Plugging Godel Metric's Line Element into Software for Covariant Einstein Tensor

[space-time]

http://en.wikipedia.org/wiki/Gödel_metric
Could someone please plug the line element for the Godel metric (seen on the above wiki page) into some software to see what comes out for the Einstein tensor in a coordinate basis (preferably the covariant version rather than the contravariant version or mixed tensor version)? I ask this because I want to check my own work for accuracy, but I can not find anywhere online that showcases the covariant form of the Einstein tensor of the Godel metric in Cartesian coordinates. Here is what I got for my Einstein tensor G_μν:

G₀₀ , G₁₁ , and G₂₂ all equal 1/2
G₀₃ and G₃₀ = e^x / 2
G₃₃= (3/4) e^2x

Everything else was 0.

In the case that I made some mistake early on, here was my metric tensor g_μν:

g₀₀= -1 / (2ω²)

g₁₁ and g₂₂ = 1 / (2ω²)

g₀₃ and g₃₀ = -e^x / (2ω²)

g₃₃ = -e^2x / (4ω²)

everything else was 0.Now that I've given you my metric and Einstein tensors, can someone plug the line element on the wiki into some software and report the Einstein tensor that it returns. Please keep in mind that these calculations are in a coordinate basis and not an orthonormal. When I tried to convert to an orthonormal basis, I ended up getting some rather strange results.

P.S. I am fine if the software returns the contravariant or mixed tensor variant of the Einstein tensor because I can always just raise some indices, but I would prefer if the answer I got back was covariant.

Thank you.

 

[Mentz114]

I have this set up already and the Einstein tensor that Maxima (ctensor) gets is the same as yours.