Investigating the interplay of the braneworld gravity and the plasma environment on the black hole shadow

This paper investigates the shadow of a rotating braneworld black hole in dispersive plasma environments, finding that plasma density and tidal charge significantly influence shadow size and shape, and uses Event Horizon Telescope observations of M87* and Sgr A* to constrain the tidal charge parameter within specific ranges depending on the plasma density.

The Gist

Imagine the universe as a giant, invisible trampoline (our 3D space). In standard physics, we think this trampoline is the whole story. But this paper explores a wilder idea: Braneworld Gravity.

Think of our universe not as the whole trampoline, but as a thin sheet of paper floating inside a giant, invisible swimming pool (the "bulk"). Gravity is unique because it can leak off the paper and swim around in the pool, while light and matter are stuck on the paper.

This "leaking" gravity leaves a fingerprint on black holes. The authors of this paper are trying to find that fingerprint by looking at the shadow of a black hole, but with a twist: they are asking, "What if the black hole is also swimming in a thick soup of plasma?"

Here is a breakdown of their findings using simple analogies:

1. The Black Hole Shadow: The "Cookie Cutter"

When a black hole sits in front of a bright background (like a glowing accretion disk), it blocks the light, creating a dark circle in the middle. This is the shadow.

• Standard View: In normal physics, the size of this cookie cutter depends only on how heavy the black hole is and how fast it's spinning.
• Braneworld View: Because gravity can leak into the "pool" (the extra dimension), the black hole has an extra "charge" called tidal charge ($q$).
  • If this charge is negative, it's like adding extra weight to the cookie cutter, making the shadow bigger.
  • If it's positive, it's like shaving off a bit of the dough, making the shadow smaller.

2. The Plasma Soup: The "Muddy Water"

Black holes aren't usually in empty space; they are surrounded by hot, swirling gas called plasma. Think of this like looking at a coin at the bottom of a swimming pool.

• Inhomogeneous Plasma (The Swirly Soup): Imagine the water is thicker near the bottom and thinner near the top.
  • Effect: This "swirly" soup acts like a magnifying glass that shrinks things. The denser the soup, the smaller the black hole shadow appears.
  • Bonus: It also makes the shadow look more perfectly round, smoothing out the wobbles caused by the black hole's spin.
• Homogeneous Plasma (The Uniform Fog): Imagine the water is the same thickness everywhere.
  • Effect: This uniform fog acts like a lens that expands things. The denser the fog, the larger the shadow appears.

3. The Detective Work: The Event Horizon Telescope (EHT)

The authors used real data from the Event Horizon Telescope (the camera that took the first picture of a black hole) looking at two famous targets: M87* (a giant black hole) and Sgr A* (the one at the center of our Milky Way).

They played a game of "What-If":

• What if the black hole has a negative tidal charge?
• What if the plasma around it is thick or thin?
• Does the math match the actual photo taken by the EHT?

4. The Big Discovery: The "Low-Density" Truth

After running thousands of simulations, they found a crucial result:

The soup is too thin to matter.

Even though plasma can change the size of the shadow, the actual plasma around M87* and Sgr A* is so incredibly thin (low density) that it's like trying to see a difference in a coin's size by looking through a single layer of mist. It's negligible.

• The Result: Because the plasma is so thin, the shape and size of the shadow are almost entirely determined by the geometry of space itself (the braneworld gravity).
• The Constraint: This allows them to put strict limits on the "tidal charge."
  • For M87*, the extra-dimensional charge must be between -1.15 and 0.45.
  • For Sgr A*, it must be between -0.65 and 0.8.

5. Why This Matters

This is a victory for the "extra dimension" theory, but with a caveat.

• If the plasma were thick (like a dense fog), it would hide the true shape of the black hole, making it impossible to tell if extra dimensions exist.
• But because the plasma is thin, the "fingerprint" of the extra dimensions is visible.
• The Catch: The data doesn't prove extra dimensions exist yet (the charge could still be zero, which is normal gravity). But it proves that if extra dimensions exist, their effect is limited to a specific range.

The Bottom Line

The authors are saying: "We looked at the black hole photos through the lens of a new gravity theory and a thick soup of gas. We found that the gas is too thin to blur the picture. So, the picture we see is a clear reflection of the black hole's true shape. This shape tells us that if our universe is floating in a higher-dimensional pool, the 'leakage' of gravity is small but measurable, and we now know exactly how big that leakage can be."

