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One thing I know about black holes is that an object gets closer to the event horizon, gravitation time dilation make it move more slower from an outside perspective, so that it looks like it take an infinite amount of time for the object to reach the event horizon. It seems like a similar process should slow the formation of the black hole itself: As the star collapses, its gravitational time dilation make itself collapse more slowly. This make me wonder, are what astronomers claim to be black holes really black holes, or are they stars that progressively make themselves more similar to one without actually reaching the stage of having an event horizon?

EDIT: Contemplating one answer, I realize the question is ambiguous. What does finite time mean in general relativity. Here is a less ambiguous question: Is there a connected solution of 3+1 dimensional general relativity with one space-like slice not have a singularity, and another space-like slice having one.

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Nice rewrite on the question. –  Carl Brannen Feb 12 '11 at 5:10
Spacelike singularities occur in uncharged nonrotating black holes. You must then distinguish between physical singularities and coordinate singularities. For example, the Schwarzchild metric has a coordinate singularity at the Schwarzchild radius that may be eliminated by a change in coordinates, but I doubt this is what you meant. –  Gordon Feb 12 '11 at 6:48
I suggest to replace "singularity" by "an intersection with an event horizon" in your rephrased question, since you want to know about black hole formation and not singularity formation. The answer is then "yes", with the Vaidya solution being the simplest example. See e.g. Fig. 4 in arxiv.org/abs/0809.2213 for its Penrose diagram. –  Daniel Grumiller Feb 12 '11 at 10:15
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4 Answers

up vote 2 down vote accepted

(this answer addresses the new question)

As a consequence of the singularity theorems, it is not only possible but (arguably) inevitable for singularities to form in a finite amount of "time" in a physically reasonable spacetime. The word "time" in this context means "proper time along a specific timelike geodesic". For example, if there is a trapped surface* in spacetime, then a singularity will appear within a finite amount of proper time (along a timelike geodesic) in the future of that surface; so, an observer sitting in a collapsing star will reach the singularity in finite time. Thus, the collapse of matter is one possible way to create a singularity "out of nothing". If your spacetime is globally hyperbolic and you foliate it by Cauchy surfaces you can say in a much more "universal" way that the singularity didn't exist at time [;t_0;] and came to exist at time [;t_1;].

I should point out that the singularities are a generic feature of physically reasonable spacetimes; take a look at the Hawking-Penrose theorem - it applies in very general situations.

Also, as the original question was about black holes and not singularities, I should advise you to make a clear distinction between the two concepts. Trapped surfaces form due to the condensation of matter (this is the famous Schoen-Yau theorem), and under a certain extra hypothesis, these surfaces will be hidden inside black holes. This extra hypothesis is the well-known (weak) Cosmic Censorship Conjecture (CCC). If it does not hold, gravitational collapse can create naked singularities, that is, singularities not "causally hidden" by the event horizon of a black hole. Much of what is known in general about black holes depend crucially on the CCC.

*A trapped surface is a two-dimensional spacelike compact surface such that the null geodesics departing from it are accelerating towards each other - mathematically, we say that the expansion of the congruence of future-directed null geodesics orthogonal to the surface is negative.

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I think the question actually was about the possibility of formation of trapped surfaces in finite time. You say "if there are trapped surfaces then there is singularity" but can the trapped surfaces formate in finite time themselves? –  Anixx Apr 10 '11 at 10:26
@Anixx Yes. See the Schoen-Yau paper on formation of apparent horizons (the relevant result is theorem 2). –  Rodrigo Barbosa Jun 24 '11 at 18:20
I don't think Schoen and Yau actually answers @Anixx 's question. It states that when sufficient mass is already concentrated, then there must exist a trapped surface. For actual dynamical formation of apparent horizons you need something like Pin Yu's recent paper (or, going historically, you have the pioneering work of Oppenheimer-Snyder on dust, Christodoulou on scalar field in spherical symmetry in the 90s, Christodoulou on vacuum gravitational collapse in 2009 etc.) –  Willie Wong Jun 25 '11 at 12:41
@Willie Wong thank for the remark. Indeed the main question here is whether sufficient mass can be concentrated in finite time to form a trapped surface since the closer mass particles approach each other, the slower they move due to gravitational time slowing effect. –  Anixx Jun 26 '11 at 16:40
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You are simply looking at it from an observer's viewpoint. Yes, looking from outside, matter tends to asymptotically approach but never reach the event horizon. If you were part of that matter spiraling into a black hole, there would be no problem reaching the horizon, crossing it, and going right down to the singularity. The event horizon is not a physical barrier. You could be free falling, and your time would not be infinitely dilated. So the answer is yes they can form easily in a finite time.

