Riemann-Geometrie und allgemeine Relativitätstheorie: Anwendungen der Tensoranalysis in der Relativitätstheorie
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Wissenschaftliche Studie aus dem Jahr 2010 im Fachbereich Physik - Theoretische Physik, Sprache: Deutsch, Abstract: In diesem Beitrag werden die mathematisch-physikalischen Grundlagen, Methoden und Grundgesetze der allgemeinen Relativitätstheorie auf der Basis der Tensoranalysis im Riemann-Raum dargelegt. Zunächst wollen wir in einer Vorbetrachtung auf die physikalischen Voraussetzungen und Effekte, sowie auf die Prinzipien der speziellen Relativitätstheorie (SRT) und der allgemeinen Relativitätstheorie (ART) eingehen. Anschlie end behandeln wir die grundlegenden Tensoren, wie Riemannscher Krümmungstensor, Ricci-Tensor und Materietensor, welche für die ART ma gebend sind. Für die mathematische Formulierung der allgemeinen Relativitätstheorie werden wir wichtige Ableitungs- und Variationsformen betrachten. Hierzu gehören die ko- und kontravariante Ableitung, die Lie-Ableitung, die Funktionsvariation, die substantielle Variation, die lokale und totale Variation, insbesondere des metrischen Tensors, sowie Variationsableitungen von Feldgrö en nach dem metrischen Tensor. Im Zusammenhang mit der Beschreibung des Materietensors wollen wir uns mit der Formulierung der Erhaltungssätze für Energie und Impuls in der allgemeinen Relativitätstheorie beschäftigen. Desweiteren behandeln wir diverse Herleitungen und die Bedeutung der Einsteinschen Feldgleichungen. Aus einer approximativen Theorie gewinnen wir die Beziehung zwischen der Newtonschen und der Einsteinschen Gravitationskonstante. Zum Abschluss gehen wir auf wichtige Lösungsmethoden und Lösungen der Einsteinschen Feldgleichungen ein.
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