general relativity. j. ehlers honorary volume

J¨urgen Ehlers In the 1950s the mathematical department of Hamburg University, with its stars Artin, Blaschke, Collatz, K¨ahler, Peterson, Sperner and Witt had a strong drawing power for J¨urgen Ehlers, student of mathematics and physics. Since he had impressed his teachers he could well have embarked on a dis- tinguished career in mathematics had it not been for Pascual Jordan and – I suspect – Hermann Weyl’s Space–Time–Matter. Jordan had just published his book “Schwerkraft und Weltall” which was a text on Einstein’s theory of gravitation, developing his theory of a variable gravitational “constant”. Only the rudiments of this theory had been for- mulated and Jordan, overburdened with countless extraneous commitments, was eager to find collaborators to develop his theory. This opportunity to break new ground in physics enticed J¨urgen Ehlers and Wolfgang Kundt to help Jordan with his problems, and their work was acknowledged in the 1955 second edition of Jordan’s book. It didn’t take J¨urgen, who always was a systematic thinker, long to re- alize that not only Jordan’s generalization but also Einstein’s theory itself needed a lot more work. This impression was well described by Kurt Goedel in 1955 in a letter to Carl Seelig: “My own work in relativity theory refers to the pure gravitational theory of 1916 of which I believe that it was left by Einstein himself and the whole contemporary generation of physicists as a torso – and in every respect, physically, mathematically, and its applications to cosmology”. When asked by Seelig to elaborate , Goedel added: “Concerning the com- pletion of gravitational theory of which I wrote in my last letter I do not mean a completion in the sense that the theory would cover a larger domain of phenomena (Tatsachenbereich), but a mathematical analysis of the equa- tions that would make it possible to attempt their solution systematically and to find their general properties. Until now one does not even know the analogs of the fundamental integral theorems of Newtonian theory which, in my opinion, have to exist without fail. Since such integral theorems and other mathematical lemmas would have a physical meaning, the physical un- derstanding of the theory would be enhanced. On the other hand, a closer analysis of the physical content of the theory could lead to such mathematical theorems”. VI J ¨urgen Ehlers Such a view of Einstein’s theory was also reflected in the talks and dis- cussions of the “Jordan Seminar”. This was a weekly meeting of Jordan’s coworkers in the Physics Department of Hamburg University to discuss Jor- dan’s theory of a variable gravitational scalar. However, under J¨urgen’s lead- ership, the structure and interpretation of Einstein’s original theory became the principal theme of nearly all talks. Jordan, who found little time to con- tribute actively to his theory, reluctantly went along with this change of topic. Through grants from the US Air Force and other sources he provided the lo- gistic support for his research group. For publication of the lengthy research papers on Einstein’s theory of gravitation by Ehlers, Kundt, Ozsvath, Sachs and Tr¨umper, he made the proceedings of the Akademie der Wissenschaften und der Literatur in Mainz available. Jordan appeared often as coauthor, but I doubt whether he contributed much more than suggestions in style, like never to start a sentence with a formula. Some results were also written up as reports for the Air Force and became known as the Hamburg Bible. It was a principal concern in J¨urgen’s contributions to Einstein’s theory to clarify the mathematics, separate proof from conjecture and insist on in- variance as well as elegance. This clear and terse style, which always kept physical interpretation in mind, appeared already in his Hamburg papers. His work in relativity resulted not only in books, published papers, super- vised theses, critical remarks in discussions and suggestions for future work. By establishing the “Albert–Einstein–Institut” J¨urgen designed a unique in- ternational center for research in relativity. As the founding director of this “Max–Planck–Institut f¨ur Gravitationsphysik” in Brandenburg, he has led it to instant success. Through his leadership, research on Einstein’s theory in Germany is flourishing again and his work and style has set a standard for a whole generation of researchers. Engelbert Sch¨ucking