Abstract
The aim of this chapter is to provide a general overview on black holes, and on black holes in 4-dimensional Einstein’s gravity in particular. Contrary to the previous chapters, often we only present the final result, without providing all the calculations.
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Notes
- 1.
Note that such a definition of black hole is not limited to Einstein’s gravity and can be applied whenever it is possible to define \(J^-({\mathscr {I}}^+)\).
- 2.
In the international system of units, the line element reads
- 3.
The definition of event horizon is provided in Sect. 10.1, but it cannot be directly applied for determining the coordinates of the event horizon from a known metric. For the black hole solutions discussed in this textbook (Schwarzschild, Reissner–Nordström, Kerr), the radial coordinate of the event horizon corresponds to the larger root of \(g^{rr} = 0\). For more general cases, the interested reader can refer to [1] and references therein. Note that the issue of the existence of an event horizon in these solutions is non-trivial. This is also proved by the fact that the Schwarzschild metric was found in 1916 and it was only in 1958 that David Finkelstein realized that there was an event horizon with specific properties.
- 4.
A naked singularity is a singularity of the spacetime that is not inside a black hole and thus belongs to the causal past of future null infinity.
- 5.
Reintroducing the speed of light c and Newton’s gravitational constants \(G_\mathrm{N}\), we have
- 6.
A closed time-like curve is a closed time-like trajectory; if there are closed time-like curves in a spacetime, massive particles can travel backwards in time. While the Einstein equations have solutions with similar properties, they are thought to be non-physical and therefore they are usually not taken into account.
- 7.
For electro-vacuum, we mean that the energy-momentum tensor on the right hand side of the Einstein equations either vanishes or is that of the electromagnetic field.
- 8.
Note that the interior metric is the time reversal of the Friedmann–Robertson–Walker solution, which will be discussed in the next chapter.
References
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Problems
Problems
10.1
Write the inverse metric \(g^{\mu \nu }\) of the Reissner–Nordström solution.
10.2
Write the inverse metric \(g^{\mu \nu }\) of the Kerr solution in Boyer–Lindquist coordinates.
10.3
Check that Eq. (10.21) is a different form of the geodesic equations.
10.4
For a Kerr black hole, \(| a_* | \le 1\). Show that the spin parameter of Earth is \(| a_* | \gg 1\).
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Bambi, C. (2018). Black Holes. In: Introduction to General Relativity. Undergraduate Lecture Notes in Physics. Springer, Singapore. https://doi.org/10.1007/978-981-13-1090-4_10
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DOI: https://doi.org/10.1007/978-981-13-1090-4_10
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