Godel Solution Metric: Shapes & Descriptions

[space-time]

Here is the Godel solution:

ds² = -dt²/(2ω²) - (e^xdzdt)/ω² - (e^2xdz²)/(4ω²) + dx²/(2ω²) + dy²/(2ω²)

Here is the metric tensor for it:
g₀₀ = -1/(2ω²)
g₀₃ & g₃₀ = -e^x/(2ω²)
g₁₁ & g₂₂ = 1/(2ω²)
g₃₃ = -e^2x/(4ω²)

Every other element is 0.

Now to my question: What shape is this metric?

To clarify what I mean:

The Schwarzschild metric describes a space-time containing a spherically symmetric static body and the Morris-Thorne traversable wormhole metric describes a space-time that contains a spherically symmetric static wormhole. "Now how can we tell that the body and the wormhole are spherically symmetric?", one might ask.The answer is because: The line elements of these two metrics have the same basis as the line element for spherical coordinates. If you were to take away the unique terms in these metrics (such as the 1 - (2GM/(rc²)) or the b² + l²), then you would have exactly the line element for spherical coordinates. In other words, these two metrics have a spherical basis.

Well what I want to know is this: What kind of basis does the Godel solution have? What shape is this supposed to be? Spherical? Cylindrical? Cubical? Triangular? Etc... What kind of space-time does this metric even describe really? I know that it contains closed time-like curves and that this universe is supposed to be infinite and rotating, but that is it. I'm not sure if those are the main points of what this metric describes, but please tell me if the object(s) in this metric have some kind of shape, what that shape is, and what this metric really describes.

On a more general note: How can I tell with any general line element what that particular space-time describes, and what the shapes of the main objects in those metrics are? As you saw above, I can tell when a metric is spherical. I can also tell cylindrical. Those two are already famous coordinate systems. How would I describe the shape and features of metrics that don't use these famous bases however?

 

[Dale]

How would I describe the shape and features of metrics that don't use these famous bases however?

The metric is the description of the shape. There are many more possible shapes than there are English words to describe shapes. This metric is an example of one that does not have a corresponding English translation that I know of.

One other tool for describing the shape is the Killing vectors. But again, there are many more symmetries than English words.