Verfügbarkeit: Exact solutions of Einstein's field equations

Exact solutions of Einstein's field equations:

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1. Verfasser:	Stephani, Hans 1935-2003 (VerfasserIn)
Format:	Buch
Sprache:	English
Veröffentlicht:	Cambridge [u.a.] Cambridge Univ. Press 2003
Ausgabe:	2. ed.
Schriftenreihe:	Cambridge monographs on mathematical physics
Schlagworte:	Einstein-Feldgleichungen Einstein, Équations du champ d' - Solutions numériques Espace et temps Ondes gravitationnelles Relativité générale (Physique) Einstein field equations > Numerical solutions General relativity (Physics) Gravitational waves Space and time Exakte Lösung Lehrbuch
Online-Zugang:	Publisher description Table of contents Inhaltsverzeichnis
Beschreibung:	Hier auch später erschienene, unveränderte Nachdrucke
Beschreibung:	XXIX, 701 S. graph. Darst.
ISBN:	0521461367 9780521467025

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Datensatz im Suchindex

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adam_text	Contents Preface xix List of tables xxiii Notation xxvii 1 Introduction 1 1.1 What are exact solutions, and why study them? 1 1.2 The development of the subject 3 1.3 The contents and arrangement of this book 4 1.4 Using this book as a catalogue 7 Part I: General methods 9 2 Differential geometry without a metric 9 2.1 Introduction 9 2.2 Differentiable manifolds 10 2.3 Tangent vectors 12 2.4 One-forms 13 2.5 Tensors 15 2.6 Exterior products and p-forms 17 2.7 The exterior derivative 18 2.8 The Lie derivative 21 2.9 The covariant derivative 23 2.10 The curvature tensor 25 2.11 Fibre bundles 27 vii viii Contents 3 Some topics in Riemannian geometry 30 3.1 Introduction 30 3.2 The metric tensor and tetrads 30 3.3 Calculation of curvature from the metric 34 3.4 Bivectors 35 3.5 Decomposition of the curvature tensor 37 3.6 Spinors 40 3.7 Conformai transformations 43 3.8 Discontinuities and junction conditions 45 4 The Petrov classification 48 4.1 The eigenvalue problem 48 4.2 The Petrov types 49 4.3 Principal null directions and determination of the Petrov types 53 5 Classification of the Ricci tensor and the energy-momentum tensor 57 5.1 The algebraic types of the Ricci tensor 57 5.2 The energy-momentum tensor 60 5.3 The energy conditions 63 5.4 The Ramich conditions 64 5.5 Perfect fluids 65 6 Vector fields 68 6.1 Vector fields and their invariant classification 68 0.1.1 Timelike unit vector fields 70 6.1.2 Geodesic null vector fields 70 6.2 Vector fields and the curvature tensor 72 6.2.1 Timelike unit vector fields 72 6.2.2 Null vector fields 74 7 The Newman—Penrose and related formalisms 75 7.1 The spin coefficients and their transformation laws 75 7.2 The Rìcci equations 78 7.3 The Bianchi identities 81 7.4 The GHP calculus 84 7.5 Geodesic null congruences 86 7.6 The Goldberg-Sachs theorem and its generalizations 87 Contents ix 8 Continuous groups of transformations; isometry and homothety groups 91 8.1 Lie groups and Lie algebras 91 8.2 Enumeration of distinct group structures 95 8.3 Transformation groups 97 8.4 Groups of motions 98 8.5 Spaces of constant curvature 101 8.6 Orbits of isometry groups 104 8.6.1 Simply-transitive groups 105 8.6.2 Multiply-transitive groups 106 8.7 Homothety groups 110 9 Invariants and the characterization of geometries 112 9.1 Scalar invariants and covariants 113 9.2 The Cartan equivalence method for space-times 116 9.3 Calculating the Cartan scalars 120 9.3.1 Determination of the Petrov and Segre types 120 9.3.2 The remaining steps 124 9.4 Extensions and applications of the Cartan method 125 9.5 Limits of families of space-times 126 10 Generation techniques 129 10.1 Introduction 129 10.2 Lie symmetries of Einstein s equations 129 10.2.1 Point transformations and their generators 