Cosmic censorship hypothesis
Encyclopedia
The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of singularities
arising in general relativity
.
Singularities that arise in the solutions
of Einstein's equations are typically hidden within event horizon
s, and therefore cannot be seen from the rest of spacetime. Singularities which are not so hidden are called naked
.
The weak cosmic censorship hypothesis conjectures that no naked singularities other than the Big Bang
singularity exist in the universe
. The hypothesis was conceived by Roger Penrose
in 1969. Cosmic censorship hypotheses should be distinguished from chronological censorship under which every closed timelike curve
passes through an event horizon, which might prevent an observer from detecting the causal violation.http://dx.doi.org/10.1007/s10701-008-9254-9
may break down, and physics
may lose its predictive power. The issue cannot be avoided, since according to the Penrose-Hawking singularity theorems
, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe is deterministic
— it's 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 a spacelike 3-dimensional hypersurface, called the Cauchy surface
). Failure of the cosmic censorship hypothesis leads to the failure of determinism, because it is yet impossible to predict the behavior of space-time in the causal future of a singularity. Cosmic censorship is not merely a problem of formal interest; some form of it is assumed whenever black hole
event horizons are mentioned.
The hypothesis was first formulated by Roger Penrose
in 1969, 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 and that can be proved to be true or false (and that is sufficiently general to be interesting).
Mathematically, the conjecture states that, for generic initial data, the maximal Cauchy development possesses a complete future null infinity.
The two conjectures are mathematically independent, as there exist spacetimes for which the weak cosmic censorship is valid but the strong cosmic censorship is violated and conversely, there exist spacetimes for which the weak cosmic censorship is violated but the strong cosmic censorship is valid.
, corresponding to a black hole of mass M and angular momentum J, can be used to derive the effective potential
for particle orbits restricted to the equator (as defined by rotation). This potential looks like:
Where
is the coordinate radius, 
and 
are the test-particle's conserved energy and angular momentum respectively (constructed from the killing vectors).
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:. 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.
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, the prize being "clothing to cover the winner's nakedness".http://www.theory.caltech.edu/people/preskill/new_naked_bet.html
which forms a counter example to many formulations of the
cosmic censorship hypothesis was found by Mark D. Roberts in 1985:
where
is a constant.
Gravitational singularity
A gravitational singularity or spacetime singularity is a location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system...
arising in general relativity
General relativity
General relativity or the general theory of relativity is the geometric theory of gravitation published by Albert Einstein in 1916. It is the current description of gravitation in modern physics...
.
Singularities that arise in the solutions
Solutions of the Einstein field equations
Where appropriate, this article will use the abstract index notation.Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations of general relativity. Solving the field equations actually gives Lorentz metrics...
of Einstein's equations are typically hidden within event horizon
Event horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
s, and therefore cannot be seen from the rest of spacetime. Singularities which are not so hidden are called naked
Naked singularity
In general relativity, a naked singularity is a gravitational singularity, without an event horizon. In a black hole, there is a region around the singularity, the event horizon, where the gravitational force of the singularity is strong enough so that light cannot escape. Hence, the singularity...
.
The weak cosmic censorship hypothesis conjectures that no naked singularities other than the Big Bang
Big Bang
The Big Bang theory is the prevailing cosmological model that explains the early development of the Universe. According to the Big Bang theory, the Universe was once in an extremely hot and dense state which expanded rapidly. This rapid expansion caused the young Universe to cool and resulted in...
singularity exist in the universe
Universe
The Universe is commonly defined as the totality of everything that exists, including all matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. Definitions and usage vary and similar terms include the cosmos, the world and nature...
. The hypothesis was conceived by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
in 1969. Cosmic censorship hypotheses should be distinguished from chronological censorship under which every closed timelike curve
Closed timelike curve
In mathematical physics, a closed timelike curve is a worldline in a Lorentzian manifold, of a material particle in spacetime that is "closed," returning to its starting point...
passes through an event horizon, which might prevent an observer from detecting the causal violation.http://dx.doi.org/10.1007/s10701-008-9254-9
Basics
Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime, causalityCausality
Causality is the relationship between an event and a second event , where the second event is understood as a consequence of the first....
may break down, and physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
may lose its predictive power. The issue cannot be avoided, since according to the Penrose-Hawking singularity theorems
Penrose-Hawking singularity theorems
The Penrose–Hawking singularity theorems are a set of results in general relativity which attempt to answer the question of when gravitation produces singularities.A singularity in solutions of the Einstein field equations is one of two things:...
, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe is deterministic
Determinism
Determinism is the general philosophical thesis that states that for everything that happens there are conditions such that, given them, nothing else could happen. There are many versions of this thesis. Each of them rests upon various alleged connections, and interdependencies of things and...
— it's 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 a spacelike 3-dimensional hypersurface, called the Cauchy surface
Cauchy surface
Intuitively, a Cauchy surface is a plane in space-time which is like an instant of time; its significance is that giving the initial conditions on this plane determines the future uniquely....
). Failure of the cosmic censorship hypothesis leads to the failure of determinism, because it is yet impossible to predict the behavior of space-time in the causal future of a singularity. Cosmic censorship is not merely a problem of formal interest; some form of it is assumed whenever black hole
Black hole
A black hole is a region of spacetime from which nothing, not even light, can escape. The theory of general relativity predicts that a sufficiently compact mass will deform spacetime to form a black hole. Around a black hole there is a mathematically defined surface called an event horizon that...
event horizons are mentioned.
The hypothesis was first formulated by Roger Penrose
Roger Penrose
Sir Roger Penrose OM FRS is an English mathematical physicist and Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute, University of Oxford and Emeritus Fellow of Wadham College...
in 1969, 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 and that can be proved to be true or false (and that is sufficiently general to be interesting).
Weak and strong cosmic censorship hypothesis
The weak and the strong cosmic censorship hypothesis 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 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 Lorentzian manifold.
The two conjectures are mathematically independent, as there exist spacetimes for which the weak cosmic censorship is valid but the strong cosmic censorship is violated and conversely, there exist spacetimes for which the weak cosmic censorship is violated but the strong cosmic censorship is valid.
Example
The Kerr MetricKerr metric
The Kerr metric describes the geometry of empty spacetime around an uncharged axially-symmetric black-hole with an event horizon which is topologically a sphere. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which...
, corresponding to a black hole of mass M and angular momentum J, can be used to derive the effective potential
Effective potential
The effective potential, or effective potential energy is a mathematical expression combining the centrifugal potential energy with the potential energy of a dynamical system. Commonly used in calculating the orbits of planets , the effective potential allows one to reduce a problem to fewer...
for particle orbits restricted to the equator (as defined by rotation). This potential looks like:
Where
is the coordinate radius, and are the test-particle's conserved energy and angular momentum respectively (constructed from the killing vectors).To preserve cosmic censorship, the black hole is restricted to the case of
. For there to exist an event horizonEvent horizon
In general relativity, an event horizon is a boundary in spacetime beyond which events cannot affect an outside observer. In layman's terms it is defined as "the point of no return" i.e. the point at which the gravitational pull becomes so great as to make escape impossible. The most common case...
around the singularity, the requirement:
must be satisfied:. This amounts to the angular momentumAngular momentum
In physics, angular momentum, moment of momentum, or rotational momentum is a conserved vector quantity that can be used to describe the overall state of a physical system...
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-Nordstrom solution, which contains a singularity at 
that is not surrounded by a horizon. A formal statement needs some set of hypotheses which exclude these situations. - CausticCaustic (mathematics)In differential geometry and geometric optics, a caustic is the envelope of rays either reflected or refracted by a manifold. It is related to the optical concept of caustics...
s 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 hypothesis.
In 1991, John Preskill
John Preskill
John Phillip Preskill is an American theoretical physicist and the Richard P. Feynman Professor of Theoretical Physics at the California Institute of Technology ....
and Kip Thorne
Kip Thorne
Kip Stephen Thorne is an American theoretical physicist, known for his prolific contributions in gravitation physics and astrophysics and for having trained a generation of scientists...
bet against
Scientific wager
A scientific wager is a wager whose outcome is settled by scientific method. They typically consist of an offer to pay a certain sum of money on the scientific proof or disproof of some currently uncertain statement. Some wagers have specific date restrictions for collection, but many are open...
Stephen Hawking
Stephen Hawking
Stephen William Hawking, CH, CBE, FRS, FRSA is an English theoretical physicist and cosmologist, whose scientific books and public appearances have made him an academic celebrity...
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, the prize being "clothing to cover the winner's nakedness".http://www.theory.caltech.edu/people/preskill/new_naked_bet.html
Counter-example
An exact solution to the scalar-Einstein equations which forms a counter example to many formulations of thecosmic censorship hypothesis was found by Mark D. Roberts in 1985:
where
is a constant.External links
- The old bet (conceded in 1997)
- The new bet