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We know how some spacetimes have "boosts", or in other words, they have spacetime geometries where travelling forward along some curve in spacetime will eventually lead you back to the event from whence you started. If such a curve is timelike, then the curve is a CTC. An example of a spacetime with such a boost is Misner space, which seems to have the boost:

(t, x) -> (tcosh(π) + xsinh(π) , xcosh(π) + tsinh(π))

This is the boost that is reported by the Misner space wiki: https://en.wikipedia.org/wiki/Misner_space

Now in the case of Misner space, that boost was just given to me by the wiki. One can see how a boost like this would indicate the existence of closed timelike curves.

However, let's look at a different metric that is known to have closed timelike curves, such as the Kerr metric:

### Kerr metric - Wikipedia

en.wikipedia.org

No such boost was reported on that wiki. I suppose that in order for someone to notice the presence of closed timelike curves within this metric, they would have to be able to somehow visualize or even graph out the spacetime geometry. Somehow, a person would have to recognize that the spacetime geometry is something akin to spherical or circular or just some shape that would allow an object to travel along some curve and then return to the same event (kind of like traveling around a globe until you eventually get back to the same point). Granted, I know that the Kerr metric describes the spacetime geometry around a spherical uncharged rotating mass, but I feel as though knowing the shape of the mass does not guarantee that the spacetime curves into the same shape.

Having said this, are there any tensors or formulas at all that would help me be able to visualize the actual geometry of a spacetime? Is there perhaps anything involving the metric tensor or covariant derivatives that would let me say "hmmm... If I propose some parameterized curve x(s) in this spacetime and traveled forward along it, then the intrinsic curvature of this spacetime would make this curve loop back on itself in a way that I would return to the initial event"?

Thanks for any assistance.