129 10.2.2 How to find the Lie point symmetries of a given differential equation 131 10.2.3 How to use Lie point symmetries: similarity reduction 132 10.3 Symmetries more general than Lie symmetries 134 10.3.1 Contact and Lie— Bäcklund symmetries 134 10.3.2 Generalized and potential symmetries 134 10.4 Prolongation 137 10.4.1 Integral manifolds of differential forms 137 10.4.2 Isovectors. similarity solutions and conservation laws 140 10.4.3 Prolongation structures 141 10.5 Solutions of the linearized equations 145 10.6 Bäcklund transformations 146 10.7 Riemann-Hilbert problems 148 10.8 Harmonic maps 148 10.9 Variational Bäcklund transformations 151 10.10 Hirota^s method 152 x Contents 10.11 Generation methods including perfect fluids 152 10.11.1 Methods using the existence of Killing vectors 152 10.11.2 Conformai transformations 155 Part II: Solutions with groups of motions 157 11 Classification of solutions with isometries or homotheties 157 11.1 The possible space-times with isometries 157 11.2 Isotropy and the curvature tensor 159 11.3 The possible space-times with proper homothetic motions 162 11.4 Summary of solutions with homotheties 167 12 Homogeneous space-times 171 12.1 The possible metrics 171 12.2 Homogeneous vacuum and null Einstein-Maxwell space-times 174 12.3 Homogeneous non-null electromagnetic fields 175 12.4 Homogeneous perfect fluid solutions 177 12.5 Other homogeneous solutions 180 12.6 Summary 181 13 Hypersurface-homogeneous space-times 183 13.1 The possible metrics 183 13.1.1 Metrics with a G6 on У3 183 13.1.2 Metrics with a G4 on V3 183 13.1.3 Metrics with a G3 on V3 187 13.2 Formulations of the field equations 188 13.3 Vacuum, Л -term and Einstein-Maxwell solutions 194 13.3.1 Solutions with multiply-transitive groups 194 13.3.2 Vacuum spaces with a G3 on V3 196 13.3.3 Einstein spaces with a G3 on V3 199 13.3.4 Einstein-Maxwell solutions with a G3 on У3 201 13.4 Perfect fluid solutions homogeneous on T3 204 13.5 Summary of all metrics with Gr on F3 207 14 Spatially-homogeneous perfect fluid cosmologies 210 14.1 Introduction 210 14.2 Robertson-Walker cosmologies 211 14.3 Cosmologies with a G4 on S3 214 14.4 Cosmologies with a G3 on S3 218 Contents xi 15 Groups G3 on non-null orbits V2· Spherical and plane symmetry 226 15.1 Metric, Killing vectors, and Ricci tensor 226 15.2 Some implications of the existence of an isotropy group h 228 15.3 Spherical and plane symmetry 229 15.4 Vacuum, Einstein-Maxwell and pure radiation fields 230 15.4.1 Timelike orbits 230 15.4.2 Spacelike orbits 231 15.4.3 Generalized Birkhoff theorem 232 15.4.4 Spherically- and plane-symmetric fields 233 15.5 Dust solutions 235 15.6 Perfect fluid solutions with plane, spherical or pseudospherical symmetry 237 15.6.1 Some basic properties 237 15.6.2 Static solutions 238 15.6.3 Solutions without shear and expansion 238 15.6.4 Expanding solutions without shear 239 15.6.5 Solutions with nonvanishing shear 240 15.7 Plane-symmetric perfect, fluid solutions 243 15.7.1 Static solutions 243 15.7.2 Non-static solutions 244 16 Spherically-symmetric perfect fluid solutions 247 16.1 Static solutions 247 16.1.1 Field equations and first integrals 247 16.1.2 Solutions 250 16.2 Non-static solutions 251 16.2.1 The basic equations 251 16.2.2 Expanding solutions without shear 253 16.2.3 Solutions with non-vanishing shear 260 17 Groups Gì and G on non-null orbits 264 17.1 Groups G-2 on non-null orbits 264 17.1.1 Subdivisions of the groups G2 264 17.1.2 Groups G2I on non-null orbits 265 17.1.3 G2II on non-null orbits 267 17.2 Boost-rotation-symmetric space-times 268 17.3 Group G on non-null orbits 271 18 Stationary gravitational fields 275 18.1 The projection formalism 275 xii Contents 18.2 The Ricci tensor on Σ3 277 18.3 Conformai transformation