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Spectral Approach to the Axisymmetric Evolution of Einstein's Vacuum

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Dissertation Spectral approach to the axisymmetric evolution of Einstein’s vacuum equations Christian Schell Diese Dissertation wird eingereicht beim Fachbereich Mathematik und Informatik der Freien Universit¨at Berlin, sie wurde angefertigt am Max-Planck-Institut f¨ur Gravitationsphysik (Albert-Einstein-Institut) in Potsdam 2017 Betreuer und Gutachter: Prof. Dr. Oliver Rinne (Hochschule f¨ur Technik und Wirtschaft, Berlin; Max-Planck-Institut f¨ur Gravitationsphysik, Potsdam; Freie Universit¨at Berlin). Externer Gutachter: Prof. Dr. J¨org Frauendiener (University of Otago, Dunedin, New Zealand). Datum der Disputation: 14. Dezember 2017. iii Abstract This thesis documents our study of Einstein’s vacuum equations in spherical polar coordinates. Open questions in this setting concern applications like the under- standing of gravitational collapse and conceptual matters such as the handling of the occurring coordinate singularity. We answer the conceptual aspects and demonstrate how they can be implemented numerically. Our choice of coordinates allows a spectral approach. As basis functions we em- ploy spin-weighted spherical harmonics. For most derivations and applications we assume hypersurface-orthogonal axisymmetry. This assumption leads to compu- tational simplifications but is not a conceptual limitation. We examine the eigenfunctions of the Laplace operator in spherical coordinates for quantities with different spin-weights and derive the consequences. A systematic investigation of the scalar wave equation in these coordinates leads to helpful insights for the regularization of the coordinate singularity at the origin and we confirm this numerically. We show that a common gauge choice in axisymmetry is inappropriate for the expansion in spin-weighted harmonics and discuss alternatives. We derive Ein- stein’s equations in axisymmetry in an appropriate gauge and solve the linearized equations exactly. A recent formulation of Einstein’s constraint equations regards them as an evolu- tionary system. We analyze the full set of equations and introduce modifications that allow us to derive two sets of locally well-posed problems. Our numerical implementation uses a hybrid discretization consisting of finite dif- ference techniques and the pseudo-spectral method. We simulate the derived equa- tions and present a successful implementation of the parabolic-hyperbolic formula- tion of the nonlinear constraints. To do so we derive several possibilities to obtain initial data at the regular origin. We demonstrate further that our implementation is able to reproduce the exact linear solution in a fully constrained scheme. The results obtained in this thesis offer a possible solution how to simulate Ein- stein’s vacuum equations numerically in spherical polar coordinates with a regular origin. We present one of the first numerical studies of an evolutionary constraint solver. v Contents Abstract v 1. Introduction and overview 1 2. Differential equations and numerical methods 7 2.1. Introduction ............................... 8 2.2. Ordinary differential equations ..................... 8 2.3. Partial differential equations ...................... 33 2.4. The spatial derivatives ......................... 51 2.5. Laplace operator for spherical coordinates .............. 62 2.6. Wave equation in spherical coordinates ................ 69 3. General relativity and its Cauchy formulation 81 3.1. Introduction ............................... 81 3.2. General relativity ............................ 82 3.3. Cauchy formulation of general relativity ............... 88 3.4. Initial data for Einstein’s constraint equations ............ 96 3.5. Symmetry-reduced situations in general relativity ..........105 4. Vacuum axisymmetry in spherical coordinates 117 4.1. Introduction ...............................117 4.2. Axisymmetry and implications .....................118 4.3. Gauges in axisymmetry .........................121 4.4. Choice of variables ...........................125 4.5. Formulation of Einstein’s equations ..................127 4.6. Exact solution to the linear problem .................134 4.7. Some analysis of Einstein’s equations .................142 5. Numerical studies in vacuum axisymmetry 167 5.1. Introduction ...............................167 5.2. Implementation and basic verification .................168 5.3. The linear mode level ..........................174 5.4. The linear 2+1-dimensional level ...................186 5.5. The nonlinear level ...........................201 vii Contents 6. Conclusion and outlook 205 A. Appendix 209 A.1. Supplementary mathematical material ................209 A.2. Nonlinear evolution equations .....................212 A.3. Nonlinear constraints ..........................218 A.4. Explicit form of the exact regular solution for ℓ =2 .........223 A.5. Nonlinear initial data for the momentum constraint .........226 Acknowledgments 237 Zusammenfassung 239 Selbst¨andigkeitserkl¨arung 241 Curriculum vitae 243 Bibliography 244 Index 267 viii Contents ix 1. Introduction and overview General relativity revolutionized our understanding of gravity and our paradigms of space, time and gravitation and, hence, represents a scientific revolution, according to the doctrine of Thomas Kuhn (1974). At present it seems to be the best-working model of gravity in reality. In its formulation many mathematical disciplines are involved including differential geometry, the theory of partial differential equations and geometric analysis. In the present thesis we want to gain a deeper understanding of its fundamental properties and to explore novel techniques to find solutions. Therefore these investigations are placed in the field of applied mathematics even though their major asset is in theoretical physics1. Similar as our characterization of mathematics and theoretical physics the following classification might not be unique but reflects our personal tendency. We want to characterize mathematical relativity as the mathematical field in which fundamental questions arising in general relativity are addressed with mathematical techniques, see Chru´sciel et al. (2010) for a sampler. On the other hand numerical mathematical relativity describes the application of numerical techniques in mathematical relativity. Under numerical relativity we understand 1There is no unique definition or characterization of mathematics but we want to advertise mathematics as an abstract science based on logical deductions and the application to fields like theoretical physics. In contrast to mathematics we want to interpret physics, in particular theoretical physics, as the natural science aiming to build models of reality and extracting predictions. Let us cite a few quotes in favor of our argumentation and supporting our point of view. We remark that there are also different standpoints. According to the Oxford dictionary (https://en.oxforddictionaries.com/definition/mathematics) mathematics is “[t]he abstract science of number, quantity, and space, either as abstract concepts (pure mathemat- ics), or as applied to other disciplines such as physics and engineering (applied mathematics)”. Courant et al. (1996, preface to the 2nd edition): “[...] mathematics is nothing but a system of conclusions drawn from definitions and postulates that must be consistent but otherwise may be created by the free will of the mathematician.” Richard Feynman (1994, page 49): “Mathematicians are only dealing with the structure of reasoning, and they do not really care what they are talking about. They do not even need to know what they are talking about [...]. But in physics you have to have an understanding of the connection of words with the real world.” Karl Popper (1972, page 246) on natural science: “It is the task of the natural scientist to search for laws which will enable him to deduce predictions.” Albert Einstein (1996, page 77): “Physics is an attempt conceptually to grasp reality as something that is considered to be independent of its being observed.” 1 1. Introduction and overview the numerical investigations of a relativistic theory including astrophysical applications. We discuss some aspects of numerical relativity and give references in sections 3.3 and the following. Accordingly, mathematical numerical relativity is the mathematical analysis of numerical relativity. It is not always possible to place a strict boundary between these fields, see Garfinkle (2017) for a recent review. We consider this thesis on that borderline between numerical mathematical and mathematical numerical relativity. General relativity admits a very elegant and beautiful formulation as a geometrical theory, see chapter 3. Einstein’s field equations are at the heart of the theory and we can write them in a very concise form, see section 3.2.2. Nevertheless in the standard formulation they are not accessible to the usual analytical and numerical techniques in a straightforward way. General relativity is reformulated in terms of the Cauchy formulation in section 3.3, such that standard theory of partial differential equations and its numerical techniques are applicable to the field equations. Einstein’s equations split into two sets of equations, evolution equations and constraints. All equations as a full set have the character of an overdetermined set of equations, i.e. not all equations need to be employed. There are different schemes depending on the amount of constraints that are incorporated in the set of equations applied. The system of constraint equations on the other hand is highly underdetermined.
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