In short: The black hole shadows are clear enough to test new physics, and the "soup" around them isn't thick enough to hide the truth.

Technical Summary

1. Problem Statement

The paper addresses the challenge of distinguishing between standard General Relativity (GR) and modified gravity theories, specifically Braneworld gravity (based on the Randall-Sundrum scenario), using observations from the Event Horizon Telescope (EHT).

• The Core Issue: While EHT images of M87* and Sgr A* confirm GR predictions, potential deviations could arise from extra dimensions. In Braneworld models, the 4D black hole geometry is modified by a tidal charge ($q$) arising from the bulk Weyl tensor. Unlike electric charge in the Kerr-Newman metric, $q$ can be negative, significantly altering the event horizon and photon dynamics.
• The Complication: Astrophysical black holes are surrounded by plasma. Plasma is dispersive, meaning it affects light propagation based on frequency. The paper investigates how inhomogeneous and homogeneous plasma environments interact with the tidal charge to modify the black hole shadow's size and shape. The goal is to determine if EHT data can constrain the tidal charge ($q$) and plasma parameters ($\alpha_i$) simultaneously.

2. Methodology

A. Theoretical Framework

• Spacetime Metric: The authors model the rotating braneworld black hole using a Kerr-Newman-like metric where the electric charge is replaced by the tidal charge $q$. The metric depends on mass $M$, spin $a$, and tidal charge $q$.
  • $q < 0$: Enhances gravitational attraction (larger horizon).
  • $q > 0$: Reduces gravitational attraction.
• Plasma Model: The study considers non-magnetized, pressureless plasma. The light ray geodesics are derived using a Hamiltonian formalism where the Hamiltonian is $H = \frac{1}{2}(g_{\mu\nu}p^\mu p^\nu + \omega_p^2)$.
  • The plasma frequency $\omega_p$ is related to electron density $n_e$.
  • The condition for light propagation is $\omega^2 \geq \omega_p^2$.

B. Three Plasma Profiles

The authors analyze three distinct plasma density profiles to study their impact:

1. Profile 1 (Inhomogeneous): Motivated by the Shapiro profile/Bondi accretion ($n_e \sim r^{-3/2}$).
   • $\omega_p^2 \propto \frac{\sqrt{r}}{\rho^2}$.
2. Profile 2 (Inhomogeneous): A toroidal profile.
   • $\omega_p^2 \propto \frac{1 + 2\sin^2\theta}{\rho^2}$.
3. Profile 3 (Homogeneous): Constant electron density.
   • $\omega_p^2 \propto \text{constant}$.

C. Shadow Calculation

• Geodesic Equations: The authors solve the Hamilton-Jacobi equation to find constants of motion (Carter constant $Q$, impact parameters $\eta, \chi$) for spherical photon orbits.
• Shadow Boundary: The shadow is defined by the critical curve separating photons that escape to infinity from those falling into the horizon. The boundary is calculated by projecting unstable spherical photon orbits onto the observer's sky using a tetrad formalism.
• Observables:
  • Angular Diameter ($\Delta\Theta$): The vertical diameter of the shadow.
  • Schwarzschild Deviation Parameter ($\delta_{sh}$): A measure of deviation from the Schwarzschild shadow size.

D. Statistical Constraints

• The authors use EHT data for M87* and Sgr A* (angular diameter and $\delta_{sh}$).
• They perform a $\chi^2$ analysis to find the allowed parameter space for $(q, \alpha_i)$, minimizing $\chi^2$ over the spin parameter $a$.
• Constraints are applied based on $1-\sigma$ error bars of EHT observations and independent estimates of electron density ($n_e$) and accretion rates ($\dot{M}$).

3. Key Contributions

1. Interplay of Tidal Charge and Plasma: The paper systematically quantifies how plasma density modifies the shadow of a braneworld black hole, a factor often neglected in pure vacuum studies.
2. Differential Effects of Plasma Profiles:
   • Inhomogeneous Plasma (Profiles 1 & 2): As plasma density increases, the shadow size decreases (contraction effect). Furthermore, high inhomogeneous plasma density tends to make the shadow more circular, reducing the deviation caused by high spin and inclination.
   • Homogeneous Plasma (Profile 3): In contrast, homogeneous plasma expands the shadow size as density increases. It does not significantly alter the circularity deviation caused by spin.
3. Degeneracy Breaking: The study demonstrates that while low-density plasma has a negligible effect (making the shadow geometry-dominated), high-density plasma creates a degeneracy where plasma effects can mimic or mask the effects of the tidal charge.
4. Joint Constraints: The authors provide the first joint constraints on the tidal charge $q$ and plasma parameters $\alpha_i$ using EHT data for both M87* and Sgr A*.

