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
Cosmic Censorship Conjecture
result in a black hole rather
than a naked singularity, it
40 years ago in the Rivista Nuovo
means that all the
Cimento Sir Roger Penrose posed one of singularities of gravitational
most important unsolved problems in
collapse are hidden within a
Theoretical Relativity, The Cosmic
black hole in such a way that
Censorship Conjecture; which in its weak distant observers cannot
version, talking in physical terms, tells us seen them.
that:
The complete gravitational
collapse
of a body always
Cosmic Censorship Conjecture
result in a black hole rather
than a naked singularity, it
40 years ago in the Rivista Nuovo
means that all the
Cimento Sir Roger Penrose posed one of singularities of gravitational
most important unsolved problems in
collapse are hidden within a
Theoretical Relativity, The Cosmic
black hole in such a way that
Censorship Conjecture; which in its weak distant observers cannot
version, talking in physical terms, tells us seen them. Let us try to
that:
understand formally what
does this mean.
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?
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.
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 (Friedman,
Schleich,Witt)
An observer cannot probe the topology
of spacetime: Any topological structure
collapses too quickly to allow light to
traverse it.
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
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,
i0 ∈
/ I− (K ) ∪ I+ (K ).
The hypothesis of the lemma 2
Energy conditions create conjugate points
From the Einstein’s equations Rab ν a ν b = 8π(Tab ν a ν b + 1/2T )
follow energy conditions of the type Rab 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 (π1 (D) 6= {0}). We consider a narrow metric
hab and U the points influenced by H+ in the spacetime with
the metric hab . If D0 = D ∪ U, π1 (D0 ) = π1 (D ∪ H+ ) = π1 (D).
The proof of the key lemma
e0, D
e the respective universal coverings, there exist
Consider D
α 6= β such that Iα− , Iβ+ , 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 ∈ Iβ+ is the final point of γ then the
closure of the past of Q, I− (Q), does not
contain α-spatial infinity iα0 . 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 iα0 ∈
/ I− (Q), then some null
geodesic, σ
e without conjugate
points, from Q meets Iα− at a point
P. Projecting this curve we obtain
a curve σ ⊂ I− (q) which enters D0
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
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