I want to see a plot of closed time-like curve in $(t,x)$.
$t$ - vertical axis
$x$ horizontal axis
(the usual setting just neglect $y$ and $z$ components of $(t,x,y,z)$).
What does it look like?
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asked Feb 7, 2012 at 1:08
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$\begingroup$ Comment to the question(v1): How about simply taking Minkowski space, and declaring that the time coordinate $t\sim t + T$ is periodic, where $T>0$ is a constant? Would that be what you have in mind? $\endgroup$– Qmechanic ♦Commented Feb 7, 2012 at 1:33
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$\begingroup$ No. I just want simplest space-time diagram where closed time-like curve is plotted. $\endgroup$– Pratik DeoghareCommented Feb 7, 2012 at 1:52
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1$\begingroup$ If you want to do it in 1+1 dimensions, then I'm not sure it's possible, other than by making an identification as @Qmechanic suggested. Any metric in 1+1 dimensions is going to be conformally flat, so removing the restriction of a flat Minkowski metric doesn't help. Going to higher dimensional cases (like Goedel), and projecting the CTCs onto two dimensions doesn't give an interesting picture presumably? $\endgroup$– twistor59Commented Feb 7, 2012 at 8:35
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$\begingroup$ @twistor59 What is the minimum number of dimensions required? $\endgroup$– Pratik DeoghareCommented Feb 7, 2012 at 9:31
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1$\begingroup$ Apparently it's possible in (2+1) dimensions, but I'm not familiar with any of these solutions. For (1+1) I think you have to do some gluing like in Deutsch Politzer $\endgroup$– twistor59Commented Feb 7, 2012 at 10:18
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2 Answers
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You cannot look at a curve through space-time and say whether or not it is time-like without considering the metric. At each point in space-time the metric defines the boundary of the local increments that are time-like. Another name for this boundary is the light cone. If, at each point along the curve, the tangent of the curve lies inside the light cone at that point, then the curve is time-like. If the curve is also closed, then it is a closed time-like curve. For solutions of Einstein’s equation with massive bodies, the metric gives rise to a warped space-time, and for any choice of coordinates the light cones will be different from point to point. But for ordinary cases, there will still not be closed time-like curves. However, for some solutions, such as within the event horizon of a spinning black hole, there are closed time-like curves.
So, a short answer to the question is draw a closed curve on the plane. Then draw little light cones along the curve. Since there are only two dimensions, each light cone looks like an x, and you can shade two opposing regions to indicate the forward and backward time-like regions of each x, which can be labeled + and -. The curve should consistently go through the center of each x from - to +, and not go through the unshaded (space-like) regions of any x. Then you can say you have a time-like curve. But without all those local light-cones, you can’t say one way or the other whether your closed curve is time-like.
Also, it would then be misleading to label the axes x and t, since you now just have general coordinates, and globally, neither one is completely time-like or space-like.
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Just draw any closed (in common sense) curve -- it will be CTC.
answered Feb 7, 2012 at 21:40