Shadow cast by a rotating and nonlinear magnetic-charged black hole in perfect fluid dark matter

Abstract: We derived an exact solution of the spherically symmetric Hayward black hole surrounded by perfect fluid dark matter (PFDM). By applying the Newman-Janis algorithm, we generalized it to the corresponding rotating black hole. Then, we studied the shadows of rotating Hayward black hole in PFDM. The apparent shape of the shadow depends upon the black hole spin $a$, the magnetic charge $Q$ and the PFDM intensity parameter $k$ ($k<0$). The shadow is a perfect circle in the non-rotating case ($a=0$) and a deformed one in the rotating case ($a\neq{0}$). For a fixed value of $a$, the size of the shadow increases with the increasing $\vert{k}\vert$, but decreases with the increasing $Q$. We further investigated the black hole emission rate. We found that the emission rate decreases with the increasing $\vert{k}\vert$ (or $Q$) and the peak of the emission shifts to lower frequency. Finally, we discussed the observational prospects corresponding to the supermassive black hole $\mathrm{Sgr\ A^{*}}$ at the center of the Milky Way.

• The paper introduces a novel model by extending spherically symmetric Hayward black holes to rotating versions using the Newman-Janis algorithm, integrating PFDM and nonlinear magnetic charge.
• The study shows that increasing PFDM intensity enlarges the black hole shadow while higher magnetic charge decreases its size, quantified by observables like shadow radius and distortion.
• Observational implications indicate that energy emission rates decline and peak at lower frequencies with parameter variations, linking theoretical constructs with empirical measurements.

Shadow Cast by a Rotating and Nonlinear Magnetic-Charged Black Hole in Perfect Fluid Dark Matter

Introduction

The study described in "Shadow cast by a rotating and nonlinear magnetic-charged black hole in perfect fluid dark matter" investigates the behavior of black holes influenced by the presence of perfect fluid dark matter (PFDM) and nonlinear magnetic charges. Employing the Newman-Janis algorithm, the research focuses on extending spherically symmetric Hayward black holes to corresponding rotating versions. Understanding the shadow of black holes is crucial for analyzing astronomical phenomena, as it provides insights into the parameters of the black hole itself.

Spherically Symmetric and Rotating Black Hole Solutions

An exact solution for a spherically symmetric Hayward black hole surrounded by PFDM was derived before generalizing it to a rotating black hole using the Newman-Janis algorithm. This technique involves modifying the Boyer-Lindquist coordinates to account for rotation, important for representing the black hole’s spinning nature. For these black holes, the spin parameter $a$, magnetic charge $Q$, and PFDM intensity parameter $k$ critically influence the structure and size of the black hole shadow. These parameters determine the metric, specifically affecting the $\Delta_r$ and $\Sigma$ terms that form the backbone of the rotating solution.

Photon Trajectories and Shadow Characteristics

Using the Hamilton-Jacobi equation and separating variables, we deduced the shadow profile of rotating Hayward black holes surrounded by PFDM. These calculations provide the photon paths around the black holes, specifically focusing on light rays that describe the shadow's boundary. It was noted that the shadow takes the form of either perfect circles (non-rotating cases) or ellipses (rotating cases), influenced by black hole parameters. Parameters $a$, $Q$, and $k$ were varied to demonstrate that PFDM enlarges the shadow's size, with higher $\vert{k}\vert$ resulting in a pronounced shadow, while increasing $Q$ reduced shadow size.

Observables and Measurement Implications

Two critical observables were introduced: the shadow radius $R_s$ and the distortion parameter $\delta_s$. $R_s$ represents the apparent size while $\delta_s$ gauges deviation from circularity, both of which vary with black hole spin, magnetic charge, and PFDM intensity. Beyond theoretical analysis, these parameters provide a pathway to observational data, supporting future astronomical endeavors like those by the EHT and VLBI systems.

Energy Emission and Observational Prospects

It’s found that the energy emission rate correlates inversely with both PFDM intensity and magnetic charge, indicating that emission peaks decline and shift to lower frequencies with increasing $k$ or $Q$. This research pioneers the linkage between high-energy observations and fundamental black hole characteristics, offering promising observational targets for instruments such as EHT and space-based VLBI platforms.

Conclusion

The exploration into rotating, nonlinear magnetic-charged black holes with PFDM provides valuable insights into their shadow characteristics and energy emission particulars. As techniques and observatories evolve, accurate black hole parameters may soon be extricated from data obtainable from the shadows they cast. This has vast implications, merging theoretical constructs with empirical observation, advancing our understanding of black holes in contexts aligning closely with cosmic scales and conditions. In extending these studies, refining techniques involving PFDM and nonlinear electromagnetic interactions will further elucidate the nature and behavior of these astronomical entities.