ) > I ( X , X ) 2 0 so that I ( Z , Z ) > 0 . This shows that know that Il
represent Z = $(X)with
I1 B x B is positive definite. With the notation of SubIernma 10.26, we then have Indl(c) 2 dim B.
10
346
10.1
MORSE IXDEX THEORY ON LORENTZIAN MANIFOLDS
for each t E [a,b ] by
From the direct sum decomposition Bo = B @ Jb(c). we obtain the equality Indo(c) = dim Bo = dim B and we also know that Indl(c)
T H E TIMELIKE MORSE INDEX THEOXY
+ dim Jb(c), i(Y) (s) =
2 dim B. Thus the proof of Proposition 10.25
will be complete if we show that Indl(c) 5 dirn B. J(t,) is a vector subspace with To establish this inequality, suppose B' SO
for a 5 s 5 t , for t 5 s 5 b.
Evidently dirn Jt(c) = dirni(Jt(c)) for any t 6 (a.b ] Recall that Indo(c) is finite by Proposition 10.25 Thus to show that c has
I j B' x 3' positive definite and dim B' = Indl(c). Suppose that dim B' > dim B . Then I is positive semidefinite on B' @ &(c)
0
only finitely many conjugate points, it suffices to prove that if Itl, t2, . .tk) is any set of pairwise distinct conjugate points t o t = a along c, then k 5 Indo(c).
that dim(B1@Jb(c)) 5
Indo(c). On the other hand, since dim B1 > dim B we obtain
To this end, set Ag = z(Jt, ( c ) )for each j = 1,2,. . . , k and A = A1 B . . . @ Ak. Then A is a vector subspace of V&(c), and ilecon~posingZ E A as Z =
c:,~ X3ZJ with A3 E W,Zg
desic segments. The proof we give here 1s modeled on the proof for Riemannian
A3, we obtain I ( Z , 2 ) = C,,lA,AII(Z,, 21). I ( Z 3 .Zl)Zi = -(&I, ZJ)l; But if t3 _< t l , we obtain I(Z2, Zl)= I(Zj. 21): I(0, &);, = 0 using (10.21, Zl 6 At, and Z,(a) = ZJ(tl) = 0. Hence i ( Z .2 ) = 0 from the symmetry of the index form. Thus I I A x A is positive semidefinite.
spaces given in Gromoll. Klingenberg, and bfeyer (1975, pp. 150-152).
Hence
This contradiction shows that dimB1 5 dimB, whence Indl(c) 5 dim B as required. 13
We are now in a position t o prove a Morse Index Theorem for tirnelike geo-
6
+
+
Theorem 10.27 (Timelike Morse Index Theorem). Let c : [a, b ] 4
A4 be a timelike geodesic segment, and for each t E [a, b] let Jt(c) denote the
as required. Therefore c : [a, b ] -+ A4 has only finitely many conjugate points
R-vector space of smooth Jacobi fields Y along c with Y (a) = Y (t) = 0. Then
in (a.b], which we denote by tl < tz < ... < t , Except for t E (tl,ta,
c has only finitely many conjugate points, and the index Ind(e) and extended
we have dim Jt(c) = 0, and thus
index Indo(c) of the index form I
:
V&(c) x V ~ ( C --+)
W are given by the
Since Jndo(c) = Ind(c) + dim Jb(c) from Proposition 10 25, !t is enough to
formulas (10.18)
, t,),
CtE(a,b dim .7t(r) i.; a finite sum
establish the equality dim Jt(c)
Ind(c) = tE(a,b)
Indo(c) =
dim Jt(c)
and (10.19)
Indo(c) =
dim Jt(c), t~(a,bj
respectively.
*
to prove Theorem 10.27. Let Zdenote the integers with the discrete topology and define f , fo : ( a ,b]
-+
Z by f (t) = ind(c 1 [a, t ] ) arid fo(t) = Indo(c / [a, t ] ) .