of Σ3 and the field equations 278 18 4 Vacuum and Einstein-Maxwell equations for stationary fields 279 18.5 Geodesic eigenrays 281 18.6 Static fields 283 18.6.1 Definitions 283 18.6.2 Vacuum solutions 284 18.6.3 Electrostatic and magnetostatic Einstein-Maxwell fields 284 18.6.4 Perfect fluid solutions 286 18.7 The conformastationary solutions 287 18.7.1 Conformastationary vacuum solutions 287 18.7.2 Conformastationary Einstein-Maxwell fields 288 18.8 Multipole moments 289 19 Stationary axisymnaetric fields: basic concepts and field equations 292 19.1 The Killing vectors 292 19.2 Orthogonal surfaces 293 19.3 The metric and the projection formalism 296 19.4 The field equations for stationary axisymmetric Einstein- Maxwell fields 298 19.5 Various forms of the field equations for stationary axisym¬ metric vacuum fields 299 19.6 Field equations for rotating fluids 302 20 Stationary axisymmetric vacuum solutions 304 20.1 Introduction 304 20.2 Static axisymmetric vacuum solutions (Weyl s class) 304 20.3 The class of solutions U = U (ω) (Papapetrou s class) 309 20.4 The class of solutions S = S (A) 310 20.5 The Kerr solution and the Tomimatsu-Sato class 311 20.6 Other solutions 313 20.7 Solutions with factor structure 316 21 Non-empty stationary axisymmetric solutions 319 21.1 Einstein-Maxwell fields 319 21.1.1 Electrostatic and magnetostatic solutions 319 21.1.2 Type D solutions: A general metric and its limits 322 21.1.3 The Kerr-Newman solution 325 Contents xiii 21.1.4 Complexification and the Newman-Janis complex trick 328 21.1.5 Other solutions 329 21.2 Perfect fluid solutions 330 21.2.1 Line element and general properties 330 21.2.2 The general dust solution 331 21.2.3 Rigidly rotating perfect fluid solutions 333 21.2.4 Perfect fluid solutions with differential rotation 337 22 Groups G4I on spacelike orbits: cylindrical symmetry 341 22.1 General remarks 341 22.2 Stationary cylindrically-symmetric fields 342 22.3 Vacuum fields 350 22.4 Einstein-Maxwell and pure radiation fields 354 23 Inhomogeneous perfect fluid solutions with symmetry 358 23.1 Solutions with a maximal Щ on Ss 359 23.2 Solutions with a maximal H3 on T3 3G1 23.3 Solutions with a G2 on S2 362 23.3.1 Diagonal metrics 3G3 23.3.2 Non-diagonal solutions with orthogonal transitivity 372 23.3.3 Solutions without orthogonal transitivity 373 23.4 Solutions with a d or a H2 374 24 Groups on null orbits. Plane waves 375 24.1 Introduction 375 24.2 Groups G3 on N3 376 24.3 Groups G2 on N2 377 24.4 Null Killing vectors (d on N1) 379 24.4.1 Non- twisting null Killing vector 380 24.4.2 Twisting null Killing vector 382 24.5 The plane-fronted gravitational waves with parallel rays (pp-waves) 383 25 Collision of plane waves 387 25.1 General features of the collision problem 387 25.2 The vacuum field equations 389 25.3 Vacuum solutions with collinear polarization 392 25.4 Vacuum solutions with non-collinear polarization 394 25.5 Einstein-Maxwell fields 397 xiv Contents 25.6 Stiff perfect fluids and pure radiation 403 25.6.1 Stiff perfect fluids 403 25.6.2 Pure radiation (null dust) 405 Part III: Algebraically special solutions 407 26 The various classes of algebraically special solutions. Some algebraically general solutions 407 26.1 Solutions of Petrov type II, D, III or iV 407 26.2 Petrov type .D solutions 412 26.3 Conformally flat solutions 413 26.4 Algebraically general vacuum solutions with geodesic and non-twisting rays 413 27 The line element for metrics with n = σ = 0 = Дії = Ru = #44, Θ + ιω ψ 0 416 27.1 The line element in the case with twisting rays [ω φ 0) 416 27.1.1 The choice of the null tetrad 416 27.1.2 The coordinate frame 418 27.1.3 Admissible tetrad and coordinate transformations 420 27.2 The line element in the case with non-twisting rays (ш = 0) 420 28 