Effective Matter Conversion in Gravitational Collapse and the Dynamical Formation of Regular Black Holes

Technical Summary: Effective Matter Conversion in Gravitational Collapse and the Dynamical Formation of Regular Black Holes

Problem Statement
The paper addresses the distinction between a regular black hole metric (which avoids curvature singularities) and a consistent dynamical source capable of generating such a geometry. While regular metrics often replace the central singularity with a finite-density de Sitter-like core, the physical interpretation of the stress-energy tensor required to support these geometries remains an open challenge. Specifically, the authors investigate whether a prescribed target density profile can be realized by a physically admissible, interacting matter system within the framework of generalized Vaidya spacetimes. A key difficulty lies in distinguishing between an algebraic decomposition of the stress tensor and a covariant dynamical evolution, particularly regarding the positivity of component densities and the satisfaction of energy conditions.

Methodology
The authors employ an inverse source reconstruction approach within the generalized Vaidya (or Husain) spacetime, described by the metric:
$$ds^2 = -\left(1 - \frac{2M(v, r)}{r}\right)dv^2 + 2dvdr + r^2d\Omega^2$$
where $M(v, r)$ is the Misner–Sharp mass function.

1. Two-Sector Decomposition: The total stress-energy tensor is partitioned into two interacting effective anisotropic sectors. One sector follows a tangential barotropic equation of state $P_b = \alpha \rho_b$, and the second is radiation-like ($P_{rad} = \rho_{rad}/3$ in the tangential sector, or an averaged pressure relation in the Kiselev-like formulation).
2. Radial Reconstruction: Given a prescribed target density $\rho_{new}$ and tangential pressure $P_{new}$ (which must satisfy the radial conservation law), the authors derive a dimensionless radial balance function $\beta(v, r)$. This function governs the interaction between the two sectors, effectively determining the radial profile required for the interacting system to mimic the target geometry.
3. Covariant Completion: The authors explicitly distinguish the radial function $\beta$ from a full covariant exchange vector. They demonstrate that specifying $\beta$ alone is insufficient to define the direction of energy transfer in time; additional information regarding the null fluxes of the individual sectors is required to construct a covariant interaction vector $Q^\mu$.
4. Energy Condition Analysis: The study rigorously checks the Null (NEC), Weak (WEC), Dominant (DEC), and Strong (SEC) energy conditions, as well as the timelike convergence condition (TCC), for both the total source and its decomposed components.

Key Contributions and Results

• Inverse Reconstruction Formalism: The paper derives explicit reconstruction formulas (e.g., Eq. 2.20 and Eq. 3.15) that determine the interaction profile $\beta(v, r)$ required to support a specific target phase. It clarifies that the algebraic split of the source is unique only when the tangential equations of state of the two sectors differ, and identifies conditions where the decomposition becomes degenerate.
• Positivity Obstruction and Vacuum-Like Completion: A central finding is that a de Sitter core ($P_t \approx -\rho$) cannot be represented by nonnegative sectors with nonnegative tangential equations of state (e.g., standard baryonic and radiative matter). The authors show that for the explicit finite-density target $\rho = \rho_c/(1+x^4)$, a decomposition into two positive-density sectors requires one sector to have a vacuum-like equation of state ($\chi = -1$). Without such a sector, the algebraic decomposition yields negative component densities near the core, indicating that a "positive baryonic-radiative" interpretation is incompatible with a regular de Sitter center.
• Explicit Dynamical Model: The authors construct a fully explicit dynamical example with a finite-density target profile.
  • Regularity: The model exhibits finite curvature invariants (Ricci scalar, Ricci tensor square, Kretschmann scalar) at the center, approaching de Sitter geometry.
  • Horizon Formation: The model demonstrates the dynamical formation of trapping horizons. A critical threshold $\rho_c r_0^2 \approx 3.173$ is identified; below this, the spacetime is horizonless and regular. Above this threshold, inner and outer apparent horizons form via a saddle-node bifurcation.
  • Energy Conditions: The total source satisfies the NEC, WEC, and DEC during monotonic accretion. However, the TCC (and thus the SEC) is violated in the central core ($x < 1$), which is necessary to evade the Penrose singularity theorem locally.
• Phase-Structured Targets: The study analyzes various target profiles, including polytropic, MIT-bag quark-matter, and condensate-inspired models. It finds that while these profiles can describe intermediate layers or specific phases, they often fail to provide a globally regular, positive-density core without matching to a different inner solution (e.g., the MIT-bag model is singular at the origin unless the integration constant vanishes).
• Cosmological Analogue: The reconstruction logic is extended to a homogeneous Friedmann–Robertson–Walker setting. The authors show that a signed reconstruction of a $\Lambda$CDM background (matter plus vacuum) does not constitute a physical matter-radiation conversion model, reinforcing the distinction between algebraic splits and physical dynamics.
• Observational Comparison: The stationary endpoint of the explicit model is analyzed for its shadow properties. The authors compute the photon sphere and shadow size, comparing the fractional deviation from the Schwarzschild limit with published Event Horizon Telescope (EHT) constraints for Sagittarius A*. They find that for specific parameter ranges (e.g., $\lambda \approx 5$), the model's shadow size falls within the 68% credible intervals of Keck and VLTI data. However, the authors emphasize that this compatibility tests the exterior density tail and does not confirm the existence of a regular core or a specific matter conversion mechanism.

Significance and Claims
The paper establishes a rigorous framework for distinguishing between the geometric regularity of a black hole and the physical admissibility of its source. Its primary significance lies in:

1. Clarifying the Inverse Problem: It provides a systematic method to determine the necessary matter interactions to support regular geometries, highlighting that the "source" is not unique and depends on the chosen decomposition.
2. Positivity Constraints: It demonstrates that constructing a regular black hole with a de Sitter core using only standard positive-energy matter sectors is algebraically impossible without introducing a vacuum-like component.
3. Local vs. Global: The authors modestly claim that their construction establishes local curvature regularity and the formation of apparent horizons. They explicitly state that the work does not prove geodesic completeness, global hyperbolicity, or the stability of the inner horizon.
4. Observational Limits: The comparison with Sgr A* data is presented as a test of the model's exterior metric deformation, not as evidence for the specific microscopic formation mechanism or the existence of the regular interior itself.

The paper concludes that while the inverse construction successfully generates locally regular, horizon-forming spacetimes satisfying standard energy conditions (except for the SEC in the core), a complete predictive model requires a closed microscopic matter theory, perturbation analysis, and a global causal extension, none of which are provided here.