bona c., palenzuela-luque c. elements of numerical relativity

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Carles Bona Carlos Palenzuela-Luque Elements of Numerical Relativity From Einstein’s Equations to Black Hole Simulations ABC Authors Carles Bona Universidad Islas Baleares Facultat Ciencias Depto. Fisica 07071 Palma de Mallorca Spain Email: cbona@uib.es Carlos Palenzuela-Luque Louisiana State University Center for Computation and Technology Baton Rouge, LA 70803 U.S.A. Email: carlos@lsu.edu Carles Bona, Carlos Palenzuela-Luque Elements of Numerical Relativity, Lect. Notes Phys. 673 (Springer, Berlin Heidelberg 2005), DOI 10.1007/b135928 Library of Congress Control Number: 2005926241 ISSN 0075-8450 ISBN-10 3-540-25779-9 Springer Berlin Heidelberg New York ISBN-13 978-3-540-25779-0 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com c  Springer-Verlag Berlin Heidelberg 2005 Printed in The Netherlands The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the author and TechBooks using a Springer L A T E X macro package Printed on acid-free paper SPIN: 11420293 56/TechBooks 543210 To Montse, my dear wife and friend Para mis padres, Francisco y Manuela, que me ense˜naron lo mas importante. Y para mi amigo Jose y mi querida Eugenia, que no han dejado que lo olvidara Preface We became involved with numerical relativity under very different circum- stances. For one of us (C.B.) it dates back to about 1987, when the current Laser-Interferometer Gravitational Wave Observatories were just promising proposals. It was during a visit to Paris, at the Institut Henri Poincar´e, where some colleagues were pushing the VIRGO proposal with such a contagious enthusiasm that I actually decided to reorient my career. The goal was to be ready, armed with a reliable numerical code, when the first detection data would arrive. Allowing for my experience with the 3+1 formalism at that time, I started working on singularity-avoidant gauge conditions. Soon, I became interested in hyperbolic evolution formalisms. When trying to get some practical appli- cations, I turned to numerical algorithms (a really big step for a theoretically oriented guy) and black hole initial data. More recently, I became interested in boundary conditions and, closing the circle, again in gauge conditions. The problem is that a reliable code needs all these ingredients to be working fine at the same time. It is like an orchestra, where strings, woodwinds, brass and percussion must play together in a harmonic way: a violin virtuoso, no matter how good, cannot play Vivaldi’s Four Seasons by himself. During that time, I have had many Ph.D. students. The most recent one is the other of us (C.P.). All of them started with some specific topic, but they needed a basic knowledge of all the remaining ones: you cannot work on the saxophone part unless you know what the bass is supposed to play at the same time. This is where this book can be of a great help. Imagine a beginning gradu- ate student armed only with a home PC. Imagine that the objective is to build a working numerical code for simple black-hole applications. This book should first provide him or her with a basic insight into the most relevant aspects of numerical relativity. But this is not enough; the book should also provide reliable and compatible choices for every component: evolution system, gauge, initial and boundary conditions, even for numerical algorithms. VIII Preface This pragmatic orientation may cause this book to be seen as biased. But the idea was not to produce a compendium of the excellent work that has been made in numerical relativity during these years. The idea is rather to present a well-founded and convenient way for a beginner to get into the field. He or she will quickly discover everything else. The structure of the book reflects the peculiarities of numerical relativity research: • It is strongly rooted in theory. Einstein’s relativity is a general-covariant theory. This means that we are building at the same time the solution and the coordinate system, a unique fact among physical theories. This point is stressed in the first chapter, which could be omitted by more experienced readers. • It turns the theory upside down. General covariance implies that no specific coordinate is more special than the others, at least not a priori. But this is at odds with the way humans and computers usually model things: as functions (of space) that evolve in time. The second chapter is devoted to the evolution (or 3+1) formalism, which reconciles general relativity with our everyday perception of reality, in which time plays such a distinct role. • It is a fertile domain, even from the theoretical point of view. The structure of Einstein’s equations allows many ways of building well-posed evolution formalisms. Chapter 3 is devoted to those which are of first order in time but second order in space. Chapter 4 is devoted instead to those which are of first order both in time and in space. In both cases, suitable numerical algorithms are provided, although the most advanced ones apply mainly to the fully first order case. • It is challenging. The last sections of Chaps. 