A Perturbation-Inspired Method of Generating Exact Solutions in General Relativity.

摘要

General relativity has a small number of known, exact solutions which model astronomically relevant systems. These models are highly idealized situations. Either perturbation theory or numerical simulations are typically needed to produce more realistic models. Numerical simulations are time-consuming and suffer from a difficulty in interpreting the results. In addition, global properties of numerical solutions are nearly impossible to uncover. On the other hand, standard perturbation methods are very difficult to implement beyond the second order, which means they barely scratch the surface of non-linear phenomena which distinguishes general relativity from Newtonian gravity. This work develops a method of finding exact solutions, inspired by perturbation theory, which have energy-momentum tensor components that approximately satisfy desired relationships. We find a spherical lump of matter which has a density profile mu ∝ r--2 in a Robertson-Walker background; it looks like a galaxy in an expanding universe. We also find a plane-symmetric perturbation of a Bianchi type I metric with a density profile mu ∝ z--2; it models a jet impacting a sheet-like structure. The former solution involves a wormhole while the latter involves a two dimensional singularity. These are both nonlinear structures which perturbation theory can never produce.