Robinson—Trautman solutions 422 28.1 Robinson—Trautman vacuum solutions 422 28.1.1 The field equations and their solutions 422 28.1.2 Special cases and explicit solutions 424 28.2 Robinson-Trautman Einstein-Maxwell fields 427 28.2.1 Line element and field equations 427 28.2.2 Solutions of type III, N and О 429 28.2.3 Solutions of type D 429 28.2.4 Type II solutions 431 28.3 Robinson-Trautman pure radiation fields 435 28.4 Robinson-Trautman solutions with a cosmological constant Л 436 29 Twisting vacuum solutions 437 29.1 Twisting vacuum solutions the field equations 437 29.1.1 The structure of the field equations 437 29.1.2 The integration of the main equations 438 29.1.3 The remaining field equations 440 29.1.4 Coordinate freedom and transformation properties 441 Contents XV 29.2 Some general classes of solutions 442 29.2.1 Characterization of the known classes of solutions 442 29.2.2 The case θζ1 = dc(G2 - dçG) φ 0 445 2 29.2.3 The case ΘςΙ = oç(G2 - ÕçG) φ 0. LM = 0 446 29.2.4 The case 1 = 0 447 29.2.5 The case I = Q = L.U 449 29.2.6 Solutions independent of ζ and ζ 450 29.3 Solutions of type Ν (Φ2 = 0 = Φ3) 451 29.4 Solutions of type III (Φ2 = 0, Φ3 φ 0) 452 29.5 Solutions of type Ό (5Φ2*4 = 2Φ§. Φ2 φ 0) 452 29.6 Solutions of type II 454 30 Twisting Einstein—Maxwell and pure radiation fields 455 30.1 The structure of the Einstein-Maxwell field equations 455 30.2 Determination of the radial dependence of the metric and the Maxwell field 456 30.3 The remaining field equations 458 30.4 Charged vacuum metrics 459 30.5 A class of radiative Einstein-Maxwell fields (ФЅ φ 0) 460 30.6 Remarks concerning solutions of the different Petrov types 461 30.7 Pure radiation fields 463 30.7.1 The field equations 463 30.7.2 Generating pure radiation fields from vacuum by changing Ρ 464 30.7.3 Generating pure radiation fields from vacuum by changing m 466 30.7.4 Some special classes of pure radiation fields 467 31 Non-diverging solutions (Kundt s class) 470 31.1 Introduction 470 31.2 The line element for metrics with Θ + ίω = 0 470 31.3 The Ricci tensor components 472 31.4 The structure of the vacuum and Einstein-Maxwell equation 473 31.5 Vacuum solutions 476 31.5.1 Solutions of types III and N 476 31.5.2 Solutions of types D and II 478 31.6 Einstein-Maxwell null fields and pure radiation fields 480 31.7 Einstein-Maxwell non-null fields 481 31.8 Solutions including a cosmologica! constant Λ 483 xvi Contents 32 Kerr-Schild metrics 485 32.1 General properties of Kerr-Schild metrics 485 32.1.1 The origin of the Kerr-Schild-Trautman ansatz 485 32.1.2 The Ricci tensor, Riemann tensor and Petrov type 485 32.1.3 Field equations and the energy-momentum tensor 487 32.1.4 A geometrical interpretation of the Kerr-Schild ansatz 487 32.1.5 The Newman-Penrose formalism for shearfree and geodesic Kerr-Schild metrics 489 32.2 Kerr-Schild vacuum fields 492 32.2.1 The case ρ = -(θ + ίω) Φ 0 492 32.2.2 The case ρ = -(Θ + ίω) = 0 493 32.3 Kerr-Schild Einstein-Maxwell fields 493 32.3.1 The case ρ = -(Θ + ίω) φ 0 493 32.3.2 The case ρ = -(Θ + ίω) = 0 495 32.4 Kerr-Schild pure radiation fields 497 32.4.1 The case ρ φ Ο, σ = 0 497 32.4.2 Thecasea^O 499 32.4.3 The case ρ = σ = 0 499 32.5 Generalizations of the Kerr-Schild ansatz 499 32.5.1 General properties and results 499 32.5.2 Non-flat vacuum to vacuum 501 32.5.3 Vacuum to electrovac 502 32.5.4 Perfect, fluid to perfect fluid 503 33 Algebraically special perfect fluid solutions 506 33.1 Generalized Robinson-Trautman solutions 506 33.2 Solutions with a geodesic, shearfree, non-expanding multiple null eigenvector 510 33.3 Type D solutions 512 33.3.1 Solutions with к — ν = 0 513 33.3.2 Solutions with κ φ 0, ν φ 0 513 33.4 Type III and type N solutions 515 Part IV: Special methods 518 34 Application of generation