Distinguish Bardeen-like black holes by Gravitational lensing

This paper investigates gravitational lensing by Bardeen-like regular black holes without Cauchy horizons, demonstrating that while weak-field predictions align with current observations of ESO 325-G004, strong-field measurements of relativistic images around Sgr A* and M87*—specifically the angular separation, flux ratio, and time delays—could distinguish these models from Schwarzschild black holes through future observations.

The Gist

The Big Idea: Fixing the "Glitch" in Black Holes

Imagine a black hole as a cosmic vacuum cleaner. According to standard physics (Einstein's General Relativity), if you suck enough matter into it, it eventually collapses into a "singularity"—a point of infinite density where the laws of physics break down. It's like a computer program crashing because it tried to divide by zero.

Physicists have proposed "Regular Black Holes" to fix this crash. These are black holes that don't have a singularity; instead, they have a smooth, dense core, like a marble instead of a mathematical point. One famous version is the Bardeen black hole. However, the original Bardeen model had a weird side effect: it created a "Cauchy horizon," which is like a glitch in the matrix that could theoretically allow time travel or make the universe unpredictable.

The New Hero: This paper studies a new, improved version called the "Bardeen-like" black hole. It fixes the singularity without creating the time-travel glitch. The big question the authors ask is: Can we tell the difference between this new "smooth" black hole and the old "crashing" Schwarzschild black hole just by looking at them?

The Detective Tool: Gravitational Lensing

Since we can't see black holes directly (they are black!), we have to look at how they bend light. This is called Gravitational Lensing.

The Analogy: Imagine holding a wine glass up to a streetlamp. The glass bends the light, creating a ring of light around the stem.

• The Lens: The black hole.
• The Light: Stars or galaxies behind it.
• The Ring: The "Einstein Ring" (a circle of light we see).

The authors used this "cosmic magnifying glass" to test their theory in two different zones: The Weak Field (far away) and The Strong Field (right next to the black hole).

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Part 1: The Weak Field (The "Far Away" Test)

The Scenario: Imagine a black hole is far away from us, like a lighthouse on a distant island. The light passing near it gets bent, but not too much.

The Discovery:
The authors found that the new "Bardeen-like" black hole bends light slightly more than the standard black hole.

• The Analogy: Think of the black hole's gravity as a trampoline. A standard black hole makes a deep dip. The new "Bardeen-like" black hole makes a dip that is just a tiny bit deeper because of its smooth core (represented by a parameter called ℓ).
• The Result: Because the dip is deeper, the ring of light (Einstein Ring) we see is slightly larger.
• Real-World Check: They tested this against a real galaxy called ESO 325-G004. The size of the ring they calculated for the new black hole matched what telescopes actually saw. This means the new theory is "safe"—it doesn't contradict what we already know.

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Part 2: The Strong Field (The "Extreme Close-Up" Test)

The Scenario: Now, imagine getting right up close to the black hole, near the "event horizon" (the point of no return). Here, gravity is so strong that light can actually loop around the black hole multiple times before escaping, like a car doing donuts on a frozen lake.

The Discovery:
This is where things get tricky. The authors looked at two famous black holes: Sgr A* (at the center of our Milky Way) and M87* (the giant one in the M87 galaxy).

1. The Shadow Size (The "Silhouette"):

   • Surprise: The size of the black hole's "shadow" (the dark circle in the middle) is exactly the same for both the new black hole and the old one.
   • Analogy: If you looked at the silhouette of a standard black hole and a "Bardeen-like" black hole from far away, they would look identical. You can't tell them apart just by the size of the hole.

2. The "Rings" Around the Shadow:

   • While the shadow size is the same, the rings of light just outside the shadow behave differently.
   • The Gap: In the new black hole, the gap between the first ring of light and the next one gets wider as the core gets "smoother" (as ℓ increases).
   • The Brightness: The first ring becomes slightly dimmer compared to the rest of the rings.

The Time Travel Test:
The authors also calculated how long it takes for light to make these loops.

