Shadow of rotating black hole surrounded by dark matter

This study generalizes a Schwarzschild black hole surrounded by dark matter to a rotating Kerr black hole using the Newman-Janis Algorithm, revealing that while dark matter has negligible effects below a critical mass threshold, exceeding this limit causes significant structural expansion that challenges current observational constraints, thereby suggesting dark matter must be absent or limited in the immediate vicinity of astrophysical black holes.

The Gist

The Big Picture: A Cosmic Detective Story

Imagine you are a detective trying to figure out what a mysterious, invisible monster looks like. You can't see the monster itself, but you can see the shadow it casts when it blocks the light from a giant flashlight behind it.

In this paper, the "monster" is a Black Hole, and the "flashlight" is the glowing gas swirling around it. The "shadow" is the dark circle we see in images taken by the Event Horizon Telescope (like the famous picture of M87*).

The scientists in this paper asked a big question: What happens to this shadow if the black hole is sitting inside a giant, invisible cloud of "Dark Matter"?

Dark Matter is the stuff that makes up most of the universe's mass, but we can't see it. It's like a ghostly fog that has weight but no light.

The Setup: Spinning in a Foggy Room

1. The Black Hole: Most black holes aren't sitting still; they are spinning incredibly fast, like a top. This spinning drags space and time around with it (a bit like a spoon stirring honey).
2. The Dark Matter: The researchers imagined this spinning black hole is surrounded by a thick layer of dark matter, like a black hole wearing a heavy, invisible winter coat.
3. The Method: They used a mathematical trick called the Newman–Janis Algorithm. Think of this as a "3D printer" for math. They took a simple, non-spinning black hole model, fed it into the printer, and "printed out" a spinning version that includes the dark matter coat.

The Discovery: The "Critical Weight"

The team ran simulations to see how much dark matter it takes to change the black hole's shadow. They found a fascinating "tipping point," which we can call the Critical Weight.

• Scenario A: The Light Coat (Low Dark Matter)
  If the dark matter cloud is light (below the critical weight), the black hole barely notices. The shadow looks almost exactly the same as a normal black hole. The dark matter is there, but it's too weak to change the game.

• Scenario B: The Heavy Coat (High Dark Matter)
  Once the dark matter gets heavy enough (crossing the critical weight), things go crazy.

  • The Shadow Swells: The shadow doesn't just get a little bigger; it explodes in size. Imagine a small coin suddenly inflating into a beach ball.
  • The Shape Changes: Normally, a spinning black hole's shadow looks like a D-shape or a heart (because the spin drags the light). But, surprisingly, the heavy dark matter coat forces the shadow back into a perfect circle. It's like the heavy coat is smoothing out all the wrinkles caused by the spin.

The "Why" and the "So What?"

Why does this happen?
Think of the black hole as a drain in a bathtub. The dark matter is like adding more water to the tub.

• If you add a cup of water, the drain looks the same.
• If you add a giant amount of water, the pressure changes everything. The "event horizon" (the point of no return) gets pushed outward, and the whole structure expands.

The Big Conclusion:
The researchers looked at real pictures from the Event Horizon Telescope. They saw that the shadows of real black holes (like M87*) are not huge beach balls; they are relatively small and fit the standard "no dark matter" model.

This leads to a powerful conclusion: If there is a lot of dark matter right next to these black holes, we would see a massive, circular shadow. Since we don't see that, it means one of two things:

1. There is no dark matter right next to the black hole (it's been pushed away).
2. If there is dark matter nearby, there is very little of it—certainly not enough to cross that "Critical Weight" threshold.

Summary Analogy

Imagine a spinning ice skater (the black hole).

• No Dark Matter: She spins fast, and her shadow is a slightly distorted oval.
• Light Dark Matter: She puts on a light scarf. Her shadow looks the same.
• Heavy Dark Matter: She puts on a massive, heavy anvil strapped to her back. Suddenly, she can't spin fast anymore, and her shadow becomes a giant, perfect circle.

The paper says: "We looked at the real ice skaters in the universe, and they aren't carrying anvils. So, either they aren't wearing anvils, or the anvils are tiny."

This helps astronomers understand where dark matter lives in our galaxy and confirms that the black holes we see are very close to the "pure" models predicted by Einstein's General Relativity.

Technical Summary

1. Problem Statement

While the Event Horizon Telescope (EHT) has provided high-resolution images of black hole (BH) shadows (e.g., M87* and Sgr A*), confirming the Kerr metric as a robust description of rotating black holes, these observations occur in realistic astrophysical environments where black holes are not isolated. They are typically embedded in dark matter (DM) halos.

• The Gap: Most existing theoretical models of BH shadows assume a vacuum Kerr geometry. There is a need to understand how the presence of a surrounding DM distribution modifies the spacetime geometry, specifically for rotating (Kerr) black holes, and how these modifications manifest in observable quantities like the shadow shape, size, and energy emission rates.
• The Goal: To generalize the Schwarzschild BH surrounded by a specific DM model to an axisymmetric Kerr BH, analyze the resulting horizon and ergosphere structures, and quantify the impact of DM mass on the BH shadow and energy emission.

