⚛️ general relativity

Kerr-Degenerate Shadows and Distinct Strong-Deflection Lensing in Rotating Hayward-like and Bardeen-like Geometries

This paper demonstrates that while rotating Hayward-like and Bardeen-like regular black hole geometries share identical Kerr-like shadow boundaries, they exhibit distinct strong-deflection lensing signatures dependent on their specific areal-radius profiles, offering a potential observational pathway to distinguish them from standard Kerr black holes.

Original authors: Chen-Hung Hsiao, Limei Yuan, Yidun Wan

Published 2026-09-23

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Chen-Hung Hsiao, Limei Yuan, Yidun Wan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the deep universe, gravity is not just a force that pulls things together; it is a warping of space and time itself. When matter is crushed into an incredibly small, dense point, this warping becomes so extreme that it creates a black hole. For decades, the standard description of these objects has been the Kerr solution, a mathematical model that predicts a black hole is defined only by its mass and how fast it spins. However, this model has a troubling flaw: at its very center, the math breaks down, predicting a point of infinite density and a ring of infinite curvature where the laws of physics cease to work. Many physicists believe this "singularity" is a sign that the model is incomplete, suggesting that a more realistic description of nature would smooth out this center, replacing the infinite point with a dense, regular core. To test this idea, scientists look at how light bends around these objects. As light from distant stars passes near a black hole, the intense gravity acts like a lens, distorting the image and creating a dark silhouette known as a shadow. By studying the shape of this shadow and the precise paths light takes as it loops around the object, astronomers hope to see if the real universe matches the standard model or if it reveals the smooth, regular cores predicted by newer theories.

A team of researchers at Fudan University in Shanghai recently set out to compare two specific theories that propose such smooth cores, known as the Hayward-like and Bardeen-like geometries. They wanted to see if these two different ideas could be told apart from each other and from the standard black hole model by looking at the light they bend. The scientists built a mathematical framework that allowed them to rotate these static, non-spinning models, creating spinning versions that could be compared directly. They found that, remarkably, if you look only at the overall shape and size of the black hole's shadow, these new models are indistinguishable from the standard one. Whether the black hole has a smooth, regular center or the traditional infinite ring, the dark silhouette it casts against the background of the universe looks exactly the same to a distant observer, provided the mass, spin, and viewing angle are identical. This means that simply taking a picture of the shadow, like the famous images of the black holes in M87 and our own galaxy, cannot tell us if the center is smooth or singular.

However, the story changes when the researchers looked deeper, past the shadow's edge, to the fine details of how light travels. While the shadow's boundary is identical, the journey light takes to get there is not. The team calculated what happens to light rays that skim very close to the black hole, looping around it multiple times before escaping to reach an observer. They found that the time it takes for these light rays to complete their loops, and the exact spacing between the multiple images they create, depends heavily on the specific structure of the object's core. For the two smooth-core models they studied, these details differ from each other and from the standard model. Specifically, the time delay between the first and second loop of light, and the brightness ratio between these images, would be measurably different if the black hole had a smooth core rather than a singular one.

The researchers applied these calculations to the two black holes we know best: the supermassive black hole at the center of the galaxy M87, and the one at the center of our own Milky Way, known as Sagittarius A*. They determined that current observations, which measure the overall size of the shadow, are consistent with all three models—the standard one and the two smooth-core versions. The data does not rule out the idea of a smooth center, but it also does not prove it. The study concludes that to actually distinguish between these theories, we need to measure things other than the shadow's size. We would need to detect the tiny differences in the time it takes for light to circle the black hole or the subtle variations in the brightness of the multiple images. While current telescopes are not yet sensitive enough to measure these tiny time delays or brightness differences with the required precision, the study provides a clear roadmap for what future instruments need to measure. If we can eventually measure these specific details with high accuracy, we may finally be able to see whether the heart of a black hole is a place where physics breaks down, or a smooth, dense region where the laws of nature hold firm.

Technical Summary: Kerr-Degenerate Shadows and Distinct Strong-Deflection Lensing in Rotating Hayward-like and Bardeen-like Geometries

Problem Statement
The paper addresses the challenge of distinguishing between rotating regular black hole geometries and the standard Kerr metric using gravitational lensing observables. Specifically, it compares two rotating extensions of static regular geometries: the Hayward-like and Bardeen-like models. While static versions of these metrics have been studied individually, their rotating extensions within a common framework have not been systematically compared to determine which observables are sensitive to the specific regularity length ( $\ell$ ) and which are degenerate with the Kerr solution. The central question is whether the shadows and strong-deflection lensing (SDL) signatures of these models can distinguish them from each other and from Kerr, particularly in parameter domains where the models lack an inner Cauchy horizon.

Methodology
The authors employ a common Kerr-like axisymmetric ansatz (based on the prescription of Azreg-Aïnou) to rotate two distinct static "seed" metrics. The seeds are defined by different areal-radius functions, $R_i(\rho)$ , where $i \in \{H, B\}$ :

Hayward-like: $R_H(\rho) = \rho + 2m\ell^2/\rho^2$
Bardeen-like: $R_B(\rho) = (\rho^2 + \ell^2)^{3/2}/\rho^2$

The study restricts analysis to the physical radial branch where $\rho \ge \rho_{0,i}$ (the minimum of $R_i$ ), ensuring the areal radius is strictly positive and excluding the Kerr ring singularity ( $\Sigma = 0$ ).

