# What is a sudden singularity?

I've seen references to some sort of black hole (or something) referred to as a sudden singularity, but I haven't seen a short clear definition of what this is for the layman.

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The best paper I could find on this is arxiv.org/abs/gr-qc/0701056 It seems a "sudden singularity" is an instantaneous change in pressure or the expansion of the universe but it seems like any non-continuous (non-differentiable) change in the expansion of space qualifies. – Brandon Enright May 4 '13 at 22:52
Why the downvote? This seems to me a perfectly reasonable question. – John Rennie May 5 '13 at 8:25
@JohnRennie: I downvoted because the question didn't show any research effort. – Ben Crowell May 5 '13 at 12:46
@BenCrowell: if a physics student were asking the question I'd agree, but for a non-physicist I haven't found any description of a sudden singularity (apart from mine of course :-) that is remotely accessible. – John Rennie May 5 '13 at 13:19
Thanks for the answers and comments. This give valuable context. I did try the Google Search before posting this, but I am happy to have an answer that is less effort than a layman parsing through the various papers. And now I see that this question is in the top 10 of Google Search, so hopefully it will help others like me in the future. – Skotch May 6 '13 at 17:45

A sudden singularity is a singularity that forms in the universe in a finite time.

This may see like a strange definition. After all, don't the singularities in black holes form in a finite time, so shouldn't they, and indeed all singularities be sudden? Actually, no! For observers like us, floating around the universe, in our frame of reference it takes an infinite time for a black hole to form.

This apparently paradoxical result has been too widely discussed, in this site and elsewhere, for it to be worth me elaborating on it here. An observer falling into the black hole will encounter the singularity in a finite time, but for observers outside the event horizon the singularity never forms. The type of big rip singularity usually referred to as a sudden singularity forms in a finite time (for all observers) because we are in effect falling towards it just like the observer falling into a black hole.

A quick footnote:

As Ben points out in his answer, my definition above is too broad because it includes singularities (like the Big Crunch) that are not sudden. as described in Nojiri's paper, sudden singularities are just one of the four types of singularities that can occur at non-zero scale factors (i.e. not the Big Bang or Big Crunch). I'm not sure how usefully this can be summarised for the lay reader since it's all a bit technical. @Skotch, I would have a look at Nojiri's paper and maybe ask a further question if you want anything in it carified.

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This is incorrect because it fails to distinguish between a Big Crunch singularity and a sudden singularity. – Ben Crowell May 5 '13 at 13:04
Hmm, good point. I've only heard the phrase used to refer to Big Rip type singularities, but maybe a Big Crunch would also be sudden singularity. – John Rennie May 5 '13 at 13:18
Actually it looks like it doesn't even refer to Big Rip scenarios -- see my answer. – Ben Crowell May 5 '13 at 13:50
Ah yes. See also arxiv.org/abs/hep-th/0501025. This paper seems to be referenced a lot in discussions of sudden singularities. – John Rennie May 5 '13 at 14:16
That's useful -- they give a more explicit definition than Barrow or Cotsakis. – Ben Crowell May 5 '13 at 14:18

People have known for a long time about the existence of cosmological models that include a Big Crunch singularity. In these models, the density of matter in the universe is great enough to cause it to recontract. These models are no longer of interest as descriptions of the actual universe, since we now know that the expansion of the universe is accelerating rather than decelerating, presumably due to dark energy with poorly known properties.

In fact, if dark energy turns out to have certain characteristics, then it's possible that the expansion will blow up at some point, so that all matter is destroyed by being torn apart. The most commonly discussed scenario of this type is called a Big Rip, but there are other scenarios as well, classified in Cotsakis 2004. A singularity like this has two properties: (1) it happens at a finite time in the future, and (2) it's a blow-up, not a recollapse like the Big Crunch. There can be cases where the cosmological scale factor $a$ blows up, and others where $a$ stays finite, but $\dot{a}$ blows up. For comparison, a Big Crunch would have $a$ going to zero.

The term "sudden singularity" seems to be used in the literature for one very specific type of blow-up. Cotsakis says, "For instance, they may correspond to ‘sudden’ singularities (see [19] for this terminology)," where reference 19 is Barrow 2004. Barrow constructs an example that is not driven by dark energy at all. The strong energy condition holds, and both the density and the scale factor are always constant. However, the pressure and curvature blow up to infinity at a finite time. He gets this behavior by not requiring any equation of state to hold between the pressure and the density. Since both Barrow and Cotsakis seem to be defining the term based on one example (the word "sudden" only appears in Barrow's title), it's a little hard to tell exactly how broad they intend this definition to be. Nojiri 2005 defines a sudden singularity as one in which the pressure blows up to plus or minus infinity at some finite time, while the scale factor and density stay finite. This implies that a sudden singularity can never occur if there is a definite equation of state.

John D. Barrow, 2004, "Sudden Future Singularities," http://arxiv.org/abs/gr-qc/0403084

Spiros Cotsakis, Ifigeneia Klaoudatou, 2004, "Future Singularities of Isotropic Cosmologies," http://arxiv.org/abs/gr-qc/0409022

Nojiri et al., 2005, "Properties of singularities in (phantom) dark energy universe," http://arxiv.org/abs/hep-th/0501025

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