Cosmic Censorship Conjecture and Topological Censorship

Cosmic Censorship Conjecture and Topological Censorship 21 settembre 2009

Cosmic Censorship Conjecture 40 years ago in the Rivista Nuovo Cimento Sir Roger Penrose posed one of most important unsolved problems in Theoretical Relativity, The Cosmic Censorship Conjecture; which in its weak version, talking in physical terms, tells us that: The complete gravitational collapse of a body always result in a black hole rather than a naked singularity, it means that all the singularities of gravitational collapse are hidden within a black hole in such a way that distant observers cannot seen them.

Oppenheimer-Snyder dust cloud The first example of one solution of Einstein equations which describes a black hole was the collapsing spherical dust cloud of Oppenheimer-Snyder (1939). Inside the cloud there is a singularity (a region where the conventional classical picture of spacetime breaks down) but this is not visible for distant observers, being shielded from view by an absolute event horizon.

Two natural questions arise, do singularities exist in more general models (i. e. models without the spherical symmetry of the Oppenheimer-Snyder cloud) and if they exist are they always shielded by an event horizon? Hawking-Penrose Singularity theorem The first question has a positive answer Theorem Assume Energy conditions, Causal structure condition, Very strong gravity region, then the spacetime is geodesically incomplete.

The Cosmic censorship conjecture says that the second question also has a positive answer. We can express the conjecture formally in the next way:

The Cosmic censorship conjecture says that the second question also has a positive answer. We can express the conjecture formally in the next way: A compact view of a suitable spacetime Weak Cosmic censorship There are no TIPs included in a -TIP, i. e. there are no naked TIPs. Particularly every future-inextendible null geodesic γ visible from q I + (i.e. such that γ I (q)) is future complete.

There is very strong theoretically evidence that some version of the Cosmic Censorship Conjecture must be valid, there are situations where the conjecture is not valid but these cases result to be non generic situations which generally are non stable (i.e. some perturbations of the initial data generates an event horizon). We have also very nice results which follows from the validity of the conjecture, one of this results is the Topological Censorship Theorem.

Topological Censorship questions follow from the next reasoning: We know that every three-manifold occurs as the spatial topology of a solution of the Einstein Equations, then why such topological structures are not part of our ordinary experience? Part of the answer is the Gannon singularity theorem Theorem (Gannon) Let (M, g) a spacetime (a 4-manifold with a Lorentzian metric) such that: M R S, S a Cauchy surface non simply connected and regular near infinity, Energy condition. Then (M, g) is null geodesically incomplete.

Gannon s theorem tell us that if there is some topology anomaly and this lives enough some singularity developes. Then if Cosmic Censorship is valid this topology anomalies should be hidden by some event horizon. On the other hand we can think that maybe topology anomalies exist but they collapse before some singularity form, but this is not possible at a big scale, because if topology change this implies (Tipler) violation of the energy conditions, which are valid for all types of matter that we know. On the other hand at quantum level energy conditions can be violated, and topology change is really possible. Then Topology censorship theorem tell us that outside all event horizons topology anomalies are permitted only at the Planck scale and they cannot grow to macroscopic size and persist for macroscopically long times.

Theorem (Topological Censorship) If an asymptotically flat spacetime with domain of outer communications D obeys Causal continuity holds at spatial infinity i 0, Energy conditions, Cosmic Censorship holds, then D is simply connected.

Topological Censorship theorem follows from the next Key Lemma Lemma (Galloway) Let D be the domain of outer communications of an asymptotically flat spacetime M, and let H + as before. If Causal continuity holds at spatial infinity i 0, Every complete null geodesic γ D H + ( where H + is the union of all future event horizons that may be present) posseses a pair of conjugate points, every future-inextendible null geodesic γ visible from q I + (i.e. such that γ I (q)) is future complete, Then D is simply connected.

The hypothesis of the lemma 1 Asymptotic flatness

The hypothesis of the lemma 1 Asymptotic flatness Causal continuity at infinity For any compact set K D, i 0 / I (K ) I + (K ).

The hypothesis of the lemma 2 Energy conditions create conjugate points From the Einstein s equations R ab ν a ν b = 8π(T ab ν a ν b + 1/2T ) follow energy conditions of the type R ab ν a ν b 0 for all null vectors ν. This condition plus hypothesis of the type that the light rays find some matter, which create some type of convergence, guarantees the existence of conjugate points.

The proof of the key lemma The proof of the lemma By way of contradiction, we will assume causal that D is non simply connected (π 1 (D) {0}). We consider a narrow metric h ab and U the points influenced by H + in the spacetime with the metric h ab. If D = D U, π 1 (D ) = π 1 (D H + ) = π 1 (D).

The proof of the key lemma Consider D, D the respective universal coverings, there exist α β such that Iα, I + β, are joined by a causal curve γ. Then the projection of γ joined with null generators of I ± form a non trivial loop.

The proof of the key lemma If Q I + β is the final point of γ then the closure of the past of Q, I (Q), does not contain α-spatial infinity iα. 0 This is because at some instant the curve γ must pass by the compact region of non trivial topology, by the causal continuity at spatial infinity then the curve never approaches a null generator that contains i 0.

The proof of the key lemma Since i 0 α / I (Q), then some null geodesic, σ without conjugate points, from Q meets I α at a point P. Projecting this curve we obtain a curve σ I (q) which enters D then σ D H +. Since every complete null geodesic contained in D H + has conjugate points, then σ should be incomplete contradicting the Cosmic censorship type hypothesis.

The proof of the Topological Censorship Theorem The proof of the Topological Censorship Theorem The proof of the topological censorship theorem follows from the previous lemma plus a result which tells us that the incomplete geodesic σ is visible not only from a point of I + but also from a point of spacetime which can signal I +.

Conclusions

The End Then Mr Einstein it seems that Mr Hawking was right, God not just play to dice but sometimes he throw them where we cannot seen them