Abstract
In Chap. 2 we begin with a historical account of the notion of black holes from Laplace to the present identification of stellar mass black holes in the galaxy and elsewhere. Next the Kruskal extension of the Schwarzschild solution is considered in full detail preceded by the pedagogical toy example of Rindler space-time. Basic concepts about Future, Past and Causality are introduced next. Conformal Mappings, the Causal Structure of infinity and Penrose diagrams are discussed and exemplified.
O radiant Dark! O darkly fostered ray
Thou hast a joy too deep for shallow Day!
George Eliot (The Spanish Gypsy)
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Notes
- 1.
Hans Jacob Reissner (1874–1967) was a German aeronautical engineer with a passion for mathematical physics. He was the first to solve Einstein’s field equations with a charged electric source and he did that already in 1916 [3]. Emigrated to the United States in 1938 he taught at the Illinois Institute of Technology and later at the Polytechnic Institute of Brooklyn. Reissner’s solution was retrieved and refined in 1918 by Gunnar Nordström (1881–1923) a Finnish theoretical physicist who was the first to propose an extension of space-time to higher dimensions. Independently from Kaluza and Klein and as early as 1914 he introduced a fifth dimension in order to construct a unified theory of gravitation and electromagnetism. His theory was, at the time, a competitor of Einstein’s theory. Working at the University of Leiden in the Netherlands with Paul Ehrenfest, in 1918 he solved Einstein field equations for a spherically symmetric charged body [4] thus extending the Hans Reissner’s results for a point charge.
References
Gibbons, G.: The man who invented black holes [his work emerges out of the dark after two centuries]. New Scientist 28, 1101 (1979)
Laplace, P.S.: Exposition du Système du Monde, 1st edn. Paris (1796)
Reissner, H.: Über die Eigengravitation des elektrischen Feldes nach der Einstein’schen Theorie. Ann. Phys. 50, 106–120 (1916). doi:10.1002/andp.19163550905
Nordström, G.: On the energy of the gravitational field in Einstein’s theory. Verhandl. Koninkl. Ned. Akad. Wetenschap., Afdel. Natuurk., Amsterdam 26, 1201–1208 (1918)
Petrov, A.Z.: Classification of spaces defined by gravitational fields. Uch. Zapiski Kazan Gos. Univ. 144, 55 (1954). English translation Petrov, A.Z, Gen. Relativ. Gravit. 22, 1665 (2000). doi:10.1023/A:1001910908054
Kruskal, M.D.: Maximal extension of Schwarzschild metric. Phys. Rev. 119, 1743 (1960)
Szekeres, G.: On the singularities of a Riemannian manifold. Publ. Math. (Debr.) 7, 285 (1960)
Ashtekar, A.: Asymptotic structure of the gravitational field at spatial infinity. Gen. Relativ. Gravit. 2 (1980)
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Frè, P.G. (2013). Extended Space-Times, Causal Structure and Penrose Diagrams. In: Gravity, a Geometrical Course. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-5443-0_2
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