# Christodoulou's paper on naked singularities in inhomogeneous dust collapse

I have been studying of late about formation of naked singularities in certain collapse scenarios in Einstein's theory. It seems to me that the canonical paper to read about how such a formation is established is the 1984 paper by Christodoulou in Communications in Mathematical Physics. ( http://www.ams.org/mathscinet-getitem?mr=742192 )

I was wondering if there is a reference which gives a more modern rewriting of the proof in that paper which say sort of highlights the generic technique of the proof which the reader can take away from there for other scenarios.

Somehow even the most recent books on Einstein's theory like the otherwise brilliant book by Choquet-Bruhat also doesn't dwell on techniques of testing in a collapse scenario whether the curvature singularity is naked or not.

I haven't seen till now any generic method or algorithm for testing this.

It would be great if someone can give me some references/explanations along these lines.

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Collapse scenarios Are you absolutely sure you want to restrict yourself to dust collapse? In the case of a spherically symmetric scalar field, there is also http://www.ams.org/mathscinet-getitem?mr=1307898 (and this paper which shows that naked singularities in the scalar field model are unstable).

There's some problem of the interpretation of the dust collapse as a physical formation. Here I'll quote Demetri from his book on vacuum collapse due to incoming gravitational waves

With the above remarks in mind the author turned to the study of the gravitational collapse of an inhomogeneous dust ball. In this case, the initial state is still spherically symmetric, but the density is a function of the distance from the center of the ball. The corresponding spherically symmetric solution had already been obtained in closed form by Tolman in 1934, in comoving coordinates, but its causal structure had not been investigated. This required integrating the equations for the radial null geodesics. A very diﬀerent picture from the one found by Oppenheimer and Snyder emerged from this study. The initial density being assumed a decreasing function of the distance from the center, so that the central density is higher than the mean density, it was found that as long as the collapse proceeds from an initial state of low compactness, the central density becomes inﬁnite before a black hole has a chance to form, thus invalidating the neglect of pressure and casting doubt on the predictions of the model from this point on, in particular on the prediction that a black hole eventually forms.

Essentially the same comment was made by Hájíček in his MathReviews on the dust collapse paper.

Modern re-writes Part of the reason that there are no modern re-writes of the proof is because of what I mentioned above, that the violation of weak cosmic censorship is unphysical (so it didn't attract that much attention). On the other hand, the basic idea behind the proof is not too hard (it is in the implementation of the analysis that is difficult). Simply speaking, in spherical symmetry, there is a lot of qualitative information that can be extracted without knowing too much details of the matter model involved (we must have matter as for vacuum, spherically symmetric space-times, we have Birkhoff's theorem). Perhaps a best modern reference is Mihalis Dafermos' paper in CQG. The most important part is that under symmetry conditions and some mild assumptions, future null infinity must be complete; in the spherically symmetric space-times, future null infinity is characterised by the area-radius $r\to\infty$ along out-going null geodesics. Thus the basic idea is to show that there exists out-going null geodesics, such that if you travel along the future direction the area radius increases without bound, and that if you travel along the past direction you will hit the singularity.

Now, in the case of the Tolman dust which was studied by Christodoulou, since the exterior of the dust cloud is glued to a Schwarzschild solution, there is a dichotomy: either the out-going null geodesic escapes the dust region before hitting the apparent horizon, or the null geodesic hits the apparent horizon first. In spherical symmetry, it is a general fact that the apparent horizon consists of space-like or expanding null portions. So once a null geodesic passes the apparent horizon it can no longer escape to infinity. On the other hand, inside the Schwarzschild region the apparent horizon agrees with the event horizon, so once the null geodesic escapes from dust without hitting the apparent horizon, it will remain in the domain of outer communications and escape to infinity.

So to demonstrate existence of naked singularity, it suffices to show that a null geodesic emanating from the first point of singularity (it doesn't make sense to consider later points, as they will no longer belong to the Cauchy development of an initial data set) will escape the dust cloud before hitting the apparent horizon. To do so one needs to estimate the size of the solution of an ODE. In spherical symmetry, the apparent horizon is characterised by $2m/r = 1$, where $m$ is the Hawking mass and $r$ is the area radius. Now the Hawking mass satisfies ordinary differential equations (with source) in spherical symmetry (see, for example, Equation 3 here; or see Section 3 of this paper.) So to estimate the size of the Hawking mass at the matching boundary, it is necessary to estimate the source terms in the ODE. This will lead to equation chasing using the Einstein equations and the matter-field equations. The rest is just being clever (in deciding in which order to estimate the various quantities) and doing the hard work of computation.

