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Stationary Black Holes in General Relativity

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Cosmological and Black Hole Apparent Horizons

Part of the book series: Lecture Notes in Physics ((LNP,volume 907))

Abstract

This chapter introduces the physics of horizons and reviews the basic stationary black holes of General Relativity (Schwarzschild, Reissner-Nordström, Kerr, and Kerr-Newman) in various coordinate systems. The various energy conditions of relativistic gravity are also discussed.

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Notes

  1. 1.

    One should be careful, however, in taking limits of the spacetime geometry based on coordinate systems: the M → 0 limit of Schwarzschild space, really, produces either Minkowski space or a Kasner metric [27] and, strictly speaking, a coordinate-free approach [55–57] is needed to make limits rigorous.

  2. 2.

    A more general family of coordinates parametrized by the parameter \(p \equiv 1 - v_{(\infty )}^{2}\) with 0 < v (∞) < 1 are introduced in [45] and discussed in [47]; it includes as special cases the Painlevé-Gullstrand coordinates for p → 1, Eddington-Finkelstein coordinates in the lightlike limit p → 0, and it is related to another family of coordinate systems characterized by p > 1 and discussed in Refs. [25, 26].

  3. 3.

    In the terminology to be introduced later, the outer horizon \(R = R_{+}\) is an event and an apparent horizon, while the inner horizon \(R = R_{-}\) is an apparent, but not an event, horizon, and is also a Cauchy horizon which is unstable [9, 59].

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  1. Physics Department, Bishop’s University, Sherbrooke, QC, Canada

    Valerio Faraoni

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Faraoni, V. (2015). Stationary Black Holes in General Relativity. In: Cosmological and Black Hole Apparent Horizons. Lecture Notes in Physics, vol 907. Springer, Cham. https://doi.org/10.1007/978-3-319-19240-6_1

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