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1	Modified Einstein Hilbert Action and the Resulting Field Equations Ahlqvist, Pontus 01 January 2007 (has links) We begin by reviewing special and general relativity in such a way as to smoothly transition into current research. We present the variational formalism of general relativity as well as the extension into the palatini formalism. This allows us to develop a theory on a metric affine manifold rather than the standard manifold in general relativity. We present a generalized action intended to replace the Einstein Hilbert action in general relativity and derive some consequences thereof. The modified field equations are derived by varying this action using the Palatini approach. The corresponding differential equations are solved thereby establishing the equivalence between the modified action and the standard action with a cosmological constant. Furthermore the metric due to a spherically symmetric distribution of mass is found and applied in calculating the bending of light in the curved space. It is deduced that no difference between the modified action and the original Einstein Hilbert action is observed thereby implying that the experiment involving the bending of light around the sun in 1919 in no way distinguishes between our modification and the original approach by Einstein and Hilbert. Physics
2	Stability of Einstein Manifolds Kröncke, Klaus January 2013 (has links) This thesis deals with Einstein metrics and the Ricci flow on compact mani- folds. We study the second variation of the Einstein-Hilbert functional on Ein- stein metrics. In the first part of the work, we find curvature conditions which ensure the stability of Einstein manifolds with respect to the Einstein-Hilbert functional, i.e. that the second variation of the Einstein-Hilbert functional at the metric is nonpositive in the direction of transverse-traceless tensors. The second part of the work is devoted to the study of the Ricci flow and how its behaviour close to Einstein metrics is influenced by the variational be- haviour of the Einstein-Hilbert functional. We find conditions which imply that Einstein metrics are dynamically stable or unstable with respect to the Ricci flow and we express these conditions in terms of stability properties of the metric with respect to the Einstein-Hilbert functional and properties of the Laplacian spectrum. / Die vorliegende Arbeit beschäftigt sich mit Einsteinmetriken und Ricci-Fluss auf kompakten Mannigfaltigkeiten. Wir studieren die zweite Variation des Einstein- Hilbert Funktionals auf Einsteinmetriken. Im ersten Teil der Arbeit finden wir Krümmungsbedingungen, die die Stabilität von Einsteinmannigfaltigkeiten bezüglich des Einstein-Hilbert Funktionals sicherstellen, d.h. die zweite Varia- tion des Einstein-Hilbert Funktionals ist nichtpositiv in Richtung transversaler spurfreier Tensoren. Der zweite Teil der Arbeit widmet sich dem Studium des Ricci-Flusses und wie dessen Verhalten in der Nähe von Einsteinmetriken durch das Variationsver- halten des Einstein-Hilbert Funktionals beeinflusst wird. Wir finden Bedinun- gen, die dynamische Stabilität oder Instabilität von Einsteinmetriken bezüglich des Ricci-Flusses implizieren und wir drücken diese Bedingungen in Termen der Stabilität der Metrik bezüglich des Einstein-Hilbert Funktionals und Eigen- schaften des Spektrums des Laplaceoperators aus. Einstein-Mannigfaltigkeiten Ricci-Fluss Variationsstabilität Einstein-Hilbert-Wirkung Einstein manifolds Ricci flow variational stability Einstein-Hilbert action Mathematics
3	Spacetime as a Hamiltonian Orbit and Geroch's Theorem on the Existence of Fermions Bergstedt, Viktor January 2020 (has links) Over a century since its inception, general relativity continues to lie at the heart of some of the most researched topics in theoretical physics. It seems likely that the coveted solutions to problems like quantum gravity are to be found in an extension of general relativity, one which may only be visible in an alternate formulation of the theory. In this thesis we consider the possibility of casting general relativity in the form of an initial value problem where spacetime is seen as the evolution of space. This evolution is shown to be constrained and of Hamiltonian type. Not all spacetimes are physically acceptable. To be compatible with particle physics, one would like spacetime to accommodate fermions. Here we can take comfort in Geroch’s theorem, which implies that any spacetime that admits a Hamiltonian formulation automatically supports the existence of fermions. We review the elements that go into the proof of this theorem. / Allmän relativitetsteori har i över hundra år legat i teoretiska fysikens framkant. Det är möjligt att lösningarna på öppna problem som kvantiseringen av gravitation går att finna i en utvidgning av allmän relativitetsteori – och kanske uppenbarar sig denna utvidgning bara ur en alternativ formulering av teorin. I den här uppsatsen formuleras allmän relativitetsteori och dess Einsteinekvationer som ett begynnelsevärdesproblem, genom vilket rumtiden kan betraktas som rummets historia. Vi visar att rummets rörelseekvationer är Hamiltons ekvationer med tvångsvillkor. Enligt partikelfysiken bör fermioner kunna finnas till i rumtiden. Härom kan vi åberopa Gerochs sats, enligt vilken rumtider som har en Hamiltonsk formulering också medger fermioner. Vi redogör för huvuddragen i beviset av Gerochs sats. 3+1 ADM black hole canonical quantization Cauchy surface causality differential geometry Einstein-Hilbert action general relativity Geroch globally hyperbolic Hamilton's equations numerical relativity parallelizable SL(2 C) spacetime spinor spin structure SO(1 3) Stiefel-Whitney class tetrad topology Physical Sciences Fysik

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