- 206

- 4

use this formula:

g

_{ab}(dx

^{a}/ds)(dx

^{b}/ds)

(where x(s) is our parameterized curve).

Assuming a (- + + +) signature, if the answer to the above summation is negative for all s then the curve is timelike.

That is simple enough. However, how do you know if a curve is closed, or how do you pick out a timelike curve that is closed?

At first, I thought you needed a periodic temporal coordinate and interval in your parameterized curve so that travel along the curve would eventually bring you back to the event from whence you started. An example of such a curve is x(s) = [ sin(s), 0, 0, 0 ] on the interval (0, 2π). However, this is wrong and you guys chewed me out for it in a thread that I made before:

### I came up with a CTC in Minkowski Space, But To What End?

If you viewed my most recent thread before this one, then you know that I have been studying curves in spacetime (timelike/spacelike/lightlike), and I have especially been looking into the CTCs (closed timelike curves) that the Godel metric is famous for. During my studies I found that I had to...

www.physicsforums.com

Apparently, a curve has to be one to one which means that for x(s) , no two values of s can yield the same event. (It was also pointed out to me in the above thread that I made a mathematical error and didn't account for the fact that the summation for the above curve in Minkowski space is not negative for all s since it is 0 at certain points).

Even if we ignore that mathematical error though, how can the curve actually loop back on itself to the beginning point if the curve has to be one to one? If it is one to one, I could see the curve being timelike, but I don't see it ever returning back to the same point.

My main reason for asking is because I just want to see a CTC in Godel spacetime already. I once tried to come up with a CTC in Godel spacetime (which is said to have CTC running through every event), and I ended up with a closed spacelike curve rather than a timelike curve. That was in this thread here:

### Is this a closed spacelike curve?

I've been refurbishing my understanding of some relativistic concepts and I've been specifically studying the concepts of spacelike, timelike and lightlike curves. According to the notes that I have been reading, curves on a Lorentzian manifold can be classified as follows: If you have a...

www.physicsforums.com

Long story short, can someone please tell me how to establish or notice the "closed" part of a closed timelike curve in something like a Godel spacetime.

If you need the line element for that metric, it is on this wiki:

### Gödel metric - Wikipedia

en.wikipedia.org

Thank you