# An Approach to Stability Analyses in General Relativity via Symplectic Geometry

## TL;DR

This paper introduces a geometric symplectic approach to analyze the stability of solutions in general relativity, simplifying the understanding of black hole formation and stability through classical mechanics analogies.

## Contribution

It develops a pedagogical, symplectic geometric framework linking stability concepts in classical mechanics to those in general relativity, including a non-linear stability analysis of Schwarzschild black holes.

## Key findings

- Simplifies stability analysis of stationary solutions in general relativity.
- Establishes a formal analogy between classical mechanics and general relativity stability.
- Performs a restricted non-linear stability analysis of Schwarzschild black hole formation.

## Abstract

We begin with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, using symplectic geometry. We then discuss what the phase space and the Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently draw a formal analogy between the notions of stability in these two theories. Our approach here is pedagogical and geometric, and considerably simplifies a formal understanding of the statements regarding the stability of stationary solutions of general relativity. In particular, the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points have simply to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow of the Hamiltonian vector field. Further, we posit that a study of the genericity of a particular black hole or naked singularity spacetime forming as an endstate of gravitational collapse is equivalent to an inquiry of how sensitive the orbits of the symplectic Hamiltonian vector field of general relativity are to changes in initial data. We demonstrate this by conducting a restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity.