The weak and the strongcosmic censorship hypotheses are two mathematicalconjectures about the structure ofgravitational singularities arising ingeneral relativity.

Singularities that arise in thesolutions ofEinstein's equations are typically hidden withinevent horizons, and therefore cannot be observed from the rest ofspacetime. Singularities that are not so hidden are callednaked. Theweak cosmic censorship hypothesis was conceived byRoger Penrose in 1969 and posits that no naked singularities exist in theuniverse.

Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime,causality may break down, andphysics may lose its predictive power. The issue cannot be avoided, since according to thePenrose–Hawking singularity theorems, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe, as described by thegeneral theory of relativity, isdeterministic:^[1] it is possible to predict the entire evolution of the universe (possibly excluding some finite regions of space hidden inside event horizons of singularities), knowing only its condition at a certain moment of time (more precisely, everywhere on aspacelike three-dimensional hypersurface, called theCauchy surface). Failure of the cosmic censorship hypothesis leads to the failure of determinism, because it is yet impossible to predict the behavior of spacetime in the causal future of a singularity. Cosmic censorship is not merely a problem of formal interest; some form of it is assumed wheneverblack hole event horizons are mentioned.^{[citation needed]}

The hypothesis was first formulated byRoger Penrose in 1969,^[2] 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,falsifiable, and sufficiently general to be interesting.^[3] Because the statement is not a strictly formal one, there is sufficient latitude for (at least) two independent formulations: a weak form, and a strong form.

The weak and the strong cosmic censorship hypotheses are two conjectures concerned with the global geometry of spacetimes.

Theweak cosmic censorship hypothesis asserts there can be no singularity visible from futurenull infinity. In other words, singularities need to be hidden from an observer at infinity by the event horizon of ablack hole. Mathematically, the conjecture states that, for generic initial data, thecausal structure is such that the maximalCauchy development possesses a complete future null infinity.

Thestrong 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 regularLorentzian manifold. Taken in its strongest sense, the conjecture suggests locally inextendibility of the maximal Cauchy development as a continuous Lorentzian manifold [very strong cosmic censorship]. This strongest version was disproven in 2018 by Mihalis Dafermos and Jonathan Luk for theCauchy horizon of anuncharged, rotating black hole.^[4]

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.

TheKerr metric, corresponding to a black hole of mass${\displaystyle M}$ and angular momentum${\displaystyle J}$, can be used to derive theeffective potential for particleorbits restricted to the equator (as defined by rotation). This potential looks like:^[5]$${\displaystyle V_{\mathrm{e}\mathrm{f}\mathrm{f}}(r,e,\ell )=-\frac{M}{r}+\frac{{\ell }^2-a^2(e^2-1)}{2r^2}-\frac{M(\ell -ae{)}^2}{r^3},a\equiv \frac{J}{M}}$$where${\displaystyle r}$ is the coordinate radius,${\displaystyle e}$ and${\displaystyle \ell }$ are the test-particle's conserved energy and angular momentum respectively (constructed from theKilling vectors).

To preservecosmic censorship, the black hole is restricted to the case of. For there to exist anevent horizon around the singularity, the requirement must be satisfied.^[5] This amounts to theangular 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'sGravity:

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${\displaystyle \ell =2Me}$. 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${\displaystyle (e^2-1)/2}$.
Solving the above effective potential equation for the maximum under the given conditions results in a maximum potential of exactly${\displaystyle (e^2-1)/2}$. 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.

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–Nordström solution, which contains a singularity at${\displaystyle r=0}$ 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 ofgravitational 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 andKip Thornebet againstStephen 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]

An exact solution to the scalar-Einstein equations${\displaystyle R_{ab}=2{\varphi }_a{\varphi }_b}$ which forms a counterexample to many formulations of the cosmic censorship hypothesis was found by Mark D. Roberts in 1985:$${\displaystyle d{s}^2=-(1+2\sigma )d{v}^2+2dvdr+r(r-2\sigma v)\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right),\phi =\frac{1}{2}\ln \left(1-\frac{2\sigma v}{r}\right),}$$where${\displaystyle \sigma }$ is a constant.^[7]

• Black hole information paradox
• Chronology protection conjecture
• Firewall (physics)
• Fuzzball (string theory)
• Thorne–Hawking–Preskill bet

• Earman, John (1995).Bangs, crunches, whimpers, and shrieks: singularities and acausalities in relativistic spacetimes(PDF). New York:Oxford University Press. See especially chapter 2.ISBN 978-0-19-509591-3.
• Wald, Robert M. (1998). "The Question of Cosmic Censorship". In Wald, Robert M. (ed.).Black holes and relativistic stars. Chicago:University of Chicago Press.ISBN 978-0-226-87034-2.
• Penrose, Roger (1979). "Singularities and time-asymmetry". In Hawking, Stephen; Israel, W. (eds.).General relativity: an Einstein centenary survey. Cambridge [Eng.] ; New York:Cambridge University Press. See especially section 12.3.2, pp. 617–629.ISBN 978-0-521-22285-3.
• Shapiro, Stuart L.; Teukolsky, Saul A. (25 February 1991)."Formation of naked singularities: The violation of cosmic censorship"(PDF).Physical Review Letters.66 (8).American Physical Society (APS):994–997.Bibcode:1991PhRvL..66..994S.doi:10.1103/PhysRevLett.66.994.ISSN 0031-9007.PMID 10043968.S2CID 7830407.Archived(PDF) from the original on 2019-12-05.
• Wald, Robert M. (1984).General relativity(PDF). Chicago:University of Chicago Press. pp. 299–308.ISBN 978-0-226-87032-8.

• The old bet (conceded in 1997)
• The new bet

• Black holes
• General relativity
• Roger Penrose
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• Conjectures

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