A (3+1)-evolutionary method in the framework of Regge Calculus, known as "Parallelisable Implicit Evolutionary Scheme", is analysed and revised so that it accounts for causality. Furthermore, the ambiguities associated with the notion of time in this evolutionary scheme are addressed and a solution to resolving such ambiguities is presented. The revised algorithm is then numerically tested and shown to produce the desirable results and indeed to resolve a problem previously faced upon implementing this scheme. An important issue that has been overlooked in "Parallelisable Implicit Evolutionary Scheme" was the restrictions on the choice of edge lengths used to build the space-time lattice as it evolves in time. It is essential to know what inequalities must hold between the edges of a 4-dimensional simplex, used to construct a space-time, so that the geometry inside the simplex is Minkowskian. The only known inequality on the Minkowski plane is the "Reverse Triangle Inequality" which holds between the edges of a triangle constructed only from space-like edges. However, a triangle, on the Minkowski plane, can be built from a combination of time-like, space-like or null edges. Part of this thesis is concerned with deriving a number of inequalities that must hold between the edges of mixed triangles. Finally, the Raychaudhuri equation is considered from the point of view of Regge Calculus. The Raychaudhuri equation plays an important role in many areas of relativistic Physics and Astrophysics, most importantly in the proof of singularity theorems. An analogue to the Raychaudhuri equation in the framework of Regge Calculus is derived. Both (2+1)-dimensional and (3+1)-dimensional cases are considered and analogues for average expansion and shear scalar are found.

A (3+1)-evolutionary method in the framework of Regge Calculus, known as "Parallelisable Implicit Evolutionary Scheme", is analysed and revised so that it accounts for causality. Furthermore, the ambiguities associated with the notion of time in this evolutionary scheme are addressed and a solution to resolving such ambiguities is presented. The revised algorithm is then numerically tested and shown to produce the desirable results and indeed to resolve a problem previously faced upon implementing this scheme. An important issue that has been overlooked in "Parallelisable Implicit Evolutionary Scheme" was the restrictions on the choice of edge lengths used to build the space-time lattice as it evolves in time. It is essential to know what inequalities must hold between the edges of a 4-dimensional simplex, used to construct a space-time, so that the geometry inside the simplex is Minkowskian. The only known inequality on the Minkowski plane is the "Reverse Triangle Inequality" which holds between the edges of a triangle constructed only from space-like edges. However, a triangle, on the Minkowski plane, can be built from a combination of time-like, space-like or null edges. Part of this thesis is concerned with deriving a number of inequalities that must hold between the edges of mixed triangles. Finally, the Raychaudhuri equation is considered from the point of view of Regge Calculus. The Raychaudhuri equation plays an important role in many areas of relativistic Physics and Astrophysics, most importantly in the proof of singularity theorems. An analogue to the Raychaudhuri equation in the framework of Regge Calculus is derived. Both (2+1)-dimensional and (3+1)-dimensional cases are considered and analogues for average expansion and shear scalar are found.

A new method for generating initial data for simulations of neutron stars in binary systems. The construction of physically relevant initial data is crucial to accurate assessment of gravitational wave signals relative to theoretical predictions. This method builds upon the Bowen-York curvature for puncture black holes. This data is evolved and compared against simulations in the literature with respect to orbital eccentricity, merger and collapse times, and emitted energy and angular momentum. The data exhibits some defects, including large central density oscillations in stars and center of mass drift in unequal-mass systems. Some approaches for improvements in potential future work are discussed.

