Linear perturbations for the vacuum axisymmetric Einstein equations

TL;DR

This paper studies linear perturbations of the vacuum Einstein equations in axial symmetry, providing explicit solutions that facilitate understanding the system's behavior and aiding future nonlinear analysis.

Contribution

It offers an explicit integral transformation solution for linearized vacuum Einstein equations in axial symmetry, aiding the analysis of their properties.

Findings

Explicit integral solutions for linear perturbations

Simplified representation suited for nonlinear analysis

Potential insights into the evolution of axially symmetric spacetimes

Abstract

In axial symmetry, there is a gauge for Einstein equations such that the total mass of the spacetime can be written as a conserved, positive definite, integral on the spacelike slices. This property is expected to play an important role in the global evolution. In this gauge the equations reduce to a coupled hyperbolic-elliptic system which is formally singular at the axis. Due to the rather peculiar properties of the system, the local in time existence has proved to resist analysis by standard methods. To analyze the principal part of the equations, which may represent the main source of the difficulties, we study linear perturbation around the flat Minkowski solution in this gauge. In this article we solve this linearized system explicitly in terms of integral transformations in a remarkable simple form. This representation is well suited to obtain useful estimates to apply in the non-linear case.