Exact Dynamical Regular Black Holes from Generalized Polytropic Matter

TL;DR

This work develops a class of exact, dynamical regular black holes sourced by matter obeying a generalized polytropic EOS, embedded in a Vaidya-type spacetime. By regularizing the Kiselev energy density via a shift $r\to r+\beta$ and enforcing center regularity, the authors obtain de Sitter cores with finite curvature invariants and a universal relation between the regularization scale and the mass function, yielding a coordinate-independent EOS $P=\alpha\rho-\zeta\rho^\gamma$. A unified density ansatz $\rho=2(1-2\alpha)D(v)/(r^n+\beta(v)^n)^{(2\alpha+2)/n}$ unifies RN-type, Hayward-type, and Bardeen-type regularizations, with the same implicit polytropic structure $P=\alpha\rho-\zeta\rho^\gamma$ and the constraint $\beta(v)/D(v)^{1/(2\alpha+2)}=\text{const}$ ensuring a mass- or charge-independent central curvature. Hayward-like solutions arise for stiff matter ($\omega=1$, $\alpha=2$) and Bardeen-like solutions for other profiles, all connected through the same framework; the parameter $\beta$ admits geometric and nonlinear electrodynamics interpretations, linking core size and possible magnetic charges. The results provide a physically motivated route to regular black holes formed by realistic gravitational collapse, with potential implications for gravitational waves and black-hole shadows, and open avenues for stability analyses and microphysical underpinnings of the generalized polytropic EOS.

Abstract

We present a class of exact, dynamical, and fully analytic solutions describing regular black holes formed via the gravitational collapse of matter obeying a generalized polytropic equation of state. Starting from a Vaidya-type geometry with a radially dependent mass function, we demonstrate that regularization of the Kiselev solutions can be achieved through a physically motivated modification of the energy density profile. This procedure leads to nonsingular spacetimes with a de~Sitter core and finite curvature invariants at the center. We show that the resulting matter content is naturally described by a generalized polytropic equation of state of the form $P=αρ-ζρ^γ$, where the polytropic index $γ$ is uniquely determined by the regularization scheme. Within this framework, we obtain exact dynamical generalizations of several well-known regular black hole solutions, including the Hayward and Bardeen spacetimes, as particular cases corresponding to specific values of the polytropic parameters. Remarkably, the requirement that the equation of state remains coordinate independent imposes a universal constraint relating the regularization scale to the mass function, which in turn guarantees the existence of a regular de~Sitter core with a curvature scale independent of the black hole mass. Our results provide a unified analytic description of Hayward-like and Bardeen-like black holes emerging from gravitational collapse, offering a consistent effective-matter interpretation rooted in generalized polytropic matter.