One of the most important predictions of General Relativity, Einstein’s theory of gravity, is the existence of gravitational radiation. The strongest source of such radiation is expected to come from the merging of black holes. Upgrades to large ground based interferometric detectors (LIGO, VIRGO, GEO 600) have increased their sensitivity to the point that the first direct observation of a gravitational wave is expected to occur within the next few years. The chance of detection is greatly improved by the use of simulated waveforms which can be used as templates for signal processing. Recent advances in numerical relativity have allowed for long stable evolution of black hole mergers and the generation of expected waveforms. openGR is a modular, open framework black hole evolution code developed at The University of Texas at Austin Center for Relativity. Based on the BSSN (strongly hyperbolic) formulation of Einstein’s equations and the moving puncture method, we are able to model the evolution of a binary black hole system through the merger and extract the gravitational radiation produced. Although we are generally interested in binary interactions, openGR is capable of handling any number of black holes. This work serves as an overview of the capabilities of openGR and a demonstration of the physics it can be used to explore. / text

The study of general relativity in higher dimensions has proven to be a fruitful avenue of research, revealing new applications of the theory, for instance in understanding strongly coupled quantum field theories through the holographic principle, and proposing an explanation of the hierarchy problem through TeV gravity scenarios. To understand the non-linear regime of higher dimensional general relativity, such as that involved in the merger of black holes, we use numerical relativity to solve the Einstein equations. In this thesis we develop and demonstrate several diagnostic tools and new initial data for use in numerical relativity simulations of higher dimensional spacetimes, and use these to investigate binary black hole systems. Firstly, we present a formalism for calculating the gravitational waves in a numerical simulation of a higher dimensional spacetime, and apply this formalism to the example of the head on merger of two equal mass black holes. In doing so, we simulate the merger of black holes in up to 10 spacetime dimensions for the first time, and investigate the dependence of the energy radiated away in gravitational waves on the number of dimensions. We also apply this formalism to the example of head on unequal mass black hole collisions, investigating the dependence of radiated energy and momentum on the number of dimensions and the mass ratio. This study complements and sheds further light on previous work on the merger of point particles with black holes in higher dimensions, and presents evidence for a link between the regime studied, and the large $D$ regime of general relativity where $D$ is the number of spacetime dimensions. We also present initial data that enables us to study black holes with initial momentum and angular momentum, putting in place the framework needed to study problems such as the scattering cross section of black holes in higher dimensions, and the nature of black hole orbits in higher dimensions. Finally, we present, and demonstrate the use of, an apparent horizon finder for higher dimensional spacetimes. This allows us to calculate a black hole's mass and spin, which characterise the black hole.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第20169号 / 理博第4254号 / 新制||理||1612(附属図書館) / 京都大学大学院理学研究科物理学・宇宙物理学専攻 / (主査)教授 柴田 大, 教授 川合 光, 教授 井岡 邦仁 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DGAM

吀e so-called "trumpet" initial data has recently received mu挀 a琀ention as a potential candidate for the natural black hole initial data to be used in 3+1 numerical relativity simulations with 1+log foliation. In this work we first derive a variant of the maximal trumpet initial data that is made to move on the numerical grid by the means of a Lorentz boost and write a numerical code that constructs this boosted trumpet initial data. We also write a numerical code for calculating the Krets挀mann scalar from the 3+1 variables, to be used in analysing the data from our simulations. With the help of those two codes, we study the behaviour of the boosted trumpet initial data when evolved with the BSSN formulation of the Einstein equations, using 1+log slicing and the Γ-driver shi昀 condition.

General relativity, one of the pillars of our understanding of the universe, has been a remarkably successful theory. It has stood the test of time for more than 100 years and has passed all experimental tests so far. Most recently, the LIGO collaboration made the first-ever direct detection of gravitational waves, confirming a long-standing prediction of general relativity. Despite this, several fundamental mathematical questions remain unanswered, many of which relate to the global existence and the stability of solutions to Einstein's equations. This thesis presents our efforts to use numerical relativity to investigate some of these questions. We present a complete picture of the end points of black ring instabilities in five dimensions. Fat rings collapse to Myers-Perry black holes. For intermediate rings, we discover a previously unknown instability that stretches the ring without changing its thickness and causes it to collapse to a Myers-Perry black hole. Most importantly, however, we find that for very thin rings, the Gregory-Laflamme instability dominates and causes the ring to break. This provides the first concrete evidence that in higher dimensions, the weak cosmic censorship conjecture may be violated even in asymptotically flat spacetimes. For Myers-Perry black holes, we investigate instabilities in five and six dimensions. In six dimensions, we demonstrate that both axisymmetric and non-axisymmetric instabilities can cause the black hole to pinch off, and we study the approach to the naked singularity in detail. Another question that has attracted intense interest recently is the instability of anti-de Sitter space. In this thesis, we explore how breaking spherical symmetry in gravitational collapse in anti-de Sitter space affects black hole formation. These findings were made possible by our new open source general relativity code, GRChombo, whose adaptive mesh capabilities allow accurate simulations of phenomena in which new length scales are produced dynamically. In this thesis, we describe GRChombo in detail, and analyse its performance on the latest supercomputers. Furthermore, we outline numerical advances that were necessary for simulating higher dimensional black holes stably and efficiently.

