Symmetries of Einstein’s Equations in Vacuum and Their Geodesics

This thesis explores symmetries of vacuum Einstein equations that are static and at least axially symmetric, i.e., Ricci-flat Lorentzian geometries that admit a timelike Killing vector field and a closed spacelike Killing vector field among their isometries. We study symmetries of the geodesics in these spacetimes as well as symmetries of the system of Einstein equations describing such spacetimes. Geodesics in three dimensions have symmetries and associated conserved quantities absent in four and higher dimensions. We employ the socalled direct method for computing the conserved quantities. For the static axisymmetric system in vacuum, we found all symmetries of the system which enabled us to explain why one cannot obtain algebraic prescriptions for generating new solutions from old ones beyond those already known. Symmetries of the geodesics in spherical symmetry show that there is no general connection between cosmological constant and projective equivalence and that one can find an appropriate coordinate system where the effect of cosmological constant disappears from the bending angle, unlike in the static coordinates.