General techniques The reason there are no references for general techniques in testing whether a singularity is naked or not is very simply, there exists no such techniques. In spherical symmetry there is the simple description I posted above, but at the end of the day the estimates strongly depend on the structure of the matter equation (its separability and what not), so can only be really dealt with in a case-by-case basis. Of course, if you are given an explicit solution of the equations, testing whether the singularity is naked is often a simple computation re-writing the solution in some sort of null coordinates. The problem is that to prove genericity or to study singularities without reference to a explicit solution, one needs analytical estimates which, like I said, depends on which equations you are studying. In fact, if you could come up with a usable, generic method of testing whether a singularity is clothed, you would be half-way there resolving the general weak cosmic censorship conjecture.

Some last comments The general issue of weak cosmic censorship is a wide open one. The problem is that the statement contains the word "generic" in it (generic initial data sets lead to blah blah blah). So while there has been quite a lot of work going into constructing solutions and verifying that those solutions contain naked singularities, these works say nothing about weak cosmic censorship. (Explicit solutions tend to be non-generic in the space of solutions, except in the case you have strong rigidity theorems like Birkhoff's theorem for spherically symmetric Einstein-vacuum/electro-vacuum space-times or the No Hair Theorem for four-dimensional stationary axi-symmetric solutions.) The only real progress to weak cosmic censorship have all been due to Christodoulou (most of the physics papers are lacking in rigour). (Interestingly, there has been more developments in strong cosmic censorship, which, despite the name, has relatively little to do with weak cosmic censorship.)

Most recently, the focus in the community seems to be that the next model to consider for cosmic censorship should be the Einstein-Vlasov model (in spherical symmetry). (Well, it has been under consideration for around 10 years now and still prohibitingly hard.) For general solutions without symmetry assumptions, there has been essentially zero work in the field. There was an attempt to reformulate the conjecture into something mathematically tractable, but not has been done with the reformulation.

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@Willie Thanks a lot for this amazingly detailed answer! When you say that "future null infinity is complete" I suppose you mean just what you said in the next line about the spatial scale growing unbounded along null geodesics. Or are you making some statement also about the structure of the future null infinity? Further is this test of yours the same as the "future in/completeness theorem" by Choquet and Costakis? (which relies on the nature of the lapse function for a regularly sliced space-time) ? –  Anirbit Sep 10 '10 at 16:57
Contd. The statement of Strong Censorship Conjecture that Choquet quotes is the one formalized by Eardley and Moncrief which states that a maximal globally hyperbolic vacuum Einstein development of generic initial data is inextensible if the initial data is asymptotically euclidean or compactly supported. In this formulation how does the existence of event horizon view point fit in? –  Anirbit Sep 10 '10 at 17:00
Contd. Your definition of the apparent horizon sounds a bit new to me. Can you kindly relate it to the definition using the Ricci tensor? Also Christodoulou talks about whether or not the null geodesic from the singularity can or cannot escape before the apparent horizon forms. (assuming equations of state such that there was no apparent horizon to begin with). Can you kindly explain this dynamical view of the apparent horizon which is a bit confusing since it seems that once I am given the metric the apparent horizon is completely known. –  Anirbit Sep 10 '10 at 17:10
Completeness of future null infinity means that, under suitable compactification, the null generators of future null infinity are complete (as null geodesics). It is not the same as the statement about spatial scale. The precise definition is given in the last paper I cited in my answer (though you should only consider the future direction, not the past one). Which paper of Choquet and Costakis are you referring to? I cannot find any paper jointly authored by them on MathSciNet. –  Willie Wong Sep 10 '10 at 18:07
As to strong cosmic censorship, the idea is thus: there can be no extension past future null infinity anyway (it is a cosmological horizon). But in the interior of the black hole region (past the endpoint of the event horizon), there may exist a Cauchy horizon (the Cauchy horizon indicates the boundary of the maximal globally hyperbolic development of some initial data set). In the Schwarzschild solution, this boundary is singular (corresponding to the r = 0 singularity). In the Reissner-Nordstrom solution, this boundary is regular (corresponding to the inner horizon). –  Willie Wong Sep 10 '10 at 18:09