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| 【獎(jiǎng)勵(lì)】 本帖被評(píng)價(jià)18次，作者pkusiyuan增加金幣 14.4 個(gè) | ||
pkusiyuan銀蟲(chóng) (正式寫(xiě)手)
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[資源]
劍橋2010年Numerical.Relativity.Solving.Einstein's.Equations.on.the.Computer
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Aimed at students and researchers entering the field, this pedagogical introduction to numerical relativity will also interest scientists seeking a broad survey of its challenges and achievements. Assuming only a basic knowledge of classical general relativity, this textbook develops the mathematical formalism from first principles, then highlights some of the pioneering simulations involving black holes and neutron stars, gravitational collapse and gravitational waves. The book contains 300 exercises to help readers master new material as it is presented. Numerous illustrations, many in color, assist in visualizing new geometric concepts and highlighting the results of computer simulations. Summary boxes encapsulate some of the most important results for quick reference. Applications covered include calculations of coalescing binary black holes and binary neutron stars, rotating stars, colliding star clusters, gravitational and magnetorotational collapse, critical phenomena, the generation of gravitational waves, and other topics of current physical and astrophysical significance. ThomasW. Baumgarte is a Professor of Physics at Bowdoin College and an Adjunct Professor of Physics at the University of Illinois at Urbana-Champaign. He received his Diploma (1993) and Doctorate (1995) from Ludwig-Maximilians-Universit¨at, M¨unchen, and held postdoctoral positions at Cornell University and the University of Illinois before joining the faculty at Bowdoin College. He is a recipient of a John Simon Guggenheim Memorial Foundation Fellowship. He has written over 70 research articles on a variety of topics in general relativity and relativistic astrophysics, including black holes and neutron stars, gravitational collapse, and more formal mathematical issues. Stuart L. Shapiro is a Professor of Physics and Astronomy at the University of Illinois at Urbana-Champaign. He received his A.B from Harvard (1969) and his Ph.D. from Princeton (1973). He has published over 340 research articles spanning many topics in general relativity and theoretical astrophysics and coauthored the widely used textbook Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects (JohnWiley, 1983). In addition to numerical relativity, Shapiro has worked on the physics and astrophysics of black holes and neutron stars, relativistic hydrodynamics, magnetohydrodynamics and stellar dynamics, and the generation of gravitational waves. He is a recipient of an IBM Supercomputing Award, a Forefronts of Large-Scale Computation Award, an Alfred P. Sloan Research Fellowship, a John Simon Guggenheim Memorial Foundation Fellowship, and several teaching citations. He has served on the editorial boards of The Astrophysical Journal Letters and Classical and Quantum Gravity. Hewas elected Fellowof both the American Physical Society and Institute of Physics (UK). 1 General relativity preliminaries 1 1.1 Einstein’s equations in 4-dimensional spacetime 1 1.2 Black holes 9 1.3 Oppenheimer–Volkoff spherical equilibrium stars 15 1.4 Oppenheimer–Snyder spherical dust collapse 18 2 The3+1 decompostion of Einstein’s equations 23 2.1 Notation and conventions 26 2.2 Maxwell’s equations in Minkowski spacetime 27 2.3 Foliations of spacetime 29 2.4 The extrinsic curvature 33 2.5 The equations of Gauss, Codazzi and Ricci 36 2.6 The constraint and evolution equations 39 2.7 Choosing basis vectors: the ADM equations 43 3 Constructing initial data 54 3.1 Conformal transformations 56 3.1.1 Conformal transformation of the spatial metric 56 3.1.2 Elementary black hole solutions 57 3.1.3 Conformal transformation of the extrinsic curvature 64 3.2 Conformal transverse-traceless decomposition 67 3.3 Conformal thin-sandwich decomposition 75 3.4 A step further: the “waveless” approximation 81 3.5 Mass, momentum and angular momentum 83 4 Choosing coordinates: the lapse and shift 98 4.1 Geodesic slicing 100 4.2 Maximal slicing and singularity avoidance 103 4.3 Harmonic coordinates and variations 111 v vi Contents 4.4 Quasi-isotropic and radial gauge 114 4.5 Minimal distortion and variations 117 5 Matter sources 123 5.1 Vacuum 124 5.2 Hydrodynamics 124 5.2.1 Perfect gases 124 5.2.2 Imperfect gases 139 5.2.3 Radiation hydrodynamics 141 5.2.4 Magnetohydrodynamics 148 5.3 Collisionless matter 163 5.4 Scalar fields 175 6 Numerical methods 183 6.1 Classification of partial differential equations 183 6.2 Finite difference methods 188 6.2.1 Representation of functions and