Gravitational lensing inside and outside of a marginally unstable photon sphere in a general, static, spherically symmetric, and asymptotically-flat spacetime in strong deflection limits

This paper extends strong-deflection-limit analysis methods to investigate gravitational lensing by rays inside and outside marginally unstable photon spheres in general static, spherically symmetric, asymptotically-flat spacetimes, applying the framework to Reissner-Nordström and Hayward models to resolve previous discrepancies in power-divergent deflection angle calculations.

The Gist

Imagine you are standing in a vast, dark forest, trying to see a lighthouse in the distance. Usually, light travels in straight lines. But if there were a giant, invisible whirlpool in the middle of the forest, it would bend the light around it, creating strange, swirling images of the lighthouse.

In the universe, Black Holes are those whirlpools. They are so heavy that they bend space and time itself. Around them exists a special zone called a Photon Sphere. Think of this as a "traffic circle" for light. If a photon (a particle of light) enters this circle, it can get stuck orbiting the black hole, going round and round like a car on a racetrack that never leaves the track.

The Problem: The "Perfectly Unstable" Track

In most black holes, this traffic circle is unstable. If a car (light) gets slightly too close to the inside edge, it falls into the black hole. If it gets slightly too far on the outside, it escapes. This is like a ball balanced on the very top of a hill; a tiny nudge sends it rolling down one side or the other.

However, in some exotic theories of the universe, there might be a special kind of black hole (or a "black hole mimic") where this traffic circle is marginally unstable. Imagine a track that is so perfectly flat at the top that the light doesn't just fall in or fly out immediately; it lingers there in a very specific, tricky way.

The problem is that our old math tools for predicting how light bends (called "Strong Deflection Limits") break down when the track is this specific. It's like trying to use a ruler to measure the thickness of a hair; the tool isn't sensitive enough, and the math gives you the wrong answer.

The Solution: A New Mathematical Lens

The author of this paper, Naoki Tsukamoto, is like a master mechanic who has built a new, ultra-precise tool to measure these tricky light paths.

1. The Old Method: Previous scientists tried to calculate how light bends near these special tracks using a formula that worked well for normal black holes but failed for these "marginally unstable" ones. They got the numbers wrong, like guessing the weight of a feather and getting it off by a few pounds.
2. The New Method: Tsukamoto took an existing method (developed by Eiroa, Romero, and Torres) and upgraded it. He applied it to two specific theoretical models:
   • Reissner-Nordström: A black hole that has an electric charge (like a giant static electricity ball).
   • Hayward: A "regular" black hole that doesn't have a singularity (a point of infinite density) at its center, making it a candidate for what a "real" black hole might look like without breaking physics.

What Did He Find?

By using his new, more precise math, Tsukamoto discovered:

• The Old Math Was Wrong: The previous calculations for how much light bends just outside this special track were incorrect. The new numbers are different and, more importantly, they actually match the real physics when you check them against the full, complex equations.
• Inside vs. Outside: Light behaves differently depending on whether it passes just inside or just outside this special track. The new method correctly calculates the bending for both sides.
• The "Shadow" Clue: When the Event Horizon Telescope (the camera that took the first picture of a black hole) looks at a black hole, it sees a dark shadow surrounded by a ring of light. This ring is made of light that has been bent by the photon sphere.
  • If the black hole is "normal," the ring looks one way.
  • If it's one of these exotic "marginally unstable" types, the ring would look slightly different (perhaps a different size or brightness).

Why Does This Matter?

We are currently trying to figure out if the objects we see in space are truly black holes or if they are "mimics"—strange, exotic objects that look like black holes but aren't.

Think of it like a police lineup. We have a suspect (the black hole), but there are many look-alikes (exotic objects).

• The Old Math was like a blurry photo; it couldn't tell the difference between the real suspect and the look-alikes.
• This New Paper provides a high-definition photo. It gives astronomers the precise numbers they need to say, "If the light bends this much, it's a real black hole. If it bends that much, it's a fake."

