Geodesic time travel in Gödel's universe

Abstract

This work is an introduction at a beginning graduate or advanced undergraduate level to Kurt Gödel's foray into cosmology. After an elementary introduction to the basics of Einstein's theory of gravitation, we simply present the Gödel's solution and the geodesic equations associated with it. This equations are then explicitly solved obtaining its full set of temporal geodesics. Armed with such explicit expressions, the geodesic time-travelling possibilities of Gödel's universe are discussed. We search for their time-like closed geodesics that, following Gödel's analysis, other people has imagined as possible routes for time-travel. We next exhibit that such time-travelling possibility do not exist in his model universe. This is done in the most straightforward way possible, framing the discussion as to serve as a simple example for students of General Relativity.