Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited

Gabriele Benomio

TL;DR

<3-5 sentence high-level summary> This paper establishes linear stability of the Schwarzschild solution to gravitational perturbations within a new geometric gauge tailored to linearised gravity. It achieves uniform boundedness and decay without any future gauge normalisation by exploiting red-shift transport structures, and it treats both gauge-invariant (Teukolsky/Regge-Wheeler) and gauge-dependent sectors with a refined initial-data normalisation. The analysis strengthens the role of horizon red-shift in stabilising the gauge-dependent part and provides a framework that aligns with, and extends, the prior DHR approach in a way that is compatible with future Kerr stability and nonlinear problems. The work thereby advances the understanding of gauge choices in black hole stability, with potential impact on sub-extremal Kerr and nonlinear stability programs.

Abstract

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the $|a|=0$ case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin $\pm 2$ Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of \cite{DHR}, which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.

A new gauge for gravitational perturbations of Kerr spacetimes II: The linear stability of Schwarzschild revisited

TL;DR

<3-5 sentence high-level summary> This paper establishes linear stability of the Schwarzschild solution to gravitational perturbations within a new geometric gauge tailored to linearised gravity. It achieves uniform boundedness and decay without any future gauge normalisation by exploiting red-shift transport structures, and it treats both gauge-invariant (Teukolsky/Regge-Wheeler) and gauge-dependent sectors with a refined initial-data normalisation. The analysis strengthens the role of horizon red-shift in stabilising the gauge-dependent part and provides a framework that aligns with, and extends, the prior DHR approach in a way that is compatible with future Kerr stability and nonlinear problems. The work thereby advances the understanding of gauge choices in black hole stability, with potential impact on sub-extremal Kerr and nonlinear stability programs.

Abstract

We present a new proof of linear stability of the Schwarzschild solution to gravitational perturbations. Our approach employs the system of linearised gravity in the new geometric gauge of \cite{benomio_kerr}, specialised to the case. The proof fundamentally relies on the novel structure of the transport equations in the system. Indeed, while exploiting the well-known decoupling of two gauge invariant linearised quantities into spin Teukolsky equations, we make enhanced use of the red-shifted transport equations and their stabilising properties to control the gauge dependent part of the system. As a result, an initial-data gauge normalisation suffices to establish both orbital and asymptotic stability for all the linearised quantities in the system. The absence of future gauge normalisations is a novel element in the linear stability analysis of black hole spacetimes in geometric gauges governed by transport equations. In particular, our approach simplifies the proof of \cite{DHR}, which requires a future normalised (double-null) gauge to establish asymptotic stability for the full system.
Paper Structure (46 sections, 35 theorems, 189 equations, 1 figure, 1 table)

This paper contains 46 sections, 35 theorems, 189 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Gauge normalisations in linear theory
  3. Linear stability in double-null gauge
  4. The main result: Linear stability in the new gauge
  5. The Kerr problem
  6. Acknowledgements
  7. Overview
  8. The Schwarzschild exterior manifold
  9. The system of linearised gravity
  10. Structural properties
  11. Initial data, well-posedness and asymptotic flatness
  12. Classes of special solutions
  13. Pure gauge solutions
  14. Reference linearised solutions
  15. Initial-data normalised solutions
  16. ...and 31 more sections

Key Result

Theorem 1.1

Consider the system of linearised gravity in a double-null gauge of DHR. Then, all appropriately initial-data normalised solutions are uniformly bounded in terms of the size of their initial data and, upon adding a suitable pure gauge solution, decay in time to a linearised Kerr solution. The additi

Figures (1)

  • Figure 1:

Theorems & Definitions (95)

  • Theorem 1.1: Linear stability of Schwarzschild in double-null gauge DHR
  • Theorem 1.2: Linear stability of Schwarzschild in the new gauge of benomio_kerr_system
  • Theorem 2.1: Linear stability of the Schwarzschild solution, first version
  • Remark 3.1
  • Definition 5.1: Smooth seed initial data
  • Proposition 5.2: Well-posedness
  • Definition 5.3: Pointwise asymptotic flatness
  • Remark 5.4
  • Remark 5.5
  • Remark 5.6
  • ...and 85 more