Cosmic censorship hypothesis
The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity.
Singularities that arise in the
Basics
Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime,
The hypothesis was first formulated by
Weak and strong cosmic censorship hypothesis
The weak and the strong cosmic censorship hypotheses are two conjectures concerned with the global geometry of spacetimes.
The weak cosmic censorship hypothesis asserts there can be no singularity visible from future null infinity. In other words, singularities need to be hidden from an observer at infinity by the event horizon of a black hole. Mathematically, the conjecture states that, for generic initial data, the causal structure is such that the maximal Cauchy development possesses a complete future null infinity.
The strong cosmic censorship hypothesis asserts that, generically, general relativity is a deterministic theory, in the same sense that classical mechanics is a deterministic theory. In other words, the classical fate of all observers should be predictable from the initial data. Mathematically, the conjecture states that the maximal Cauchy development of generic compact or asymptotically flat initial data is locally inextendible as a regular
The two conjectures are mathematically independent, as there exist spacetimes for which weak cosmic censorship is valid but strong cosmic censorship is violated and, conversely, there exist spacetimes for which weak cosmic censorship is violated but strong cosmic censorship is valid.
Example
The Kerr metric, corresponding to a black hole of mass and angular momentum , can be used to derive the
To preserve cosmic censorship, the black hole is restricted to the case of . For there to exist an event horizon around the singularity, the requirement must be satisfied.[5] This amounts to the angular momentum of the black hole being constrained to below a critical value, outside of which the horizon would disappear.
The following thought experiment is reproduced from Hartle's Gravity:
Imagine specifically trying to violate the censorship conjecture. This could be done by somehow imparting an angular momentum upon the black hole, making it exceed the critical value (assume it starts infinitesimally below it). This could be done by sending a particle of angular momentum . Because this particle has angular momentum, it can only be captured by the black hole if the maximum potential of the black hole is less than .
Solving the above effective potential equation for the maximum under the given conditions results in a maximum potential of exactly . Testing other values shows that no particle with enough angular momentum to violate the censorship conjecture would be able to enter the black hole, because they have too much angular momentum to fall in.
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"; the canonical example of such a spacetime is perhaps the "superextremal" Reissner–Nordströmsolution, which contains a singularity at that is not surrounded by a horizon. A formal statement needs some set of hypotheses which exclude these situations.
- 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 hypotheses.
In 1991, John Preskill and Kip Thorne bet against Stephen Hawking that the hypothesis was false. Hawking conceded the bet in 1997, due to the discovery of the special situations just mentioned, which he characterized as "technicalities". Hawking later reformulated the bet to exclude those technicalities. The revised bet is still open (although Hawking died in 2018), the prize being "clothing to cover the winner's nakedness".[6]
Counter-example
An exact solution to the scalar-Einstein equations which forms a counterexample to many formulations of the cosmic censorship hypothesis was found by Mark D. Roberts in 1985:
See also
- Black hole information paradox
- Chronology protection conjecture
- Firewall (physics)
- Fuzzball (string theory)
- Thorne–Hawking–Preskill bet
References
- ^ Earman, J. (2007). "Aspects of Determinism in Modern Physics" (PDF). The Philosophy of Physics. pp. 1369–1434. Archived (PDF) from the original on 2014-05-22.
- Bibcode:1969NCimR...1..252P.
- ^ "A Bet on a Cosmic Scale, And a Concession, Sort Of". New York Times. February 12, 1997.
- ^ Hartnett, Kevin (17 May 2018). "Mathematicians Disprove Conjecture Made to Save Black Holes". Quanta Magazine. Retrieved 29 March 2020.
- ^ ISBN 0-8053-8662-9)
- ^ "New bet on naked singularities". 5 February 1997. Archived from the original on 6 June 2004.
- S2CID 121601921.
Further reading
- Earman, John (1995). Bangs, Crunches, Whimpers, and Shrieks: Singularities and Acausalities in Relativistic Spacetimes. Oxford University Press. See especially chapter 2. ISBN 0-19-509591-X.
- Penrose, Roger (1994). "The Question of Cosmic Censorship". In Wald, Robert (ed.). Black Holes and Relativistic Stars. University of Chicago Press. ISBN 0-226-87034-0.
- Penrose, Roger (1979). "Singularities and time-asymmetry". In Hawking; Israel (eds.). General Relativity: An Einstein Centenary Survey. See especially section 12.3.2, pp. 617–629. ISBN 0-521-22285-0.
- Shapiro, Stuart L.; Teukolsky, Saul A. (1991-02-25). "Formation of naked singularities: The violation of cosmic censorship" (PDF). Physical Review Letters. 66 (8). American Physical Society (APS): 994–997. (PDF) from the original on 2019-12-05.
- Wald, Robert (1984). General Relativity. University of Chicago Press. pp. 299–308. ISBN 0-226-87033-2.
External links
- The old bet (conceded in 1997)
- The new bet