Diese Doktorarbeit behandelt neue Methoden der numerischen Evolution von Systemen mit binären Schwarzen Löchern. Wir analysieren und vergleichen Evolutionen von verschiedenen physikalisch motivierten Anfangsdaten und zeigen Resultate der ersten Evolution von so genannten 'Thin Sandwich' Daten, die von der Gruppe in Meudon entwickelt wurden. <br /> Zum ersten Mal wurden zwei verschiedene Anfangsdaten anhand von dreidimensionalen Evolutionen verglichen: die Puncture-Daten und die Thin-Sandwich Daten. Diese zwei Datentypen wurden im Hinblick auf die physikalischen Eigenschaften während der Evolution verglichen. <br /> Die Evolutionen zeigen, dass die Meudon Daten im Vergleich zu Puncture Daten wesentlich mehr Zeit benötigen bevor sie kollidieren. Dies deutet auf eine bessere Abschätzung der Parameter hin. Die Kollisionszeiten der numerischen Evolutionen sind konsistent mit unabhängigen Schätzungen basierend auf Post-Newtonschen Näherungen die vorhersagen, dass die Schwarzen Löcher ca. 60% eines Orbits rotieren bevor sie kollidieren. / This thesis presents new approaches to evolutions of binary black hole systems in numerical relativity. We analyze and compare evolutions from various physically motivated initial data sets, in particular presenting the first evolutions of Thin Sandwich data generated by the Meudon group. <br /> For the first time two different quasi-circular orbit initial data sequences are compared through fully 3d numerical evolutions: Puncture data and Thin Sandwich data (TSD) based on a helical killing vector ansatz. The two different sets are compared in terms of the physical quantities that can be measured from the numerical data, and in terms of their evolutionary behavior. <br /> The evolutions demonstrate that for the latter, "Meudon" datasets, the black holes do in fact orbit for a longer amount of time before they merge, in comparison with Puncture data from the same separation. This indicates they are potentially better estimates of quasi-circular orbit parameters. The merger times resulting from the numerical simulations are consistent with independent Post-Newtonian estimates that the final plunge phase of a black hole inspiral should take 60% of an orbit.

