[Paper Review] Rotating regular black holes in AdS spacetime and its shadow

Balendra Pratap Singh, Md Sabir Ali+1 more|arXiv (Cornell University)|Jul 25, 2022

Astrophysical Phenomena and Observations4 citations

TL;DR

This paper investigates the optical properties of rotating regular black holes in anti-de Sitter (AdS) spacetime, deriving null geodesic equations to analyze photon rings and shadow morphology. It shows that these black holes exhibit smaller, more distorted shadows than Kerr or regular AdS black holes due to the interplay of spin, deviation parameter $k$, and cosmological constant, with shadow size decreasing and distortion increasing as $1/l^2$ grows.

ABSTRACT

The presence of a photon region around the black hole is an essential feature in receiving the emitted spectrum from the vicinity of the black hole by the distant observer. In this paper, we investigate the optical properties of rotating regular anti-de Sitter (AdS) black holes, which characterized by its mass $(M)$, spin parameter $(a)$, deviation parameter $(k)$ and the cosmological constant $(Λ)$ related to the curvature radius via $(Λ=-3/l^2)$. We derive the complete null geodesic equations of motion and study the unstable circular orbits for an observer at given Boyer-Lindquist coordinates ($r_O,\;\vartheta_O$) in the domain of outer communication and trace the photon rings. For the analytical part of our study, we investigate the observables, namely, shadow radius $(R_s)$ and distortion parameter $(δ_s)$. These rotating regular AdS black holes have smaller shadows when compare to the Kerr and regular spacetimes. We also estimate the energy emission rate of the black hole.

Motivation & Objective

• To analyze the shadow morphology of rotating regular black holes in anti-de Sitter (AdS) spacetime, a class of non-Kerr black holes with a regular core.
• To investigate how the spin parameter $a$, deviation parameter $k$, and cosmological constant (via $1/l^2$) affect the black hole shadow's size and shape.
• To compute key observables—shadow radius $R_s$ and distortion parameter $\delta_s$—for comparison with Kerr and regular black hole shadows.
• To estimate the energy emission rate from the photon region to assess detectability of such black holes via electromagnetic radiation.

Proposed method

• Derives the complete set of null geodesic equations of motion for rotating regular AdS black holes in Boyer-Lindquist coordinates.
• Identifies unstable circular photon orbits to trace photon rings and determine the shadow boundary.
• Analytically computes the shadow radius $R_s$ and distortion parameter $\delta_s$ using the formalism of Grenzebach et al. (2014) for finite observer positions.
• Uses the limiting geometric cross-section $\sigma_{\text{lim}} \approx \pi R_s^2$ to estimate the energy emission rate of the black hole.
• Constructs the temperature of the nonsingular-AdS black hole as a correction to the Kerr-AdS temperature, incorporating the deviation parameter $k$.
• Performs numerical analysis and plots shadow contours, $R_s$, $\delta_s$, and energy emission rate across parameter space ($a$, $k$, $1/l^2$).

Experimental results

Research questions

• RQ1 How does the presence of a deviation parameter $k$ affect the size and shape of the black hole shadow in rotating regular AdS spacetime?
• RQ2 How does the cosmological constant (via $1/l^2$) influence the shadow radius $R_s$ and distortion parameter $\delta_s$ compared to Kerr and regular AdS black holes?
• RQ3 How does the energy emission rate from the photon region vary with $1/l^2$ and other black hole parameters?
• RQ4 What is the relative size and distortion of the shadow compared to the Kerr and regular AdS black hole shadows for the same spin parameter $a$?
• RQ5 How do the observables $R_s$ and $\delta_s$ scale with changes in $a$, $k$, and $1/l^2$?

Key findings

• The shadow radius $R_s$ increases with increasing $1/l^2$, while the distortion parameter $\delta_s$ also increases, indicating a more elongated shadow for larger curvature radius.
• For fixed $a$ and $k$, the shadow of rotating regular AdS black holes is smaller and more distorted than in the Kerr or regular AdS spacetime.
• The shadow radius $R_s$ decreases with increasing deviation parameter $k$, indicating that stronger regularity effects shrink the shadow.
• The energy emission rate peaks at a finite frequency and decreases with increasing $1/l^2$, suggesting reduced high-energy emission for larger AdS curvature.
• Numerical results in Table 1 show that for $a = 0.6M$, $k = 0.2M$, $R_s/M$ increases from 0.89 to 1.28 as $1/l^2$ increases from 0.05 to 0.20, while $\delta_s$ increases from 0.11 to 0.27.
• The shadow becomes more distorted and larger in size compared to the Kerr black hole, especially as $1/l^2$ increases, indicating a significant deviation from Kerr-like morphology.