Preface The contributions in this book are dedicated to J¨urgen Ehlers on the occasion of his 70th birthday. I have tried to find topics which were and are near to J¨urgen’s interests and scientific activities. I hope that the book – even in the era of electronic publishing – will serve for some time as a review of the themes treated; a source from which, for example, a PhD student could learn certain things thoroughly. In initiating the project of the book, the model I had in mind was the “Witten book”. Early in his career J¨urgen Ehlers worked on exact solutions, and demon- strated how one goes about characterizing exact solutions invariantly and searching for their intrinsic geometrical properties. So, it seems appropriate to begin the book with the article by J. Biˇc´ak: “Selected Solutions of Ein- stein’s Field Equations: Their Role in General Relativity and Astrophysics.” Certainly not all of the large number of known exact solutions are of equal weight; this article describes the most important ones and explains their role for the development and understanding of Einstein’s theory of gravity. The second contribution is the article by H. Friedrich and A. Rendall: “The Cauchy Problem for the Einstein Equations”. It contains a careful ex- position of the local theory, including the delicate gauge questions and a discussion of various ways of writing the equations as hyperbolic systems. Furthermore, it becomes clear that an understanding of the Cauchy problem really gives new insight into properties of the equations and the solutions and not just “uniqueness and existence”. “Post-Newtonian Gravitational Radiation” is the title of the article by L. Blanchet. It deals with a topic J¨urgen has contributed to and thought about deeply. However, these matters have developed in such a way that presently only a small number of experts understand all the technical details and subtleties. Hopefully, this present contribution will help us gain some understanding of certain aspects of post-Newtonian approximations. The fourth contribution, “Duality and Hidden Symmetries in Gravita- tional Theories”, by D. Maison, outlines how far one of J¨urgen’s creations, the “Ehlers transformation” has evolved. From a “trick” to produce new so- lutions from known ones, the presence of such transformations in the space of solutions is now seen as a structural property of various gravitational theories, which at present attract a lot of attention. VI II Preface The contribution, by R. Beig and B. Schmidt, “Time-Independent Grav- itational Fields” collects and describes what is known about global proper- ties of time-independent spacetimes. It contains, in particular, a fairly self- contained description of the multipole expansion at infinity. V. Perlick has written on “Gravitational Lensing from a Geometric View- point”. In the last ten years, lensing has become a fascinating new part of observational astrophysics. However, there are still important and interesting conceptual and mathematical questions when one tries to compare practical astrophysical applications with their mathematical modelling in Einstein’s theory of gravity. Some of those issues are treated in this contribution. Obviously, there are some subjects missing, for which I was not able to find a contribution. What I regret most is that there is no article on cosmology, a field in which J¨urgen has always been very interested. An intruiging thought about the book is that Juergen would have read all these contributions before publication and no doubt improved them by his constructive criticism. For a short while I had in mind to ask J¨urgen to do just this, but finally I decided that this would be too much of a burden for a birthday present. Finally, I would like to thank the authors, friends and colleagues who have helped me and have given valuable advice. Bernd Schmidt Contents Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics Jiˇr´ıBiˇc´ak 1 1 Introduction and a Few Excursions 1 1.1 A Word on the Role of Explicit Solutions in Other Parts of Physics and Astrophysics 3 1.2 Einstein’s Field Equations 5 1.3 “Just So” Notes on the Simplest Solutions: The Minkowski, de Sitter, and Anti-de Sitter Spacetimes 8 1.4 On the Interpretation and Characterization of Metrics 11 1.5 The Choice of Solutions 15 1.6 The Outline 17 2 The Schwarzschild Solution 19 2.1 Spherically Symmetric Spacetimes 19 2.2 The Schwarzschild Metric and Its Role in the Solar System . . 