Regularized Black Hole Solution from a New String Cloud Source

TL;DR

The paper addresses the challenge of constructing a regular black hole within a Letelier–Alencar string-cloud spacetime by introducing a Dagum-type regulator that smooths the core. This approach yields a geometry transitioning from a string-cloud exterior to an AdS-like core with finite curvature and a well-defined horizon structure, while allowing detailed thermodynamic and shadow analyses. Key findings include entropy dependence solely on the regularization scale, Regnéy non-extensive entropy removing standard phase transitions, and shadow bounds that are compatible with current EHT measurements for Sgr A* and M87*. The work links microscopic core regularization to macroscopic observables, providing a physically motivated, observationally testable regular black hole model in the presence of a string cloud.

Abstract

We construct a new family of regular black hole solutions supported by the novel Letelier-Alencar string cloud and regularized through a rational Dagum-type distribution. The regulator smooths the matter profile and ensures finite curvature invariants, yielding a geometry that interpolates between a string-cloud exterior and an anti--de Sitter core. We analyze the energy conditions, identifying where the null, weak, dominant and strong conditions hold or fail across the core and exterior. The parameter space for horizon formation is mapped and the thermodynamic propertie -- mass, Hawking temperature, entropy and heat capacity -- are derived; notably, the entropy depends only on the regularization scale while the string parameter modifies temperature and heat capacity. Employing Rényi non-extensive entropy and the topological thermodynamics approach, we show the non-extensive deformation stabilizes the system and removes the standard phase transition. Finally, we compute the shadow radius and derive constraints compatible with current Event Horizon Telescope bounds for Sgr~A* and M87*.

Figures (13)

• Figure 1: Left panel: Kretschmann scalar as a function of $r$, varying the string parameter $a$, with $r_0=0.5$. Right panel: The same quantity, now varying the scale $r_0$, with $a=0.2$. It was considered $M=1.0$.
• Figure 2: Parameter space $(M,a)$ of the regularized new string cloud black hole. The shaded region indicates the values of the parameters yielding black hole solutions with event horizons, bounded by the critical line where the horizon degenerates. Beyond this boundary, the spacetime describes no black hole. The regularization core radius is set to $r_0=0.1$ (left panel) and $r_0=0.5$ (right panel).
• Figure 3: Contours of the energy density $\rho(r)$. In the left panel, we set $r_{0}=0.7$ and $M=1$; larger $a$ expands the inner region with $\rho<0$. In the right panel, we set $a=0.4$ and $M=1$; increasing $r_{0}$ moves the $\rho$ peak to larger $r$ and reduces its amplitude. The line $\rho=0$ marks the WEC threshold.
• Figure 4: Contours of the tangential WEC ($\text{WEC}_t$). In the left panel, we set $r_{0}=0.7$ and $M=1$; larger $a$ drives $\rho+p_t$ more negative at fixed $r$ and shifts the $\rho+p_t=0$ curve outward. In the right panel, we set $a=0.4$ and $M=1$; increasing $r_{0}$ raises $\rho+p_t$ and moves its maximum to larger $r$. The line $\rho+p_t=0$ marks the boundary of the WEC$_t$ domain.
• Figure 5: Contours of the SEC. In the top-left panel, we set $r_{0}=0.7$ and $M=1$; larger $a$ widens the negative–SEC region and shifts the $\mathrm{SEC}=0$ curve outward. In the top-right panel, we set $a=0.4$ and $M=1$; increasing $r_{0}$ raises $\mathrm{SEC}$ and moves its zero to larger $r$, shrinking the negative band. In the bottom panel, the radial profiles of SEC with $r_{0}=0.7$ and $M=1$ for $a=\{0.2,0.4,0.6\}$; this panel shows the delimited region of SEC violation.