# Difference between revisions of "Penrose cosmic censorship"

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− | A (future) singularity in a [[Space-time|space-time]] corresponds to a causal geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120090/p1200901.png" /> (cf. also [[Geodesic line|Geodesic line]]) which is future inextensible and whose affine parameter remains finite (cf. [[Naked singularity|Naked singularity]]). Penrose's weak (respectively, strong) cosmic censorship hypothesis states that any | + | A (future) singularity in a [[Space-time|space-time]] corresponds to a causal geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120090/p1200901.png" /> (cf. also [[Geodesic line|Geodesic line]]) which is future inextensible and whose affine parameter remains finite (cf. [[Naked singularity|Naked singularity]]). Penrose's weak (respectively, strong) cosmic censorship hypothesis states that any "physically realistic" space-time with a future-inextensible incomplete geodesic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120090/p1200902.png" /> that lies in the past of future infinity (respectively, of a point of space-time) is unstable with respect to a (still to be specified) natural topology of space-times [[#References|[a9]]], [[#References|[a10]]]. Here, "future infinity" refers to the conformal future boundary of a weakly asymptotically flat space-time in the sense of Penrose [[#References|[a5]]] or a variant thereof. The meanings of "physically realistic" and "stable" are not specified. This is due to the existence of counter-examples which are not entirely unrealistic from a physical point of view but still have properties which seem to be very special (see, for instance, [[#References|[a6]]]). There are various variants for both the weak and the strong cosmic censorship hypothesis. An especially important version of strong cosmic censorship is the hypothesis that any physically reasonable and qualitatively stable space-time is globally hyperbolic. The deepest general result on weak cosmic censorship has been obtained by R.P.A.C. Newman [[#References|[a7]]], [[#References|[a8]]], who shows that "persistent curvature" enforces a version of weak cosmic censorship. A more direct version of the cosmic censorship theorem, which is, however, only applicable for space-times very close to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120090/p1200903.png" />-dimensional flat Lorentzian space, has been obtained by D. Christodoulou and S. Klainerman [[#References|[a4]]]. Christodoulou has also investigated subclasses of spherically symmetric scalar field space-times and has obtained very detailed results with regard to cosmic censorship in these classes of space-times [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]. |

It should be remarked that spherically symmetric space-times are highly non-generic and that therefore his qualitative results may not hold in the general case. | It should be remarked that spherically symmetric space-times are highly non-generic and that therefore his qualitative results may not hold in the general case. | ||

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====References==== | ====References==== | ||

− | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> D. Christodoulou, "The formation of black holes and singularities in spherically symmetric gravitational collapse" ''Commun. Pure Appl. Math.'' , '''XLIV''' (1991) pp. 339–373 {{MR|1090436}} {{ZBL|0728.53061}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Christodoulou, "Bounded variation solutions of the spherically symmetric Einstein-scalar field equations" ''Commun. Pure Appl. Math.'' , '''XLVI''' (1993) pp. 1131–1220 {{MR|1225895}} {{ZBL|0853.35122}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> D. Christodoulou, "Examples of naked singularity formation in the gravitational collapse of a scalar field" ''Ann. of Math.'' , '''140''' (1994) pp. 607–653 {{MR|1307898}} {{ZBL|0822.53066}} </TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top"> D. Christodoulou, S. Klainerman, "The global nonlinear stability of Minkowski space" , Princeton Univ. Press (1992) {{MR|}} {{ZBL|0827.53055}} {{ZBL|0733.35105}} </TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top"> S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) {{MR|0424186}} {{ZBL|0265.53054}} </TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top"> M. Kriele, "A stable class of spacetimes with naked singularities" P. Chruschiel (ed.) A. Krolak (ed.) , ''Mathematics of Gravitation, Lorentzian Geometry and Einstein Equations'' , '''47:1''' , Banach Centre (1997) pp. 169–178 {{MR|1466517}} {{ZBL|0892.53041}} </TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top"> R.P.A.C. Newman, "Censorship, strong curvature, and asymptotic causal pathology" ''Gen. Rel. Grav.'' , '''16''' (1984) pp. 1163–1176 {{MR|0777399}} {{MR|0777400}} {{MR|0779324}} {{ZBL|0549.53065}} </TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top"> R.P.A.C. Newman, "Persistent curvature and cosmic censorship" ''Gen. Rel. Grav.'' , '''16''' (1984) pp. 1177–1187 {{MR|0779324}} {{MR|0777400}} {{ZBL|0559.53045}} {{ZBL|0549.53066}} </TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top"> R. Penrose, "Gravitational collapse: the role of general relativity" ''Rivista del Nuovo Cimento'' , '''1''' (1969) pp. 252–276 {{MR|2388164}} {{MR|1915236}} {{ZBL|}} </TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top"> R. Penrose, "Gravitational collapse" C. DeWitt-Morette (ed.) , ''IAU Symposium 64 on Graviational Radiation and Gravitational Collapse'' , Reidel (1974) pp. 82–91 {{MR|2388164}} {{MR|1915236}} {{MR|0584259}} {{MR|0264959}} {{MR|0172678}} {{ZBL|1030.83516}} {{ZBL|1001.83040}} {{ZBL|1043.83018}} {{ZBL|0954.83012}} {{ZBL|0125.21206}} </TD></TR></table> |