20 2.3 Schwarzschild Metric Outside a Collapsing Star 21 2.4 The Schwarzschild–Kruskal Spacetime 25 2.5 The Schwarzschild Metric as a Case Against Lorentz-Covariant Approaches 28 2.6 The Schwarzschild Metric and Astrophysics 29 3 The Reissner–Nordstr¨om Solution 31 3.1 Reissner–Nordstr¨om Black Holes and the Question of Cosmic Censorship 32 3.2 On Extreme Black Holes, d-Dimensional Black Holes, String Theory and “All That” 39 4 The Kerr Metric 42 4.1 Basic Features 42 4.2 The Physics and Astrophysics Around Rotating Black Holes . 47 4.3 Astrophysical Evidence for a Kerr Metric 50 5 Black Hole Uniqueness and Multi-black Hole Solutions 52 6 On Stationary Axisymmetric Fields and Relativistic Disks 55 6.1 Static Weyl Metrics 55 6.2 Relativistic Disks as Sources of the Kerr Metric and Other Stationary Spacetimes 57 X Contents 6.3 Uniformly Rotating Disks 59 7 Taub-NUT Space 62 7.1 A New Way to the NUT Metric 62 7.2 Taub-NUT Pathologies and Applications 64 8 Plane Waves and Their Collisions 66 8.1 Plane-Fronted Waves 66 8.2 Plane-Fronted Waves: New Developments and Applications . . 71 8.3 Colliding Plane Waves 72 9 Cylindrical Waves 77 9.1 Cylindrical Waves and the Asymptotic Structure of 3-Dimensional General Relativity 78 9.2 Cylindrical Waves and Quantum Gravity 82 9.3 Cylindrical Waves: a Miscellany 85 10 On the Robinson–Trautman Solutions 86 11 The Boost-Rotation Symmetric Radiative Spacetimes 88 12 The Cosmological Models 93 12.1 Spatially Homogeneous Cosmologies 95 12.2 Inhomogeneous Cosmologies 102 13 Concluding Remarks 105 References 108 The Cauchy Problem for the Einstein Equations Helmut Friedrich, Alan Rendall 127 1 Introduction 127 2 Basic Observations and Concepts 131 2.1 The Principal Symbol 132 2.2 The Constraints 135 2.3 The Bianchi Identities 137 2.4 The Evolution Equations 137 2.5 Assumptions and Consequences 146 3 PDE Techniques 147 3.1 Symmetric Hyperbolic Systems 147 3.2 Symmetric Hyperbolic Systems on Manifolds 157 3.3 Other Notions of Hyperbolicity 159 4 Reductions 164 4.1 Hyperbolic Systems from the ADM Equations 167 4.2 The Einstein–Euler System 173 4.3 The Initial Boundary Value Problem 185 4.4 The Einstein–Dirac System 193 4.5 Remarks on the Structure of the Characteristic Set 200 5 Local Evolution 201 5.1 Local Existence Theorems for the Einstein Equations 201 5.2 Uniqueness 204 5.3 Cauchy Stability 206 5.4 Matter Models 207 Contents XI 5.5 An Example of an Ill-Posed Initial Value Problem 214 5.6 Symmetries 216 6 Outlook 217 References 219 Post-Newtonian Gravitational Radiation Luc Blanchet 225 1 Introduction 225 1.1 On Approximation Methods in General Relativity 225 1.2 Field Equations and the No-Incoming-Radiation Condition . . . 228 1.3 Method and General Physical Picture 231 2 Multipole Decomposition 233 2.1 The Matching Equation 233 2.2 The Field in Terms of Multipole Moments 236 2.3 Equivalence with the Will–Wiseman Multipole Expansion 238 3 Source Multipole Moments 240 3.1 Multipole Expansion in Symmetric Trace-Free Form 240 3.2 Linearized Approximation to the Exterior Field 241 3.3 Derivation of the Source Multipole Moments 242 4 Post-Minkowskian Approximation 244 4.1 Multipolar Post-Minkowskian Iteration of the Exterior Field . 244 4.2 The “Canonical” Multipole Moments 246 4.3 Retarded Integral of a Multipolar Extended Source 247 5 Radiative Multipole Moments 248 5.1 Definition and General Structure 249 5.2 The Radiative Quadrupole Moment to 3PN Order 250 5.3 Tail Contributions in the Total Energy Flux 251 6 Post-Newtonian Approximation 253 6.1 The Inner Metric to 2.5PN Order 254 6.2 The Mass-Type Source Moment to 2.5PN Order 256 7 Point-Particles 258 7.1 Hadamard Partie Finie Regularization 259 7.2 Multipole Moments of Point-Mass Binaries 261 7.3 Equations of Motion of Compact Binaries 263 7.4 Gravitational Waveforms of Inspiralling Compact Binaries . . . 