Photon Spheres and shadow of modified black-hole entropies

This paper establishes a correspondence between corrected black hole entropy and the metric function to calculate resulting photon sphere and shadow characteristics, which are then constrained by Event Horizon Telescope observations of Sgr A* to test deviations from the Bekenstein-Hawking area law.

The Gist

Imagine a black hole not just as a cosmic vacuum cleaner, but as a giant, invisible whirlpool in the fabric of space. Around this whirlpool, there is a very specific "no-fly zone" for light. If a photon (a particle of light) gets too close, it doesn't fall in immediately; instead, it gets trapped in a tight, unstable circle, like a satellite orbiting a planet but with no engine to keep it stable. This ring of trapped light is called the photon sphere.

If you were to take a picture of this black hole from far away, you wouldn't see the black hole itself (since it's black). Instead, you'd see a dark circle in the middle, surrounded by a glowing ring of light. This dark circle is called the shadow. The size of this shadow depends entirely on the size of that "no-fly zone" (the photon sphere).

The Big Question
For decades, scientists have used a standard rule (the Bekenstein-Hawking law) to calculate how much "disorder" or entropy a black hole has. They assumed this entropy is directly proportional to the black hole's surface area, like the amount of paint needed to cover a ball.

However, modern physics suggests that at the tiniest scales (quantum gravity), this rule might be slightly wrong. The surface of the black hole might be "fractal" or "rough" rather than perfectly smooth, or the rules of statistics might be different. This means the entropy could be "corrected" by adding some extra mathematical terms.

The Experiment
The authors of this paper asked: If we change the rules for how we calculate a black hole's entropy, how does that change the shape of space around it, and does that change the size of the shadow we see?

They didn't just guess; they built a bridge between two worlds:

1. Thermodynamics: The rules of heat and entropy.
2. Geometry: The shape of space and time (gravity).

They started with the "First Law of Thermodynamics" (a fundamental rule about energy) and asked, "If the entropy is corrected, what must the shape of space look like to make the math work?" They found that different types of "entropy corrections" create different shapes of space, which in turn change the size of the photon sphere and the black hole's shadow.

The Three "Flavors" of Correction
The paper tested three different theories about how entropy might be corrected, treating them like three different recipes for a cake:

1. The "Rough Surface" Recipe (Barrow Entropy):

   • The Idea: Imagine the black hole's surface isn't smooth like a marble, but rough like a piece of coral.
   • The Result: As the "roughness" increases, the photon sphere gets smaller, but the shadow gets larger. It's like the light gets squeezed into a tighter circle, but the dark hole behind it appears bigger.

2. The "Statistical Shift" Recipe (Rényi Entropy):

   • The Idea: This changes how we count the possibilities of the black hole's internal states, similar to how a crowd behaves differently than a single person.
   • The Result: This does the opposite of the rough surface. As the correction gets stronger, the photon sphere gets larger, and the shadow gets smaller.

3. The "Hybrid" Recipe (Sharma-Mittal Entropy):

   • The Idea: This is a mix of the previous two ideas, with two knobs you can turn.
   • The Result: Depending on which knob you turn, you can get results that look like the "Rough Surface" or the "Statistical Shift." One knob makes the shadow bigger, the other makes it smaller.

Checking Against Reality
The authors didn't just do math on paper; they compared their results to real-world data. In 2019 and 2024, the Event Horizon Telescope (EHT) took actual pictures of the black hole at the center of our galaxy, Sagittarius A*. They measured the size of the shadow very precisely.

The team used these real measurements as a ruler. They asked: "How much 'roughness' or 'statistical shift' can we add to our black hole models before the predicted shadow size no longer matches the EHT photo?"

The Conclusion
The paper found that:

• Different entropy corrections predict different shadow sizes.
• The EHT observations act as a strict filter. They allow only very small amounts of these "corrections."
• If the corrections were too large, the black hole's shadow would look different than what we actually see.

In short, by looking at the size of a black hole's shadow, we can test the fundamental laws of physics. The paper shows that while the universe might have "rough edges" or "weird statistics" at the quantum level, they must be very subtle, otherwise, the black hole's shadow would look wrong compared to our telescopes.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of how corrections to the Bekenstein-Hawking entropy-area law (arising from quantum gravity effects like Loop Quantum Gravity, String Theory, or non-extensive statistics) manifest in the macroscopic spacetime geometry of black holes. Specifically, the authors investigate how different generalized entropy models (Barrow, Rényi, and Sharma-Mittal) alter the photon sphere radius and the black hole shadow size. With the advent of Event Horizon Telescope (EHT) observations of Sagittarius A* (Sgr A*), there is an urgent need to constrain the parameters of these modified entropy models using observational data to test deviations from General Relativity.

2. Methodology

The authors employ a thermodynamic-geometric correspondence approach to derive modified spacetime metrics from corrected entropy functions.