• The Result: Light takes a tiny bit longer to loop around the "Bardeen-like" black hole than the standard one.
• The Catch: This time difference is incredibly small—like the difference between a blink of an eye and a blink of an eye plus a nanosecond. Current telescopes aren't fast enough to measure this yet, but future ones might be.

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The Verdict: Can We Spot the Difference?

The Short Answer: Not yet, but maybe soon.

• Current Tech: The Event Horizon Telescope (EHT), which took the famous pictures of black holes, is amazing. But it can only measure the size of the shadow. Since the shadow size is the same for both types of black holes, the EHT can't tell them apart right now.
• The Future: To see the difference, we need to see the fine details (the rings of light) with much higher resolution.
  • Imagine trying to read the text on a coin from a mile away. The EHT can tell you "there is a coin there." Future telescopes (like the next-gen EHT) might be able to read the "text" on the coin (the rings and time delays).

Why Does This Matter?

If we can eventually measure these tiny differences, we can prove that black holes don't have "singularities" (infinite points) and that the laws of physics hold up even in the most extreme places in the universe. It would be like finding out that the "glitch" in the universe's code was just a bug that has finally been patched.

Summary in a Nutshell:
The authors built a new, smoother model of a black hole. They checked if it bends light differently than the old model.

1. Far away: It bends light a tiny bit more (making rings slightly bigger).
2. Close up: The shadow looks the same, but the rings of light around it are spaced differently.
3. Conclusion: Our current telescopes can't see the difference yet, but future super-powerful telescopes might be able to prove that black holes are "smooth" rather than "broken."

Technical Summary

1. Problem Statement

Regular black holes (RBHs) are theoretical solutions designed to resolve the central curvature singularity found in standard General Relativity (e.g., Schwarzschild black holes). However, many well-known RBH models, such as the original Bardeen and Hayward black holes, introduce Cauchy horizons (inner horizons), which lead to issues with determinism and stability (the strong cosmic censorship conjecture).

Recent work by Calzà et al. (2025) proposed a "Bardeen-like" regular black hole that resolves the singularity without generating a Cauchy horizon. The central problem addressed in this paper is: Can the distinct geometry of this specific Bardeen-like black hole be observationally distinguished from a standard Schwarzschild black hole using modern or future gravitational lensing data?

2. Methodology

The authors analyze the gravitational lensing signatures of the Bardeen-like black hole in two distinct regimes: the Weak-Field Limit and the Strong-Field Limit (SDL).

• Theoretical Model:
  The spacetime metric is defined by a mass function $M(\rho)$ and an areal radius $r(\rho)$ dependent on a regularization parameter $\ell$ (the core scale) and ADM mass $m$:
  $$M(\rho) = \frac{m\rho^3}{(\rho^2 + \ell^2)^{3/2}}, \quad r(\rho) = \frac{(\rho^2 + \ell^2)^{3/2}}{\rho^2}$$
  The condition $\ell < \frac{4m}{3\sqrt{3}}$ ensures the existence of an event horizon without a Cauchy horizon. Setting $\ell=0$ recovers the Schwarzschild metric.

• Weak-Field Analysis:

  • The deflection angle $\hat{\alpha}$ is calculated via an integral over the null geodesics.
  • An expansion in powers of $1/r_0$ (closest approach distance) is performed to derive the deflection angle as a function of the impact parameter $b$.
  • The Einstein ring radius ($\theta_E$) is derived for a source-lens-observer alignment and compared against observational data from the galaxy ESO 325-G004.

• Strong-Field Analysis (SDL):

  • The authors employ the Strong Deflection Limit (SDL) formalism (Bozza et al.) to study photons orbiting near the photon sphere.
  • Key observables are computed:
    1. Asymptotic position ($\theta_\infty$): The limiting angular position of the infinite sequence of relativistic images.
    2. Angular separation ($s$): The gap between the first relativistic image and the rest.
    3. Relative flux ratio ($r_{mag}$): The brightness ratio between the first image and the sum of all subsequent images.
    4. Time delay ($\Delta T_{2,1}$): The time difference between the arrival of the first and second relativistic images.
  • These observables are calculated numerically for the supermassive black holes Sgr A* and M87* using their known masses and distances.

