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Master's Thesis the Rotating Mass Shell in the General Theory of Relativity

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Master's Thesis the Rotating Mass Shell in the General Theory of Relativity Master’s Thesis The rotating mass shell in the general theory of relativity Florian Atteneder Graz, October 2019 supervised by Univ.-Prof. Dr.rer.nat Reinhard Alkofer Abstract The model of a rotating mass shell (RMS) was initially introduced to judge if rotation has only relative meaning. It comprises a general relativistic description of a spacetime with an energy-matter content that is assembled in a rotating quasi-spherical shell with zero radial extension. The concept of relativity of rotation goes back to Mach’s principle and latest results obtained by perturbation theory (PT) calculations have shown that it is indeed realized in such a spacetime. However, because this conclusion was based on PT, its validity is limited to slowly RMSs. This thesis pursues a numerical treatment of the problem which can provide insight into the validity of Mach’s idea of relativity of rotation also for rapidly RMSs. The mathematical formulation of the RMS problem involves a splitting of spacetime into a region that is flat and another region that is asymptotically flat, where the latter is used as a reference to define relative rotation. Under these assumptions, a RMS forms at the common boundary of these two regions. On the basis of previous work, we formulate Einstein’s equations as a free-boundary value problem and solve them numerically using a pseudo-spectral method. As a result we obtain a three-parameter solution that is characterized by the shell’s polar radius R, its total gravitational mass M and angular momentum J. The existence of the solution is enough to positively answer the question if Mach’s idea of relativity of rotation can be extended for rapidly RMSs. 2 Kurzfassung Das Modell einer rotierenden Massenschale (RMS) wurde entwickelt um zu testen, ob Rotation nur eine relative Bedeutung besitzt. Dies geschieht im Sinne einer allgemein relativistischen Beschreibung einer Raumzeit in welcher der Energie-Masse Gehalt durch eine rotierende und quasi-sphärischen Schale mit verschwindender radialer Ausdehnung betrachtet wird. Das Konzept von relativer Rotation geht auf das Mach’sche Prinzip zurück und jüngste Ergebnisse einer störungstheoretischen Behandlung des RMS Prob- lems zeigten, dass dies tatsächlich realisiert werden kann in einer solchen Raumzeit. Da diese Schlussfolgerung jedoch auf einer stöhrungstheoretischen Rechnung gründete, ist diese auf langsam rotierende Massenschale beschränkt. In dieser Arbeit wird ein numerischer Zugang des Problems entwickelt, welcher klären soll, ob Machs Idee der rel- ativen Rotation auch auf schnell rotierende Massenschalen ausgedehnt werden kann. Die mathematische Formulierung des RMS Problems erfolgt durch eine Teilung der Raumzeit in einen flachen sowie asymptotisch-flachen Anteil, wobei letzterer als Referenze zurBes- timmung der relativen Rotation herangezogen wird. Aus diesen Annahmen folgt, dass sich die RMS an der gemeinsamen Grenze der beiden Teilbereiche bildet. Aufbauend auf einer vorrangegangenen Arbeit formulieren wir die Einstein Gleichungen als ein freies Randwertproblem und lösen dieses numerisch mittels einer pseudo-spektralen Methode. Als Ergebnis erhalten wir eine numerische Lösung, welche durch die drei Parameter polarer Schalenradius R, gravitative Gesamtmasse M und den Drehimpuls J charak- terisiert ist. Die Existenz der numerischen Lösung erlaubt eine positive Beanwortung der Frage nach der Realisierung der Mach’schen Idee von relativer Rotation für schnell rotierende Massenschalen. 3 Acknowledgement I would like to thank my supervisor Univ.-Prof. Dr.rer.nat Reinhard Alkofer for introducing me to the topic of rotating mass shells. I am also thankful to Dr.rer.nat. Helios Sanchis-Alepuz for co-supervising this project. Many thanks must go to Tobias Benjamin Russ. I very much appreciate the time he took to explain clarifying details to me about his previous work on rotating mass shells. I would also like to thank Fabian and Martin for proofreading the first draft of this work. Lastly, I am very thankful for the financial support by the Paul-Urban-Stipendienstiftung of the University of Graz. 4 To my family. Contents Notation8 1. Introduction9 2. The system under study 11 2.1. Previous work................................. 11 2.2. Assumptions.................................. 12 2.3. Coordinates and metric tensor........................ 13 2.4. Einstein’s field equations........................... 16 2.4.1. Solution strategy........................... 16 2.4.2. Vacuum field equations........................ 17 2.5. Boundary and regularity conditions..................... 18 2.5.1. Behavior at spatial infinity...................... 