We now show that (10.20) holds if we establish the left ~oritinuityof f arid
xCtE(a,bj
Proof. We first show th2t dim Jt(c) is a finite sum. We know that dim Jt(c) 2 1 if arid only if c(t) is a conjugate point of t = a along c. We may also define embeddings i : Jt(c)
-
I#(.)
the right continuity of So. liere we mean that iirntnrtf ( t n ) Emtnlt fo(tn) = fo(t). Using (10.15) it follows that f (tj- fo(t)
=
f ( t ) a11d
= - aim Jt(c) =
0 if t $! (ti, t z , . . .,t,). Assuming we have shown f is left continuous and f U is right cont~nuous,we also have f (t3+l)= fO(t3)f ~ each r 3 = 1 , 2 , . . . ,T - 1.
348
10
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
10.1
THE TIMELIKE MORSE INDEX THEORY
349
-
points. Choose a partition a = to < tl < ..- < tk-l < t k = t (unrelated to the above enumeration of conjugate points) such that It, - tgtll < S for
Thus
z = 0,1,2,. . . , k - 1. Let J C [a, b] be an open interval containing -teach with It - tk-11 < E for ail t E J . FOTeach t E J , let J"(c,) denote the
finite-dimensional R-vector subspace of V&(c/ [a,t]) consisting of all vector fields
Y
E 'ybi(cj[a,t]) such that Yl[t,,t,+i] for each j = 0.1.. . . k - 2
By Theorem 10.22 the index form is negative definite if c has no conjugate
and Y I [tk-1, t] are Jacobi fields. By Sublemma 10.26, f ( t ) is the index of I
points, so that f ( t ) = 0 for all t < t l . Hence f(tl) = 0 since f is left
restricted to J"(ct) x
continuous. Thus
CtE(a,bl dim Jc(c) = fo(tr).
Since fo is constant on [t,, b],
y(ct) and
fo(t) is the extended index of 1 restricted to
T(ct) x T(ct). Now let E = N(c(t1)) x N(c(t2)) x
we have fo(t,) = jo(b). Hence
. . - x N ( ~ ( t k - ~ )The ) set E is ciosed
since each N(c(t,)) = (v E T,(t,)M : (v,cl(t,)) = 0) is a closed set of spacelike tangent vectors. We may define a Euclidean product metric (( , )) : E x E -+ R by ( ( v , ~ ) )= ~ f i T , ' ( v w,) , , for v = (vi, 212, - ., ,uk-1). w = which establishes (20.20). We have thus reduced the proof of the Morse Index Theorem to showing
(ulw2 , ,... ,wk-l)E E. Then by Remark i0.2, S = {v E E : compact.
lltrll
= 1) is
that f is left continuous and fo is right continuous. First note that f and fo
If Y E T(ct), then Y ( a ) = Y(t) = 0 by definition. Also since c j [t,,t,+l]
are nondxreasing (i.e., f ( t )1 f (8) if t 2 s). For suppose we fix t l , t2 E (a, b]
has no conjugate points, for any v E N(c(t,)) and w f N ( c ( ~ , + ~there ) ) is a
with tl 5 tz. Let cl = c 1 [a,tl] and
unique Jacobi field Y along c with Y(t,) = v and Y(t,+l) = ZL. Since (Y,cl)l,
6.2
= cl [a,tz]. We then have an R-linear
is an aEne function of t and (v. cf(t,)) = (w, ~ ' ( t , + ~ = ) ) 0. it follows that
embedding i : v$(c~) + ~ d ( c 2 given ) by
(Y, c') 1, = 0 for all t. Hence the map
4t : ?(ct) This map has the property that I(Y, Y) = I(i(Y), i(Y)), where the indexes are calculated with respect to cl and c2, respectively. Thus if A C @(cl) is a vec-
+E
defined for t E J by
