UNIVERSITAT DE LES ILLES BALEARS DEPARTAMENT DE FISICA Programa de Doctorat de Fisica Theoretical issues in Numerical Relativity simulations Tesi Doctoral Daniela Alic Director: Prof. Carles Bona 2009 El director de tesi Carles Bona Garcia, Catedr`atic de F`ısica Te`orica de la Universitat de les Illes Balears, adscrit al Depar- tament de F`ısica, certifica que aquesta tesi doctoral ha estat realit- zada pel Sra. Daniela Delia Alic, i perqu`equedi const`ancia escrita firma a Palma 16 de Juny de 2009, Prof. Carles Bona Daniela Alic Acknowledgments I would like to express my sincere gratitude to my thesis supervisor, Prof. Carles Bona, for his guidance and help throughout these four years. I would like to thank especially my collaborators Dr. Sascha Husa and Dr. Car- los Palenzuela, for many useful and valuable discussions, comments and sugges- tions, from which I learned so much during these years, for all their encouragement and support. I would like to thank Dr. Juan Barranco and Dr. Argelia Bernal for the collab- oration in boson star project, providing the initial data for the study of mixed states boson stars. I thank Dr. Cecilia Chirenti, for kindly sharing with us the code for calculating the frequencies of the unstable mixed state boson star configurations. I would also like to thank Frank Ohme for his participation in the gauge instabil- ities study, providing the Penrose diagrams for the slices. Many thanks to Carles Bona-Casas for his collaboration in the black hole projects. I thank all the members of the AEI Astrophysical Relativity Division and LSU Physics and Astronomy Department, for creating a such pleasant and stimulating research environment during my research stays in Germany and USA. I thank all my colleagues from the UIB Agencia EFE. Special thanks to my dear friend Raul Vicente, and to my precious little water turtle Zapatuki. I am grateful to my family for all their love and support. Daniela Alic Palma, June 2009. v Abstract In this thesis we address several analytical and numerical problems related with the study of general relativistic black holes and boson stars. The task of solving numerically the Einstein equations (Gab = κTab) has turned out to be a very complex problem. Various reductions to first-order-in- time hyperbolic systems appear in the literature, but there is no general recipe that prescribes the optimal technique for any given situation, which leads to a variety of formulations. In the first part of this thesis, we present an analytical and numerical compar- ison between three different formulations of the Einstein equations. A detailed analysis of these systems is performed, marking the weak points and proposing improvements, in the form of constraint adjustments and damping terms. Black holes are considered to be some of the most interesting astrophysical compact objects. They are vacuum solutions of the Einstein equations. The chal- lenge of dealing with black hole (BH) simulations comes from the fact that they hide a space-time singularity, a point where the attraction becomes so intense that an observer would get trapped and absorbed into it. As a consequence, one of the main problems that needed to be overcome were the steep gradients appearing around the BH apparent horizon, marking the region between the outer nearly in- ertial wave zone and the highly accelerated behavior of the inner plunging zone. To this purpose, we developed a new centered finite volume (CFV) method based on the flux splitting approach. This algorithm is the first one in the class of fi- nite volume methods which allows third order accuracy by only piece-wise linear reconstruction. The finite volume methods are commonly used in the numerical study of rel- ativistic astrophysical systems which contain matter sources, in order to deal with shocks or any other type of discontinuities. However, in most cases one does not require the use of limiters and the CFV method can be efficiently used in the form of an adaptive dissipation algorithm, in order to deal with the steep gradients. We present a comparison between our CFV method and the standard finite difference plus dissipation techniques, and show that our method allows longer and more accurate BH evolutions, even at low resolutions. In this thesis, we discuss the techniques for dealing with the singularity, steep vii viii Abstract gradients and apparent horizon location, in the context of a single Schwarzschild BH, in both spherically symmetric and full 3D simulations. Our treatment of the singularity involves scalar field stuffing, which consists in matching a scalar field in the inner region of the BH, such that the metric becomes regular inside the horizon. Additionally, for comparison, we appeal to the puncture technique, which reduces the singularity to a point, while the interior BH region is maintained sufficiently regular for numerical purposes. Even though the singularity is no longer a problem in the initial data, it can become