Ju¨rgen Ehlers In the 1950s the mathematical department of Hamburg University, with its stars Artin, Blaschke, Collatz, Ka¨hler, Peterson, Sperner and Witt had a strongdrawingpowerforJu¨rgenEhlers,studentofmathematicsandphysics. 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. Jordanhadjustpublishedhisbook“Schwerkraft und Weltall”whichwas 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 Ju¨rgen Ehlers and Wolfgang Kundt to helpJordanwithhisproblems,andtheirworkwasacknowledgedinthe1955 second edition of Jordan’s book. It didn’t take Ju¨rgen, 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”. WhenaskedbySeeligtoelaborate,Goedeladded:“Concerningthecom- 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 othermathematicallemmaswouldhaveaphysicalmeaning,thephysicalun- derstanding of the theory would be enhanced. On the other hand, a closer analysisofthephysicalcontentofthetheorycouldleadtosuchmathematical theorems”. VI Ju¨rgen 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’stheoryofavariablegravitationalscalar.However,underJu¨rgen’slead- 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- tributeactivelytohistheory,reluctantlywentalongwiththischangeoftopic. 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 Tru¨mper, 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 Ju¨rgen’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” Ju¨rgen designed a unique in- ternational center for research in relativity. As the founding director of this “Max–Planck–Institutfu¨rGravitationsphysik”inBrandenburg,hehasledit 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 Schu¨cking Preface ThecontributionsinthisbookarededicatedtoJu¨rgenEhlersontheoccasion of his 70th birthday. I have tried to find topics which were and are near to Ju¨rgen’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 themestreated;asourcefromwhich,forexample,aPhDstudentcouldlearn certain things thoroughly. In initiating the project of the book, the model I had in mind was the “Witten book”. Early in his career Ju¨rgen 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;thisarticledescribesthemostimportantonesandexplainstheirrole 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 reallygivesnewinsightintopropertiesoftheequationsandthesolutionsand not just “uniqueness and existence”. “Post-Newtonian Gravitational Radiation” is the title of the article by L. Blanchet. It deals with a topic Ju¨rgen 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 Ju¨rgen’s creations, the “Ehlers transformation” has evolved. From a “trick” to produce new so- lutionsfromknownones,thepresenceofsuchtransformationsinthespaceof solutionsisnowseenasastructuralpropertyofvariousgravitationaltheories, which at present attract a lot of attention. VIII 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.Perlickhaswrittenon“GravitationalLensingfromaGeometricView- point”. In the last ten years, lensing has become a fascinating new part of observationalastrophysics.However,therearestillimportantandinteresting 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,therearesomesubjectsmissing,forwhichIwasnotabletofind a contribution. What I regret most is that there is no article on cosmology, a field in which Ju¨rgen has always been very interested. AnintruigingthoughtaboutthebookisthatJuergenwouldhavereadall these contributions before publication and no doubt improved them by his constructive criticism. For a short while I had in mind to ask Ju¨rgen to do just this, but finally I decided that this would be too much of a burden for a birthday present. Finally,Iwouldliketothanktheauthors,friendsandcolleagueswhohave 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´ı Biˇc´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 KalEuza–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 XII 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 Ju¨rgen Ehlers – Bibliography ................................. 427 Selected Solutions of Einstein’s Field Equations: Their Role in General Relativity and Astrophysics Jiˇr´ı Biˇc´ak Institute of Theoretical Physics, Charles University, Prague 1 Introduction and a Few Excursions Theprimarypurposeofallphysicaltheoryisrootedinreality,andmostrela- tivistspretendtobephysicists.Wemayoftenbemembersofdepartmentsof mathematicsandourworkorientedtowardsthemathematicalaspectsofEin- stein’stheory,buteventhoseofuswhoholdapermanentpositionon“scri”, are primarily looking there for gravitational waves. Of course, the builder of thistheoryanditsfieldequationswasthephysicist.Ju¨rgenEhlershasalways been very much interested in the conceptual and axiomatic foundations of physicaltheoriesandtheirrigorous,mathematicallyelegantformulation;but he has also developed and emphasized the importance of such areas of rela- tivityaskinetictheory,themechanicsofcontinuousmedia,thermodynamics and,morerecently,gravitationallensing.Feynmanexpressedhisviewonthe 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;heneverwantsthearbitraryforcecaseinndimensions.Soacertain 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,physicistsarestillrightlyimpressedbyspecialexplicitformulae. Explicitsolutionsenableustodiscriminatemoreeasilybetweena“physical” and“pathological”feature.Wherearetheresingularities?Whatistheirchar- 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– Nordstro¨m 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