# Closed Trajectories in wormhole

This might seem trivial but if I want to go about finding closed trajectories in a wormhole with a metric: $$ds^2=-dt^2+dr^2+(b^2+r^2)(d\theta^2+sin^2\theta d\phi^2)$$ how should I approach that? Is there any conditions that has to be followed? I have just started learning these, and had this doubt if they existed and if they do how to find them.I did try searching about them in net but was not able to find anything that would help me understand it better. I am familiar with finding the geodesic of this. How should I proceed from there?Any suggestion or help would be helpful.

ps:This is not my homework in case you are wondering. I am just curious to learn about these.

• I notice that this is tagged with the closed timelike curve tag. The Ellis wormhole, as are all the Morris-Thorne type wormholes, does not contain any closed timelike curves. – Slereah Jul 1 '18 at 8:23

The metric shown is independent of the time coordinate $t$ and of the azimuthal coordinate $\phi$. I presume $b$ is some function of the radial coordinate $r$. That means you have two Killing vectors which provide as conserved quantities the energy and the angular momentum of the body.
Conservation of angular momentum confines the particle motion to a plane, that you may assume to be the equatorial plane, $\theta = \pi / 2$, implying $d \theta = 0$.
Another condition is given from the 4-velocity constraint $g_{\mu \nu} u^\mu u^\nu = -1$.
Plugging what above in the metric, you get a relation $dr / d\tau = f(r, b)$, where $\tau$ is the proper time.
Note: In case of a light ray, you have $g_{\mu \nu} u^\mu u^\nu = 0$, and instead of the proper time you should use an affine parameter $\lambda$.