Titlebook: An Introduction to Mathematical Relativity; José Natário Textbook 2021 The Editor(s) (if applicable) and The Author(s), under exclusive li

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,Einführung in die Untersuchung,field equations, following Wald (General relativity, University of Chicago Press, 1984). We start by studying the Klein–Gordon equation, as a prototypical wave equation, and the Maxwell equations, where the issues of constraints on the initial data and gauge freedom also arise. We then sketch the pr

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Befunde der empirischen Untersuchung,ress, 1984), we start by defining the Komar mass for stationary spacetimes. We then discuss field theory and introduce the Einstein–Hilbert action as a means of motivating the definition of the ADM mass. Finally, we prove the (Riemannian) positive mass theorem and the (Riemannian) Penrose inequality

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Singularity Theorems,h timelike and null geodesics cannot be continued. The singularity theorems of Hawking and Penrose, proved in this chapter, show that this is a generic phenomenon: any sufficiently small perturbation of these singular solutions will still be singular.

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Contributions to Management ScienceIn this initial chapter we give a very short introduction to special and general relativity for mathematicians. In particular, we relate the index-free differential geometry notation used in mathematics to the index notation used in physics. As an exercise in index gymnastics, we derive the contracted Bianchi identities.

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https://doi.org/10.1007/978-3-8349-8859-1In this chapter we present a number of exact solutions of the Einstein field equations, construct their Penrose diagrams, and analyze their matching across a timelike hypersurface. These solutions will be used as examples or counter-examples when discussing the theorems in the subsequent chapters.

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Exact Solutions,In this chapter we present a number of exact solutions of the Einstein field equations, construct their Penrose diagrams, and analyze their matching across a timelike hypersurface. These solutions will be used as examples or counter-examples when discussing the theorems in the subsequent chapters.

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