techniques to general relativity 518 34.1 Methods using harmonic maps (potential space symmetries) 518 34.1.1 Electrovacuum fields with one Killing vector 518 34.1.2 The group SU(2.1) 521 Contents xvii 34.1.3 Complex invariance transformations 525 34.1.4 Stationary axisymmetric vacuum fields 526 34.2 Prolongation structure for the Ernst equation 529 34.3 The linearized equations, the Kinnersley-Chitre В group and the Hoenselaers-Kinnersley-Xanthopoulos transformations 532 34.3.1 The field equations 532 34.3.2 Infinitesimal transformations and transformations preserving Minkowski space 534 34.3.3 The Hoenselaers-Kinnersley-Xanthopoulos transfor¬ mation 535 34.4 Bäcklund transformations 538 34.5 The Belinski-Zakharov technique 543 34.6 The Riemann-Hilbert problem 547 34.6.1 Some general remarks 547 34.6.2 The Neugebauer-Meinel rotating disc solution 548 34.7 Other approaches 549 34.8 Einstein-Maxwell fields 550 34.9 The case of two space-like Killing vectors 550 35 Special vector and tensor fields 553 35.1 Space-times that admit constant vector and tensor fields 553 35.1.1 Constant vector fields 553 35.1.2 Constant tensor fields 554 35.2 Complex recurrent, conformally recurrent, recurrent and symmetric spaces 556 35.2.1 The definitions 556 35.2.2 Space-times of Petrov type D 557 35.2.3 Space-times of type N 557 35.2.4 Space-times of type О 558 35.3 Killing tensors of order two and Killing-Yano tensors 559 35.3.1 The basic definitions 559 35.3.2 First integrals, separability and Killing or Killing- Yano tensors 560 35.3.3 Theorems on Killing and Killing-Yano tensors in four- dimensional space-times 561 35.4 Collineations and conformai motions 564 35.4.1 The basic definitions 564 35.4.2 Proper curvature collineations 565 35.4.3 General theorems on conformai motions 565 35.4.4 Non-conformally flat solutions admitting proper conformai motions 567 xviii Contents 36 Solutions with special subspaces 571 36.1 The basic formulae 571 36.2 Solutions with flat three-dimensional slices 573 36.2.1 Vacuum solutions 573 36.2.2 Perfect fluid and dust solutions 573 36.3 Perfect fluid solutions with conformally flat slices 577 36.4 Solutions with other intrinsic symmetries 579 37 Local isometric embedding of four-dimensional Riemannian manifolds 580 37.1 The why of embedding 580 37.2 The basic formulae governing embedding 581 37.3 Some theorems on local isometric embedding 583 37.3.1 General theorems 583 37.3.2 Vector and tensor fields and embedding class 584 37.3.3 Groups of motions and embedding class 586 37.4 Exact solutions of embedding class one 587 37.4.1 The Gauss and Codazzi equations and the possible types of nab 587 37.4.2 Conformally flat perfect fluid solutions of embedding class one 588 37.4.3 Type D perfect fluid solutions of embedding class one 591 37.4.4 Pure radiation field solutions of embedding class one 594 37.5 Exact solutions of embedding class two 596 37.5.1 The Gauss-Codazzi-Ricci equations 596 37.5.2 Vacuum solutions of embedding class two 598 37.5.3 Conformally flat solutions 599 37.6 Exact solutions of embedding class ρ > 2 603 Part V: Tables 605 38 The interconnections between the main classification schemes 605 38.1 Introduction 605 38.2 The connection between Petrov types and groups of motions 606 38.3 Tables 609 References 615 Index 690
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author	Stephani, Hans 1935-2003
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genre_facet	Lehrbuch
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illustrated	Illustrated
indexdate	2024-07-09T19:03:10Z
institution	BVB
isbn	0521461367 9780521467025
language	English
lccn	2002071495
oai_aleph_id	oai:aleph.bib-bvb.de:BVB01-009884584
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physical	XXIX, 701 S. graph. Darst.
publishDate	2003
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publisher	Cambridge Univ. Press
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series2	Cambridge monographs on mathematical physics
spelling	Stephani, Hans 1935-2003 Verfasser (DE-588)1025772032 aut Exact solutions of Einstein's field equations Hans Stephani ... 2. ed. Cambridge [u.a.] Cambridge Univ. Press 2003 XXIX, 701 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Cambridge monographs on mathematical physics Hier auch später erschienene, unveränderte Nachdrucke Einstein-Feldgleichungen Einstein, Équations du champ d' - Solutions numériques Espace et temps Ondes gravitationnelles Relativité générale (Physique) Einstein field equations Numerical solutions General relativity (Physics) Gravitational waves Space and time Einstein-Feldgleichungen (DE-588)4013941-4 gnd rswk-swf Exakte Lösung (DE-588)4348289-2 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Einstein-Feldgleichungen (DE-588)4013941-4 s Exakte Lösung (DE-588)4348289-2 s DE-604 http://www.loc.gov/catdir/description/cam031/2002071495.html Publisher description http://www.loc.gov/catdir/toc/cam031/2002071495.html Table of contents Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009884584&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk
spellingShingle	Stephani, Hans 1935-2003 Exact solutions of Einstein's field equations Einstein-Feldgleichungen Einstein, Équations du champ d' - Solutions numériques Espace et temps Ondes gravitationnelles Relativité générale (Physique) Einstein field equations Numerical solutions General relativity (Physics) Gravitational waves Space and time Einstein-Feldgleichungen (DE-588)4013941-4 gnd Exakte Lösung (DE-588)4348289-2 gnd
subject_GND	(DE-588)4013941-4 (DE-588)4348289-2 (DE-588)4123623-3
title	Exact solutions of Einstein's field equations
title_auth	Exact solutions of Einstein's field equations
title_exact_search	Exact solutions of Einstein's field equations
title_full	Exact solutions of Einstein's field equations Hans Stephani ...
title_fullStr	Exact solutions of Einstein's field equations Hans Stephani ...
title_full_unstemmed	Exact solutions of Einstein's field equations Hans Stephani ...
title_short	Exact solutions of Einstein's field equations
title_sort	exact solutions of einstein s field equations
topic	Einstein-Feldgleichungen Einstein, Équations du champ d' - Solutions numériques Espace et temps Ondes gravitationnelles Relativité générale (Physique) Einstein field equations Numerical solutions General relativity (Physics) Gravitational waves Space and time Einstein-Feldgleichungen (DE-588)4013941-4 gnd Exakte Lösung (DE-588)4348289-2 gnd
topic_facet	Einstein-Feldgleichungen Einstein, Équations du champ d' - Solutions numériques Espace et temps Ondes gravitationnelles Relativité générale (Physique) Einstein field equations Numerical solutions General relativity (Physics) Gravitational waves Space and time Exakte Lösung Lehrbuch
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