Using a result by Koch (1988 Trans. Am. Math. Soc. 307 827–41) we realize Gödelʼs universe as the total space of a principal -bundle over a strictly pseudo-convex CR manifold M3 and exploit the analogy between and Feffermanʼs metric (Fefferman 1976 Ann. Math. 103 395–416; 104 393–4) to show that for any -invariant wave map Φ of into a Riemannian manifold N, the corresponding base map is subelliptic harmonic, with respect to a canonical choice of contact form θ on M3. We show that the subelliptic Jacobi operator of ϕ has a discrete Dirichlet spectrum on any bounded domain supporting the Poincaré inequality on and Kondrakov compactness, i.e. compactness of the embedding . We exhibit an explicit solution to the wave map system on , of index for any bounded domain . Mounoudʼs distance (Mounoud 2001 Differ. Geom. Appl. 15 47–57) is bounded below by a constant depending only on the rotation frequency of Gödelʼs universe, thus giving a measure of the bias of from being Fefferman like in the region .