Improved decay for solutions to the linear wave equation on a Schwarzschild black hole

TL;DR

The paper advances the understanding of linear wave decay on Schwarzschild backgrounds by proving near-shorizon and finite-$r$ decay rates that beat the previously known $v_+^{-1}$ bound: $|\phi| \lesssim v_+^{-3/2+\delta}$ and $|\partial_t\phi| \lesssim v_+^{-2+\delta}$ for any $\delta>0$ on compact $r$-regions and along the horizon. The authors introduce a novel vector field commutator $S$ analogous to the Minkowski scaling vector field, derive a controlled equation for $\psi=S\phi$, and blend energy, conformal, and redshift techniques (including a horizon-specific $Y$ field) to propagate decay from a compact region to the horizon. The approach builds on Dafermos–Rodnianski and Klainerman–Sideris methodologies, handling trapping at the photon sphere and the near-horizon geometry with refined estimates and a bootstrap argument. The results sharpen our understanding of linear decay in Schwarzschild spacetimes and have potential implications for nonlinear stability analyses (e.g., wave maps and Kerr stability), where precise decay rates feed into global existence and scattering arguments.

Abstract

We prove that sufficiently regular solutions to the wave equation $\Box_gφ=0$ on the exterior of the Schwarzschild black hole obey the estimates $|φ|\leq C_δv_+^{-{3/2}+δ}$ and $|\partial_tφ|\leq C_δ v_+^{-2+δ}$ on a compact region of $r$ and along the event horizon. This is proved with the help of a new vector field commutator that is analogous to the scaling vector field on Minkowski spacetime. This result improves the known decay rates in the region of finite $r$ and along the event horizon.

Figures (4)

• Figure 1: Schwarzschild spacetime
• Figure 2: Regions of integration
• Figure 3: The region $\mathcal{R}$
• Figure 4: The region $\mathcal{P}$

Theorems & Definitions (50)

• Remark 1
• Proposition 1
• Proposition 2
• Remark 2
• Proposition 3
• Proposition 4
• Remark 3
• proof
• Theorem 5: Dafermos-Rodnianski
• Lemma 6