Key methodological steps include:

Geometric Analysis: Determining horizon structures, Komar charges, and curvature invariants. The Einstein tensor is treated as an effective stress-energy tensor to test energy conditions.
Shadow Calculation: Solving the null geodesic equations to find the critical impact parameters ( $\xi_c, \eta_c$ ) for unstable spherical photon orbits. The shadow boundary is projected onto an observer's screen.
Strong-Deflection Limit (SDL): Applying the Bozza formalism to the equatorial prograde photon trajectories. The authors calculate the logarithmic divergence coefficients ( $\bar{a}_+, \bar{b}_+$ ), image separations ( $s_+$ ), relative magnitudes ( $r_{mag}^+$ ), and time delays ( $\Delta T_{2,1}^+$ ).
Observational Comparison: Comparing calculated shadow sizes and SDL observables against Event Horizon Telescope (EHT) data for M87* and Sgr A*, using specific mass and distance priors.

Key Contributions and Results

Regularity and Horizon Structure:
- Both rotating geometries are free of scalar curvature singularities on the adopted branch because the minimum areal radius $R_{0,i} > 0$ excludes the Kerr ring locus.
- The models exhibit three distinct parameter regimes based on spin ( $a$ ) and regularity length ( $\ell$ ): two horizons, one outer horizon, or no horizon.
- For $\ell > 0$ , the effective stress tensor violates the Null Energy Condition (NEC) and Weak Energy Condition (WEC) in the equatorial plane, as demonstrated by a negative radial-null contraction.
- The asymptotic Komar mass and angular momentum match the Kerr values ( $m$ and $am$), but finite-radius values differ.
Shadow Degeneracy:
- Main Finding: The ideal black hole shadow boundary is identical to that of the Kerr metric for the same mass ( $m$ ), spin ( $a$ ), and observer inclination ( $\theta_o$ ).
- This degeneracy arises because the critical impact parameters depend only on the areal radius $R_p$ and the metric functions expressed in terms of $R_p$ , which take the standard Kerr form. The specific relation between coordinate radius $\rho$ and areal radius $R$ (determined by $\ell$ ) affects the coordinate location of the photon sphere but not the celestial coordinates $(X, Y)$ of the shadow.
- Consequently, shadow size comparisons with EHT data (M87* and Sgr A*) are consistent with current bounds but cannot constrain the regularity length $\ell$ or distinguish between the Hayward-like, Bardeen-like, and Kerr models.
Strong-Deflection Lensing Distinctions:
- While the shadow boundary is degenerate, SDL observables are partially degenerate and retain sensitivity to the specific geometry.
- Common quantities: The critical impact parameter $u_m^+$ and the limiting image position $\theta_\infty^+$ are identical to Kerr and independent of $\ell$ .
- Distinguishing quantities: The coefficients $\bar{a}_+$ and $\bar{b}_+$ , image separation $s_+$ , relative magnitude $r_{mag}^+$ , and subleading time-delay corrections depend explicitly on $\ell$ and the derivative of the areal radius $R_i'(\rho)$ .
- Quantitative Differences:
  - At fixed spin (e.g., $a/m=0.6$ ), the separation $s_+$ and time delays differ between the models. For M87*, the time delay difference between regular models and Kerr is on the order of hours (~~7.7 h), while the difference between Hayward-like and Bardeen-like is smaller (~~19 min).
  - The relative magnitude $r_{mag}^+$ differs by approximately 0.08 magnitudes between the two regular models, whereas the difference from Kerr is ~0.8 magnitudes.
Observational Consistency:
- The calculated area-equivalent shadow diameters for M87* and Sgr A* fall within the approximate intervals reported by the EHT. However, this consistency is a check of scale rather than a confirmation of the specific metric, as the Kerr prediction yields the same shadow size.

Significance and Claims
The paper claims that within the adopted ansatz and radial branches:

Shadow Boundaries are Insufficient: The black hole shadow is a "Kerr-degenerate" observable for this class of rotating regular geometries. Measuring the shadow size alone cannot test the regularity length $\ell$ or distinguish these models from Kerr.
SDL Observables are Promising: Strong-deflection lensing observables (image separation, magnification, and time delay) retain information about the radial metric profile. These quantities offer a potential pathway to distinguish between Hayward-like and Bardeen-like geometries and to test them against Kerr, provided future measurements achieve sufficient precision (e.g., ~0.01 $\mu$ as for angular separation or ~0.08 mag for photometry).
Limitations: The authors emphasize that these results apply strictly to the specific rotating extension and radial branches chosen. They do not claim these results apply to all rotating regular black holes. Furthermore, distinguishing these models observationally requires realistic source modeling, radiative transfer calculations, and handling of instrumental uncertainties, which are beyond the scope of this geometric analysis. The violation of energy conditions is noted as a feature of the effective geometry description rather than a definitive physical constraint on all regular spacetimes.

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