tor subspace on which the index form of cl is positive (semi) definite, then i ( A ) is a vector subspace of V&(cz) on which the index form of c2 is positive (serni) definite and dim A = dim i ( A ) . Thus f (tl) = Ind(c I
[a,tl]) < Ind(c I [O,tz]) =
f jt2) and similarly fo(tr) 5 Soft2). Thus fo and f are nondecreasing. To obtain the continuity properties of f and fo, we fix an arbitrary
E
( a ,b] and study the continuity off and fo at ;using the same approximation techniques as in the proof of Sublemma 10.26. Since c([a, b ] ) is a compact
subset of M , there is a constant 5 > 0 such that for any sl,sz E la, b] with Is1 - szl
< 6 and sl 5
82,
the geodesic segment c 1 [sl,s2]has no conjugate
is an isomorphism. For each t t J we also have a quadratic form Qt : E x E -4 R given by QQt (u, v) = I(#;' (u), #tf(v)). Sublemma 10.26 implies that jo(t) the extended index of the quadratic form &t on E x E and f ( t ) is the index ofQtonExEforeacht~J. Each Qt may then be used to define a map Q . E x E x J -+ W given by Q(u.v,t) = Qt(u, v). We want to show that Q is continuous in order to prove that f and fo have the d ~ i r e dcontinuity properties. To this end, let
10
350
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
B = ( Y / [a,tkPl] : Y E
4 :B
-+
Z(c,)).
T H E TIMELIKE MO=E
10.1
Then B is isomorphic to E via the mapping
E given by
INDEX THEORY
351
that Q,,, / A, x A, is positive semidefinite. Let (a' (n),a2(n),. . . . ak(n)) be an orthonormal basis of A, for each n. Thus al(n), uz(n), . . . . ak(n) are contained in the compact subset S of E. By the compactness of S, we may assume
a,(n)
-+
a, 6
S for
that the vectors
Thus
(51,
each j
By continuity of the innel product, it follows
ar, . . . ,a,) form an orthonormal subset of S. Let A =
span{al, az.. . .,ak), which is thus a k-dimensional subspace of E. Given u E
A. we may write u ~ ( n -)-I u as n
-+
=
k
A3a3. Let u(n) =
k C,=l Xla,(n).
Obviously
- R.
oo. Thus using the continuity of Q : E x E x J
me
obtain where
are the Jacobi fields along c given by and YVat
QZ(u,u)= lim Q(u(n),u(n).s,)
20
11-00
I
since Q,- 1 A, x A, is positive semidefinite for each n. Hence Qt A x A is positive semidefinite. Thus f o ( t )
2 dim A = k
as required.
O
and
YV,t= $';'(v)
We cnnclude this section with an application of the Timelike Morse Index
I [tk-1, t l .
Theorem to the structure of the cut locus of future one-connected, globally Since the map (u, v) -+ I(p-'(u). $-'(v)) from E x E
4
W is a bilinear form,
hyperbolic space-times [cf. Beem and Ehrlich (1979c, Section 8)j.
the map ( u , ~ , t+ ) I($-1(u),6-1(v)) is certainly continuous. By Proposition 10.16, the map (u,v,t) 4 (Xu,t,~:,,)/:k-l is continuous. This establishes the continuity of Q : E x E x 3 -+
W.
We are finally ready t o show that fo is right continuous and f is left contlnuous at the arbitrary
t€
(a, b].
Since Sublemma 10.26 implies that f -+
R
directed timelike curves from p to q are homotoplc through smooth future directed timelike curves with fixed endpoints p and q.