## Revision as of 17:00, 15 April 2012

A (future) singularity in a space-time corresponds to a causal geodesic (cf. also Geodesic line) which is future inextensible and whose affine parameter remains finite (cf. Naked singularity). Penrose's weak (respectively, strong) cosmic censorship hypothesis states that any "physically realistic" space-time with a future-inextensible incomplete geodesic that lies in the past of future infinity (respectively, of a point of space-time) is unstable with respect to a (still to be specified) natural topology of space-times [a9], [a10]. Here, "future infinity" refers to the conformal future boundary of a weakly asymptotically flat space-time in the sense of Penrose [a5] or a variant thereof. The meanings of "physically realistic" and "stable" are not specified. This is due to the existence of counter-examples which are not entirely unrealistic from a physical point of view but still have properties which seem to be very special (see, for instance, [a6]). There are various variants for both the weak and the strong cosmic censorship hypothesis. An especially important version of strong cosmic censorship is the hypothesis that any physically reasonable and qualitatively stable space-time is globally hyperbolic. The deepest general result on weak cosmic censorship has been obtained by R.P.A.C. Newman [a7], [a8], who shows that "persistent curvature" enforces a version of weak cosmic censorship. A more direct version of the cosmic censorship theorem, which is, however, only applicable for space-times very close to -dimensional flat Lorentzian space, has been obtained by D. Christodoulou and S. Klainerman [a4]. Christodoulou has also investigated subclasses of spherically symmetric scalar field space-times and has obtained very detailed results with regard to cosmic censorship in these classes of space-times [a1], [a2], [a3].

It should be remarked that spherically symmetric space-times are highly non-generic and that therefore his qualitative results may not hold in the general case.

A consequence of weak cosmic censorship would be that all singularities are contained in black holes. A black hole is a maximal subset of space-time which does not intersect the past of future infinity. One of the simplest examples of space-times containing a black hole is the Schwarzschild space-time, which models the exterior of a non-rotating spherically symmetric star (cf. also Schwarzschild metric).

#### References

[a1] | D. Christodoulou, "The formation of black holes and singularities in spherically symmetric gravitational collapse" Commun. Pure Appl. Math. , XLIV (1991) pp. 339–373 MR1090436 Zbl 0728.53061 |

[a2] | D. Christodoulou, "Bounded variation solutions of the spherically symmetric Einstein-scalar field equations" Commun. Pure Appl. Math. , XLVI (1993) pp. 1131–1220 MR1225895 Zbl 0853.35122 |

[a3] | D. Christodoulou, "Examples of naked singularity formation in the gravitational collapse of a scalar field" Ann. of Math. , 140 (1994) pp. 607–653 MR1307898 Zbl 0822.53066 |

[a4] | D. Christodoulou, S. Klainerman, "The global nonlinear stability of Minkowski space" , Princeton Univ. Press (1992) Zbl 0827.53055 Zbl 0733.35105 |

[a5] | S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973) MR0424186 Zbl 0265.53054 |

[a6] | M. Kriele, "A stable class of spacetimes with naked singularities" P. Chruschiel (ed.) A. Krolak (ed.) , Mathematics of Gravitation, Lorentzian Geometry and Einstein Equations , 47:1 , Banach Centre (1997) pp. 169–178 MR1466517 Zbl 0892.53041 |

[a7] | R.P.A.C. Newman, "Censorship, strong curvature, and asymptotic causal pathology" Gen. Rel. Grav. , 16 (1984) pp. 1163–1176 MR0777399 MR0777400 MR0779324 Zbl 0549.53065 |

[a8] | R.P.A.C. Newman, "Persistent curvature and cosmic censorship" Gen. Rel. Grav. , 16 (1984) pp. 1177–1187 MR0779324 MR0777400 Zbl 0559.53045 Zbl 0549.53066 |

[a9] | R. Penrose, "Gravitational collapse: the role of general relativity" Rivista del Nuovo Cimento , 1 (1969) pp. 252–276 MR2388164 MR1915236 |

[a10] | R. Penrose, "Gravitational collapse" C. DeWitt-Morette (ed.) , IAU Symposium 64 on Graviational Radiation and Gravitational Collapse , Reidel (1974) pp. 82–91 MR2388164 MR1915236 MR0584259 MR0264959 MR0172678 Zbl 1030.83516 Zbl 1001.83040 Zbl 1043.83018 Zbl 0954.83012 Zbl 0125.21206 |

**How to Cite This Entry:**

Penrose cosmic censorship.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Penrose_cosmic_censorship&oldid=24525