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Date: March 24, 1997
Christodoulou Offers Strong Support for Cosmic Censorship, Says Naked Singularities Can't Really Exist
Princeton, N.J.--Princeton Mathematician Demetrios Christodoulou says that British physicist Stephen Hawking shouldn't have conceded his bet on the cosmic censorship conjecture. In a paper to be published in the Annals of Mathematics, Christodoulou marshals what may be the strongest support to date for cosmic censorship.
At issue is the question of whether naked singularities exist or not. Hawking bet two Caltech physicists (John Preskil and Kip Thorne) that they do not. Hawking's concession was reported in a Feb. 12 New York Times front-page article. Hawking was persuaded to pay up on the strength of findings originating in a 1993 paper "Universality and Scaling in Gravitational Collapse of a Massless Scalar Field" by Matthew Choptuik of the University of Texas at Austin.
A singularity occurs when the quantities in a theory "blow up" or become infinite and therefore is a sign that the theory doesn't work.
The theory relevant to cosmic censorship is Einstein's theory of general relativity and its insight into gravity in terms of the dynamic relationship between mass and spacetime.
According to relativity theory, the denser an object, the more it curves the surrounding space. But when a star at least 1.6 times the mass of the sun dies, it collapses, and its density increases to the point that space becomes so curved that tidal forces (i.e., gravity) affecting particles become infinite. To understand whatever is going on at that point of singularity, more than the theory of general relativity is required.
But if the singularity is always hidden rather than being "naked" and therefore observable, then what's going on at the singularity need not be accommodated by general relativity theory. It is the challenge to that theory posed by the possibility of naked singularities that interests general relativity theorists such as Hawking and Christodoulou.
Some 20 years ago the British mathematician Roger Penrose made the cosmic censorship conjecture, which states that in the universe black holes enshroud singularities so that no information about singularities can reach an outside observer. The tremendous gravitational forces of collapsing stars confine even light particles within black holes so that there is no way for anyone outside a black hole to learn from observation anything about what's going on inside. The alternative to cosmic censorship is a naked singularity.
Both Choptuik's paper and a published paper by Christodoulou himself, "Examples of Naked Singularity Formation in the Gravitational Collapse of a Scalar Field," show that naked singularities are a possibility. (Christodoulou submitted that paper to the Annals of Mathematics in 1992; it was published in 1994.)
But what Christodoulou shows in his forthcoming paper is that the possibility is mathematical only and that the probability of an actual naked singularity forming is zero. "So," says Christodoulou, "that means that naked singularities cannot really--in a physical sense--form." And it means too that Christodoulou's work supports the cosmic censorship conjecture and that Hawking shouldn't have paid up.
"Suppose you have a hill and a ball," says Christodoulou, "and roll the ball up the hill. If you send the ball up with a small velocity, then the ball will climb and at some point begin to roll back down. If you send it up with lots of velocity, the ball will go up and over the top of the hill. But if you send the ball with exactly the correct velocity, it is going to take an infinite time to rise to the top of the hill, and at any point in time other than infinity it will still be rising, and somehow at infinity in time it will just get to the top of the hill.
"So the ball doesn't really get to the top if it gets to the top in an infinity of time. And this situation only happens for a very precise value of the velocity, but in physics what's interesting is never such a hairline case because you can never have a precise value of velocity." (That last statement is an allusion to Heisenberg's uncertainty principle which says that an observer can know the position of a particle or its momentum, but not both at the same time.)
"The fact that there is a certain exceptional set of parameters which mathematically would give rise to these singularities is definitely interesting," says Christodoulou, "because if there were no such parameters, the whole mathematical treatment would have been a much easier problem. You have to take that set of parameters into account and prove that it is of zero probability."
"Proof" is the key word that differentiates Choptuik's and Christodoulou's work on naked singularities. And it's Christodoulou who has the proof. "Only a general mathematical theorem can enable you to say that in all situations of collapsing matter singularities either will not form or will form preceded by black holes. Everything else is of zero probability. This is essentially what my theorem says."
Christodoulou's analysis turned up another kind of singularity which is a mathematical possibility. He calls it a collapsed light cone and reported it in his 1994 paper. Like the naked singularity the probability of it really existing is zero.
Note: Robert Wald, professor of physics at the University of Chicago, will give a review talk on the cosmic censorship conjecture at the April American Physical Society meeting (April 18-21) in Washington, D.C.