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Quasinormal modes of near-extremal Reissner-Nordström-de Sitter spacetimes

Peter Hintz

TL;DR

The paper establishes a rigorous link between near-extremal QNMs of subextremal Reissner–Nordström–de Sitter spacetimes and the quasinormal spectrum of the extremal near-horizon geometry ${\mathrm{AdS}}^2\times{\mathbb S}^2$. By constructing a unified geometric framework that couples the exterior and near-horizon limits via a blown-up total space and a spectral family, the authors prove that all QNMs with size $O(\kappa_{\rm C})$ are $\kappa_{\rm C}$-scaled NH modes, specifically $-i(\lambda_\ell^{+}(\mu)+n)$, with multiplicities $2\ell+1$ and resonant states localized near the event horizon. The analysis leverages Fredholm theory for the NH spectral family, zero-energy estimates at extremality, and a Grushin problem to connect the NH spectrum to near-extremalQNM poles. The results provide a rigorous justification for the predominance of near-horizon QNMs in the near-extremal regime and clarify the limitations of prior heuristic predictions for higher angular modes, with implications for strong cosmic censorship in these spacetimes. The methods extend the conic and AdS-like analyses to a charged, de Sitter setting, offering a blueprint for charged and rotating generalizations.

Abstract

We study quasinormal modes (QNMs) for the Klein-Gordon equation on Reissner-Nordström-de Sitter black holes with near-extremal charge. We locate all QNMs of size $\mathcal{O}(κ_{\rm C})$ where $κ_{\rm C}$ is the surface gravity of the Cauchy horizon (which vanishes at extremality): they are well-approximated by $κ_{\rm C}$ times QNMs of the near-horizon geometry ${\rm AdS}^2\times{\mathbb S}^2$ of the extremal limit.

Quasinormal modes of near-extremal Reissner-Nordström-de Sitter spacetimes

TL;DR

The paper establishes a rigorous link between near-extremal QNMs of subextremal Reissner–Nordström–de Sitter spacetimes and the quasinormal spectrum of the extremal near-horizon geometry . By constructing a unified geometric framework that couples the exterior and near-horizon limits via a blown-up total space and a spectral family, the authors prove that all QNMs with size are -scaled NH modes, specifically , with multiplicities and resonant states localized near the event horizon. The analysis leverages Fredholm theory for the NH spectral family, zero-energy estimates at extremality, and a Grushin problem to connect the NH spectrum to near-extremalQNM poles. The results provide a rigorous justification for the predominance of near-horizon QNMs in the near-extremal regime and clarify the limitations of prior heuristic predictions for higher angular modes, with implications for strong cosmic censorship in these spacetimes. The methods extend the conic and AdS-like analyses to a charged, de Sitter setting, offering a blueprint for charged and rotating generalizations.

Abstract

We study quasinormal modes (QNMs) for the Klein-Gordon equation on Reissner-Nordström-de Sitter black holes with near-extremal charge. We locate all QNMs of size where is the surface gravity of the Cauchy horizon (which vanishes at extremality): they are well-approximated by times QNMs of the near-horizon geometry of the extremal limit.

Paper Structure

This paper contains 21 sections, 22 theorems, 192 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Setup and main result.
  3. Near-extremality, extremality, near-horizon limit
  4. Related works on QNMs and resonance expansions
  5. Outlook
  6. Outline
  7. Geometric singular analysis of the extremal charge limit
  8. Extremal RNdS
  9. Near-horizon geometry
  10. Combination via geometric singular analysis
  11. Massive waves on the near-horizon geometry
  12. Asymptotics of waves at the conformal boundary
  13. Fredholm theory for the spectral family
  14. Zero energy estimate on extremal RNdS
  15. QNMs on near-extremal RNdS: proof of Theorem \ref{['ThmI']}
  16. ...and 6 more sections

Key Result

Theorem 1.1

Fix $0<r_{\rm e}<r_{\rm c}$ and define the quantity For $\mu\geq 0$ and $\ell\in\mathbb{N}_0$, define and define the set of QNMs for the massive scalar wave equation on the near-horizon geometry byWe do not make the dependence of this set on $r_{\rm e},r_{\rm c}$ explicit in the notation. Let $C_0>0$ with $C_0\neq\lambda_\ell^+(\mu)+n$ for all $\ell,n\in\mathbb{N}_0$. Then, in the Hausdorff dist