3. Key Contributions and Results

A. Weak-Field Regime

• Deflection Angle Correction: The deflection angle receives a positive $\ell$-dependent correction term ($\propto +\ell^2/b^2$).
  • Contrast: Standard Bardeen black holes typically exhibit a negative correction (reducing deflection). The Bardeen-like model enhances deflection, effectively mimicking a slight increase in effective mass at large distances.
• Einstein Ring: The theoretical Einstein ring radius is slightly larger than the Schwarzschild prediction.
• Observational Consistency: Using data from ESO 325-G004, the predicted ring radius for the Bardeen-like model falls well within the $1\sigma$ observational error bars ($2.85 \pm 0.40$ arcsec). Current data cannot rule out the model, but higher resolution could constrain $\ell$.

B. Strong-Field Regime (Sgr A and M87)**

• Photon Sphere & Shadow:
  • The critical impact parameter ($u_{ps}$) and the photon sphere radius remain identical to the Schwarzschild value ($r_{ps} = 3m$) regardless of $\ell$.
  • Consequently, the asymptotic position of the relativistic images, $\theta_\infty$, is independent of $\ell$.
  • Significance: The black hole shadow size predicted by this model is indistinguishable from Schwarzschild based solely on shadow radius, but differs from standard Bardeen models (where $\theta_\infty$ decreases with $\ell$).
• SDL Coefficients:
  • The coefficient $\bar{a}$ (governing the divergence rate) increases with $\ell$.
  • The coefficient $\bar{b}$ (constant term) decreases with $\ell$.
• Observables ($s$ and $r_{mag}$):
  • Angular Separation ($s$): Increases with $\ell$. For Sgr A*, $s$ ranges from ~33 to 65 nanoarcseconds (nas); for M87*, ~25 to 49 nas.
  • Relative Flux ($r_{mag}$): Decreases with $\ell$, ranging from ~6.82 (Schwarzschild) down to ~5.91 for high $\ell$.
• Time Delays: The time delay $\Delta T_{2,1}$ between the first two relativistic images increases with $\ell$. However, the deviation from the Schwarzschild value is minute compared to the total delay.

4. Significance and Future Outlook

• Distinguishing Feature: The primary method to distinguish this Bardeen-like black hole from a Schwarzschild black hole is not the shadow size ($\theta_\infty$), but rather the angular separation ($s$) and flux ratio ($r_{mag}$) of the relativistic images.
• Observational Requirements:
  • Current Event Horizon Telescope (EHT) resolution is insufficient to resolve the separation $s$ (which is in the nanoarcsecond range).
  • Future facilities, such as the next-generation EHT (ngEHT), aiming for resolutions of $\sim 10$ nas, could potentially resolve the first relativistic image from the inner photon ring.
  • Measuring $s$ and $r_{mag}$ would allow constraints on the regularization parameter $\ell$.
• Theoretical Implications: Since regular black holes can arise as vacuum solutions in generally covariant gravity theories, detecting these signatures would provide a direct test of alternative gravity theories in the strong-field regime.
• Microlensing: The authors note that recent work suggests microlensing of regular black holes yields higher magnification, offering a potentially more viable near-future target for detection than direct imaging of relativistic images.

Conclusion

The paper establishes that Bardeen-like regular black holes (free of Cauchy horizons) produce unique gravitational lensing signatures distinct from Schwarzschild black holes. While the shadow size remains unchanged, the enhanced deflection in the weak field and the increased angular separation/decreased flux ratio in the strong field provide specific, testable predictions. Current observations of ESO 325-G004, Sgr A*, and M87* are consistent with this model, but definitive discrimination requires future high-resolution interferometry capable of resolving nanoarcsecond-scale features.