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Numerical Relativity
Solving Einstein’s Equations on the Computer
Aimed at students and researchers entering the field, this pedagogical introduction
to numerical relativity will also interest scientists seeking a broad survey of its
challengesandachievements.Assumingonlyabasicknowledgeofclassicalgeneral
relativity, this textbook develops the mathematical formalism from first principles,
thenhighlightssomeofthepioneeringsimulationsinvolvingblackholesandneutron
stars,gravitationalcollapseandgravitationalwaves.
The book contains 300 exercises to help readers master new material as it is
presented.Numerousillustrations,manyincolor,assistinvisualizingnewgeomet-
ric concepts and highlighting the results of computer simulations. Summary boxes
encapsulatesomeofthemostimportantresultsforquickreference.Applicationscov-
eredincludecalculationsofcoalescingbinaryblackholesandbinaryneutronstars,
rotating stars, colliding star clusters, gravitational and magnetorotational collapse,
criticalphenomena,thegenerationofgravitationalwaves,andothertopicsofcurrent
physicalandastrophysicalsignificance.
ThomasW.BaumgarteisaProfessorofPhysicsatBowdoinCollegeandanAdjunct
ProfessorofPhysicsattheUniversityofIllinoisatUrbana-Champaign.Hereceived
his Diploma (1993) and Doctorate (1995) from Ludwig-Maximilians-Universita¨t,
Mu¨nchen, 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
JohnSimonGuggenheimMemorialFoundationFellowship.Hehaswrittenover70
researcharticlesonavarietyoftopicsingeneralrelativityandrelativisticastrophysics,
including black holes and neutron stars, gravitational collapse, and more formal
mathematicalissues.
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.fromPrinceton(1973).Hehaspublishedover340researcharticlesspanning
many topics in general relativity and theoretical astrophysics and coauthored the
widelyusedtextbookBlackHoles,WhiteDwarfsandNeutronStars:ThePhysicsof
CompactObjects(JohnWiley,1983).Inadditiontonumericalrelativity,Shapirohas
workedonthephysicsandastrophysicsofblackholesandneutronstars,relativistic
hydrodynamics,magnetohydrodynamicsandstellardynamics,andthegenerationof
gravitationalwaves.HeisarecipientofanIBMSupercomputingAward,aForefronts
ofLarge-ScaleComputationAward,anAlfredP.SloanResearchFellowship,aJohn
SimonGuggenheimMemorialFoundationFellowship,andseveralteachingcitations.
He has served on the editorial boards of The Astrophysical Journal Letters and
ClassicalandQuantumGravity.HewaselectedFellowofboththeAmericanPhysical
SocietyandInstituteofPhysics(UK).
Numerical Relativity
Solving Einstein’s Equations on the Computer
THOMAS W. BAUMGARTE
BowdoinCollege
AND
STUART L. SHAPIRO
UniversityofIllinoisatUrbana-Champaign
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore,
São Paulo, Delhi, Dubai, Tokyo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521514071
© T. Baumgarte and S. Shapiro 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-72937-9 eBook (NetLibrary)
ISBN-13 978-0-521-51407-1 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Preface pagexi
Suggestionsforusingthisbook xvii
1 Generalrelativitypreliminaries 1
1.1 Einstein’sequationsin4-dimensionalspacetime 1
1.2 Blackholes 9
1.3 Oppenheimer–Volkoffsphericalequilibriumstars 15
1.4 Oppenheimer–Snydersphericaldustcollapse 18
2 The3+1decompostionofEinstein’sequations 23
2.1 Notationandconventions 26
2.2 Maxwell’sequationsinMinkowskispacetime 27
2.3 Foliationsofspacetime 29
2.4 Theextrinsiccurvature 33
2.5 TheequationsofGauss,CodazziandRicci 36
2.6 Theconstraintandevolutionequations 39
2.7 Choosingbasisvectors:theADMequations 43
3 Constructinginitialdata 54
3.1 Conformaltransformations 56
3.1.1 Conformaltransformationofthespatialmetric 56
3.1.2 Elementaryblackholesolutions 57
3.1.3 Conformaltransformationoftheextrinsic
curvature 64
3.2 Conformaltransverse-tracelessdecomposition 67
3.3 Conformalthin-sandwichdecomposition 75
3.4 Astepfurther:the“waveless”approximation 81
3.5 Mass,momentumandangularmomentum 83
4 Choosingcoordinates:thelapseandshift 98
4.1 Geodesicslicing 100
4.2 Maximalslicingandsingularityavoidance 103
4.3 Harmoniccoordinatesandvariations 111
v
vi Contents