This thesis is concerned with the development of numerical methods using finite difference techniques for the discretization of initial value problems (IVPs) and initial boundary value problems (IBVPs) of certain hyperbolic systems which are first order in time and second order in space. This type of system appears in some formulations of Einstein equations, such as ADM, BSSN, NOR, and the generalized harmonic formulation. For IVP, the stability method proposed in [14] is extended from second and fourth order centered schemes, to 2n-order accuracy, including also the case when some first order derivatives are approximated with off-centered finite difference operators (FDO) and dissipation is added to the right-hand sides of the equations. For the model problem of the wave equation, special attention is paid to the analysis of Courant limits and numerical speeds. Although off-centered FDOs have larger truncation errors than centered FDOs, it is shown that in certain situations, off-centering by just one point can be beneficial for the overall accuracy of the numerical scheme. The wave equation is also analyzed in respect to its initial boundary value problem. All three types of boundaries - outflow, inflow and completely inflow that can appear in this case, are investigated. Using the ghost-point method, 2n-accurate (n = 1, 4) numerical prescriptions are prescribed for each type of boundary. The inflow boundary is also approached using the SAT-SBP method. In the end of the thesis, a 1-D variant of BSSN formulation is derived and some of its IBVPs are considered. The boundary procedures, based on the ghost-point method, are intended to preserve the interior 2n-accuracy. Numerical tests show that this is the case if sufficient dissipation is added to the rhs of the equations. / Diese Doktorarbeit beschäftigt sich mit der Entwicklung numerischer Verfahren für die Diskretisierung des Anfangswertproblems und des Anfangs-Randwertproblems unter Einsatz von finite-Differenzen-Techniken für bestimmte hyperbolischer Systeme erster Ordnung in der Zeit und zweiter Ordnung im Raum. Diese Art von Systemen erscheinen in einigen Formulierungen der Einstein'schen-Feldgleichungen, wie zB. den ADM, BSSN oder NOR Formulierungen, oder der sogenanten verallgemeinerten harmonischen Darstellung. Im Hinblick auf das Anfangswertproblem untersuche ich zunächst tiefgehend die mathematischen Eigenschaften von finite-Differenzen-Operatoren (FDO) erster und zweiter Ordnung mit 2n-facher Genaugigkeit. Anschließend erweitere ich eine in der Literatur beschriebene Methode zur Stabilitätsanalyse für Systeme mit zentrierten FDOs in zweiter und vierter Genauigkeitsordung auf Systeme mit gemischten zentrierten und nicht zentrierten Ableitungsoperatoren 2n-facher Genauigkeit, eingeschlossen zusätzlicher Dämpfungsterme, wie sie bei numerischen Simulationen der allgemeinen Relativitätstheorie üblich sind. Bei der Untersuchung der einfachen Wellengleichung als Fallbeispiel wird besonderes Augenmerk auf die Analyse der Courant-Grenzen und numerischen Geschwindigkeiten gelegt. Obwohl unzentrierte, diskrete Ableitungsoperatoren größere Diskretisierungs-Fehler besitzen als zentrierte Ableitungsoperatoren, wird gezeigt, daß man in bestimmten Situationen eine Dezentrierung des numerischen Moleküls von nur einem Punkt bezüglich des zentrierten FDO eine höhere Genauigkeit des numerischen Systems erzielen kann. Die Wellen-Gleichung in einer Dimension wurde ebenfalls im Hinblick auf das Anfangswertproblem untersucht. In Abhängigkeit des Wertes des sogenannten Shift-Vektors, müssen entweder zwei (vollständig eingehende Welle), eine (eingehende Welle) oder keine Randbedingung (ausgehende Welle) definiert werden. In dieser Arbeit wurden alle drei Fälle mit Hilfe der 'Ghost-point-methode' numerisch simuliert und untersucht, und zwar auf eine Weise, daß alle diese Algorithmen stabil sind und eine 2n-Genauigkeit besitzen. In der 'ghost-point-methode' werden die Evolutionsgleichungen bis zum letzen Punkt im Gitter diskretisiert unter Verwendung von zentrierten FDOs und die zusätzlichen Punkte die am Rand benötigt werden ('Ghost-points') werden unter Benutzung von Randwertbedingungen und Extrapolationen abgeschätzt. Für den Zufluß-Randwert wurde zusätzlich noch eine andere Implementierung entwickelt, welche auf der sogenannten SBP-SAT (Summation by parts-simulatanous approximation term) basiert. In dieser Methode werden die diskreten Ableitungen durch Operatoren angenähert, welche die 'Summation-by-parts' Regeln erfüllen. Die Randwertbedingungen selber werden in zusätzlichen Termen integriert, welche zu den Evolutionsgleichnungen der Punkte nahe des Randes hinzuaddiert werden und zwar auf eine Weise, daß die 'summation-by-parts' Eigenschaften erhalten bleiben. Am Ende dieser Arbeit wurde noch eine eindimensionale (kugelsymmetrische) Version der BSSN Formulierung abgeleitet und einige physikalisch relevanten Anfangs-Randwertprobleme werden diskutiert. Die Randwert-Algorithmen, welche für diesen Fall ausgearbeitet wurden, basieren auf der 'Ghost-point-Methode' and erfüllen die innere 2n-Genauigkeit solange genügend Reibung in den Gleichungen zugefügt wird.