f(t)and
This concept, a Lorentzian analogue for simple connectivity, has been stud-
is contirnions
ied in Avez (1963), Smith (1960a), and Flaherty (1975a, p. 395). The van-
is finite-valued, we may choose a subspace A of E with dimA = Q ( u , u , t ) > 0 for all u E A, u f 0 Since Q : E x E x .J
Definition 10.28. (Future One-Connected Space-Tzme) A space-time ( M , g ) is said to be juizlkre one-connected if for all p,q E M, any two future
and S1 = S n A = (u E A : l/ull = 1) is compact, there is a neighborhood 30
ishing of the Lorentzian fundamental group implies that ( M , g )is future orie-
of Fin J such that Q(zl.u, t ) > 0 for all t E Jo and u E S1. Hence Qt 1 A x A
connected. However, the simple connectivity of M as a topological space does
is positive definite for dl t E Jo. Thus f(t)
not imply that ( M ,g j is future one-connected, as the following example of Ge-
nondecreasing, it follows that f ( t )= f
2 f (r)for all t f Jo. Since f is (z)for all t E Jo with t 5 z Hence f
1s
left continuous.
roch shows. Consider B3 with coordinates (x, y, t ) and the Lorentzian metric ds2 = -dt2
It remains to show the right continuity of fo at Let (s,) be a sequence Since Jo is nondecreasing and integerIn J with s, > t for all n and s, valued, we may suppose that fo(s,) = k for all n. By the monotonicity of -+
fo, we then have f o ( t ) 5 k. It thus rernains to &how that f o f t )
> k.
To
accomplish this, choose for each n a k-dimensional subspace A, of E such
+ dz2 + dy2. Let T = ((z, 0,O)
E W3 : x
7: 0) and set M
= B3- T
with the induced Lorentzian metric from (W3, ds2) Then M is simply connected. On the other hand, let p = (2,0, -1) and q = (2.0,1). Then p and q may be joined by future directed timelike curves yl and sides of the positive z axis (cf. Figure 10.1) But
-/I
lying on opposite and 72 are not homotopic
through fcture directed timelike curves starting at p and ending at, q since such
352
10
10.1 THE TIMELIKE MORSE INDEX THEORY
MORSE INDEX THEORY ON LORENTZISN MANIFOLDS
353
proposition 10.29. Let ( M , g ) be future one-connected and globally hyperbolic. F i x p 6 M and suppose that every future directed timelike geodesic segment starting at p has index zero or index greater than or equal to two. Given q E I+@)such that q is not conjugate to p along any future directed timelike geodesic from p to q, there is exact1.y one future directed timelike geodesic from p to q of index zero, namely, the unlque maximal geodeslc from p to q. We are now ready to prove the Lorentzian analogue for globally hyperbolic, future one-connected space-times of a theorem of Cheeger and Ebin (1975. Theorem 5.11) on the cut locus of a complete Riemannian manifold which generalized a theorem of Crittenden (1962) for simply connected Lie groups w ~ t hbi-invariant Riemannian metrics. The global hyperbolicity is used in Theorem 10.30 to guarantee the existence of maximal geodesic segments joining chronologically related points. Here the order of a conjugate point t = to to
p along the t~rnelikegeodesic y with $0) = p is defined as dim J,, (y) lrecali Definition 10.241. T h e o r e m 10.30. Let (M,g ) be future one-connected and globally hyperbolic. Suppose that for p € A4,the first future conjugate point ofp along every timelike geodesic y with y(0) = p is of order tu-o or greater. Then the future timelike cut locus of p and the locus of first future timelike co~~jugate p o ~ n t of s
FIGURE 10.1. A space-time M = W3 - T which is simply connected but not future one-connected is shown. The future directed timelike curves yl and 7 2 from p to q are not homotopic through timelike curves with endpoints p and q.
p coincide. Thus all future timelike geodesics from p maximize up to the first
future conjugate point.
Proof. All geodesics will be unit speed and future timelike during the course of this proof. It suffices to show that ~f y : [O, t ] -+ M wlth y(0) = p has index zero and y(t) is not conjugate to p along y 1 10,t ', then y 1s maximal. Since
a homotopy would have to go around the point (0. 0 , 0), which would introduce
the set of singula points of expp is closed, it follows from Theorem 10.27 that
spacelike curves.
there exist
Note that if (M,g) is future one-connected. then the path space of smooth timelike curves from p to q is connected. Thus Lemma 4.11-(2) of Cheeger and Ebin (1975, p. 85) and the standard path space Morse theory [cf. Everson and Talbot (1976), Uhlenbeck (1975), Woodhouse (1976)]imply the followmg proposition.