4.4 Quasi-isotropicandradialgauge 114
4.5 Minimaldistortionandvariations 117
5 Mattersources 123
5.1 Vacuum 124
5.2 Hydrodynamics 124
5.2.1 Perfectgases 124
5.2.2 Imperfectgases 139
5.2.3 Radiationhydrodynamics 141
5.2.4 Magnetohydrodynamics 148
5.3 Collisionlessmatter 163
5.4 Scalarfields 175
6 Numericalmethods 183
6.1 Classificationofpartialdifferentialequations 183
6.2 Finitedifferencemethods 188
6.2.1 Representationoffunctionsandderivatives 188
6.2.2 Ellipticequations 191
6.2.3 Hyperbolicequations 200
6.2.4 Parabolicequations 209
6.2.5 Meshrefinement 211
6.3 Spectralmethods 213
6.3.1 Representationoffunctionsandderivatives 213
6.3.2 Asimpleexample 214
6.3.3 Pseudo-spectralmethodswithChebychevpolynomials 217
6.3.4 Ellipticequations 219
6.3.5 Initialvalueproblems 223
6.3.6 Comparisonwithfinite-differencemethods 224
6.4 Codevalidationandcalibration 225
7 Locatingblackholehorizons 229
7.1 Concepts 229
7.2 Eventhorizons 232
7.3 Apparenthorizons 235
7.3.1 Sphericalsymmetry 240
7.3.2 Axisymmetry 241
7.3.3 Generalcase:nosymmetryassumptions 246
7.4 Isolatedanddynamicalhorizons 249
8 Sphericallysymmetricspacetimes 253
8.1 Blackholes 256
8.2 Collisionlessclusters:stabilityandcollapse 266
8.2.1 Particlemethod 267
8.2.2 Phasespacemethod 289
Contents vii
8.3 Fluidstars:collapse 291
8.3.1 Misner–Sharpformalism 294
8.3.2 TheHernandez–Misnerequations 297
8.4 Scalarfieldcollapse:criticalphenomena 303
9 Gravitationalwaves 311
9.1 Linearizedwaves 311
9.1.1 Perturbationtheoryandtheweak-field,
slow-velocityregime 312
9.1.2 Vacuumsolutions 319
9.2 Sources 323
9.2.1 Thehighfrequencyband 324
9.2.2 Thelowfrequencyband 328
9.2.3 Theverylowandultralowfrequencybands 330
9.3 Detectorsandtemplates 331
9.3.1 Ground-basedgravitationalwave
interferometers 332
9.3.2 Space-baseddetectors 334
9.4 Extractinggravitationalwaveforms 337
9.4.1 Thegauge-invariantMoncriefformalism 338
9.4.2 TheNewman–Penroseformalism 346
10 Collapseofcollisionlessclustersinaxisymmetry 352
10.1 Collapseofprolatespheroidstospindlesingularities 352
10.2 Head-oncollisionoftwoblackholes 359
10.3 Diskcollapse 364
10.4 Collapseofrotatingtoroidalclusters 369
11 Recastingtheevolutionequations 375
11.1 Notionsofhyperbolicity 376
11.2 RecastingMaxwell’sequations 378
11.2.1 GeneralizedCoulombgauge 379
11.2.2 First-orderhyperbolicformulations 380
11.2.3 Auxiliaryvariables 381
11.3 Generalizedharmoniccoordinates 381
11.4 First-ordersymmetrichyperbolicformulations 384
11.5 TheBSSNformulation 386
12 Binaryblackholeinitialdata 394
12.1 Binaryinspiral:overview 395
12.2 Theconformaltransverse-tracelessapproach:Bowen–York 403
12.2.1 Solvingthemomentumconstraint 403
12.2.2 SolvingtheHamiltonianconstraint 405
12.2.3 Identifyingcircularorbits 407
viii Contents
12.3 Theconformalthin-sandwichapproach 410
12.3.1 Thenotionofquasiequilibium 410
12.3.2 Quasiequilibriumblackholeboundaryconditions 413
12.3.3 Identifyingcircularorbits 419
12.4 Quasiequilibriumsequences 421
13 Binaryblackholeevolution 429
13.1 Handlingtheblackholesingularity 430
13.1.1 Singularityavoidingcoordinates 430
13.1.2 Blackholeexcision 431
13.1.3 Themovingpuncturemethod 432
13.2 Binaryblackholeinspiralandcoalescence 436
13.2.1 Equal-massbinaries 437
13.2.2 Asymmetricbinaries,spinandblackholerecoil 445
14 Rotatingstars 459
14.1 Initialdata:equilibriummodels 460
14.1.1 Fieldequations 460
14.1.2 Fluidstars 461
14.1.3 Collisionlessclusters 471
14.2 Evolution:instabilitiesandcollapse 473
14.2.1 Quasiradialstabilityandcollapse 473
14.2.2 Bar-modeinstability 478
14.2.3 Blackholeexcisionandstellarcollapse 481
14.2.4 Viscousevolution 491
14.2.5 MHDevolution 495
15 Binaryneutronstarinitialdata 506
15.1 Stationaryfluidsolutions 506
15.1.1 Newtonianequationsofstationaryequilibrium 508
15.1.2 Relativisticequationsofstationaryequilibrium 512
15.2 Corotationalbinaries 514
15.3 Irrotationalbinaries 523
15.4 Quasiadiabaticinspiralsequences 530
16 Binaryneutronstarevolution 533
16.1 Peliminarystudies 534
16.2 Theconformalflatnessapproximation 535
16.3 Fullyrelativisticsimulations 545
17Binaryblackhole–neutronstars:initialdataandevolution562
17.1 Initialdata 565
17.1.1 Theconformalthin-sandwichapproach 565
17.1.2 Theconformaltransverse-tracelessapproach 572
