Symmetries of Free Fall in Homogeneous Gravitational Fields

In Einstein\u27s general theory of relativity freely falling test particles follow geodesics of the spacetime geometry. Some geodesics have symmetries, known as affine collineations. Mathematically, these affine collineations are transformations that preserve the connection defined by the metric, without preserving the metric. Physically, they change the notion of lengths and angles, while preserving the notion of parallelism. Associated with each affine collineation are two conserved quantities. Previously these quantities were understood to be non-Noetherian, however we show that they can be derived from a direct application of Noether\u27s theorem. We calculate all affine collineations and their corresponding conservation laws for all of the homogeneous solutions to the Einstein Field Equations in vacuum, with perfect fluid sources, and with homogeneous electromagnetic sources. We also calculate all curvature collineations and explain their relationship to geodesic deviation and affine collineations