The math shows that it's possible to have a curved 4-dimensional Minkowski space such that there's a continuous bijection from it to $R^4$ and a closed light-like curve and some closed timelike curves all of which are in the future light cone of the closed light-like curve. Also according to the Wikipedia article Black hole, a closed time-like curve might be possible in a black hole. I'm wondering if in general relativity, it's possible to start with a state of the universe that will evolve into a state with a closed light-like curve with no solution to the state of the universe anywhere in the future light cone of that closed light-like. Maybe such a light-like curve would nucleate the diappearance of space at the speed of light and such a light-like curve already formed in a rotating black hole but since nothing can escape from a black hole, the nucleated disappearance of space never escaped the black hole.
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$\begingroup$ The 'related column' leads to this answer by John: physics.stackexchange.com/q/101748 $\endgroup$– RobCommented Apr 21, 2018 at 23:13
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$\begingroup$ @Rob That question has only one answer and it doesn't answer my question. I'm not sure it answers that question either but it seems very obvious from what I wrote in my question that it is asking a different thing than the other question. $\endgroup$– TimothyCommented Apr 21, 2018 at 23:26
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$\begingroup$ Though it has nothing in particular to do with closed timelike (or null) curves, the defining property of a "singularity" is pretty much that the theory yields "no solution as to what state it will evolve into". $\endgroup$– hmakholm left over MonicaCommented Apr 22, 2018 at 9:26
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$\begingroup$ I was already aware of that property of the singularity of a Schwarzchild black hole. I wanted to know if there can be a singularity that is a closed light-like curve with no solution to its future light cone and with the region near it outside of its future light cone being very smooth and the singlularity can't go through an object faster than light devouring it in the process like the singularity of a Schwarzchild black hole does. $\endgroup$– TimothyCommented Apr 23, 2018 at 2:25
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$\begingroup$ I'm not sure about the question, does the closed timelike curves inside a Kerr black hole satisfy you? $\endgroup$– RexcirusCommented Apr 23, 2018 at 13:06
1 Answer
Your question appears to be unanswerable being based on a misinterpretation of what "closed time-like curve" means. Wikipedia states :"The photon sphere is a spherical boundary of zero thickness in which photons that move on tangents to that sphere would be trapped in a circular orbit about the black hole." A circular orbit is not necessarily a closed curve though. It would be a closed curve, if it ends up where it starts. That this isn't the case as is shown here photon sphere Fig.3.469.
Closed time-like and light-like curves (which are not geodesics) exist in Gödel's universe.
Let me add that if you drop "closed" then your question "I'm wondering if in general relativity, it's possible to start with a state of the universe that will evolve into a state with a (closed) light-like curve with no solution to the state of the universe anywhere in the future light cone of that closed light-like." would refer to the circular orbit of photons (which are light-like curves) in the photon sphere of a static black hole. The evolution of an astrophysical black hole, the collapse of a spherical symmetric cloud of matter, was described by Oppenheimer & Snyder. But once the event horizon is formed the future light cone at $r=3M$ (photon sphere) is tilted toward the black hole with no further evolution because the Schwarzschild spacetime is static outside the event horizon.
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$\begingroup$ I think I know what I'm talking about. I said that there exists such a Minowski space, not that there exists such a Minkowski space that follows the laws of general relativity. I know that when a photon makes an orbit around a black hole in the photon sphere which is outside of the event horizon, it just ends up at the same position, and not at the same point in space time. What I figured out is that it is mathematically possible to have a very wierd Minkowski space that is not like the space outside the event horizon of a black hole. $\endgroup$– TimothyCommented Apr 23, 2018 at 1:56
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$\begingroup$ I'm not sure what you want to say with your last sentence. Have you a reference? Note, the Minkowski metric is flat. $\endgroup$– timmCommented Apr 23, 2018 at 7:06