Teukolsky on slowly-rotating Kerr-de Sitter in the vanishing $Λ$ limit

TL;DR

The authors establish Λ-uniform energy and decay estimates for solutions to the Teukolsky equations on slowly rotating Kerr–de Sitter spacetimes, proving integrated local energy decay and $r^p$-weighted bounds that hold up to the cosmological horizon at $r\sim\Lambda^{-1/2}$. By transforming the Teukolsky system into a wave–transport framework via a Chandrasekhar-type transformation, they isolate a generalized Regge–Wheeler equation coupled to transport equations, enabling robust control over horizon redshift, trapping, and the far-field region. The analysis yields a Kerr–de Sitter stability mechanism that remains uniform as $\Lambda\to0$, allowing recovery of the known Kerr Teukolsky estimates in the vanishing cosmological constant limit. The work combines non-integrable horizontal-structure methods, strategic use of almost-Killing vector fields, and microlocal (pseudo-differential) techniques to treat trapping and the exterior region in a unified, $\Lambda$-independent fashion. This provides a rigorous route to understanding how perturbations decay for rotating black holes with small but positive cosmological constant and links results across Kerr–de Sitter and Kerr spacetimes.

Abstract

We prove energy, Morawetz and $r^p$-weighted estimates for solutions to the Teukolsky equation set on a slowly-rotating Kerr-de Sitter background. The main feature of our estimates is their uniformity with respect to the cosmological constant $Λ>0$ (thus allowed to tend to $0$), while they hold on the whole domain of outer communications, extending up to $r\sim Λ^{-\frac{1}{2}}$. As an application of our result, we recover well-known corresponding estimates for solutions to Teukolsky on a slowly-rotating Kerr background in the limit $Λ\to 0$.

Figures (4)

• Figure 1: Penrose diagrams of Kerr-de Sitter (on the left) and Kerr (on the right). The stationary region (also known in the literature as the domain of outer communication) is shaded in gray in both figures, $r_{{\mathcal{H}},\Lambda}$ and $r_{\overline{{\mathcal{H}}},\Lambda}$ are the largest positive roots of $\Delta$ when $\Lambda>0$, and $r_{{\mathcal{H}},0}$ is the largest root of $\Delta$ when $\Lambda=0$.
• Figure 2: Penrose diagram of ${\mathcal{M}}_{\mathrm{tot}}$: ${\mathcal{M}}$ is in gray while ${\mathcal{M}}_e$ is in brown.
• Figure 3: A Penrose diagram depicting $\partial\mathcal{M}(\tau_1,\tau_2)$.
• Figure 4: A Penrose diagram depicting $\partial\mathcal{M}_e(\tau_{\overline{\mathcal{H}}},\tau)$.

Theorems & Definitions (321)

• Conjecture 1.1: Stability of $\mathcal{B}_\Lambda$
• Conjecture 1.2: Vanishing $\Lambda$ black hole stability
• Theorem 1.3: Rough statement of MAINTHEOREM
• Corollary 1.4: Rough statement of theo comparaison
• Remark 1.5
• Definition 2.1
• Remark 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5