A Special & General Relativity: CTC Metrics in Cylindrical Coordinates

[YRC]

I am studying metrics that exhibit CTCs. I was looking at a few different metrics...
Tipler's solution
Godel metric
Kerr metric
For starters to compare them, I am trying to convert said metrics into cylindrical coordinates. Thanks in advance for any help

 

[PeterDonis]

I am trying to convert said metrics into cylindrical coordinates.

What have you found in sources that you have looked at? For the last two, at any rate, it should be easy to find expressions for the metrics in cylindrical coordinates.

 

[YRC]

What have you found in sources that you have looked at? For the last two, at any rate, it should be easy to find expressions for the metrics in cylindrical coordinates.

I understand that the coordinate transform for the Godel metric would look like this, $x^{\alpha}=(t, x, y, z)=(t, r\cos{\phi}, r\sin{\phi}, z)$ for cartesian and cylindrical coordinates. The cylindrical metric itself is given by $g_{t,t}=c^2$,$g_{r,r}=1/(1+(r/2a)^2)$, $g_{\phi,\phi}= -r^2(1-(r/2a)^2)$, $g_{z,z}=-1$ and $g_{t,\phi}=r^2*c/(\sqrt{2}a), g_{\phi,t}=r^2*c/(\sqrt{2}a)$ right? That's what I found for Godel, For the Tipler cylinder I found it in cylindrical coordinates... ds^2 = H(dr^2+ dz^2 ) + Ldϕ^2 + 2Mdϕdt − Fdt^2 , What about the Kerr metric that is commonly expressed in Boyer-Lindquist coordinates?

Mentors' note: The Latex was posted without the required delimiters. We've edited in the delimiters but have not otherwise cleaned up the formatting. OP, would you please finish this cleanup?

 

[vanhees71]

Note: The LaTeX Guide is in the link directly under the editing window:

https://www.physicsforums.com/help/latexhelp/

 

[Demystifier]

I am studying metrics that exhibit CTCs.

I don't know if this helps, but any (locally Minkowski) metric can exhibit CTC. For example the flat metric
$$ds^2=dt^2-dx^2$$
contains CTC's if we interpret it as a spacetime with cylindrical topology in which $t$ and $t+2\pi$ are identified. The point is that cylinder admits a flat metric, so when we say that the metric is flat, we have not excluded a possibility that it is a cylinder.

EDIT: It's not valid for any metric, but it's valid for any metric with timelike Killing vector.

 

[YRC]

I don't know if this helps, but any (locally Minkowski) metric can exhibit CTC. For example the flat metric
$$ds^2=dt^2-dx^2$$
contains CTC's if we interpret it as a spacetime with cylindrical topology in which $t$ and $t+2\pi$ are identified. The point is that cylinder admits a flat metric, so when we say that the metric is flat, we have not excluded a possibility that it is a cylinder.

Yes thanks, the Tipler cylinder metric does reduce to the Minkowski line element in cylindrical coordinates when the angular velocity is 0. I was wondering about using the Kerr metric to describe the exterior of the rotating Tipler cylinder. How would that work? Would that work? In a configuration where we have a metric describing the negative energy density required, and this exterior, would CTCs arise in a confined region bounded by the cylinder?