Figures (8)

  • Figure 1.1: Parameter space of subextremal RNdS black holes (computed using HintzKNdSStability). At the thick dashed line at the top, the charge is extremal but the mass is not; thus $r_{\rm C}=r_{\rm e}<r_{\rm c}$. (We exclude the circle on the top right, where $r_{\rm C}=r_{\rm e}=r_{\rm c}$.) We study RNdS black holes with parameters in a small neighborhood of this dashed line.
  • Figure 1.2: On the left: the radii $r_{\rm C},r_{\rm e},r_{\rm c}$ as functions of the charge $Q\in[0,Q_{\rm ext}]$ for $\Lambda=0.05$, $\mathfrak{m}=1$ where $Q_{\rm ext}\approx 1.00893$ is the extremal charge. On the right: the graph of $F$ for the near-extremal parameters $\Lambda=0.05$, $\mathfrak{m}=1$, $Q=0.9$.
  • Figure 1.3: On the left: the set ${\mathrm{QNM}}_{\mathrm{NH}}(0)$ of near-horizon QNMs for $r_{\rm e}=1$, $r_{\rm c}=2.82$; the corresponding extremal RNdS parameters satisfy $\Lambda\mathfrak{m}^2\approx 0.14$. The values of $\ell,n$ identify the QNM $-i(\lambda^+_\ell(0)+n)$. In the middle: the set ${\mathrm{QNM}}_{\mathrm{NH}}(0)$ for $r_{\rm e}=1$, $r_{\rm c}=11$, and thus $\Lambda\mathfrak{m}^2\approx 0.02$. On the right: illustration of \ref{['EqIConv']} for $\mu=0$. The QNMs of near-extremal RNdS are equal to $\kappa_{\rm C}$ times small perturbations of the near-horizon QNMs (indicated by the blue intervals), while the red QNM $0$ is independent of the RNdS parameters. (For scalar field masses $\mu>0$, there is no QNM at $0$.)
  • Figure 1.4: We fix $r_{\rm e}=1$, $r_{\rm c}=2.82$, $\mu=0.1$, and consider $\ell=0$, $n=1$. On the left: the resonant state $u_{0,1}(z)$ (see \ref{['EqNHQNMResState']}) for the massive wave equation on the near-horizon geometry corresponding to the near-horizon QNM $-i(\lambda_0^+(\mu)+1)\approx -2.138$. On the right: illustration of the resonant state $u_\epsilon$ for the Klein--Gordon equation on RNdS with parameters $r_{\rm C}=r_{\rm e}-2\epsilon$, $\epsilon=0.05\ll r_{\rm e}$. We are showing here the approximation $u_{0,1}(2\frac{r-r_{\rm C}}{r_{\rm e}-r_{\rm C}}-1)$ of $u_\epsilon$.
  • Figure 1.5: On the left: Penrose diagram of subextremal RNdS. On the right: Penrose diagram of the near-horizon geometry ${\mathrm{AdS}}^2\times\mathbb{S}^2$. The level sets of the function $z$ meet at the point $i^+$ on the conformal boundary, and $t_*\to\infty$ as one approaches $i^+$. In the shaded regions on both sides, the metrics are close to being constant multiples of one another upon relating $t_*,r$ and $\mathfrak{t}_*,z$ as indicated after \ref{['EqIz']}.
  • ...and 3 more figures

Theorems & Definitions (55)

  • Theorem 1.1: Main result, abridged version
  • Remark 1.2: Comparison: zero mass limit
  • Remark 1.3: QNMs of extremal RNdS
  • Conjecture 1.4: More precise asymptotics of QNMs
  • Conjecture 1.5: Shallow QNMs
  • Remark 2.1: $g_{\mathrm{NH}}$ and the near-horizon geometry of extremal RNdS
  • Remark 2.2: $g_{\mathrm{NH}}$ and the Einstein--Maxwell equations
  • Definition 2.3: Total space
  • Definition 2.4: q-vector fields on the total space
  • Definition 2.5: q-differential operators
  • ...and 45 more