Testing the Nature of Rotating Black Hole Shadows Surrounded by a Thin Accretion Disk within Rastall Gravity

This paper employs ray-tracing simulations to demonstrate that in Rastall gravity, increasing the structure parameter $\gamma$ enlarges the shadow radius of a rotating black hole while simultaneously reducing its distortion, thereby offering a potential observational method to constrain the theory's parameters.

The Gist

Imagine the universe as a giant, cosmic movie set. For a long time, we thought the "director" of gravity was a man named Einstein, and his script (General Relativity) was perfect. But recently, astronomers have started noticing tiny glitches in the movie—things that don't quite fit the script. This has led scientists to ask: "Is there a different director, or a different script, that explains these glitches better?"

This paper is like a team of special effects artists testing a new script called Rastall Gravity. They want to see if this new script changes how the universe's biggest stars, Black Holes, look on camera.

Here is a simple breakdown of what they did and what they found:

1. The Setting: A Spinning Black Hole with a "Ghost"

In this study, the scientists aren't just looking at a normal black hole. They are looking at a spinning one surrounded by a thin, glowing disk of hot gas (like a cosmic pizza dough spinning around a fire).

In the old script (Einstein's), the gravity is fixed. But in this new Rastall script, there are two "knobs" or dials that can be turned:

• The Rastall Knob ($\mu$): This controls how much the matter (the gas) and the shape of space itself talk to each other.
• The Structure Knob ($\gamma$): This controls the specific "flavor" of the matter surrounding the black hole.

Think of these knobs like the settings on a video game. If you turn them, the physics of the world changes slightly.

2. The Experiment: The "Flashlight" Test

To see what happens when you turn these knobs, the scientists used a computer program that acts like a digital flashlight.

They imagined an observer (a camera) floating in space looking at the black hole. They traced the path of light rays backwards from the camera to the black hole.

• The Shadow: Some light rays get sucked into the black hole and disappear. This creates a dark circle in the middle, called the shadow.
• The Ring: Other light rays get bent around the black hole like a rollercoaster before escaping to the camera. This creates a bright ring of light around the dark shadow.

3. What Happened When They Turned the Knobs?

The scientists turned the knobs to different settings and watched how the shadow changed. Here is the magic they found:

• Turning up the "Structure Knob" ($\gamma$):

  • The Shadow Gets Bigger: Imagine the black hole's shadow is a hole in a piece of paper. As they turned this knob, the hole got wider. The area where light gets trapped expanded.
  • The Shape Gets Rounder: Black holes usually look a little squashed or "D-shaped" because they are spinning fast. But as they turned this knob, the shadow became more like a perfect circle. It was as if the new gravity "smoothed out" the wrinkles caused by the spin.

• Turning up the "Rastall Knob" ($\mu$):

  • Similar to the other knob, this also made the shadow bigger and the surrounding ring of light expand outward.
  • It also changed the colors of the light coming from the gas disk. In the real world, gas moving toward us looks bluer (blueshifted), and gas moving away looks redder (redshifted). The new gravity settings made the "blue" side of the image much more dominant and spread out.

4. The Real-World Check: Did it Match the Photos?

The scientists didn't just make pretty pictures; they checked them against real photos taken by the Event Horizon Telescope (EHT). This is the actual telescope that took the first pictures of real black holes (M87* and Sgr A*).

They compared their new, "knob-adjusted" shadows with the real photos.

• The Result: The new script worked! The shadows they calculated with the Rastall gravity knobs fit perfectly within the range of the real photos taken by the EHT.
• The Takeaway: This means that Rastall gravity is a valid possibility. It's a script that could explain the universe just as well as Einstein's, and maybe even better in some ways.

Summary

Think of this paper as a cosmic tuning fork. The scientists took a new theory of gravity, tuned its dials, and saw how it changed the "silhouette" of a black hole. They found that by adjusting these dials, they could make the black hole's shadow bigger and rounder, and they proved that this new theory fits the actual photos we have taken of black holes in space.

It doesn't mean Einstein was wrong, but it does mean there might be other ways to write the laws of gravity that look just as good on the big screen.

Technical Summary

Technical Summary: Testing the Nature of Rotating Black Hole Shadows Surrounded by a Thin Accretion Disk within Rastall Gravity

Problem Statement
While General Relativity (GR) has successfully described gravitational interactions across various scales, it faces theoretical challenges regarding dark energy, dark matter, and spacetime singularities. Rastall gravity offers a modification to GR by relaxing the strict conservation of the energy-momentum tensor ($\nabla_\alpha T^{\alpha\beta} = 0$), allowing for a non-minimal coupling between matter and geometry characterized by a coupling parameter $\mu$ and a structure parameter $\gamma$. Although rotating black hole (BH) solutions in Rastall gravity have been constructed, their observational signatures—specifically the morphology of the BH shadow and accretion disk emission under realistic illumination—remain relatively unexplored. This paper investigates how the parameters of Rastall gravity ($\mu$, $\gamma$) and the BH spin ($a$) influence the optical appearance of rotating black holes, aiming to determine if these deviations from GR can be constrained by current observational data.