5 and 6 contain front- edge developments on constraint-preserving boundary conditions and gauge pathologies, respectively. These are very active research topics, where new developments will soon improve on the ones presented here. The prudent reader is encouraged to look for updates of these front-edge areas in the current scientific literature. A final word. Numerical relativity is not a matter of brute force. Just a PC, not a supercomputer, is required to perform the tests and applications proposed here. Numerical relativity is instead a matter of insight. Let wisdom be with you. Palma de Mallorca, Carles Bona April 2005 Carlos Palenzuela Luque Contents 1 The Four-Dimensional Spacetime 1 1.1 Spacetime Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 The Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.2 General Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.3 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.1.5 Symmetries of the Curvature Tensor . . . . . . . . . . . . . . . . . 6 1.2 General Covariant Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.1 The Stress-Energy Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Structure of the Field Equations . . . . . . . . . . . . . . . . . . . . 9 1.3 Einstein’s Equations Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.1 Symmetries. Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 12 1.3.2 Exact Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.3.3 Analytical and Numerical Approximations . . . . . . . . . . . . 16 2 The Evolution Formalism 19 2.1 Space Plus Time Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.1.1 A Prelude: Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . 20 2.1.2 Spacetime Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.3 The Eulerian Observers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.2 Einstein’s Equations Decomposition . . . . . . . . . . . . . . . . . . . . . . . 25 2.2.1 The 3+1 Form of the Field Equations . . . . . . . . . . . . . . . . 25 2.2.2 3+1 Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 2.2.3 Generic Space Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3 The Evolution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.1 Evolution and Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.3.2 Constraints Conservation . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3.3 Evolution Strategies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.4 Gravitational Waves Degrees of Freedom . . . . . . . . . . . . . . . . . . . 34 2.4.1 Linearized Field Equations . . . . . . . . . . . . . . . . . . . . . . . . . 34 X Contents 2.4.2 Plane-Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.4.3 Gravitational Waves and Gauge Effects . . . . . . . . . . . . . . 37 3FreeEvolution 41 3.1 The Free Evolution Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1 The ADM System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.2 Extended Solution Space . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.1.3 Plane-Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Robust Stability Test-Bed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.2.1 The Method of Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.2.2 Space Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 3.2.3 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.3 Pseudo-Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.1 Extra Dynamical Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.3.2 The BSSN System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3.3 Plane-Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 3.4 The Z4 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 3.4.1 General Covariant Field Equations . . . . . . . . . . . . . . . . . . 59 3.4.2 The Z4 Evolution System . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.4.3 Plane-Wave Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.4.4 Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 4 First Order Hyperbolic Systems 67 4.1 First Order Versions of Second Order Systems . . . . . . . . . . . . . . 67 4.1.1 Introducing Extra First Order Quantities . . . . . . . . . . . . . 67 4.1.2 Ordering Ambiguities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.1.3 The First Order Z4 System . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.4 Symmetry Breaking: The KST System . . . . . . . . . . . . . . . 71 4.2 SystemsofBalance Laws 73 4.2.1 Fluxes and Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 Flux-Conservative Space Discretization . . . . . . . . . . . . . . . 74 4.2.3 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3 Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.3.1 Weak and Strong Hyperbolicity . . . . . . . . . . . . . . . . . . . . . 79 4.3.2 High-Resolution Shock-Capturing Numerical Methods . . 82 4.3.3 The Gauge-Waves Test-Bed . . . . . . . . . . . . . . . . . . . . . . . . . 