Structural stability and general relativity

TL;DR

This paper reviews recent advances in applying bifurcation theory and structural stability analysis to various phenomena in general relativity, providing a classification of solutions and their topological transformations.

Contribution

It introduces a comprehensive framework for understanding stability and bifurcations in Einstein's equations using versal unfoldings and applies it to key models in cosmology and black hole physics.

Findings

Bifurcation theory classifies stable perturbations of Einstein's equations.

Versal unfoldings describe topological changes in solutions.

Applications include Friedmann models and black hole spacetimes.

Abstract

We review recent developments in structural stability as applied to key topics in general relativity. For a nonlinear dynamical system arising from the Einstein equations by a symmetry reduction, bifurcation theory fully characterizes the set of all stable perturbations of the system, known as the `versal unfolding'. This construction yields a comprehensive classification of qualitatively distinct solutions and their metamorphoses into new topological forms, parametrized by the codimension of the bifurcation in each case. We illustrate these ideas through bifurcations in the simplest Friedmann models, the Oppenheimer-Snyder black hole, the evolution of causal geodesic congruences in cosmology and black-hole spacetimes, crease flow on event horizons, and the Friedmann-Lema\^itre equations. Finally, we list open problems and briefly discuss emerging aspects such as partial differential equation stability of versal families, the general relativity landscape, and potential connections between gravitational versal unfoldings and those of the Maxwell, Dirac, and Schr\"{o}dinger equations.