In 1949, Kurt Gödel proposed a solution to Einstein's field equations that exhibited a rotation of matter (with a nonvanishing cosmological constant). This solution allows the existence of closed timelike curves, which implies the possibility of time travel. In this worksheet, we demonstrate the use of DifferentialGeometry and Tensor packages to calculate the Ricci tensor, Ricci scalar, and Einstein tensor. We also apply the calculus of variations to calculate the null geodesics (trajectories of light rays) in Gödel's universe. The notation is based on the presentation in the monograph of Stephen Hawking and George Ellis.