In 1949, Kurt Gödel found a solution of Einstein's field equations and gave it to Einstein on his birthday [1]. This solution describes a homogeneous, anisotropic, and rotationally symmetric spacetime in which matter takes the form of a pressurefree perfect fluid with negative cosmological constant, where closed timelike world lines exist. Theoretically, traveling along such a world line allows an observer to travel into their own past. For an astounding and detailed visualization of how an observer would perceive such a travel, see [2, 3]. The manipulate graphic reproduces the famous illustration contained in the book by Hawking and Ellis [4], showing photons emitted by an observer on the axis of symmetry at the bottom; the photons spiral out, expand, and reconverge at the top. Lightlike geodesics are shown as well as light cones that tip as you increase producing the closed timelike curves. Further interpretations can be found in [5].
In cylindricalpolar coordinates , Gödel's metric is given by , where is a constant. The coordinate does not appear in the metric. [1] W. Rindler, “Gödel, Einstein, Mach, Gamow, and Lanczos: Gödel’s Remarkable Excursion into Cosmology,” American Journal of Physics, 77(6), 2009 pp. 498–510. doi: 10.1119/1.3086933. [2] M. Buser, E. Kajari, and W. P. Schleich, "Visualization of the Gödel Universe," New Journal of Physics, 15, 2013 013063. doi: 10.1088/13672630/15/1/013063. [4] S. Hawking and G. F. R. Ellis, The Large Scale Structure of SpaceTime, Cambridge: Cambridge University Press, 1973 section 5.7. [5] I. Németi, J. X. Madarász, H. Andréka, and A. Andai, "Visualizing Ideas about GödelType Rotating Universes," in GödelType Spacetimes: History and New Developments (M. Scherfner and M. Plaue, eds.). Kurt Gödel Society: Collegium Logicum X, 2010 pp. 77–127. www.mathinst.hu/pub/algebraiclogic/godunivisurevised.pdf.
