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Topologia, Geometria e Buracos Negros

Abstract : This is an introductory text on (Algebraic) Topology and Geometry, focusing on the general theory of spherically symmetric black holes. Written in Brazilian Portuguese. The text start with some category theory, and then we introduce the homotopy category of topological spaces. We talk briefly about basic aspects of homotopy theories, including higher homotopy groups, generalized cohomology, etc. Next we introduce general fiber bundles, sketch the classication theorem for principal bundles and show the correspondence between principal bundles and vector bundles. We study the problem of building global sections by means of obstruction theory methods. In the sequence we define connections in principal G-bundles as integrable distribution, show the relation with Lie(G)-valued forms an analyze the interplay between curvature and holonomy (Ambrose-Singer theorem). The correspondence between horizontal equivariant Lie(G)-valued forms on principal bundles P-->M and Ad(P)-valued forms on M is established and the Bianchi-type identities are discussed. General Relativity is introduced in terms of this geometrical background. It is proved the standard obstruction theorem on existence of Lorentzian metrics. The notions of singular spacetime are presented. We then focused on spherically symmetric black holes, following first something like "The Mathematical Theory of Black Holes", by Chandrasekhar, where we prove the "No-Hair Theorem". The remaining of the text is on Newman-Penrose formalism applied to the problem of determining the normal modes of a black hole perturbation.
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Fábio Dadam, Yuri Ximenes Martins. Topologia, Geometria e Buracos Negros. 2016, 978-3841718242. ⟨hal-02908935⟩

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