2. Methodology

The authors employ a multi-step theoretical framework combining General Relativity (GR) and specific DM modeling:

• DM Model: They adopt a piecewise mass function model (originally proposed for Schwarzschild BHs in Ref. [28]) where the DM is distributed in a shell around the BH. The mass function $m(r)$ is defined in three regions:

  1. $r < r_s$: Pure BH mass $M$.
  2. $r_s \le r \le r_s + \Delta r_s$: A smooth transition region where DM density increases.
  3. $r > r_s + \Delta r_s$: Constant total mass $M + \Delta M$.
     The model assumes DM is non-luminous and electromagnetically inert.

• Newman–Janis Algorithm (NJA): To generate the rotating solution from the static Schwarzschild-DM metric, the authors use the NJA.

  1. Transform the static metric to Eddington–Finkelstein coordinates.
  2. Apply a complex coordinate transformation ($r \to r + ia\cos\theta$) to introduce the spin parameter $a$.
  3. Reconstruct the metric in Boyer–Lindquist coordinates.
  4. Verify that the resulting metric satisfies the Einstein field equations with the corresponding DM energy-momentum tensor (density and pressure profiles).

• Geodesic Analysis:

  • Derive the null geodesic equations using the Hamilton–Jacobi formalism.
  • Identify the conditions for unstable circular photon orbits (the photon sphere) by solving $R(r) = 0$ and $dR/dr = 0$.
  • Calculate the impact parameters ($\xi, \eta$) which define the boundary of the shadow.

• Shadow Construction:

  • Map the photon trajectories to celestial coordinates ($\alpha, \beta$) for a distant observer.
  • Define observables: Shadow radius ($R_{sh}$) and distortion parameter ($\delta_s$) using the Hioki-Maeda framework.
  • Calculate the energy emission rate based on the high-energy absorption cross-section (approximated by the average shadow area) and the Hawking temperature.

3. Key Contributions

• Generalization to Rotating Spacetime: Successfully constructed a rotating Kerr-like metric surrounded by a piecewise DM halo, extending previous static models.
• Critical Mass Threshold Discovery: Identified a critical DM mass ($\Delta M_{crit}$) beyond which the spacetime structure undergoes a phase transition. Below this threshold, the BH structure remains largely similar to the vacuum Kerr case; above it, the event horizon and ergosphere expand dramatically (by orders of magnitude).
• Circularization Effect: Demonstrated that while BH spin typically distorts the shadow into a "D-shape," a sufficiently massive DM halo counteracts this, forcing the shadow back toward a near-perfect circular shape.
• Thermodynamic Constraints: Linked the DM mass to the Hawking temperature and energy emission rate, showing that high DM mass drastically suppresses thermal radiation.

4. Key Results

A. Spacetime Structure (Horizons and Ergosphere)

• Event Horizon: For low DM mass ($\Delta M < \Delta M_{crit}$), the horizon position is minimally affected. However, once $\Delta M$ exceeds the critical value (e.g., $\Delta M/M \approx 88.3$ for extremal spin $a/M=1$), the event horizon radius $r_+$ expands significantly (e.g., from $\sim 2M$ to $>100M$).
• Ergosphere: Similar to the horizon, the ergosphere expands and shifts outward as DM mass increases. For high DM masses, the ergosphere boundaries can merge or shift so drastically that the geometric configuration is fundamentally altered compared to the vacuum Kerr case.

B. Shadow Observables

• Shadow Radius ($R_{sh}$):
  • Low DM Mass: The shadow radius increases slightly and linearly with $\Delta M$.
  • High DM Mass: Once the critical threshold is crossed, the shadow radius grows by orders of magnitude, directly correlating with the expanded event horizon.
• Shadow Distortion ($\delta_s$):
  • In vacuum Kerr, high spin ($a \to M$) causes significant distortion (asymmetry).
  • DM Effect: The presence of DM acts as a "circularizing" agent. As $\Delta M$ increases, the distortion parameter $\delta_s$ decreases monotonically. For very high DM masses, the shadow becomes almost perfectly circular, regardless of the BH spin. This suggests DM can mask the spin-induced asymmetry.

C. Energy Emission Rate

• The energy emission rate is proportional to the shadow area and the Hawking temperature ($T$).
• Spin Effect: Increasing spin $a$ lowers $T$, reducing emission.
• DM Effect:
  • Below the critical mass, DM has a negligible effect on emission.
  • Above the critical mass: The expansion of the horizon leads to a drastic drop in Hawking temperature. Consequently, the energy emission rate is strongly suppressed, effectively approaching zero.

5. Significance and Implications

• Observational Constraints: The paper concludes that the "structural expansion" caused by high DM mass would result in BH shadows and emission rates that are inconsistent with current EHT observations (which show shadows consistent with the vacuum Kerr metric size and shape).
• Astrophysical Conclusion: To remain consistent with observations, the localized dark matter mass in the immediate vicinity of supermassive black holes (like M87* and Sgr A*) must either:
  1. Be absent in the immediate environment, or
  2. Remain below the critical threshold identified in the model.
• Methodological Utility: The study provides a robust framework for using BH shadows as indirect probes of dark matter distributions in strong gravity regimes, distinguishing between standard GR predictions and those modified by matter halos.

In summary, this work establishes that while dark matter can subtly modify black hole shadows at low densities, a significant accumulation of dark matter would produce observable signatures (massive shadow expansion and circularization) that are currently ruled out by EHT data, thereby placing strict upper limits on the density of dark matter spikes around astrophysical black holes.