On well-posedness, linear perturbations and mass conservation for axisymmetric Einstein equation

TL;DR

This paper investigates the well-posedness and properties of a gauge formulation of axisymmetric Einstein equations, analyzing linear perturbations and mass conservation to gain insights into the nonlinear system and black hole stability.

Contribution

It introduces a gauge with a conserved mass integral for axisymmetric Einstein equations and studies linear perturbations to understand the system's structure and stability.

Findings

Mass integral is conserved and positive definite.

Linear perturbations reduce to a simple, insightful system.

Numerical evidence supports well-posedness and expected behavior.

Abstract

For axially symmetric solutions of Einstein equations there exists a gauge which has the remarkable property that the total mass can be written as a conserved, positive definite, integral on the spacelike slices. The mass integral provides a nonlinear control of the variables along the whole evolution. In this gauge, Einstein equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. As a first step in analyzing this system of equations we study linear perturbations on flat background. We prove that the linear equations reduce to a very simple system of equations which provide, thought the mass formula, useful insight into the structure of the full system. However, the singular behavior of the coefficients at the axis makes the study of this linear system difficult from the analytical point of view. In order to understand the behavior of the solutions, we study the numerical evolution of them. We provide strong numerical evidence that the system is well-posed and that its solutions have the expected behavior. Finally, this linear system allows us to formulate a model problem which is physically interesting by itself, since it is connected with the linear stability of black holes solutions in axial symmetry. This model can contribute significantly to solve the nonlinear problem and at the same time it appears to be tractable.