الملخص الإنجليزي
Geodesic equations for a space-time metric is a set of four coupled non-linear second order ordinary differential equations (ODEs) in four unknown functions. A Lie point symmetry of a system of differential equations is a map from the solution space of the system to itself. It is known that for a first order ODE,a symmetry (if exists) provides an integrating factor and hence solves the ODE. In the case of higher order ODEs, a symmetry if exists, reduces the order of the ODE by one. For finding the Lie point symmetries of ODEs one uses the symmetry conditions which are four identities and establish the determining equations. The solution of these equations then provides the generators of the symmetries.
In this work it is focused to find the Lie point symmetries of the geodesic equations for the Gödel metric. The determining equations in this case reduce to a system of seventy coupled partial differential equations (PDEs) (of first and second order), in five unknown functions of five variables. A complete solution of these equations gives all Lie point symmetries of the geodesic system. The maximal set of Lie point symmetries for such a system of PDEs is unknown and it varies from case to case. Here a solution of the system of the determining equations is found by considering that all the unknown functions are dependent on four variables instead of five. This gives a solution which generates ten Lie point symmetries of the geodesic equations. These symmetry generators are then shown to satisfy a ten-dimensional Lie algebra which contains a seven-dimensional solvable Lie algebra.