18 2.5.2. Behavior on the axis of symmetry and across the equatorial plane 21 2.5.3. Conditions on the shell........................ 23 3. Numerical treatment 24 3.1. Introduction to pseudo-spectral methods.................. 24 3.2. Series expansions............................... 25 3.2.1. Ansätze for U; Ω;V .......................... 25 3.2.2. Analytic solution for W ....................... 27 3.2.3. Ansatz for f .............................. 28 3.3. Putting everything on the computer..................... 28 3.3.1. Guess and extrapolation....................... 31 4. Results 33 4.1. Centrifugal deformation and differential rotation.............. 33 4.2. Surface-stress-energy tensor......................... 36 4.3. Error estimation............................... 37 5. Conclusion 44 5.1. Discussion................................... 44 5.2. Summary and Outlook............................ 45 A. Thin shell description 47 A.1. Definition and notation............................ 47 A.2. Description of the shell............................ 47 6 A.3. Induced metric and first junction conditions................ 48 A.4. Extrinsic curvature and surface-stress-energy tensor............ 49 A.4.1. Eigenvalues and eigenvectors..................... 50 A.4.2. Energy conditions........................... 51 B. Komar integrals 53 C. Details on numerical implementation 55 C.1. Rational Chebyshev functions........................ 55 C.2. Collocation points............................... 56 C.3. Orthogonality relations............................ 57 C.4. Code and external libraries.......................... 58 D. Perturbation theory 59 Bibliography 63 7 Notation Throughout this work we apply the metric sign convection ( + ++). Greek indices α; β; γ; ::: run over 4 dimensional spacetime coordinates.− Latin indices a; b; c; ::: run over 3 dimensional hypersurface coordinates. @µ represents partial derivative operation and when acting on a tensor it is abbreviated α... α... as @µ X β::: = X β::: ,µ. µ represents covariant derivative operation and when acting on a tensor it is abbrevi- r α... α... ated as µX β::: = X β::: ;µ. Geometricr units are employed, that is, G = c = 1 is used. For the definition of the Riemann and Ricci tensor we follow closely the convention that is employed in [1,2]. 8 1. Introduction General relativity (GR) was published by Einstein in 1915 [3]. It was developed to circumvent the shortcomings of a Newtonian description of gravity, which suffered from incompatibility with the previously established theory of special relativity [4]. GR is a geometric theory in which the gravitational interaction is mediated via the curvature of space and time, or spacetime. The content of this theory is embodied in Einstein’s field equations. They relate the spacetime’s curvature to the energy and momentumof the matter and radiation content. For the most general case, these equations are ten coupled, nonlinear, second-order partial differential equations (PDEs). Solving Einstein’s equations for a rotating mass shell (RMS), utilizing numerical meth- ods, is the topic of this work. A RMS may be imagined as a rotating and mas- sive quasi-spherical shell of zero radial extension. The model was introduced first by Einstein [2, section 4.1], and the reason why he was interested in studying it, was that it shall allow calculating what is nowadays known as (frame) dragging effects, see [5, chapter XII.5] for an example. These effects refer to the gravitational influence of the RMS on local inertial frames. Ultimately, according to [1,2,6], based on such a calculation it can be made plausible that rotation has only a relative meaning in such a spacetime and that [1] "(...) it should be impossible to decide, in principle, whether an observer rotates relative to the fixed stars or all the stars and galaxies in the universe rotate relative to him." This idea often goes under the name Mach’s principle of relativity of rotation. In par- ticular, it is this principle one was interested initially to test with the RMS model and to check whether it is included in GR or not. This is a controversy topic, because no precise definition of Mach’s principle has been given so far and for further details onthis topic we refer to [2] and the references listed there. On the trail of Einstein’s RMS model, over the years many physicists have worked on refining his initial perturbation theory (PT) calculations. The most recent results found in the literature were conducted by Pfister et al. [1], with a revision of this work published in [2, appendix B]. Their conclusion states that Mach’s idea of relativity of rotation is completely realized in general relativity within the RMS model. The initial question that has led to the introduction of the RMS has thus been answered. So why would one bother and reexamine their work using numerical methods? Firstly, the conclusion of Pfister
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