A class of $d$-dimensional regular black holes: Shadows, Thermodynamics and Gravitational collapse

TL;DR

This work constructs a $d$-dimensional family of regular black holes with a de Sitter core sourced by nonlinear electrodynamics and magnetic charge, connecting stellar polytropic collapse (via a parameter $n$) to known 4D solutions such as Hayward and Bardeen. It analyzes horizons, energy conditions, photon spheres, and shadows, showing that shadows persist up to a degenerate charge $q_{ ext{deg}}$ and that increasing dimension generally reduces differences from Schwarzschild–Tangherlini, while preserving regularity and rich thermodynamic behavior including possible phase transitions. The paper also generalizes the Oppenheimer–Snyder–Datt collapse to $d$ dimensions, detailing the evolution of the stellar surface and horizons through Israel junction conditions, and demonstrating how the collapse dynamics slow with higher dimensions. Overall, the results provide a coherent higher-dimensional framework for regular BHs and their observational and dynamical implications, with tighter constraints on magnetic charge and polytropic index from shadow and thermodynamic considerations.

Abstract

This paper examines a recently introduced class of regular black holes that can form from the collapse of a polytropic star with an arbitrary polytropic index. This class has a de Sitter core and reduces to the Bardeen and Hayward black holes when the polytropic index is chosen appropriately. We demonstrate that this class of black holes is sourced by a nonlinear electrodynamics Lagrangian in $d$ dimensions and that its regularity stems from the presence of magnetic charge. We analyze the energy conditions and study the photon spheres analytically and the shadows numerically. Then, we compare our results with observations. Additionally, we present the thermodynamic properties of this class of black holes, including their temperature, entropy, and heat capacity. We also examine their thermodynamic stability. Finally, we generalize the Oppenheimer-Snyder-Datt collapse scenario to this $d$-dimensional class of black holes and study stellar collapse into them, as well as the evolution of the star's surface, the apparent horizon, and the event horizon.

Figures (8)

• Figure 1: The plot of $A_d( \tilde{r})$ and the corresponding effective potential for a massless particle as a function of radius ($\tilde{r}$ is scaled by $m$ as $\tilde{r} = r/m$) for different dimensions and for the cases $n = 1$ (i.e., Hayward), $n = 3/2$ (i.e., Bardeen), $n = 2$ and $n = 5/2$. The magnetic charge $q$ is set to different values in each dimension to make the plots comparable. The value of the rescaled magnetic charge $\tilde{q}$ is chosen so that the BH has two distinct roots. This ensures that the BH is neither extremal nor horizonless. The dashed line represents the Schwarzschild BH in $4$ dimensions. We have omitted other dimensions for the ST BH plot due to the indistinguishability of the curves. The same $\tilde{q}$ values are used for the effective potential plot.
• Figure 2: SEC and DEC$_3$ as a function of the dimensionless radius for different polytropic indices and dimensions. The solid and dashed lines represent SEC and DEC$_3$, respectively. All other energy conditions are satisfied.
• Figure 3: Schematic illustration of the horizons, the BH photon sphere (BHPS), and the compact object photon sphere (COPS), as well as the stability of the photon spheres.
• Figure 5: Shadow of the regular $4$-dimensional metric \ref{['met4']} compared to the shadow boundaries of $Sgr A^*$ and $M87^*$ for various polytropic indices. The dashed line corresponds to the Schwarzschild BH.
• Figure 6: The numerical result of the horizon temperature as a function of the horizon radius for different polytropic index values and dimensions with $q = 0.3$. The solid and dashed lines represent the regular BH \ref{['ad1']} and the ST BH, respectively. The horizon temperature begins at zero at the extremal radius. As $r_{\text{h}}$ increases, the temperature decreases for all dimensions.