# Is it possible to have a state of the universe with no solution to what state it will evolve into in general relativity?

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.

– Rob
Commented Apr 21, 2018 at 23:13
• @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. Commented Apr 21, 2018 at 23:26
• 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". Commented Apr 22, 2018 at 9:26
• 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. Commented Apr 23, 2018 at 2:25
• I'm not sure about the question, does the closed timelike curves inside a Kerr black hole satisfy you? Commented Apr 23, 2018 at 13:06

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.