I have recently derived both the purely covariant Riemann tensor as well as the purely covariant Weyl tensor for the Gödel solution to Einstein's field equations. Here is a wiki for the Gödel metric if you need it: http://en.wikipedia.org/wiki/Gödel_metric There you can see the line element I was working with. Now I will give you my Ricci tensor: R00 = 1 R03 and R30 = ex R33 = e2x Every other Ricci tensor element is 0. Now for my Riemann tensor Rabcd: R0110 & R1001 = -1/(4ω2) R1010 & R0101 = 1/(4ω2) R0330 & R3003 = -e2x/(8ω2) R3030 & R0303 = e2x/(8ω2) R0113 & R1301 & R1031 & R3110 = -ex/(4ω2) R1013 & R0131 & R3101 & R1310 = ex/(4ω2) R1331 & R3113 = -3e2x/(8ω2) R3131 & R1313 = 3e2x/(8ω2) Every other element was 0. Now for the Weyl tensor Cabcd: C0110 & C1001 = -1/(12ω2) C1010 & C0101 = 1/(12ω2) C0220 & C2002 = 1/(6ω2) C2020 & C0202 = -1/(6ω2) C0330 & C3003 = -e2x/(24ω2) C3030 & C0303 = e2x/(24ω2) C0113 & C1301 & C1031 & C3110 = -ex/(12ω2) C1013 & C0131 & C3101 & C1310 = ex/(12ω2) C0223 & C2302 & C2032 & C3220 = ex/(6ω2) C2023 & C0232 & C3202 & C2320 = -ex/(6ω2) C1221 & C2112 = 1/(12ω2) C2121 & C1212 = -1/(12ω2) C1331 & C3113 = -e2x/(6ω2) C3131 & C1313 = e2x/(6ω2) C2332 & C3223= 5e2x/(24ω2) C3232 & C2323= -5e2x/(24ω2) If you'd like to verify my math, feel free to put this in some software, because I don't have any. Anyway, I learned that the R00 element of the Ricci tensor tells you how the volume of objects traveling along geodesics change, while the Weyl tensor tells you how the shape of objects change as they travel along geodesics due to space-time curvature. Using my Gödel metric as an example, would someone please explain how I get the info about how objects change shape using this Weyl tensor? In other words, what info does the value of C0110 for example give me (as well as the other elements)? Thank you.