Integrable singularities obstruct Cauchy horizons and mass inflation

Technical Summary: Integrable Singularities Obstruct Cauchy Horizons and Mass Inflation

Problem Statement
The paper addresses the dynamical instability of black-hole interiors admitting an inner (Cauchy) horizon, a phenomenon known as mass inflation. In standard general relativity, the collision of ingoing and outgoing perturbations near a Cauchy horizon leads to an unbounded blueshift and exponential growth of the Misner–Sharp mass, driven by the Dray–'t Hooft–Redmount (DTR) relation. This mechanism is widely viewed as a dynamical realization of the Strong Cosmic Censorship conjecture. While most analyses assume the existence of an inner horizon and study its nonlinear fate, this work investigates a complementary question: Can the formation of such a horizon be structurally obstructed by specific matter content itself? The author focuses on static, spherically symmetric interiors that admit an integrable singularity at the center (where curvature invariants diverge, but volume-weighted integrals remain finite) and determine under what conditions the standard Poisson–Israel–Ori amplification channel becomes kinematically inoperative.

Methodology
The study employs a combination of static analysis and nonlinear dynamical evolution within spherical symmetry, assuming the Null Energy Condition (NEC).

1. Static Setup: The author considers anisotropic fluid sources with energy density $\rho \sim A/r^2$ (a marginal profile) near the origin, where $A > 1/(8\pi)$. This profile ensures a finite central compactness $q(0^+) = 4\pi A > 1/2$, placing the center in the trapped region. They introduce a "screening" condition where the local density $\rho(r)$ is bounded by one-third of the averaged density $\bar{\rho}(r)$ throughout the trapped region ($\rho \le \bar{\rho}/3$).
2. Compactness Gap Analysis: Using the Einstein equations, they derive the behavior of the compactness function $q(r) = m(r)/r$. They prove that under the screening condition, $q(r)$ decreases monotonically from a super-critical value at the center to $1/2$ at the unique event horizon. This establishes a strictly positive "compactness gap" ($\Delta_\epsilon$) in the trapped region, preventing a second zero of the lapse function $f(r) = 1 - 2m(r)/r$ (which would correspond to a Cauchy horizon).
3. Dynamical Propagation: The author extends this static result to dynamical spacetimes using double-null coordinates. They demonstrate that the compactness gap is preserved under nonlinear evolution if the NEC and the dynamical screening condition hold.
4. Quantitative Bounds: They apply this gap to three specific models of mass inflation:
   • Generalized Vaidya geometry (ingoing radiation).
   • Counter-streaming null dust on a fluid background.
   • The Ori thin-shell model (continuous influx crossing an outgoing shell).

Key Contributions and Results

• Structural Obstruction to Cauchy Horizons: The paper proves that for a class of screened, marginally divergent density profiles, the compactness $q(r)$ is strictly greater than $1/2$ everywhere inside the event horizon (except at the horizon itself). Consequently, the lapse function $f(r)$ cannot vanish a second time, ruling out the existence of an inner Cauchy horizon in the static limit.
• Propagation of the Gap (Theorem 1): The author shows that the condition $|f| \ge \Delta_\epsilon$ is dynamically stable. Under the NEC and the screening condition, the quantity $2m/r$ is non-decreasing along both future null directions in a trapped region. Thus, the gap preventing the lapse from approaching zero is preserved during evolution, preventing the formation of an inner horizon dynamically.
• Bounding the Blueshift (Theorem 2a): In the presence of a compactness gap, the advanced time $v$ spent traversing the trapped region is bounded. Specifically, $dv/dr \le 2/\Delta_\epsilon$. This contrasts sharply with the Cauchy horizon case, where the advanced time diverges, leading to unbounded blueshift.
• Suppression of Mass Inflation (Theorem 2b, 2c): The mass-inflation mechanism relies on the nonlinear coupling between ingoing and outgoing fluxes, governed by a coefficient proportional to $2/(r|f|)$.
  • In standard mass inflation, as $r \to r_-$ (inner horizon), $|f| \to 0$, causing the coupling to diverge and the mass to grow exponentially ($m \sim e^{\kappa_- v}$).
  • With a compactness gap, $|f|$ is uniformly bounded from below by $\Delta_\epsilon$. This bounds the coupling coefficient, preventing exponential growth.
  • For the Ori model, the e-folding rate of the mass amplification is bounded by the influx luminosity rather than the inner-horizon surface gravity $\kappa_-$. The growth is at most linear or power-law, not exponential.

Significance and Claims
The paper claims to identify a specific structural condition—the existence of a compactness gap—that renders the Poisson–Israel–Ori mass-inflation mechanism inoperative. Unlike previous works that modify black-hole interiors via effective field theories or require full regularity, this work operates within classical general relativity but restricts the matter content to a specific class of integrable singularities.

The author emphasizes that their result is a "modest" structural inquiry: they do not claim these interiors describe realistic collapse scenarios, but rather demonstrate that if such matter content exists, the standard channel for mass inflation is kinematically blocked. The significance lies in showing that the instability is not inevitable for all trapped regions; it specifically requires the degeneracy of the lapse function ($f \to 0$) within the trapped region. By ensuring $|f|$ remains bounded away from zero via the compactness gap, the exponential amplification of the Misner–Sharp mass is structurally excluded. The work complements recent findings on regular black holes by providing a unified, static sufficient condition that applies to an entire class of interiors without needing to solve the nonlinear dynamics for each specific model individually.