86 5 Boundary Conditions 93 5.1 The Initial-Boundary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.1.1 Causality Conditions: The 1D Case . . . . . . . . . . . . . . . . . . 94 5.1.2 1D Energy Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 5.1.3 The Multi-Dimensional Case: Symmetric-Hyperbolic Systems 97 5.2 Algebraic Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.2.1 The Modified-System Approach . . . . . . . . . . . . . . . . . . . . . 102 [...]... continuous set of values of f is replaced by a discrete (and finite) set of N numbers The adimensional parameter related to the order of the numerical approximation is just 1/N , independently of the physics of the problem This is why one can apply numerical approximations to any physical situations, without having to restrict oneself to any particular dynamical regime The discrete set of values {f (n)... General Relativity, the physical theory of Gravitation, models spacetime in a geometrical way: as a four-dimensional manifold The concept of manifold is just a generalization to the multidimensional case of the usual concept of a two-dimensional surface This will allow us to apply the well known tools of differential geometry, the branch of mathematics which describes surfaces, to the study of spacetime... on the particular numerical approach which is being used: • In the Spectral Methods approach, the values f (n) correspond to the coefficients of the development of the function f in a series of some specific set of basis functions {φ(n) }, namely N f (n) φ(n) f= (1.58) 1 • In the Finite Elements approach, the values f (n) correspond rather to the integrals of the function f over a set of finite domains... absolute value of the determinant of the metric This allows to rearrange terms in (1.32) so that the principal part can be written as 1 √ (1.37) √ ∂ρ [ g (Γ ρ − δ ρ Γ σ )] , µν µ σν g and the remaining quadratic terms are now Γρ Γσ − Γρ Γλ , ρµ σν λµ ρν (1.38) instead of (1.34) and (1.35), respectively On the other side, from the Numerical Relativity point of view, the balance law structure of (1.32) is... group-theoretical point of view, we can identify φ with the parameter labelling a continuous group of transformations (rotations around one axis in this case), which is usually known as a ‘Lie group’ These transformations can be interpreted as mapping every spacetime point P into a continuous set of points, one for every value of φ This set of image points of a single one P defines an orbit of the group As... from the fact that General Relativity laws are formulated in a completely general coordinate system (that is where the name of ‘General’ Relativity comes from) Special Relativity, instead, makes use of inertial reference frames, where the formulation of the physical laws is greatly simplified This means that one has to learn how to distinguish between the genuine features of spacetime geometry and the... must prescribe suitable coordinate conditions before any numerical calculation can be made The mathematical properties of the resulting complete system will of course depend of this choice of the coordinate gauge We will come back to this point later 1.1.3 Covariant Derivatives The very concept of derivative intrinsically involves the comparison of field values at neighboring points The prize one has... case of the Finite Elements approach when the volumes Vn tend to zero, so that the set of (normalized) basis functions {φ(n) } tends to a set of Dirac delta functions Although all these approaches are currently used to deal with the space variables, time evolution in Numerical Relativity is usually dealt with finite differences (1.60) This can be interpreted as describing spacetime by a series of snapshots,... General Relativity is not adapted to our experience from everyday life The most intuitive concept is not that of spacetime geometry, but rather that of a time succession of space geometries This ‘flowing geometries’ picture could be easily put into the computer, by discretizing the time coordinate, in the same way that the continuous time flow of the real life is coded in terms of a discrete set of photograms... complicated) field equations that translate the manifestly covariant ones (1.31) in terms of these three-dimensional pieces General covariance will then become a hidden feature of the resulting ‘3+1 equations’ The equations themselves will no longer be covariant under C Bona and C Palenzuela Luque: Elements of Numerical Relativity, Lect Notes Phys 673, 19–39 (2005) c Springer-Verlag Berlin Heidelberg 2005 . Löhneysen, Karlsruhe, Germany I. Ojima, Kyoto, Japan D. Sornette, Nice, France, and Los Angeles, CA, USA S. Theisen, Golm, Germany W. Weise, Garching, Germany J. Wess, München, Germany J. Zittartz,. Augsburg, Germany G. Hasinger, Garching, Germany K. Hepp, Zürich, Switzerland W. Hillebrandt, Garching, Germany D. Imboden, Zürich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Lipowsky, Golm, Germany H Lecture Notes in Physics Editorial Board R. Beig, Wien, Austria W. Beiglböck, Heidelberg, Germany W. Domcke, Garching, Germany B G. Englert, Singapore U. Frisch, Nice, France P. Hänggi, Augsburg,