The Godel Metric is a solution of Einstein's Equation that has swirling dust particles (representing galaxies) and a cosmological constant, which is a parameter that can be tuned make the universe expand, contract, or stay stable. One interesting thing about the Godel Metric is that it allows the existence of closed time-like curves. Closed time-like curves in the space time of General Relativity are paths that allow an observer to start at a certain time, travel along this path and end up at the same time she started at! These paths can be further extended to allow the possibly of traveling back in time.

One other such solution of Einstein's Equation that allows the existence of closed time-like curves is the Kerr vacuum, which models a rotating uncharged black hole. The Kerr vacuum has closed time-like curves within the event horizon of the black hole.

However, it is important to note that although these solutions exist in theory, scientists do not believe that they exist in reality, and time travel will stay the topic of science fiction. The Godel Metric, as well as most other solutions of Einstein's Equation that allow closed time-like curves, are only mathematical toy models that do not exist in nature.

The Kerr metric mentioned above is thought to be a good representation of the real space time around a rotating black hole. However, the closed time-like curves found in the Kerr solution are only found inside the event horizon of the black hole, where we unfortunately cannot travel. Additionally, even though closed time-like curves exist in the Kerr metric, it is thought that these are just artifacts left over because General Relativity does not take quantum mechanics into account. A full quantum theory of gravity will most likely not contain closed time-like curves, as we cannot have them exist in reality.

So, although General Relativity has solutions that suggest time travel would be possible, these are merely mathematical curiosities, and any theory must always pass a "common sense" test, and unfortunately commonsense tells us that time travel is not physically possible.