Gauge fixing for the simulation of black hole spacetimes [Elektronische Ressource] / vorgelegt von Erik Schnetter

Sujets

• Physik
• Astronomie

Informations

• Publié par: eberhard_karls_universitat_tubingen
• Publié le: 01 janvier 2003
• Nombre de lectures: 6
• Langue: English
• Poids de l'ouvrage: 1 Mo

Extrait

Gaugefixingforthesimulation
ofblackholespacetimes
Dissertation
zurErlangungdesGradeseinesDoktors
derNaturwissenschaften
derFakultat¨ fur¨ MathematikundPhysik
derEberhard–Karls–Universitat¨ zuTubingen¨
vorgelegtvon
ErikSchnetter
ausLetmathe
2003Tagdermundlichen¨ Prufung:¨ 13. Juni2003
Dekan: Prof. Dr. HerbertMuther¨
1. Berichterstatter: Prof. Dr. HannsRuder
2. Priv. Doz. Dr. J or¨ gFrauendiener
2Abstract
I consider the initial boundary value problem of nonlinear general
relativistic vacuum spacetimes, which today cannot yet be evolved
numerically in a satisfactory manner. Specifically, I look at gauge
conditions, classifying them into gauge evolution conditions and
gauge fixing conditions. In this terminology, a gauge fixing condi
tion is a condition that removes all gauge degrees of freedom from
asystem,whereasagaugeevolutionconditiondeterminesonlythe
time evolution of the gauge condition, while the gauge condition
itself remains unspecified. I find that most of today’s gauge condi
tionsareonlygaugeevolutionconditions.
I present a system of evolution equations containing a gauge fix
ing condition, and describe an efficient numerical implementation
using constrained evolution. I examine the behaviour of
this system for several test problems, such as linear gravitational
waves or nonlinear gauge waves. I find that the system is robustly
stableandsecond orderconvergent. Ithenapplyittomorerealistic
configurations, such as Brill waves or single black holes, where the
systemisalsostableandaccurate.
34Acknowledgements
Ihavegreatlyenjoyedalltheresearchthathasgoneintothisthesis.
ThisenjoymentcomesnotleastfromtheinteractionthatIhavehad
with my supervisors and colleagues, and the liberal and inspiring
atmospheresthattheycreatedintheirdepartments.
It was certainly luck that brought me to the Institut fur¨ Theore
tische Astrophysik in Tubingen.¨ In countless discussions with my
colleaguesItherelearnedthejoythatliesinacademicdiscourseand
inthediscoveryofnewandoldideas. MysupervisorHannsRuder
hastherareabilityofcapturinghislisteners’wholeattentionbyex
plaining physics in a most interesting manner. He also encouraged
metospendsometimeabroad.
The Deutsche Akademische Austauschdienst then granted me a
stipend that allowed me to spend one year with Pablo Laguna at
the Department of Astronomy in Penn State. He introduced me to
numerical relativity, his research inspired the topic of this thesis,
and his Latin mentality taught me how to defend my ideas. Mijan
HuqandDeirdreShoemakerpatientlyansweredmyquestions,and
IhadmanyinterestingdiscussionswithmyfellowstudentsBernard
KellyandKenSmith,especiallyonThursdays.
After returning to Tubingen,¨ Hanns Ruder gave me very gener-
ous financial support, and also gave me the paternal nudging that
is necessary to finish a thesis. I would also like to thank Jor¨ g Frau
endienerandUteKrausfortheircompetentadvice,andDanielKo
bras and Stefan Kunze for making the important hardware in our
departmentwork.
It is almost by their nature that these acknowledgements are in
5complete,andIapologisetothemanypeoplewhomIhavetothank
inpersoninstead.
6Contents
Abstract 3
Acknowledgements 5
Table of contents 7
1. Introduction 11
2. Simulating spacetimes 15
2.1. TheADMformalism . . . . . . . . . . . . . . . . . . . 15
2.2. Constrainedevolution . . . . . . . . . . . . . . . . . . 17
2.3. Picturingspacetimes . . . . . . . . . . . . . . . . . . . 18
2.4. Interpretingthequantities . . . . . . . . . . . . . . . . 21
2.5. Instabilities . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6. Mygoalinthisthesis . . . . . . . . . . . . . . . . . . . 25
3. Gauge conditions 27
3.1. Commongaugeconditions . . . . . . . . . . . . . . . 28
3.2. Gaugeevolutionandgaugefixing . . . . . . . . . . . 29
3.3. Goodgaugefixingconditions . . . . . . . . . . . . . . 32
