Black Holes
This extended chapter of almost hundred pages on black hole physics is a central part of the book. We begin with Israel’s original demonstration of the statement that a static black hole solution of Einstein’s vacuum equation has to be spherically symmetr
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Black Holes
In my entire scientific life (. . .) the most shattering experience has been the realization that an exact solution of Einstein’s equations of general relativity, discovered by the New Zealand mathematician Roy Kerr, provides the absolutely exact representation of untold numbers of massive black holes that populate the Universe. —S. Chandrasekhar (1975)
8.1 Introduction All stars rotate more or less rapidly. When a horizon is formed during gravitational collapse, a Schwarzschild black hole is thus never produced. One expects, however, that the horizon will quickly settle down to a stationary state as a result of the emission of gravitational waves. The geometry of the stationary black hole is of course no longer spherically symmetric. It is remarkable that we know all stationary black hole solutions of Einstein’s vacuum equations. Surprisingly, they are fully characterized by just two parameters, namely the mass and angular momentum of the hole. These quantities can be determined, in principle, by distant observers. Thus when matter disappears behind a horizon, an exterior observer sees almost nothing of its individual properties. One can no longer say for example how many baryons formed the black hole. A huge amount of information is thus lost. The mass and angular momentum completely determine the external field, which is known analytically (Kerr solution). This led J.A. Wheeler to say “A black hole has no hair”, and the previous statement is now known as the no-hair-theorem. The proof of this fact is an outstanding contribution of mathematical physics, and was completed only in the course of a number of years by various authors (W. Israel, B. Carter, S. Hawking and D. Robinson). A first decisive step was made by W. Israel (see [244, 245]) who was able to show that a static black hole solution of Einstein’s vacuum equation has to be spherically symmetric and, therefore, agree with the Schwarzschild solution. We shall give Israel’s proof in Sect. 8.2. In a second paper Israel extended this result to black hole solutions of the coupled Einstein–Maxwell N. Straumann, General Relativity, Graduate Texts in Physics, DOI 10.1007/978-94-007-5410-2_8, © Springer Science+Business Media Dordrecht 2013
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system. The Reissner–Nordstrøm 2-parameter family (see Sect. 4.9) turned out to exhaust all static electrovac black holes. It was then conjectured by Israel, Penrose and Wheeler that in the stationary case the electrovac black holes should all be given by the 3-parameter Kerr–Newman family, discussed later in this chapter. After a number of steps by various authors this conjecture could finally be proven. For a very readable text book presentation of the proof we refer to [246]. With this black hole uniqueness theorem it was natural to conjecture further generalizations for more complicated matter models, for instance for the Einstein– Yang–Mills system. Surprisingly, it turned out that generalized no-hair conjectures do not always hold (for reviews, see [247] and [248]). While this dev
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