Near-extremal asymptotics and strong cosmic censorship for black holes immersed in a Chaplygin-like dark fluid

Technical Summary: Near-Extremal Asymptotics and Strong Cosmic Censorship for Black Holes Immersed in a Chaplygin-like Dark Fluid

Problem Statement
The paper investigates the validity of the Strong Cosmic Censorship (SCC) conjecture for electrically neutral black holes immersed in a Chaplygin-like dark fluid (CDF). Unlike charged black holes where the inner Cauchy horizon is supported by electromagnetic fields, these CDF black holes possess an inner Cauchy horizon supported by matter. The central problem is determining whether generic perturbations render the Cauchy horizon singular (preserving SCC) or remain regular enough to allow extension beyond it (violating SCC).

In asymptotically de Sitter spacetimes, exterior perturbations decay exponentially, competing with the infinite blueshift at the inner horizon. The stability of the Cauchy horizon is quantified by the parameter $\beta = \alpha / \kappa_-$, where $\alpha$ is the spectral gap (the slowest exterior decay rate) and $\kappa_-$ is the Cauchy-horizon surface gravity. According to the Christodoulou formulation, SCC is potentially violated if $\beta > 1/2$ for smooth initial data, implying the field retains sufficient regularity ($H^1$) at the horizon.

Methodology
The authors employ a combination of analytic near-extremal asymptotics and global numerical calculations:

1. Analytic Parametrization: The authors derive a closed-form parametrization for the extremal (cold) and Nariai horizon boundaries of the CDF black hole geometry. This allows for an explicit analytic treatment of the near-extremal limit where the event horizon ($r_+$) and Cauchy horizon ($r_-$) coalesce.
2. Near-Extremal Asymptotics: Using the derived parametrization, they analytically determine the scaling of horizon splitting, Cauchy-horizon surface gravity ($\kappa_-$), and the leading near-extremal (NE) quasinormal mode (QNM) spectrum. They establish the behavior of the conformal weight $h_\ell$ in the near-horizon $AdS_2 \times S^2$ throat.
3. Numerical Spectral Analysis: The authors compute the full QNM spectrum using a Chebyshev pseudospectral method. They map the exterior region to a compact interval and solve the resulting quadratic matrix eigenvalue problem. These results are cross-checked against direct integration and Wentzel–Kramers–Brillouin (WKB) approximations for high angular multipoles.
4. Spectral Competition: They analyze the competition between three QNM families relevant to the spectral gap: photon-sphere (PS) modes, de Sitter (dS) modes, and near-extremal (NE) modes. The spectral gap $\alpha$ is defined as the infimum of the imaginary parts of these frequencies.

Key Contributions and Results

• Analytic Control of Extremal Geometry: The paper provides explicit analytic expressions for the horizon splitting and surface gravity near the extremal limit. It demonstrates that the fundamental NE mode scales such that $\beta_{NE} \to 1$ as the system approaches extremality (the "cold" limit), while higher angular multipoles ($\ell > 0$) retain explicit dependence on the extremal CDF geometry via a conformal weight $h_\ell(B)$.
• Spectral Gap Determination: The study identifies that the spectral gap is determined by the lowest of the three QNM families (dS, PS, or NE). While the NE family dominates very close to extremality (where $\beta \to 1$), the threshold for SCC violation ($\beta = 1/2$) is often crossed earlier by dS or PS modes depending on the effective cosmological scale.
• SCC Phase Diagram: The authors map the SCC phase diagram across the three-horizon domain. They find that potential SCC violation is confined to a narrow region near extremality.
  • The mode controlling the SCC threshold ($\beta = 1/2$) transitions sequentially: from de Sitter modes at low effective cosmological scales, to photon-sphere (eikonal) modes at intermediate scales, and back to de Sitter modes near the triple-horizon (ultracold) endpoint.
  • The normalized width of the violation region ($\Delta \tilde{q}$) varies nonmonotonically, reaching a maximum of approximately 1.47% in the PS-controlled regime.
• Comparison with PFDM: The results are compared with perfect-fluid dark matter (PFDM) black holes. While both models exhibit narrow near-extremal violation regions, the specific sequence of mode dominance and the behavior of the violation width differ due to the distinct matter profiles.

Significance
The paper claims to provide a rigorous analytic and numerical framework for studying SCC in matter-supported black hole geometries where the asymptotic de Sitter scale and the strong-field geometry are controlled by the same matter sector. By deriving exact closed-form parametrizations for the degenerate horizons, the authors make the near-extremal scaling of the CDF geometry analytically accessible.

The primary significance lies in demonstrating that SCC violation in this specific matter-supported context is a "narrow" phenomenon, restricted to a small parameter range near extremality. The work clarifies how the competition between different QNM families dictates the regularity of the Cauchy horizon, showing that the outcome is highly sensitive to the specific geometry and the perturbing sector. The results reinforce the view that SCC violation is not a generic feature of all near-extremal black holes but depends critically on the interplay between the decay rates of exterior modes and the blueshift at the inner horizon.