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"If you were part of that matter spiraling into a black hole, there would be no problem reaching the horizon, crossing it, and going right down to the singularity." - why if the time of the black hole existence is finite? The BH will simply evaporate befgore you reach the horizon. Probably you got this quote from a very old book that does not accout for BH evaporation. –  Anixx Jun 25 '11 at 11:01
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To begin with, there is a connected solution of 3+1 GR in which particles fall to the singularity in finite time. In particular, Gullstrand-Painleve coordinates do this. The big difference with Schwarzschild coordinates is that the speed of light depends on direction: light moves into a black hole faster than it moves out. See:

For the formation of a black hole in these coordinates, see:

Phys.Rev.D79:101503,2009, J. Ziprick, G. Kunstatter, Spherically Symmetric Black Hole Formation in Painlevé-Gullstrand Coordinates

For the generalization of Gullstrand-Painleve coordinates to the rotating black hole, see the very readable paper that gives an intuitive explanation for what is going on, see:
Am.J.Phys.76:519-532,2008, Andrew J. S. Hamilton, Jason P. Lisle, The river model of black holes http://arxiv.org/abs/gr-qc/0411060

Note: that the above paper is peer reviewed and shows that yes indeed, particles falling past the event horizon travel with speeds greater than the 1 (in GP coordinates). In GR, the speeds of objects depend on the choice of coordinates. Consequently, this exceeding of the speed 1 is not equivalent to exceeding the speed of light. In GP coordinates, a light beam moving towards the singularity inside the event horizon also moves at speed greater than 1. Consequently, there is no violation of special relativity.

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Thanks for giving refrences. –  Itai Bar-Natan Feb 12 '11 at 19:05
Another reference I like (g) is my paper on Gullstrand-Painleve coordinates, which shows how to write them as F=ma: Int.J.Mod.Phys.D18:2289-2294,2009, "The Force of Gravity in Schwarzschild and Gullstrand-Painleve Coordinates", arxiv.org/abs/0907.0660 –  Carl Brannen Feb 12 '11 at 20:17
Gullstrand-Painleve coordinates use local time of the free-falling object. On the event horizon this time corresponds to the infinite time of the external observer. Thus the infalling object still cannot reach horizon in finite time. Also in Gullstrand-Painleve coordinates the infalling object reaches the local speed of light at the horizon and even higher speed inside which is impossible for any object that has internal structure. For an object that has mass its kinetic energy will also exceed that of its mass which contradicts the conservation of energy. –  Anixx Apr 10 '11 at 10:43
So yes, such coordinates are possible. No, no massive object can follow this path. –  Anixx Apr 10 '11 at 10:46
@Anixx; Re: "no massive object can follow this path." Rather than arguing the point, I'm going to note that I'm the only amateur who's ever won an honorable mention at the annual gravitation essay contest, and that the topic of my paper was GP coordinates: arxiv.org/abs/0907.0660 For more information on GP coordinates, see "The River Model of Black Holes" cited in the answer. –  Carl Brannen Apr 10 '11 at 21:08
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What astronomers claim to be black holes are objects that "progressively make themselves more similar to [a black hole] without actually reaching the stage of having an event horizon", as they reckon. That's assuming that GR is valid, since all such claims depend on GR's equations. Plenty of books on GR note that black holes are perhaps better named "frozen stars" from a distant observer's perspective.

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Who ever is +1-ing this is a joker. –  Killercam May 31 '13 at 14:30
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