> 0 such that if a which is not
a critical value of F, the topological space F - I ( - m , b) 1s
-
y, = y, and hence 7, is maximal. Thus 7 is maximal as a limit of maximal
homotopically equivalent to the space F-"-w.
geodesics. t
one cell of dimension equal to the index of the cor~espondingcritical polnt 1s
a ) with celis adjoined, where
adjoined for each critical point of F with critical talue in ( a ,b).
10.2
The Timelike Path Space of a Globally Hyperbolic
Space-time
We will show in this section that the Lorentzian arc length functional is a homotopic Morse function for C(,,,) provided that p and q are nonconjugate
directed timelike curves joining two chronologically related points in a globally
along any nonspacelike geodesic. This result is analogous to Morse's result [cf. Milnor (1963, Theorems 16.3 and 17.3)) for complete Riemannian mani-
hyperbolic space-tirne following Uhlenbeck (1975) [cf. also Woodhouse (1976)j.
folds. Namely, if p and q are any two dlstinct points not conjugate along any
Both of these treatmerits are modeled on Milnor's exposition (1963,pp. 8&92)
geodesic, then the space R(,,,) of piecewise smooth curves from p to q has the
In this section we discuss the Morse theory of the path space of future
of the Morse theory for the path space of a complete Riemannian manifold in
homotopy type of a countable CW-complex with a cell of dimension X for each
which the full path space is approximated by piecewise smooth geodesics. A
geodesic from p to q of index A.
different approach has been given to the Morse theory of nonspaeelike curves
Given that p and q must be nonconjugate along any geodesic for L to be
in globally hyperbolic space-time by Everson and Talbot (1976, 1978). They
a Morse function, it is of interest to know that such pairs of points exist. As
use a result of Clarke (1970) that any four-dimensional globally hyperbolic
for Riemannian spaces, conjugate points in an arbitrary Lorentzian manifold
space-time may be isometrically embedded in a high-dimensional hlinkowski
may be viewed as singularities of the differentia! of the exponential mapping.
space-time to give a Wilbert manifold structure to a subclass of timelike curves
Hence Sard's Theorem [cf. Hirsch (1976, p 69)] may be applied. Here a subset
in M .
X of a manifold is said to have measure zevo if for every chart (li.4): the set O(U n X ) E R" has Lebesgue measure zero in Wn,n = dim A 4 Also a subset
We now turn to Uhlenbeck's treatment of the Morse theory of the path space of piecewise smooth timelike curves joining points p ( Z )1') - g(Z1(z'),B(z))
=
[T ( 2 ' ( ~ ) , Z ( z- jg )( A] '~( z ) , Z ( z l ) )- 3(A1(z),X'(z)) - 3(A1(z1),A(zj).
Theorem 10.69. Let following are equivalent.
O : [a,b ] -,M
be a null geodesic segment Then the
(1) The segment ,t3 has no conjugate points to t = a in (a,b ] .
( 2 ) f(W,W ) 4 0 for dl W E Xo(P), W # [pi:.
388
10
MORSE INDEX TWEGRY ON LOFtENTZIAN MANIFOLDS
10.3
Proof. We have already shown ( 1 ) + (2) in the proof of Proposition 10.68.
To show (2)
+ ( I ) , we suppose @(a)is conjugate
to P(to) with O < t o 5 b
and produce a nontrivial vector class W E X o ( p ) with?(W, W )2 0. If ,/3(a) is conjugate to P(tof, we know that there is a nontrivial Jaeobi cl=
Z E X(P)
with
T a E NULL MORSE INDEX THEORY
from @(a)to P(b) provided that t = a 1s conjugate to some to E ( a ,6) along p. We thus give a separate proof of this result m Theorem 10.72 [cf. Hawking
and Ellis (1973, pp. 115-116), Bolts (1977, pp. 117-121); It
Now let a : jO. 1) x
z(t)
a f (Y,Y).
