[Paper Review] Coupled gravitational and electromagnetic perturbations of Reissner-Nordström spacetime in a polarized setting II - Combined estimates for the system of wave equations

Elena Giorgi|arXiv (Cornell University)|Apr 16, 2018

Cosmology and Gravitation Theories5 references3 citations

TL;DR

This paper establishes combined energy-Morawetz and $r^p$-weighted estimates for a system of coupled wave equations governing gravitational and electromagnetic perturbations in Reissner-Nordström spacetime under polarized symmetry. It proves that for small charge, the perturbations decay uniformly, with key terms controlled via integrated energy bounds and weighted norms, establishing a crucial step toward linear stability of charged black holes under axially symmetric perturbations.

ABSTRACT

In a previous paper on coupled gravitational and electromagnetic perturbations of Reissner-Nordström spacetime in a polarized setting, we derived a system of wave equations for two independent quantities, one related to the Weyl curvature and one related to the Ricci curvature of the perturbed spacetime. We analyze here the system of coupled wave equations, deriving combined energy-Morawetz and $r^p$-estimates for the system, in the case of small charge.

Motivation & Objective

• To derive combined energy-Morawetz and $r^p$-weighted estimates for a system of coupled wave equations arising from linearized Einstein-Maxwell equations in Reissner-Nordström spacetime.
• To analyze the stability of the Reissner-Nordström solution under polarized, axially symmetric perturbations with small charge.
• To control coupling terms and lower-order quantities in the wave system using integrated energy and weighted norms.
• To establish decay of perturbations in the trapping, redshift, and far-field regions via a unified framework of estimates.
• To provide foundational estimates required for the final state conjecture on the nonlinear stability of Kerr-Newman black holes.

Proposed method

• Derives a system of two coupled wave equations for $rak{q}$ (related to Weyl curvature) and $rak{q}^{f F}$ (related to Ricci curvature), with coupling terms arising from electromagnetic and gravitational interactions.
• Applies a partition of spacetime into three regions: trapping ($r o r_+$), redshift ($r o r_{ ext{red}}$), and far-field ($r o rown$), each requiring distinct estimates.
• Uses $r^p$-weighted energy estimates with $p e 1$ to handle the degeneracy in the trapping region and to control lower-order terms.
• Employs transport estimates for lower-order quantities ($rak{q}^{f F}$-related $ ilde{rak{q}}_3$, $ ilde{rak{q}}_1$, $ ilde{rak{q}}_0$) to bound their contributions in the energy hierarchy.
• Applies a multiplier method with a vector field $ ilde{rak{q}}$ to control the coupling terms, particularly the $e imes C_1[rak{q}^{f F}]$ and $e imes C_2[rak{q}]$ terms.
• Combines non-degenerate energy bounds with weighted $L^2$ norms to absorb error terms, especially those involving $e^2$-order lower-order terms ($L_1[rak{q}^{f F}]$, $L_2[rak{q}^{f F}]$).

Experimental results

Research questions

• RQ1 Can a unified framework of $r^p$-estimates and energy-Morawetz bounds be constructed for a system of coupled wave equations in Reissner-Nordström spacetime?
• RQ2 How do electromagnetic and gravitational perturbations interact in the linearized regime, and can their coupling be controlled via energy estimates?
• RQ3 What is the decay behavior of perturbations in the trapping, redshift, and far-field regions for small-charge Reissner-Nordström spacetime?
• RQ4 Can lower-order terms arising from curvature components be controlled using transport and weighted estimates in the context of coupled systems?
• RQ5 To what extent do $e^2$-order terms in the coupling affect the overall stability of the system?

Key findings

• The paper establishes a complete set of $r^p$-weighted estimates for $p e 1$ in the trapping region, overcoming the degeneracy of the wave operator.
• It proves that the combined energy-Morawetz and $r^p$-estimates control the full system, with the error terms bounded by $e imes ext{energy} + e^2 imes ext{weighted norms}$.
• Lower-order terms $L_1[rak{q}^{f F}]$ and $L_2[rak{q}^{f F}]$ are controlled via transport estimates and absorption into non-degenerate energy norms.
• The $e^2$-order terms $e^2 L_1[rak{q}]$, $e^2 L_2[rak{q}^{f F}]$, and $e^2 N_1[rak{q}, rak{q}^{f F}]$ are absorbed into the non-degenerate bulk norms, ensuring stability.
• The final estimate shows that the total energy and weighted norms are bounded by initial data and lower-order terms: $E[rak{q}]( au) + E^1[rak{q}^{f F}]( au) ext{ and } ext{weighted norms} ext{ are controlled by } E_p[ ext{initial data}] + ext{lower-order terms}$.
• The result holds for all $p e 1$ in the range $ au_1 o au_2$, with $p e 1$ essential for avoiding degeneracy in the trapping region.