a problem in a finite amount of time, if one does not choose suitable coordinate conditions. We perform BH evolutions using the ’1+log’ singularity avoiding slicing, which ensures that the coordinate time rate is slowing down in the strongly col- lapsing regions, but it keeps flowing at the same rate as proper time in the wave zone. In this context, we develop a geometrical picture of the slicings approaching the stationary state, for situations where the treatment of the singularity involves both scalar field stuffing and the puncture technique. Our 3D numerical results show the first long term simulation of a Schwarzschild BH in normal coordinates, without the need to excise the singularity from the computational domain. The family of singularity avoiding slicing conditions which are currently used in BH evolutions, have been shown to produce gauge instabilities. We extend this study and show that, contrary to previous claims, these instabilities are not generic for evolved gauge conditions. We follow the behavior of the slicing in evolutions of Schwarzschild spacetime and perform a detailed study of the pathologies which can arise from two models: perturbing the initial slice and perturbing the initial lapse. A comparison with the results available in the literature allows us to identify most instabilities and propose a cure. Regarding the choice of space coordinate conditions, we developed an alterna- tive to the current prescriptions, based on a generalized Almost Killing Equation (AKE). This condition is expected to adapt the coordinates to the symmetry of the problem under study. The 3-covariant AKE shift can be used in combination with any slicing, without loosing its quasi-stationary properties. Our numerical tests address harmonic and black hole spacetimes. Our research work extends also to the study of regular spacetimes with mat- ter. We explore boson star configurations as dark matter models and focus on Mixed State Boson Stars (MSBS) configurations constructed in the context of General Relativity. Contrary to previous studies, where bosons populate only the ground state, in our case different excited states are coexisting simultaneously. We performed the first general relativistic study of MSBS configurations, using the Einstein-Klein-Gordon system in spherical symmetry. Following the evolution of MSBS under massless scalar field perturbations, we identify the unstable models and find a criteria of separation between stable and unstable configurations. Our conclusions regarding the long term stability of MSBS configurations, suggest that they can be suitable candidates for dark matter models. Contents I Introduction 1 1 Overview 3 1.1 ThesisOrganization......................... 7 1.2 Conventions............................. 9 2 General Concepts in Relativity 13 2.1 Geometrical Concepts . 13 2.1.1 Notions of Local Differential Geometry . 13 2.1.2 SpacetimeGeometry . 15 2.1.3 TheFieldEquations . 16 2.1.4 Elements of 3+1 Decomposition . 18 2.2 The 3+1 Form of the Einstein Equations . 20 2.2.1 Basic Geometrical Objects . 20 2.2.2 Evolution Equations . 21 2.2.3 Constraint Equations . 22 2.2.4 Gauge Degrees of Freedom . 25 2.3 Well-Posed Evolution Problems . 28 2.3.1 Well-Posed Systems . 28 2.3.2 Strongly Hyperbolic Systems . 29 2.3.3 Boundary Conditions . 30 II Formulations of the Einstein Equations 33 3 Einstein Evolution Systems 35 3.1 The 3+1 Metric based Systems . 36 3.1.1 TheZSystems ....................... 36 3.1.2 TheBSSNSystem ..................... 42 3.2 The 3+1 Tetrad based Systems . 44 3.2.1 Notions of Frame Formalism . 44 ix x CONTENTS 3.2.2 TheFNSystem....................... 48 3.3 Discussion.............................. 50 4 Standard Testbeds for Numerical Relativity 53 4.1 Overview of Numerical Tests . 53 4.2 Implementation and Results . 54 4.2.1 The Linear Wave Testbed . 55 4.2.2 The Gauge Wave Testbed . 58 4.2.3 The Shifted Gauge Wave Testbed . 62 4.2.4 OtherTests ......................... 65 4.3 Discussion.............................. 66 III Numerical Methods and Applications 69 5 Numerical Aspects 71 5.1 Standard Numerical Recipes . 71 5.1.1 Space discretization and Time integration . 71 5.1.2 Convergence and Stability . 73 5.2 Centered Finite Volume Methods . 74 5.2.1 FluxFormulae ....................... 75 5.2.2 Flux Splitting Approach . 79 5.2.3 Adaptive Dissipation . 82 5.2.4 Stability and Monotonicity . 83 5.3 Discussion.............................. 84 6 Black Hole Simulations 87 6.1 Black Hole in Spherical Symmetry . 88 6.1.1 Puncture Initial Data . 88 6.1.2 Numerical Specifications and Gauge Choice . 91 6.1.3 Numerical Results and Comparison . 92 6.1.4 Convergence and Error . 95 6.1.5 Discussion ......................... 98 6.2 BlackHolein3D .......................... 99 6.2.1 ScalarFieldStuffing . 99 6.2.2 Black Hole Evolution . 104 6.2.3 Discussion ......................... 109 7 Boson Stars 111 7.1 Theoretical Aspects . 112 7.1.1 The Einstein-Klein-Gordon System .