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It's been known since Oppenheimer and Snyder's work in 1939 that it's easy to get a naked (i.e., timelike) singularity in models of spherically symmetric gravitational collapse, for forms of matter such as dust that obey all the standard energy conditions. (A review article on this kind of thing is Joshi 2011.) Whether this is physically realistic, stable with respect to perturbations, and so on is a different question, but not relevant for the purposes of this question.

Now it seems to me that the formation of a naked singularity by gravitational collapse is an example of topology change. Spacelike slices before the collapse have the trivial topology, while slices after the collapse have a hole in them at the singularity. (This is not the case for a black hole singularity, since a black hole singularity is spacelike.)

Topology change in GR has also been studied for a long time, and the classic reference seems to be Geroch 1967, which is summarized in Borde 1994. Geroch proves that topology change always involves both acausality and violation of the weak energy condition (WEC).

This confuses me, because doesn't dust satisfy the WEC? I'm sure I'm misunderstanding something, but I don't know what it is.

Borde, 1994, "Topology Change in Classical General Relativity," http://arxiv.org/abs/gr-qc/9406053

Geroch 1967, http://adsabs.harvard.edu/abs/1967JMP.....8..782G , paywalled

Joshi and Malafarina, "Recent developments in gravitational collapse and spacetime singularities," 2011, https://arxiv.org/abs/1201.3660

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  • $\begingroup$ Note that the theorems of Geroch all assume that the boundaries of the cobordism are compact, ie they correspond to the case of closed universes, this is not the case as far as I know for the various dust collapse scenarios leading to naked singularities $\endgroup$ – Slereah Mar 19 '18 at 9:16
  • $\begingroup$ (in particular, if there is a naked singularity, odds are good the final cobordant surface will not be compact) $\endgroup$ – Slereah Mar 19 '18 at 9:27
  • $\begingroup$ Good point, but Borde seems to show that the restriction to a closed universe is inessential. See pp. 15-17. I think the resolution of the paradox may be simply the following. Let T=the proposition that topology change occurs, C=causality violation occurs, CC=causal compactness holds, and W=weak energy condition fails. Reading the abstract, it sounds like he's claiming T⟹C∧W, but I think he's actually proving that T∧CC⟹C∧W. For Oppenheimer-Snyder collapse to a naked singularity, CC fails. $\endgroup$ – Ben Crowell Mar 20 '18 at 2:41
  • $\begingroup$ He does indeed specify causal compactness, which is never true for nakedly singular spacetimes I believe $\endgroup$ – Slereah Mar 20 '18 at 9:36
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It's been known since Oppenheimer and Snyder's work in 1939 that it's easy to get a naked (i.e., timelike) singularity in models of spherically symmetric gravitational collapse, for forms of matter such as dust that obey all the standard energy conditions. (A review article on this kind of thing is Joshi 2011.) Whether this is physically realistic, stable with respect to perturbations, and so on is a different question, but not relevant for the purposes of this question.

This is where you're going awry. IMHO it's very important to appreciate what Oppenheimer and Snyder were saying in their 1939 paper on continued gravitational contraction. Also read Einstein's 1939 paper on a stationary system with spherical symmetry consisting of many gravitating masses. Einstein said this: “it is noted that g44 = (1 - μ/2r / 1 + μ/2r)² vanishes for r = μ/2. This means that a clock kept at this place would go at the rate zero. Further it is easy to show that both light rays and material particles take an infinitely long time (measured in "coordinate time") in order to reach the point r = μ/2 when originating from a point r > μ/2”.

Now it seems to me that the formation of a naked singularity by gravitational collapse is an example of topology change. Spacelike slices before the collapse have the trivial topology, while slices after the collapse have a hole in them at the singularity. (This is not the case for a black hole singularity, since a black hole singularity is spacelike).

Yes, it would be a topology change.

Topology change in GR has also been studied for a long time, and the classic reference seems to be Geroch 1967, which is summarized in Borde 1994. Geroch proves that topology change always involves both acausality and violation of the weak energy condition (WEC).

Yes, it would be a violation all right.

This confuses me, because doesn't dust satisfy the WEC? I'm sure I'm misunderstanding something, but I don't know what it is.

You're misunderstanding the frozen star. That topology change hasn't happened yet, and it never ever will.

Note though that this doesn't mean that black holes never form. They form from the inside out. Since we're talking about frozen stars, I shall employ a hailstone as an analogy: you are a water molecule. You cannot pass through the surface of a hailstone. But you alight upon it, and you are surrounded by other water molecules. Then you are buried by other water molecules. So you don't pass through the surface, the surface passes through you.

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  • $\begingroup$ @John Rennie: "Rob Jeffries is a professor of astrophysics at Keele University. You have grossly underestimated the technical sophistication of his question". The question was asked by Ben Crowell, and my answer goes to the heart of the matter. I give a hyperlink to Oppenheimer and Snyder's paper, which is crucial to this issue. Google on Oppenheimer Snyder frozen star. $\endgroup$ – John Duffield Apr 11 '18 at 16:33

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