(10.64)
Proof. This may be established just as in Theorem 10.23 using (10.45) and
Theorem 10.69.
U
Now f (t) = g(V. T)l(t,o)is a piecewise smooth function with f ( 0 ) = f ( 1 ) = 0. Hence if f ( t )
Since the canonical variation (10.5) of Section 10.1 applied to a vector fie
W E V'(@ is not necessarily an admissible variation of fi in the sense of D inition 10.58, Theorem 10.69 do- not imply the existence of a timelike c
Then
# 0 for some t
6 10,11, there exists a to E (O,1) with f l ( t o ) > 0.
390
10
10.3
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
in contradiction to (10.65). Hence we obtain as a first necessary condition for
THE NULL MORSE INDEX THEORY
we obtain
a to be an admissible deformation of ,6 that
for all t E [0,I]. Thus V E V-(0). Consequently
This calculation yields the following lemma. for all t E [O, I]. It then follows from (10.67) that the neighboring curves a, of
Lemma 10.11. A sufficient condition for the plecewise smooth proper vari[0,1] x ( - e , ~ )-+ A4 of the null geodes~cD [0,I] -+ M to he an CL.
the variation will be timelike provided that the variation vector field satisfies
ation
(10.66), (10.67). and the condition that d 2 / d s 2 ( g ( T ,T))l(t,O) < -c < 0 for all
adrnlsslble varlatlon
t
E (0:1). As above,
a11 s
[I
e , the curves cx,(t) = a(t s ) are tlrnellke for s j O] for
# 0 suficientlysmall is that the vanation vector field V ( t )= ~ ~ . , d / d $ l ( ~ , ~ )
satisfies the condltlons (10.68) Hence
(10.69)
g ( ~ ( t ) , p ' (= t ) 0) d - ( g ( T T ) ) / ( t , O=) 0 ds
for all t E {O,I]: for
f
E [O,l:; and
for all t E (O,1) at which V is smooth. We are now ready to prove the desired result [cf. Hawking and Ellis (1973, p. 115)) Using the identities Theorem 10.72. Let
P : [0,1]-+M be a null geodesic. If ,!3(to)is conju-
gate to F(0) along /3 for some to E (0. I ) , then there is a timelike curve from
a(of tool-1). Proof. We uiill suppose that to > 0 is the first conjugate point of B(0) along
<
and
ValatTllt), = 'ValatP1lt= 0 ,
0. It is enough to show that for some t2 wlth to < ta 1. there is a future directed timelike curve from $(O) to $(t2).For then we have B(O) < B(tz)<
392
10
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
P(1) whence P(O) < P ( 1 )
10.3
Thus there exists a timelike curve from B(0) to
Because W is a Jacobi class, we obtain
P(1). Since P(to) is conjugate to B(0) dong ,D, there exists a smooth nontrivial 3acobi class W E X(3) with W ( 0 ) = {fl(O)]and W ( t o ) = [$'(to)]. W-e may write
o = 3 (w", a(wpl)pt,%) = fIi
+ 2ft3(w,@-) + f g ( W , @) + f 3 (R(*,pl)P'.
= f"+2f1?j(lV'.*)+ But since g($, @) =
where @I is a smooth -vector class along /? with g(%,
I&') = 1 and f : [O, I ] --+
B
a smooth function. Since to is the first conjugate point along P, f (0) = f ( t o ) =
0 , and changing t%' to -T@
if necessary, we may assume that f ( t ) > 0 for
all t E (0,to). Since W is a nontrivial Jacobi class and W(to)= [$'/(to)]. we have Url(to) # [P1(to)].Thus from the formula W1(to) = fl(to)I&'(to) +
THE NULL lvIORSE INDEX THEORY
fh.