The Bottom Line

This paper is a technical manual for a better telescope. It doesn't just say "light bends"; it gives the exact recipe for how light bends around the most tricky, unstable cosmic traps. By fixing the math, it helps us prepare for future space observations that might finally prove whether the mysterious objects at the center of our galaxy are truly black holes or something even stranger.

Technical Summary

1. Problem Statement

The paper addresses a specific gap in the theory of gravitational lensing within strong gravitational fields. While the behavior of light rays near standard photon spheres (unstable circular orbits) and antiphoton spheres (stable circular orbits) is well-understood, the case of a marginally unstable photon sphere presents unique mathematical challenges.

• The Issue: In standard strong deflection limits (where the impact parameter $b \to b_m$), the deflection angle $\alpha$ typically diverges logarithmically ($\alpha \sim -\bar{a} \ln|b/b_m - 1|$). However, when a photon sphere and an antiphoton sphere degenerate into a single marginally unstable photon sphere, the deflection angle diverges as a power law ($\alpha \sim |b/b_m - 1|^{-1/6}$).
• Previous Limitations: Existing methods (such as Bozza's semi-analytic approach or the author's previous semi-analytic work [Ref 133]) failed to correctly calculate the coefficients for this power-law divergence. Specifically, previous calculations yielded incorrect coefficients ($\bar{c}_+$) for rays deflected just outside the marginally unstable sphere, leading to non-convergence with exact numerical solutions.
• Context: This is crucial for interpreting data from the Event Horizon Telescope (EHT) and future space missions, as rays deflected near these spheres contribute to the "photon ring" structure of black hole shadows. Distinguishing between standard black holes and "black hole mimickers" (like naked singularities or regular black holes) requires precise modeling of these strong-field effects.

2. Methodology

The author extends the numerical method originally proposed by Eiroa, Romero, and Torres (ERT) [Ref 117] to handle the specific case of marginally unstable photon spheres in general static, spherically symmetric, and asymptotically-flat spacetimes.

• Metric Framework: The study considers a general line element:
  $$ds^2 = -A(r)dt^2 + B(r)dr^2 + C(r)(d\vartheta^2 + \sin^2\vartheta d\varphi^2)$$
• Conditions for Marginal Instability: The photon sphere radius $r_m$ is defined where the effective potential $V(r)$ has a triple root. This requires:
  1. $D_m = 0$ (Circular orbit condition)
  2. $D'_m = 0$ (Marginal stability condition)
  3. $D''_m > 0$ (Ensuring the specific nature of the instability)
     Where $D(r) = C'/C - A'/A$.
• Deflection Angle Formulation: Instead of the logarithmic divergence, the deflection angle $\alpha$ in the strong deflection limit ($b \to b_m \pm 0$) is modeled as:
  $$\alpha(b) = \frac{\bar{c}_{\pm}}{|b/b_m - 1|^{1/6}} + \bar{d}_{\pm}$$
  Here, $\bar{c}_{\pm}$ and $\bar{d}_{\pm}$ are coefficients to be determined for rays just outside ($+$) and just inside ($-$) the sphere.
• Numerical Extraction: The author derives formulas to extract $\bar{c}_{\pm}$ and $\bar{d}_{\pm}$ numerically by taking limits of the exact deflection angle integral and its derivative as the closest approach distance $r_0 \to r_m$.
  $$\bar{c}_{\pm} = \lim_{r_0 \to r_m \pm 0} \left[ \mp 2 |r_0 - r_m|^{3/2} \frac{d\alpha}{dr_0} \right]$$
  $$\bar{d}_{\pm} = \lim_{r_0 \to r_m \pm 0} \left[ \alpha \pm 2 |r_0 - r_m| \frac{d\alpha}{dr_0} \right]$$
• Application: The method is applied to two specific spacetimes:
  1. Reissner-Nordström (RN) Naked Singularity: With specific charge parameters creating a marginally unstable sphere.
  2. Hayward Spacetime: A regular black hole model with a marginally unstable sphere.

