Cosmic censorship hypothesis
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In general relativity, the cosmic censorship hypothesis (or CCH for abbreviation) is a conjecture about the nature of singularities in spacetime. The CCH proposes that singularities are always hidden within event horizons, and therefore cannot be seen from the rest of spacetime. Singularities which are not so hidden are called naked. (This version of the cosmic censorship hypothesis is known as the weak cosmic censorship hypothesis.)
Contents 
Basics
The fundamental concern is that since the physical behavior of singularities is unknown, if singularities can be seen from the rest of spacetime, causality may break down, and physics may lose its predictive power. The issue cannot be avoided, since according to the PenroseHawking singularity theorems, singularities are inevitable in physically reasonable situations.
The hypothesis was first formulated by Roger Penrose in 1969, and it is not stated in a completely formal way. In a sense it is more of a research program proposal: part of the research is to find a proper formal statement that is physically reasonable and that can be proved to be true or false (and that is sufficiently general to be interesting).
Problems with the concept
There are a number of difficulties in formalizing the hypothesis:
 there are technical difficulties with properly formalizing the notion of a singularity.
 it is not difficult to construct spacetimes which have naked singularities, but which are not "physically reasonable". A formal statement needs some set of hypotheses which exclude these situations.
 It could happen that an observer inside the event horizon could see the singularity. The version of the conjecture which excludes this case is the strong cosmic censorship hypothesis.
 Caustics may occur in simple models of gravitational collapse, and can appear to lead to singularities. These have more to do with the simplified models of bulk matter used, and in any case have nothing to do with general relativity, and need to be excluded.
 Computer models of gravitational collapse have shown that naked singularities can arise, but these models rely on very special circumstances (such as spherical symmetry). These special circumstances need to be excluded by some hypothesis.
In 1991, John Preskill and Kip Thorne bet against Stephen Hawking that the hypothesis was false. They won the bet (for a Tshirt to cover the winner's nakedness) due to the discovery of the special situations just mentioned. Hawking later reformulated the bet to exclude those technicalities. The revised bet is still open.
The strong cosmic censorship hypothesis
While the weak cosmic censorship hypothesis asserts that any observer who has observed a singularity is destined to fall into it, it does not give a timeframe for this to happen. As such, for classical general relativity to be a complete theory, an observer of a naked singularity should still have a theory to explain what is observed; Penrose thus formulated a stronger version of the cosmic censorship hypothesis (known as the strong cosmic censorship hypothesis) that asserts that no singularity is ever visible to any observer.
References
 Earman, John: Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes (1995), see especially chapter 2 (ISBN 019509591X)
 Penrose, Roger: "The Question of Cosmic Censorship", Chapter 5 in Black Holes and Relativistic Stars, Robert Wald (editor), (1994) (ISBN 0226870340)
 Penrose, Roger: "Singularities and timeasymmetry", Chapter 12 in General Relativity: An Einstein Centenary Survey (Hawking and Israel, editors), (1979), see especially section 12.3.2, pp. 617629 (ISBN 0521222850)
 Shapiro, S. L., and Teukolsky, S. A.: "Formation of Naked Singularities: The Violation of Cosmic Censorship", Physical Review Letters 66, 994997 (1991)
 Wald, Robert, General Relativity, 299308 (1984) (ISBN 0226870332)
See also
External links
 The bet (http://www.theory.caltech.edu/people/preskill/new_naked_bet.html)