Regular black holes with Minkowskian cores: causal structure and observational degeneracy

Technical Summary: Regular Black Holes with Minkowskian Cores

Problem Statement
General Relativity (GR) predicts that gravitational collapse inevitably leads to spacetime singularities where geometric quantities diverge. While the event horizon hides these singularities from external observers (the Cosmic Censorship hypothesis), the theory remains conceptually incomplete within the singular region. Various Regular Black Hole (RBH) models have been proposed to replace the singularity with a finite core, typically approximating a de Sitter geometry. However, these models often introduce internal Cauchy horizons, which are dynamically unstable, and it remains unclear whether the specific nature of the interior geometry (e.g., de Sitter vs. Minkowskian) significantly alters observable signatures. Recent observational advances, such as the Event Horizon Telescope (EHT) images and gravitational wave detections, offer a potential testbed for these deviations, yet studies suggest that different internal structures may produce nearly indistinguishable shadows. This work addresses the construction of a new class of RBHs with Minkowskian cores and investigates whether their distinct internal causal structures and geometric mechanisms can be distinguished from standard singular black holes via optical observations.

Methodology
The authors construct a family of static and rotating regular black hole solutions using the Gravitational Decoupling (GD) method.

1. Static Case: Starting with the Schwarzschild vacuum as a seed metric, the authors introduce an additional energy-momentum source $\theta_{\mu\nu}$ via the GD formalism. This involves deforming the metric functions $\nu(r)$ and $\lambda(r)$ using a coupling constant $\alpha$ and deformation functions $f(r)$ and $g(r)$.
2. Constraints: To ensure physical viability, the model imposes:
   • Horizon Structure: The condition $e^\nu = e^{-\lambda}$ is enforced to ensure a well-defined causal structure with horizons.
   • Energy Conditions: The solution is constrained to satisfy the Weak Energy Condition (WEC).
   • Regularity: The deformation is specifically tuned to remove the central singularity, requiring the metric to approach Minkowski space ($e^\nu \to 1$) as $r \to 0$, rather than the de Sitter behavior ($e^\nu \to 1 - \Lambda r^2$) common in other RBH models.
3. Rotating Case: The static solution is extended to an axially symmetric rotating geometry using the Gurses-Gursey metric (a Kerr-Schild type extension) within the GD framework. The mass function $M$ is replaced by a radial profile $\tilde{m}(r)$ derived from the static solution.
4. Observational Analysis:
   • Shadow Calculation: The photon sphere and critical impact parameters are calculated to determine the black hole shadow boundary.
   • Ray-Tracing: The authors employ a ray-tracing algorithm (using the Gradus.jl package for the rotating case) to simulate the appearance of geometrically thin, optically thin accretion disks. They model three specific emission profiles (GLM1, GLM2, GLM3) peaking at different radii (near the horizon or the ISCO) to generate synthetic images for both face-on and highly inclined ($\iota = 87^\circ$) observers.

Key Contributions and Results

• Minkowskian Core Construction: The paper successfully constructs a static RBH where the central geometry is strictly Minkowskian ($e^\nu \approx 1 + O(r^4)$ as $r \to 0$). This contrasts with the standard de Sitter cores found in Bardeen or Hayward geometries. The regularization is achieved not via an effective vacuum energy density, but through a matter distribution whose energy density vanishes smoothly at the center.
• Causal Structure: The solutions exhibit a two-horizon structure (outer event horizon and inner Cauchy horizon) for specific ranges of the deformation parameter $\alpha$. The authors demonstrate that the inner horizon in this model arises from a competition between the Schwarzschild term and an exponentially localized geometric deformation, rather than from a de Sitter vacuum region. This establishes that regularity and the existence of Cauchy horizons are logically independent in this framework.
• Extremal Configurations: The model admits extremal black hole solutions where the inner and outer horizons merge, separating the black hole branch from a no-horizon (naked) branch.
• Observational Degeneracy:
  • Shadows: Despite the fundamental differences in the interior geometry (Minkowskian vs. de Sitter vs. singular), the gravitational shadows of these regular black holes are found to be virtually indistinguishable from those of Schwarzschild (static) and Kerr (rotating) black holes with the same asymptotic mass and spin parameters.
  • Accretion Disk Images: Full ray-traced simulations of thin accretion disks show that the observed intensity profiles, photon ring structures, and Doppler asymmetries (for inclined observers) remain remarkably similar to standard Kerr/Schwarzschild predictions. The morphological features are robust against variations in the deformation parameter $\alpha$, even as the system approaches extremality.

Significance
The paper concludes that the internal causal structure and the specific mechanism of regularization (Minkowskian vs. de Sitter) do not necessarily imprint distinct signatures on global observables accessible from spatial infinity. The results highlight a significant observational degeneracy: markedly different black hole interiors can produce nearly identical optical appearances, including both the shadow boundary and the detailed radiative structure of accretion disks. This suggests that current and near-future horizon-scale imaging (e.g., EHT) may be insufficient to probe the deep internal structure of compact objects or to rule out specific regular black hole models based solely on shadow or image morphology. The work reinforces the idea that while regular black holes can be constructed to satisfy energy conditions and avoid singularities, their external phenomenology may remain consistent with classical General Relativity predictions, leaving the nature of the black hole interior as an open question for future theoretical and observational developments.