Methodology
The authors employ a comprehensive numerical framework to model the optical signatures of a rotating Rastall black hole:

1. Spacetime Geometry: The study utilizes a rotating Kerr-like metric derived in Rastall gravity, characterized by the mass $M$, spin parameter $a$, structure parameter $\gamma$, and Rastall coupling parameter $\mu$. The metric function $\Delta(r)$ determines the horizon structure.
2. Photon Dynamics: The motion of photons is analyzed using the Hamilton-Jacobi formalism. The authors derive the null geodesic equations and identify the unstable circular photon orbits (photon sphere) to determine the critical impact parameters ($\xi, \zeta$) that define the shadow boundary.
3. Ray-Tracing Simulation: A backward ray-tracing algorithm is implemented to map photon trajectories from a distant observer's image plane to the emission region. The study considers two distinct illumination scenarios:
   • Celestial Light Source: A uniform background illumination (celestial sphere) to visualize the shadow and Einstein ring.
   • Thin Accretion Disk: A physically motivated model where the disk consists of neutral plasma moving on equatorial timelike geodesics. The study distinguishes between prograde and retrograde accretion flows.
4. Observables: Key physical quantities are calculated, including the shadow radius ($R_d$), the distortion parameter ($\delta_d$), redshift distributions (blueshifted vs. redshifted emission), and lensing bands (direct, lensed, and photon ring images).
5. Observational Constraints: The theoretical predictions for the angular diameter of the shadow are compared against Event Horizon Telescope (EHT) measurements of M87* and Sgr A* to assess the viability of the model parameters.

Key Contributions and Results

• Shadow Morphology and Parameters:

  • The shadow radius ($R_d$) exhibits a pronounced dependence on the Rastall parameters. Increasing the structure parameter $\gamma$ leads to a gradual enlargement of the shadow radius, indicating an expansion of the photon capture region.
  • Conversely, the distortion parameter ($\delta_d$) decreases as $\gamma$ increases. This implies that higher values of $\gamma$ suppress the asymmetry induced by rotation, resulting in a shadow boundary that is progressively more circular and less deformed.
  • The Rastall parameter $\mu$ also modifies the shadow, with both $\mu$ and $\gamma$ leaving distinct signatures on the shadow's size and position.

• Accretion Disk Imaging:

  • Under celestial illumination, increasing $\mu$ or $\gamma$ causes the Einstein ring to expand outward while the central dark shadow region decreases in size. The characteristic "D-shaped" petals of the shadow become more prominent.
  • Under thin accretion disk illumination (both prograde and retrograde), increasing $\mu$ and $\gamma$ reduces the size of the central dark region and enhances the surrounding luminous structures.
  • Redshift Distribution: A significant finding is the impact on frequency shifts. As $\mu$ increases, the blueshifted region of the accretion flow expands and becomes the dominant feature on the observer's screen, while the redshifted emission becomes more localized. This effect is observed in both direct and lensed images for prograde and retrograde flows.
  • Lensing Bands: The lensed bands surrounding the shadow become narrower and more confined as $\mu$ increases, whereas the photon ring remains relatively stable across the parameter space.

• Observational Constraints (EHT):

  • The study compares the predicted angular diameter of the shadow ($D$) with EHT data for M87* and Sgr A*.
  • The results indicate that the predicted shadow diameter increases monotonically with $\gamma$. For the tested ranges of $\mu$ (0.02 and 0.05) and $\gamma$, the theoretical predictions fall within the 1$\sigma$ and 2$\sigma$ confidence intervals of the EHT observations for both black holes.
  • This suggests that the parameter space explored in this Rastall gravity model is consistent with current high-resolution observations.

Significance and Claims
The paper claims that the combined effects of the Rastall coupling parameter $\mu$ and the structure parameter $\gamma$ leave distinct, observable imprints on the optical appearance of rotating black holes. Specifically, the study demonstrates that:

1. Shadow observations serve as an effective tool for constraining the parameter space of rotating black holes in Rastall gravity.
2. The parameters $\mu$ and $\gamma$ significantly alter the shadow radius, distortion, and the distribution of redshifted/blueshifted emission in accretion disks.
3. The model remains viable within the current observational bounds provided by the EHT for M87* and Sgr A*, supporting the potential of Rastall gravity as a framework for investigating deviations from General Relativity in the strong-field regime.

The authors conclude that while the model is consistent with current data, future studies could further explore these effects in thick disk accretion, polarized imaging, and hotspot scenarios.