a(g(@, I@))'
= $ ( l ) '= 0, we obtain the formllla
f" =
-fh. Returning to consideration of the vector class 2, first note that by choice and z ( t l ) = [P1(tl)].In view of the constants a and b, we have z ( 0 ) = [P1(0)] --/I -of formula (10.70), we wish to have g ( Z , Z + R ( Z , P 1 ) P 1> ) 0 also. Setting
r ( t ) = b(eat - 1) + f ( t ) remembering that 7j(lvi/', I% = '0. ) and dderentiating ij(Z,Z1' + x ( Z ,pl)pl) = r(rU+ r h ) = r[beat(n2+ h ) - hh + fl' + fh].
yields
Since f" = -f h. we obtain
f (to)@'(to) = f'(to)l.i/(to), we obtain fl(to) # 0. Hence we may choose tl E (to. 11 such that W ( t )# $'(t)] and f (f) < O for all t E (to, tl]. With tl as above, the idea of the proof 1s now to show that there exists a t2 E (to, t l ] such that there is an admissible proper variation cu : [0,t2]x (-e, €1 M of ,B I [O, tz]. Then the neighboring curves a, of the variation a will be timelike curves from P ( O ) to O ( t 2 ) for s $ 0 . This will then imply that F ( O ) < P(1) as required. To this end, we want to deform W 1 [0,t l ] to a vector class Z 1 [I). t2] - -11 with 7.j ( 2. Z ot)@l) > 0 so that if Z E V L ( P )is a n appropriate lift of Z E X($ / [0,t z ] )and a is a variation of ,B 1 [O: tz] with variation vector field Z , then conditions (10.69) and (10.70) of Lemma 10.71 will be satisfied. Consider a vector class of the form
-
+a@,
with b = -f (tl)(eatl- I)-'
E R and a
> O in R chosen such that
+
Noiv b = -f (tl)(eatl- 1) > O as f ( t l ) < 0. so the expression b(eat[a2 --I/ -h ( t ) ]- h ( t ) ) > 0 for all t E [0,t l ] . Thus g ( Z , Z R ( Z , ,LY)O1)It> O provided
+
r ( t ) > 0. Since f (t)> 0 for t E (O,to), we have r(t) > 0 for t E (0,to]. B y continuity, there is some t z > to with r ( t ) > 0 for t E [to, t z ) and r ( t z ) = 0. If t2
> t l r then in fact t 2 = t l since r ( t l ) = 0 by construction, and we let tz = tl
below. If tz
< tl, then the vector class Z / 10,tz] will satisfy Z ( t a ) = [Pr(t2)] --,I +R ( Z .P')F1)lt > 0 for all t E (0,t a ) We now
since r ( t z )= 0 and also g ( Z , Z work with ,B 1 [O, t 2 ] .
- z.
Let 2 E V L ( PI [0,t z ] )satisfy a ( 2 ) = Since z(0)= [,LY(O)] and Z ( t 2 ) = [ p t ( t 2 ) ]we , have Z(0) = pP1(0)and Z(t2) = XP1(t-,)for some constants p. X E R. Set Z = 2 - p,8' + [ ( j i - X)/t2]t01.Then Z ( 0 ) = Z ( t a )= 0 and 7i(Z) = 2. Consequently
(a2+ glb(h(t) : t E [O, t l ] ) )> 0, for all t E ( 0 ,t 2 ) . Chowe a constant
where
h(;) = g(*
+ K(I@,B')P', .1,
(10.73)
e
E
> O SO that
< glb {g ( Z 1 ' ( t + ) R ( z ( t ) . P 1 ( t ) ) P ' f tZ ) .( t ) ) : t G
[?, 21)
MORSE INDEX THEORY ON LORENTZIAN MANIFOLDS
10
394
10.3
which is possible in view of (10.72). Now define a function p : [O: tP]-+ R by -€t
o i t i 2 ,
€(t-2)
h4
View more...
Comments
