do they exist in reality or in nature?
We have no evidence that they exist in our universe. There is also a conjecture, which I think is widely considered plausible, http://en.wikipedia.org/wiki/Chronology_protection_conjecture that a spacetime that doesn't already have CTCs can't acquire them.
so they might not be permitted in the universe?
Unless (a) the chronology protection conjecture is false, or (b) CTCs naturally existed as a feature of the universe starting from the big bang. There are cosmologies like the Godel metric http://en.wikipedia.org/wiki/Gödel_metric that have CTCs and always have had them. Observations of the CMB anisotropy rule out the Godel metric as a model of our universe, but they don't necessarily rule out all possible cosmologies that have CTCs.
what could possibly rule out ctc's all together?
Closed timelike curves are predicted to exist around the ring singularity in Kerr-Newman metric-
though this is technically a hypothesis (CTCs) within a hypothesis (Cauchy horizon) within a theory (black hole/event horizon).
The CTCs are also within a boundary called the turnaround radius which some predict is supposed to send an infalling object out through a (very hypothetical) white hole-
http://casa.colorado.edu/~ajsh/phys5770_08/bh.pdf [Broken] (page 23)
so even at this level of prediction, there seems to be some level of protection from reaching the CTCs (though a white hole would be just as elusive as CTCs, it's normally best to say that the Cauchy horizon is the barrier of predictability).
The Kerr Newman metric is probably unlikely to exist in our universe, though - it's basically idealized and inherently unstable. From what I've read it's so unstable that it can't self-consistently describe a single particle falling into the inner horizon - such a particle would acquire infinite blueshift, infinite energy - and distort the geometry into something that wasn't a Kerr Newman metric.
Here are some good review articles on CTCs:
does the kerr metric exist?
The metric exists, it is an exact solution of the field equations. What pervect stated is that it probably doesn't exist *in our universe*.
An exact, symmetric, treatment says if 10 hunters in a circle fire toward the center at the same time, you get a metal ball stationary in the center. You want to try this some time? Pervect is saying the Kerr metric is idealized in a similar sense. Presumably, even less likely than the proposed method of manufacturing ball bearings.
i don't get it exist but it does not exist in our universe? also won't quantum gravity rule out all closed timelike curves?
In mathematics "exists" means that there is a solution to an equation. It doesn't mean that it really happens.
PAllen's example of the hunters exists as a solution of Newton's laws, but it has never happened on our planet.
We don't have a working theory of quantum gravity, so we don't know for sure. For speculation on this point, take a look at the articles listed in #8.
i see so your saying they exist mathematicly that does not mean it exist in reality?
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