Geodesics in Gödel-Synge spaces

Abstract

Our purpose is to study the geodesic lines in the form: ds2=dx2 + 2h(x4)dx1dx2 + g(x4)dx22 – dx23 – dx24, and to compare with the work of S. Chandrasekhar and J. P. Wright on geodesics in Gödel’s universe. We will give, first, an isometric embedding of the form ds2=dx2 + 2h(x4)dx1dx2 + g(x4)dx22 – dx23 – dx24 in a pseudo Euclidean space of 10 dimensions, from which some properties on closed time-like curves can be deduced. Finally, we give an isometric embedding in a 10-dimensional pseudo Euclidean space of Gödel’s space in cylindrical coordinates and deduce some consequences