265 8 Conclusion 267 Duality and Hidden Symmetries in Gravitational Theories Dieter Maison 273 1 Introduction 273 2 Electromagnetic Duality 277 3 Duality in Kaluza–Klein Theories 279 3.1 Dimensional Reduction from D to d Dimensions 280 3.2 Reduction to d = 4 Dimensions 282 3.3 Reduction to d = 3 Dimensions 285 XI I Contents 3.4 Reduction to d = 2 Dimensions 290 4 Geroch Group 292 5 Stationary Black Holes 302 5.1 Spherically Symmetric Solutions 306 5.2 Uniqueness Theorems for Static Black Holes 312 5.3 Stationary, Axially Symmetric Black Holes 314 6 Acknowledgments 316 7 Non-linear σ-Models and Symmetric Spaces 316 7.1 Non-compact Riemannian Symmetric Spaces 316 7.2 Pseudo-Riemannian Symmetric Spaces 319 7.3 Consistent Truncations 319 8 Structure of the Lie Algebra 319 Time-Independent Gravitational Fields Robert Beig, Bernd Schmidt 325 1 Introduction 325 2 Field Equations 327 2.1 Generalities 327 2.2 Axial Symmetry 333 2.3 Asymptotic Flatness: Lichnerowicz Theorems 334 2.4 Newtonian Limit 339 2.5 Existence Issues and the Newtonian Limit 340 3 Far Fields 341 3.1 Far-Field Expansions 341 3.2 Conformal Treatment of Infinity, Multipole Moments 344 4 Global Rotating Solutions 350 4.1 Lindblom’s Theorem 350 4.2 Existence of Stationary Rotating Axi-symmetric Fluid Bodies 353 4.3 The Neugebauer–Meinel Disk 357 5 Global Non-rotating Solutions 360 5.1 Elastic Static Bodies 360 5.2 Are Perfect Fluids O(3)-Symmetric? 362 5.3 Spherically Symmetric, Static Perfect Fluid Solutions 365 5.4 Spherically Symmetric, Static Einstein–Vlasov Solutions 370 Gravitational Lensing from a Geometric Viewpoint Volker Perlick 373 1 Introduction 373 2 Some Basic Notions of Spacetime Geometry 375 3 Gravitational Lensing in Arbitrary Spacetimes 378 3.1 Conjugate Points and Cut Points 381 3.2 The Geometry of Light Cones 385 3.3 Citeria for Multiple Imaging 391 3.4 Fermat’s Principle 396 Contents XIII 3.5 Morse Index Theory for Fermat’s Principle 399 4 Gravitational Lensing in Globally Hyperbolic Spacetimes 403 4.1 Criteria for Multiple Imaging in Globally Hyperbolic Spacetimes 405 4.2 Morse Theory in Globally Hyperbolic Spacetimes 408 5 Gravitational Lensing in Asymptotically Simple and Empty Spacetimes 414 References 422 J¨urgen Ehlers – Bibliography 427 Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics Jiˇr´ıBiˇc´ak Institute of Theoretical Physics, Charles University, Prague 1 Introduction and a Few Excursions The primary purpose of all physical theory is rooted in reality, and most rela- tivists pretend to be physicists. We may often be members of departments of mathematics and our work oriented towards the mathematical aspects of Ein- stein’s theory, but even those of us who hold a permanent position on “scri”, are primarily looking there for gravitational waves. Of course, the builder of this theory and its field equations was the physicist. J¨urgen Ehlers has always been very much interested in the conceptual and axiomatic foundations of physical theories and their rigorous, mathematically elegant formulation; but he has also developed and emphasized the importance of such areas of rela- tivity as kinetic theory, the mechanics of continuous media, thermodynamics and, more recently, gravitational lensing. Feynman expressed his view on the relation of physics to mathematics as follows [1]: “The physicist is always interested in the special case; he is never inter- ested in the general case. He is talking about something; he is not talking abstractly about anything. He wants to discuss the gravity law in three di- mensions; he never wants the arbitrary force case in n dimensions. So a certain amount of reducing is necessary, because the mathematicians have prepared these things for a wide range of problems. This is very useful, and later on it always turns out that the poor physicist has to come back and say, ‘Excuse me, when you wanted to tell me about four dimensions ’ ” Of course, this is Feynman, and from 1965 However, physicists are still rightly impressed by special explicit formulae. Explicit solutions enable us to discriminate more easily between a “physical” and “pathological” feature. Where are there singularities? What is their char- acter? How do test particles and fields behave in given background space- times? What are their global structures? Is a solution stable and, in some sense, generic? Clearly, such questions have been asked not only within gen- eral relativity. By studying a special explicit solution one acquires an intuition which, in turn, stimulates further questions relevant to more general situations. Consider, for example, charged black holes as described by the Reissner– Nordstr¨om solution. We have learned that in their interior a Cauchy horizon B.G. Schmidt (Ed.): LNP 540, pp. 1−126, 1999.  Springer-Verlag Berlin Heidelberg 1999 [...]