4. Key Results

A. Impact on Shadow Geometry

• Tidal Charge ($q$):
  • $q < 0$ (Negative): Increases the shadow diameter.
  • $q > 0$ (Positive): Decreases the shadow diameter.
• Plasma Density ($\alpha$):
  • Inhomogeneous profiles: Higher $\alpha$ $\rightarrow$ Smaller shadow.
  • Homogeneous profile: Higher $\alpha$ $\rightarrow$ Larger shadow.

B. Constraints on M87*

• Low Density Limit ($\alpha \approx 0$):
  • Using stellar dynamics mass estimates ($M \approx 6.2 \times 10^9 M_\odot$), the allowed tidal charge range is $-1.15 \lesssim q \lesssim 0.45$.
  • This range includes the standard Kerr scenario ($q=0$) but allows for negative tidal charges (extra-dimensional signatures).
• Plasma Constraints:
  • EHT angular diameter data imposes tighter constraints on plasma parameters than theoretical propagation limits.
  • For $q \approx -2$, the allowed plasma parameters are ruled out if $\alpha_1 \gtrsim 10$, $\alpha_2 \gtrsim 17$, or $\alpha_3 \gtrsim 0.56$.
• Realistic Astrophysical Limits:
  • Using observed electron density ($n_e \sim 10^4 - 10^7 \text{ cm}^{-3}$) and accretion rates, the actual plasma parameters are extremely small ($\alpha \sim 10^{-10} - 10^{-7}$).
  • Conclusion: For M87*, the plasma contribution is negligible. The observed shadow is dominated by the background spacetime geometry. The constraints on $q$ remain $-1.15 \lesssim q \lesssim 0.45$.

C. Constraints on Sgr A*

• Low Density Limit ($\alpha \approx 0$):
  • Using Keck team mass estimates: $-0.65 \lesssim q \lesssim 0.7$.
  • Using GRAVITY collaboration mass estimates: $-0.3 \lesssim q \lesssim 0.8$.
• Plasma Constraints:
  • Unlike M87*, for Sgr A*, the Schwarzschild deviation parameter ($\delta_{sh}$) provides tighter constraints than the angular diameter ($\Delta\Theta$).
  • Similar to M87*, realistic electron density estimates ($n_e \lesssim 10^5 \text{ cm}^{-3}$) yield $\alpha \sim 10^{-11} - 10^{-8}$.
  • Conclusion: The background metric dominates the shadow size for Sgr A* as well.

D. General Trends

• High Density Scenarios: If a black hole were surrounded by high-density plasma, the shadow size would be governed by a complex interplay of geometry and plasma. In such cases, joint constraints from plasma density estimates and shadow observations are crucial to disentangle the effects.
• Spin Constraints: The EHT observables ($\Delta\Theta$ and $\delta_{sh}$) fail to constrain the spin $a$ significantly, likely due to weak dependence at low inclination angles.

5. Significance

• Validation of GR/Braneworld: The study confirms that for the current low-density plasma environments of M87* and Sgr A*, the EHT observations are consistent with standard GR ($q=0$) but also permit specific ranges of negative tidal charges, offering a window to test extra-dimensional gravity.
• Methodological Advance: It highlights the necessity of incorporating realistic plasma models when interpreting black hole shadows. While plasma is negligible for current EHT targets, it could be dominant for other sources or future high-density observations.
• Future Outlook: The paper emphasizes that future multi-band EHT observations and improved polarimetric modeling are essential to further constrain the plasma environment, thereby tightening the bounds on the tidal charge and providing more robust tests of higher-dimensional gravity.

In summary, the paper establishes that while plasma modifies the black hole shadow, the low density of the plasma surrounding M87* and Sgr A* means the background spacetime geometry is the primary determinant of the observed shadow. Consequently, current EHT data constrains the braneworld tidal charge to a specific range that includes standard GR but allows for detectable extra-dimensional signatures.