Shadow dependent phenomenology framework for rotating black hole metric

This paper proposes a thermodynamic-optical duality framework that re-parameterizes black hole mass using the observable shadow radius, allowing for a direct comparison between classical lensing and quantum evaporation to distinguish between Kerr and modified gravity models (MOG and Horndeski) using EHT observations.

The Gist

Imagine you are looking at a distant, dark silhouette in the middle of a bright, glowing fog. You can’t see the object itself—it’s too dark—but you can see the exact size and shape of the "hole" it leaves in the fog.

In astrophysics, this is a Black Hole Shadow. For years, scientists have struggled with a problem: to understand a black hole, they usually need to know its "bare mass" (how much it weighs). But mass is invisible. It’s like trying to figure out how heavy a person is just by looking at the shadow they cast on a wall—it’s a guess based on assumptions.

This paper, written by researchers from De La Salle University, proposes a clever new mathematical "bridge" to solve this.

The Core Idea: The "Shadow-First" Approach

Instead of starting with the invisible mass and trying to predict the shadow, the authors flip the script. They say: "Let’s treat the shadow as the only thing we actually know for sure, and use it to calculate everything else."

They call this Thermodynamic-Optical Duality.

Think of it like this: Imagine you are looking at a closed, steaming cup of coffee. You can't see the liquid inside, and you don't know its exact weight. However, you can see the size of the dark circle of the cup (the Shadow) and you can measure how much steam is rising from it (the Hawking Radiation/Temperature).

The researchers created a mathematical formula that allows you to use the size of that dark circle to instantly calculate how hot the coffee is and how much it's "evaporating," without ever needing to weigh the cup.

Testing the "Rules of the Universe"

The researchers didn't just do this for standard black holes (the "Kerr" model, which follows Einstein's General Relativity). They tested it against "Modified Gravity" models—theories that suggest Einstein might be slightly wrong and that there might be extra "invisible forces" (like scalar or vector fields) acting on the black hole.

By using their "Shadow-First" method, they found that different theories leave different "fingerprints" on the shadow:

1. The Standard Model (Kerr): This is the baseline. If Einstein is 100% right, the shadow size and the heat of the black hole follow a very predictable, simple relationship.
2. The "Repulsive" Model (Kerr-MOG): Imagine a black hole that has a "shield" of repulsive force around it. This model predicts that the black hole will bend light more than Einstein expected, but it will actually be colder and emit less heat. It’s like a heavy object that somehow acts "lighter" to the light passing by.
3. The "Hairy" Model (Horndeski): Some theories suggest black holes have "hair"—extra layers of energy (scalar fields) clinging to them. This model predicts a "logarithmic" effect, meaning the shadow behaves in a very specific, slightly weird way that acts like a unique signature. It can make the black hole appear much "hotter" than Einstein’s math would suggest.

Why does this matter?

Right now, we have the Event Horizon Telescope (EHT), which has actually taken pictures of black hole shadows (like the famous one of M87*).

Before this paper, if a scientist saw a shadow, they had to make a lot of guesses about the mass to figure out if the gravity was "weird." Now, thanks to this framework, they can take the actual measurement from the telescope and plug it directly into these formulas.

It’s like having a universal translator. Instead of guessing what a black hole is made of, we can look at its shadow and immediately see if it’s following Einstein’s rules or if it’s wearing "modified gravity" clothing. It gives astronomers a much faster, more accurate way to hunt for new laws of physics.

Technical Summary

Technical Summary: Shadow-Dependent Phenomenology Framework for Rotating Black Hole Metrics

Authors: Nikko John Leo S. Lobos and Emmanuel T. Rodulfo
Subject Area: General Relativity, Black Hole Thermodynamics, Modified Gravity, Observational Astrophysics

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1. The Problem: Phenomenological Disconnect

In standard black hole physics, there exists a fundamental "disconnect" between two critical regimes:

• The Macroscopic/Optical Regime: Observables like the black hole shadow (measured by the Event Horizon Telescope) and gravitational lensing (weak deflection angles) are typically parameterized by the intrinsic bare mass ($M$). However, $M$ is a theoretical parameter that cannot be directly measured; it must be inferred through environmental assumptions.
• The Semiclassical/Thermodynamic Regime: Hawking temperature ($T_H$) and luminosity ($L$) are expressed as functions of the unobservable bare mass and the coordinate-dependent event horizon radius ($r_h$).

Because both regimes rely on the unobservable $M$, it is difficult to directly link the optical footprint of a black hole (the shadow) to its quantum evaporation properties, making it hard to test modified gravity theories using purely observational data.

2. Methodology: Thermodynamic-Optical Duality

The authors propose a novel mathematical framework that elevates the observable shadow radius ($R_{sh}$) from a secondary output to the central, unifying phenomenological variable. The methodology follows three main steps:

• Diffeomorphic Inversion (Mass-Shadow Mapping): Using the Implicit Function Theorem (IFT), the authors prove that a unique, non-degenerate mapping exists to invert the relationship between mass and shadow. This allows them to express the unobservable mass $M$ entirely as a function of the observable $R_{sh}$ and the spin $a$: $M = M(R_{sh}, a)$.
• Topological Lensing Computation: By applying the Gauss-Bonnet theorem to the optical spatial manifold, they derive the weak deflection angle ($\hat{\alpha}$) as a topological function of the shadow radius, bypassing the need for $M$.
• Thermodynamic Re-parameterization: They substitute the mass-shadow inversion into the equations for surface gravity and Hawking radiation. This transforms the thermodynamic variables ($T_H$ and $L$) into functions of the macroscopic shadow boundary.

3. Key Contributions and Results

The framework was tested against three distinct spacetime geometries to demonstrate its predictive power:

A. Standard Kerr Spacetime (The Baseline)

• Result: The authors derive a closed-form scaling law where the integrated semiclassical luminosity is inversely proportional to the square of the shadow radius: $L \propto R_{sh}^{-2}$.
• EHT Application: Using M87* data, they show that for a supermassive black hole, the Hawking temperature and luminosity are extremely suppressed ($T_H \approx 10^{-15}$ K), whereas for primordial black holes, the same framework predicts high-energy emission.

B. Kerr-MOG Spacetime (Scalar-Tensor-Vector Gravity)

• Mechanism: This theory introduces a repulsive vector field ($\alpha$).
• Result: The authors reveal an inverse correlation between classical and quantum signatures. For a fixed shadow size, the MOG field amplifies the classical weak deflection angle (lensing) while simultaneously suppressing the Hawking temperature and quantum luminosity. This breaks the degeneracy between mass and modified gravity parameters.

C. Rotating Horndeski Spacetime (Scalar-Tensor Theory)

• Mechanism: This theory introduces "scalar hair" ($h$) via a logarithmic potential.
• Result: Unlike MOG, the Horndeski scalar hair introduces a logarithmic augmentation to the lensing profile and increases the Hawking emission. Under current EHT limits, the scalar hair can drive a deviation of up to ~52% in Hawking luminosity compared to the Kerr baseline.

4. Significance and Implications

• Observational Efficiency: The framework provides a computationally efficient way to test the Kerr Hypothesis by anchoring theoretical predictions to empirical interferometric boundaries (like those from the EHT) rather than theoretical mass estimates.
• Model Discrimination: By establishing how different fundamental fields (vector vs. scalar) "fingerprint" the relationship between lensing and thermodynamics, the framework allows astronomers to distinguish between different modified gravity theories.
• Multimessenger Link: It bridges the gap between strong-field optical observations (shadows) and semiclassical quantum effects, providing a unified path for probing fundamental physics in the strong-field regime.