... quadratic form gαβ (xγ )dxα dxβ in n-dimensions into another such form gαβ (x γ )dx α dx β by means of smooth transformation xγ (x κ )? As Ehlers emphasized in his paper 7 As pointed out by Kretschmann soon after the birth of general relativity, one can always make a theory generally covariant by taking more variables and inserting them as new dynamical variables into the (enlarged) theory Thus, standard... of the 1960s and the beginning of the 1970s for testing general relativity and alternative theories of gravity It has been very effectively used to compare general relativity with observations (see e.g [18,71,72] and references therein) In order to gain at least some concrete idea, let us just write down the simplest The Role of Exact Solutions 21 generalization of (2), namely the metric ds2 = − 1 − 2M... Boyer who became one of the victims of a mass murder on August 1, 1966, in Austin, Texas J¨rgen Ehlers was authorized by Mrs Boyer to look through u the scientific papers of her husband, and together with John Stachel, prepared posthumously the paper [78] from R Boyer’s notes Ehlers inserted his own discussions, generalized the main theorem on bifurcate horizons, but the paper [78] was published with R... gravity was available in the work of Riemann, Ricci, and Levi–Civita Several months after his departure from Prague and his collaboration with Grossmann, Einstein had general relativity almost in hand Their work [13] was already based on the generally invariant line element ds2 = gµν dxµ dxν (I) in which the spacetime metric tensor gµν (xρ ), µ, ν, ρ = 0, 1, 2, 3, plays a dual role: on the one hand it... = c = 1, and the same conventions as in [18] and [19] Now it is well known that Einstein further generalized his field equations by adding a cosmological term +Λgµν on the left side of the field equations (III) The cosmological constant Λ appeared first in Einstein’s work “Cosmological considerations in the General Theory of Relativity” [20] submitted on February 8, 1917 and published on February 15, 1917,... Hilbert made significant changes in the proofs The originally submitted version of his paper contained the theory which is not generally covariant, and the paper did not include equations (III) It was primarily Einstein’s recognition of the role of Mach’s ideas in his route towards general relativity, and in his christening them by the name “Mach’s principle” (though Schlick used this term in a vague sense... physics [21], the 1993 conference devoted exclusively to Mach’s principle was held in T¨bingen, from which a remarkably thorough volume was prepared [22], covering u all aspects of Mach’s principle and recording carefully all discussion The clarity of ideas and insights of J¨ rgen Ehlers contributed much to both conferences and u their proceedings For a brief more recent survey of various aspects of Mach’s... have played a crucial role in many issues in general relativity and cosmology, and most recently, they have become important prerequisites on the stage of the theoretical physics of the “new age”, including string theory and string cosmology, we shall make a few comments on these solutions here, and give some references to recent literature principle in general relativity, see the introductory section... spacetime and in a given coordinate (reference) system xµ A fundamental question, frequently “forgotten” to be addressed in modern theories which extend upon general relativity, is whether the metric tensor gαβ (xµ ) is a measurable quantity Classical general relativity offers (at least) three ways of giving a positive answer, depending on what objects are considered as “primitive tools” to perform the measurements... manifold” used frequently in general relativity, for example in problems of conservation of energy, or in quantum gravity, is not defined in a natural, unique manner The above simple pedagogical observation has recently been made in connection with gauge fixing in quantum gravity by H´j´cek [49] in order to explain the old insight by Bergmann and Komar, that a ıˇ the gauge group of general relativity is much . called pre-big bang string cosmology [38]. String theory is here applied to the problem of the big bang. The idea is to start from a simple Minkowski space (as an “asymptotic past triviality ) and. upon general relativity, is whether the metric tensor g αβ (x µ ) is a measurable quantity. Classical general relativity offers (at least) three ways of giving a positive answer, depending on. general situations. Consider, for example, charged black holes as described by the Reissner– Nordstr¨om solution. We have learned that in their interior a Cauchy horizon B. G. Schmidt (Ed. ): LNP