• Thermodynamic Framework:

  • They start with the First Law of Black Hole Thermodynamics: $dM = T dS$.
  • Constraints: The derivation assumes fixed black hole mass ($M$) and a fixed event horizon position ($r_+ = 2M$ for Schwarzschild).
  • Entropy-Geometry Correspondence: Instead of solving Einstein's equations with a specific matter source, they reverse the process. They assume the corrected entropy $S$ induces a backreaction on the metric. They derive the modified metric function $f(r, \alpha, \beta, \dots)$ by enforcing that the Hawking temperature derived from the metric ($T = f'(r_+)/4\pi$) satisfies the thermodynamic relation $T = (\partial M / \partial S)$.
  • The modified metric is constructed as $f(r)_{mod} = \frac{\partial S_{BH}}{\partial S} f(r)_{Schwarzschild}$, ensuring it reduces to the standard Schwarzschild metric when correction parameters vanish.

• Optical Analysis:

  • Using the derived static, spherically symmetric metrics, they analyze the motion of massless photons via the Lagrangian formalism.
  • They calculate the effective potential $V_{eff}(r)$ for photon propagation.
  • The photon sphere radius ($r_{ph}$) is determined by the condition for unstable circular null geodesics: $V'_{eff}(r_{ph}) = 0$.
  • The critical impact parameter ($b_{ph}$), which defines the angular size of the black hole shadow, is calculated as $b_{ph} = r_{ph} / \sqrt{f(r_{ph})}$.

• Observational Constraints:

  • Theoretical results for $b_{ph}$ are compared against the EHT observational data for Sgr A* (specifically the range $4.55 \le b_{ph} \le 5.22$ at $1\sigma$ and $4.21 \le b_{ph} \le 5.56$ at $2\sigma$).
  • This comparison is used to constrain the free parameters in the entropy correction models.

3. Key Contributions

• Systematic Metric Derivation: The paper provides a unified framework to derive the spacetime metric directly from various non-extensive entropy corrections (Barrow, Rényi, Sharma-Mittal) while maintaining fixed horizon and mass constraints.
• Differential Optical Signatures: The study reveals that different entropy corrections lead to distinct and sometimes opposite optical behaviors. This challenges the assumption that all quantum corrections behave similarly regarding photon spheres.
• Parameter Constraints: The authors provide specific numerical constraints on the correction parameters ($\Delta$, $\lambda$, $\alpha$, $\beta$) based on current EHT data, offering a pathway to test these theoretical models observationally.

4. Detailed Results

A. Barrow Entropy (Fractal Horizon)

• Correction: Introduces a fractal dimension parameter $\Delta$ ($0 \le \Delta \le 1$).
• Effect on Geometry: As $\Delta$ increases:
  • The photon sphere radius ($r_{ph}$) decreases.
  • The critical impact parameter ($b_{ph}$) increases.
  • The effective potential maximum decreases.
• Constraints: Based on EHT data, the parameter is constrained to $0 \le \Delta \le 0.004$ ($1\sigma$) and $0 \le \Delta \le 0.062$ ($2\sigma$).

B. Rényi Entropy (Non-extensive)

• Correction: Introduces a parameter $\lambda$ (related to $\chi = 12\pi\lambda M^2$).
• Effect on Geometry: As $\lambda$ (or $\chi$) increases:
  • The photon sphere radius ($r_{ph}$) increases.
  • The critical impact parameter ($b_{ph}$) decreases.
  • The effective potential maximum decreases.
• Contrast: This behavior is exactly opposite to the Barrow entropy case regarding the photon sphere radius and impact parameter.
• Constraints: Constrained to $0 \le \chi \le 0.37$ ($1\sigma$) and $0 \le \chi \le 0.597$ ($2\sigma$).

C. Sharma-Mittal Entropy (Generalized)

• Correction: Introduces two parameters, $\alpha$ and $\beta$ (dimensionless forms $x$ and $y$).
• Complex Behavior: The optical properties depend on the interplay between $x$ and $y$:
  • Fixed $x$, varying $y$: Similar to Rényi entropy. As $y$ increases, $r_{ph}$ increases and $b_{ph}$ decreases.
  • Fixed $y$, varying $x$: Similar to Barrow entropy. As $x$ increases, $r_{ph}$ decreases and $b_{ph}$ increases.
• Constraints: The allowed ranges for $x$ and $y$ are interdependent. For example, with $x=0.3$, $y$ is constrained to $0 \le y \le 0.365$ ($1\sigma$). As $x$ increases, the permissible range for $y$ shifts and expands.

5. Significance and Conclusion

• Testing Quantum Gravity: The study demonstrates that black hole shadows are sensitive probes of the microscopic structure of spacetime. Different entropy corrections (representing different quantum gravity hypotheses) produce unique "optical fingerprints."
• Opposite Trends: A crucial finding is that not all corrections lead to larger photon spheres (a common conjecture for some quantum-corrected black holes). Barrow entropy reduces the sphere size, while Rényi and specific Sharma-Mittal regimes increase it. This highlights the necessity of distinguishing between specific entropy models rather than treating "quantum corrections" as a monolithic effect.
• Future Outlook: The paper suggests that as VLBI (Very Long Baseline Interferometry) resolution improves, these constraints will tighten, potentially ruling out specific entropy models or confirming the presence of non-extensive thermodynamic effects in the strong-field regime of black holes. The methodology can be extended to rotating (Kerr) black holes and multi-parameter scenarios.

In summary, the paper establishes a rigorous link between corrected thermodynamic entropy and observable optical phenomena, using EHT data to place the first quantitative bounds on the parameters of Barrow, Rényi, and Sharma-Mittal entropy models.