Sachs equations and plane waves III: Microcosms

This article examines the structure of plane wave spacetimes (of signature $(1,n+1)$, $n\ge 2$) that are homogeneous (the isometry group is transitive) and geodesically complete -- which we call microcosms. In general, a plane wave is shown to determine a smooth positive curve in the Lagrangian Grassmannian associated with the $2n$ dimensional symplectic vector space of Jacobi fields. We show how to solve the Sachs equations in full generality for microcosms and, moreover, we relate the power series expansions canonical solutions to the Sachs equations on a general plane wave to Bernoulli-like recursions. It is shown that for microcosms, the curve in the Lagrangian Grassmannian associated to a microcosm is an orbit of a one parameter group in $\operatorname{Sp}(2n,\mathbb R)$. We also give an effective method for determining the orbit. Finally, we specialize to the case of $n=2$, and give analytical formulae for the solutions to the Sachs equations and the associated one-parameter group orbit.