3. Key Contributions

• Extension of ERT Method: Successfully generalized the Eiroa, Romero, and Torres numerical method to handle the power-law divergence characteristic of marginally unstable photon spheres, covering both the interior ($b < b_m$) and exterior ($b > b_m$) regions.
• Correction of Previous Errors: The paper identifies and corrects errors in the author's previous semi-analytic calculations (Ref 133) and confirms discrepancies found by Sasaki [Ref 134]. The previous semi-analytic method yielded an incorrect coefficient $\bar{c}_+$ for the RN spacetime.
• Validation: The new numerical results are shown to converge correctly to the exact deflection angle (calculated without approximation) in the strong deflection limit, whereas the previous semi-analytic results did not.
• Comprehensive Observable Calculations: The paper calculates not just deflection angles, but also observable quantities such as Einstein ring diameters, angular separations, and magnifications for both interior and exterior rays.

4. Key Results

The study provides precise numerical values for the coefficients $\bar{c}_{\pm}$ and $\bar{d}_{\pm}$ for the RN and Hayward spacetimes.

A. Reissner-Nordström Spacetime:

• Exterior ($+$): $\bar{c}_+ \approx 6.67748$, $\bar{d}_+ \approx -5.59011$.
  • Correction: This corrects the previous semi-analytic value of $\bar{c}_+ \approx 5.49892$.
• **Interior ($-$):**$\bar{c}_- \approx 11.5658$, $\bar{d}_- \approx -5.64071$.
• Comparison: The results align closely with the analytical findings of Sasaki [Ref 134], confirming the validity of the numerical approach.

B. Hayward Spacetime:

• Exterior ($+$): $\bar{c}_+ \approx 6.01324$, $\bar{d}_+ \approx -5.60958$.
  • Comparison: Matches Chiba and Kimura [Ref 136] for $\bar{c}_+$ but provides the previously missing $\bar{d}_+$.
• **Interior ($-$):**$\bar{c}_- \approx 10.4154$, $\bar{d}_- \approx -5.67374$.
• Correction: The previous semi-analytic value of $\bar{c}_+ \approx 4.95196$ is shown to be invalid.

C. Observables:

• The paper calculates the angular diameter of Einstein rings ($2\theta_{E1\pm}$) and the separation from the shadow edge ($\bar{s}_{\pm}$).
• RN Spacetime: The exterior ring diameter is $\sim 37.6 \, \mu\text{as}$, while the interior ring is significantly smaller ($\sim 6.1 \, \mu\text{as}$).
• Hayward Spacetime: The exterior ring is $\sim 47.0 \, \mu\text{as}$, and the interior ring is $\sim 26.0 \, \mu\text{as}$.
• Magnification: The magnification of the first-order image ($n=1$) is significantly higher than higher-order images, making them potentially detectable.

5. Significance

• Distinguishing Compact Objects: The study provides a robust framework to distinguish between standard black holes and exotic compact objects (like naked singularities or regular black holes) that possess marginally unstable photon spheres. The specific power-law divergence and the resulting ring sizes/magnifications serve as unique signatures.
• Future Observations: The calculated angular scales (e.g., ring diameters of $\sim 34\text{--}47 \, \mu\text{as}$ for RN and Hayward cases) are within the potential resolution range of near-future space-based interferometers (targeting $< 10 \, \mu\text{as}$).
• Methodological Robustness: By demonstrating that the extended ERT method yields results consistent with exact numerical integration, the paper establishes a reliable tool for analyzing gravitational lensing in complex spacetimes where semi-analytic approximations fail.
• Interior vs. Exterior: A critical insight is the treatment of rays inside the marginally unstable sphere. The paper shows these rays can produce distinct, potentially brighter images compared to exterior rays, a feature often overlooked in previous studies.

In conclusion, this paper resolves inconsistencies in the theoretical modeling of gravitational lensing near marginally unstable photon spheres, providing corrected coefficients and a validated numerical method essential for the interpretation of high-resolution black hole imaging data.