Sachs equations and plane waves III: Microcosms

TL;DR

This paper analyzes the structure of homogeneous, geodesically complete plane wave spacetimes called microcosms, providing solutions to Sachs equations, relating them to symplectic geometry, and offering methods to determine their orbits, especially for the case n=2.

Contribution

It introduces the concept of microcosms, solves Sachs equations in full generality for these spacetimes, and relates solutions to symplectic group orbits, including explicit formulas for n=2.

Findings

Microcosms are characterized by orbits of a one-parameter group in Sp(2n,R).

Solutions to Sachs equations can be expressed via Bernoulli-like recursions.

Explicit analytical formulas are provided for the case n=2.

Abstract

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.