Linearised perturbation of constant mass aspect function foliation in Schwarzschild black hole spacetime

TL;DR

This paper analyzes how small linear perturbations of a specific foliation in Schwarzschild spacetime affect the asymptotic geometry and physical quantities like energy and momentum at null infinity, revealing a 4-dimensional space of perturbations with particular properties.

Contribution

It identifies a 4-dimensional linear space of initial perturbations that preserve the roundness of the asymptotic geometry at null infinity in Schwarzschild spacetime.

Findings

Linearised perturbations of the Bondi energy and mass vanish.

The linearised perturbations of linear momentum can be fully realized within the identified space.

A 4-dimensional space of initial perturbations preserves the asymptotic roundness.

Abstract

We study the linearised perturbation of the constant mass aspect function foliation in a Schwarzschild black hole spacetime. In particular, we investigate the linearised perturbation of the asymptotic geometry of the foliation at null infinity. We show that there is a 4-dimensional linear space for the linearised perturbation of the initial leaf inside the event horizon, corresponding to which the linearised perturbation of the asymptotic geometry of the foliation at null infinity is preserved to be round. For such a linearised perturbation of the initial leaf in this 4-dimensional linear space, we calculate the corresponding linearised perturbation of the energy-momentum vector and the Bondi mass at null infinity. We show that the linearised perturbations of the Bondi energy and the Bondi mass both vanish, and every possible linearised perturbation of the linear momentum can be achieved by a linearised perturbation of the initial leaf in the 4-dimensional linear space.