derivatives 188 6.2.2 Elliptic equations 191 6.2.3 Hyperbolic equations 200 6.2.4 Parabolic equations 209 6.2.5 Mesh refinement 211 6.3 Spectral methods 213 6.3.1 Representation of functions and derivatives 213 6.3.2 A simple example 214 6.3.3 Pseudo-spectral methods with Chebychev polynomials 217 6.3.4 Elliptic equations 219 6.3.5 Initial value problems 223 6.3.6 Comparison with finite-difference methods 224 6.4 Code validation and calibration 225 7 Locating black hole horizons 229 7.1 Concepts 229 7.2 Event horizons 232 7.3 Apparent horizons 235 7.3.1 Spherical symmetry 240 7.3.2 Axisymmetry 241 7.3.3 General case: no symmetry assumptions 246 7.4 Isolated and dynamical horizons 249 8 Spherically symmetric spacetimes 253 8.1 Black holes 256 8.2 Collisionless clusters: stability and collapse 266 8.2.1 Particle method 267 8.2.2 Phase space method 289 Contents vii 8.3 Fluid stars: collapse 291 8.3.1 Misner–Sharp formalism 294 8.3.2 The Hernandez–Misner equations 297 8.4 Scalar field collapse: critical phenomena 303 9 Gravitational waves 311 9.1 Linearized waves 311 9.1.1 Perturbation theory and the weak-field, slow-velocity regime 312 9.1.2 Vacuum solutions 319 9.2 Sources 323 9.2.1 The high frequency band 324 9.2.2 The low frequency band 328 9.2.3 The very low and ultra low frequency bands 330 9.3 Detectors and templates 331 9.3.1 Ground-based gravitational wave interferometers 332 9.3.2 Space-based detectors 334 9.4 Extracting gravitational waveforms 337 9.4.1 The gauge-invariant Moncrief formalism 338 9.4.2 The Newman–Penrose formalism 346 10 Collapse of collisionless clusters in axisymmetry 352 10.1 Collapse of prolate spheroids to spindle singularities 352 10.2 Head-on collision of two black holes 359 10.3 Disk collapse 364 10.4 Collapse of rotating toroidal clusters 369 11 Recasting the evolution equations 375 11.1 Notions of hyperbolicity 376 11.2 Recasting Maxwell’s equations 378 11.2.1 Generalized Coulomb gauge 379 11.2.2 First-order hyperbolic formulations 380 11.2.3 Auxiliary variables 381 11.3 Generalized harmonic coordinates 381 11.4 First-order symmetric hyperbolic formulations 384 11.5 The BSSN formulation 386 12 Binary black hole initial data 394 12.1 Binary inspiral: overview 395 12.2 The conformal transverse-traceless approach: Bowen–York 403 12.2.1 Solving the momentum constraint 403 12.2.2 Solving the Hamiltonian constraint 405 12.2.3 Identifying circular orbits 407 viii Contents 12.3 The conformal thin-sandwich approach 410 12.3.1 The notion of quasiequilibium 410 12.3.2 Quasiequilibrium black hole boundary conditions 413 12.3.3 Identifying circular orbits 419 12.4 Quasiequilibrium sequences 421 13 Binary black hole evolution 429 13.1 Handling the black hole singularity 430 13.1.1 Singularity avoiding coordinates 430 13.1.2 Black hole excision 431 13.1.3 The moving puncture method 432 13.2 Binary black hole inspiral and coalescence 436 13.2.1 Equal-mass binaries 437 13.2.2 Asymmetric binaries, spin and black hole recoil 445 14 Rotating stars 459 14.1 Initial data: equilibrium models 460 14.1.1 Field equations 460 14.1.2 Fluid stars 461 14.1.3 Collisionless clusters 471 14.2 Evolution: instabilities and collapse 473 14.2.1 Quasiradial stability and collapse 473 14.2.2 Bar-mode instability 478 14.2.3 Black hole excision and stellar collapse 481 14.2.4 Viscous evolution 491 14.2.5 MHD evolution 495 15 Binary neutron star initial data 506 15.1 Stationary fluid solutions 506 15.1.1 Newtonian equations of stationary equilibrium 508 15.1.2 Relativistic equations of stationary equilibrium 512 15.2 Corotational binaries 514 15.3 Irrotational binaries 523 15.4 Quasiadiabatic inspiral sequences 530 16 Binary neutron star evolution 533 16.1 Peliminary studies 534 16.2 The conformal flatness approximation 535 16.3 Fully relativistic simulations 545 17 Binary black hole–neutron stars: initial data and evolution 562 17.1 Initial data 565 17.1.1 The conformal thin-sandwich approach 565 17.1.2 The conformal transverse-traceless approach 572 Contents ix 17.2 Dynamical simulations 574 17.2.1 The conformal flatness approximation 574 17.2.2 Fully relativistic simulations 578 18 Epilogue 596 A Lie derivatives, Killing vectors, and tensor densities 598 A.1 The Lie derivative 598 A.2 Killing vectors 602 A.3 Tensor densities 603 B Solving the vector Laplacian 607 C The surface element on the apparent horizon 609 D Scalar, vector and tensor spherical harmonics 612 E Post-Newtonian results 616 F Collisionless matter evolution in axisymmetry: basic equations 629 G Rotating equilibria: gravitational field equations 634 H Moving puncture representions of Schwarzschild: analytical results 637 I Binary black hole puncture simulations as test problems 642 References 647 Index 684 |
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