3.4. Mygaugefixingcondition . . . . . . . . . . . . . . . . 33
4. The system of equations 37
4.1. Thevariables . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2. Enforcingthegaugeandtheconstraints . . . . . . . . 38
7Contents
5. Boundary conditions 41
5.1. Outerboundary . . . . . . . . . . . . . . . . . . . . . . 41
5.1.1. Locationoftheouterboundary . . . . . . . . . 41
5.1.2. Physicalboundaryconditions . . . . . . . . . . 42
5.1.3. Periodicity . . . . . . . . . . . . . . . . . . . . . 43
5.1.4. Gaugeandconstraintboundaryconditions . . 44
5.2. Excisionboundary . . . . . . . . . . . . . . . . . . . . 46
5.2.1. Locationandshapeoftheexcisionboundary . 48
5.2.2. Boundaryconditions . . . . . . . . . . . . . . . 49
6. Initial data 53
6.1. Datawithoutblackholes . . . . . . . . . . . . . . . . . 54
6.1.1. Minkowskimetric(flatspace) . . . . . . . . . . 54
6.1.2. WeakBondiwave(linearplanarwave) . . . . 54
6.1.3. Gaugepulse(nonlineargaugewave) . 55
6.1.4. Brillwave . . . . . . . . . . . . . . . . . . . . . 56
6.2. Singleblackholedata . . . . . . . . . . . . . . . . . . . 57
6.2.1. Kerr–Schildcoordinates . . . . . . . . . . . . . 57
6.2.2. Painleve–Gullstrand´ coordinates . . . . . . . . 59
6.2.3. Harmoniccoordinates . . . . . . . . . . . . . . 59
6.2.4. Coordinatetransformations . . . . . . . . . . . 60
6.3. Multipleblackholedata . . . . . . . . . . . . . . . . . 62
6.3.1. Brill–Lindquistdata . . . . . . . . . . . . . . . 62
6.3.2. SuperposedKerr–Schilddata . . . . . . . . . . 63
7. Code 67
7.1. TheTigercode . . . . . . . . . . . . . . . . . . . . . . . 67
7.2. Initialdata . . . . . . . . . . . . . . . . . . . . . . . . . 68
7.3. Constraintsolvers . . . . . . . . . . . . . . . . . . . . . 69
7.4. Timeintegration . . . . . . . . . . . . . . . . . . . . . . 70
7.5. Boundaryconditions . . . . . . . . . . . . . . . . . . . 71
7.5.1. Outerboundary . . . . . . . . . . . . . . . . . . 71
7.5.2. Excision . . . . . . . . . . . . . . . . 71
7.6. Analysisroutines . . . . . . . . . . . . . . . . . . . . . 72
8Contents
8. Results 75
8.1. Convergencetests . . . . . . . . . . . . . . . . . . . . . 75
8.1.1. Staticandstationarytests: blackholes . . . . . 75
8.1.2. Dynamiclineartest: weakBondiwave . . . . 80
8.1.3.nonlineartest: gaugepulse. . . . . . 84
8.2. Stabilitytest . . . . . . . . . . . . . . . . . . . . . . . . 86
8.3. Brillwave . . . . . . . . . . . . . . . . . . . . . . . . . 88
8.4. Kerr–Schildblackhole . . . . . . . . . . . . . . . . . . 100
9. Conclusion 105
A. Equations 107
A.1. TheADMformalism . . . . . . . . . . . . . . . . . . . 107
A.1.1. Variables . . . . . . . . . . . . . . . . . . . . . . 107
A.1.2. Timeevolution . . . . . . . . . . . . . . . . . . 108
A.1.3. Riccitensor . . . . . . . . . . . . . . . . . . . . 108
A.1.4. Constraints . . . . . . . . . . . . . . . . . . . . 109
A.2. TheTGRsystem . . . . . . . . . . . . . . . . . . . . . . 109
A.2.1. Variables . . . . . . . . . . . . . . . . . . . . . . 110
A.2.2. Timeevolution . . . . . . . . . . . . . . . . . . 110
A.2.3. Constraints . . . . . . . . . . . . . . . . . . . . 112
A.2.4. Enforcingtheconstraints . . . . . . . . . . . . 112
A.2.5. Gaugecondition . . . . . . . . . . . . . . . . . 113
A.2.6. Enforcingthegaugecondition . . . . . . . . . 114
A.2.7. Determininglapseandshift . . . . . . . . . . . 115
B. Numerics 117
B.1. Spatialdiscretisation . . . . . . . . . . . . . . . . . . . 117
B.2. Timeintegration . . . . . . . . . . . . . . . . . . . . . . 117
B.2.1. Artificialviscosity . . . . . . . . . . . . . . . . 118
B.3. Ellipticintegration . . . . . . . . . . . . . . . . . . . . 119
B.3.1. Variabletransformations . . . . . . . . . . . . . 120
B.3.2. Ellipticsolvers . . . . . . . . . . . . . . . . . . 121
B.4. Codingequations . . . . . . . . . . . . . . . . . . . . . 122
9Contents
C. De nitions 125
C.1. Glossaryofterms . . . . . . . . . . . . . . . . . . . . . 125
C.2. Abbreviations . . . . . . . . . . . . . . . . . . . . . . . 126
C.3. Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Bibliography 131
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