Kerr-like black holes shadow surrounded by dark matter halos: Comparison between various dark matter profiles

This paper investigates how various dark matter halo profiles (King, Hernquist, and Moore) affect the shadows of rotating Kerr-like black holes and concludes that while the presence of dark matter increases the shadow radius compared to a vacuum Kerr black hole, the specific halo profile has a negligible impact on the shadow's size and shape, rendering it an ineffective tool for distinguishing between different dark matter distributions in galactic centers.

The Gist

Imagine the universe as a giant, dark ocean. In the middle of this ocean, there are massive whirlpools called black holes. We know these whirlpools exist because we've taken pictures of their "shadows" (the dark silhouette they cast against the bright background of space).

For a long time, scientists have studied these shadows assuming the black hole is floating in empty space. But in reality, black holes don't live in empty space; they are surrounded by a thick, invisible fog called Dark Matter. Think of Dark Matter like a giant, invisible cloud of mist that clumps around the galaxy, holding stars in place with its gravity.

This paper asks a simple question: Does this invisible "fog" of Dark Matter change the shape or size of the black hole's shadow?

To find out, the authors looked at three different theories about what this "fog" looks like:

1. The King Profile: Imagine a cloud that is very dense in the middle and fades out smoothly, like a fluffy marshmallow.
2. The Hernquist Profile: A cloud that is dense in the center but drops off very quickly, like a steep hill.
3. The Moore Profile: A cloud that gets incredibly dense right at the very center, like a sharp spike.

They also considered that black holes spin (like a top), which makes the shadow look a bit like a D-shape rather than a perfect circle.

The Experiment

The scientists used complex math (like a recipe called the "Newman-Janis algorithm") to build a model of a spinning black hole sitting inside each of these three different types of Dark Matter clouds. They then calculated what the shadow would look like from a distance.

What They Found

Here is the surprising result, explained simply:

• The Shadow Gets Bigger (But Just a Tiny Bit): When they added the Dark Matter "fog" to their models, the shadow of the black hole did get slightly larger. It's like if you put a spinning top inside a thick jar of honey; the honey pushes back a little, making the area the top affects slightly bigger.
• The Shape Doesn't Change: Even though the shadow got a tiny bit bigger, the shape of the shadow didn't change in a way that we could easily spot. Whether the Dark Matter was a "fluffy marshmallow" (King), a "steep hill" (Hernquist), or a "sharp spike" (Moore), the shadow looked almost exactly the same.
• The "Fog" is Hard to See: The most important finding is that the difference between a black hole with Dark Matter and a black hole without it is so small that our current telescopes can't tell the difference. It's like trying to tell the difference between a clear glass of water and a glass of water with a single grain of sand in it—you just can't see it.

The Big Conclusion

The authors conclude that looking at a black hole's shadow is not a good way to figure out what kind of Dark Matter is around it.

Even though the math shows the shadow changes slightly depending on the type of Dark Matter, the change is so tiny that it doesn't matter in practice. Whether the Dark Matter is "cuspy" (spiky in the middle) or "cored" (fluffy in the middle), the shadow looks the same.

In short: Black holes are great at hiding their surroundings. No matter what kind of invisible "fog" (Dark Matter) they are swimming in, their shadow looks almost identical to a black hole floating in empty space. So, we can't use the shadow to solve the mystery of what Dark Matter actually is.

Technical Summary

Technical Summary: Kerr-like Black Hole Shadows Surrounded by Dark Matter Halos

Problem Statement
While the existence of supermassive black holes (BHs) at galactic centers is well-established, the majority of matter in these galaxies consists of dark matter (DM). Previous studies have investigated the effects of DM on BH properties, such as null geodesics, accretion disks, and thermodynamics. However, obtaining exact analytical solutions for rotating BHs immersed in specific DM halo profiles remains challenging due to the unknown interaction behavior of DM particles. While static solutions for BHs in various DM halos exist, the rotating (Kerr-like) solutions and their corresponding shadows for the King, Hernquist, and Moore DM profiles had not been previously investigated. This paper addresses this gap by constructing these rotating solutions and analyzing how the characteristic density ($\rho_s$), characteristic radius ($r_s$), and spin parameter ($a$) influence the event horizons, ergoregions, and shadow radii.

Methodology
The authors employ a two-step analytical approach:

1. Metric Construction: Starting with static, spherically symmetric BH solutions immersed in King, Hernquist, and Moore DM halos (derived from specific density profiles $\rho(r)$), the authors apply the Newman–Janis algorithm. This technique extends the static metrics to rotating (Kerr-like) metrics in Boyer-Lindquist coordinates. The resulting line elements incorporate a mass function $m(r)$ that accounts for both the central BH mass $M$ and the DM halo contribution.
2. Geodesic and Shadow Analysis: Using the Hamilton-Jacobi formalism, the authors derive the equations of motion for photons (null geodesics) in these spacetimes. They identify the conditions for unstable circular photon orbits to determine the celestial coordinates ($\alpha, \beta$) defining the BH shadow. The shadow radius ($R_{sh}$) is calculated by comparing prograde and retrograde photon orbits. The study focuses primarily on the King profile for detailed analysis, noting that Hernquist and Moore yield similar physical results.

Key Contributions

• New Analytical Solutions: The paper provides the first derivation of rotating BH solutions surrounded by King, Hernquist, and Moore DM halos using the Newman–Janis algorithm.
• Systematic Parameter Analysis: The study systematically maps the influence of DM parameters ($\rho_s, r_s$) and the spin parameter ($a$) on the inner and outer horizons, the ergoregion shape, and the shadow radius.
• Comparative Profile Study: The authors compare the shadows of BHs in these specific profiles against the standard Kerr BH (no DM) and other DM profiles (e.g., NFW, Burkert, Dehnen) to assess the distinguishability of different DM distributions via shadow imaging.

Results

• Horizons and Ergoregion: The presence of DM increases the radius of the event horizon compared to a Kerr BH with $\rho_s = 0$. Specifically, increasing the characteristic density ($\rho_s$) or radius ($r_s$) expands the event horizon, while increasing the spin parameter ($a$) reduces the outer horizon radius. The ergoregion (the region between the event horizon and the static limit) is similarly affected by these parameters.
• Shadow Radius: The shadow radius of rotating BHs immersed in DM is larger than that of a standard Kerr BH. Increasing $\rho_s$ and $r_s$ leads to an increase in the shadow size. The distortion of the shadow increases with higher spin parameters ($a$) and when the observer is closer to the equatorial plane.
• Profile Independence: Crucially, the study finds that while the presence of DM increases the shadow size, the magnitude of this effect is negligible. The differences in shadow size and shape between the King, Hernquist, Moore, and other profiles (both "cuspy" and "cored") are extremely small.
• Indistinguishability: The variation in shadow radius is not unique to a specific profile. The authors note that for some models (e.g., PFDM, Dehnen with $\gamma=1$), the shadow radius might even slightly decrease relative to the Kerr case, but the overall deviation from the vacuum Kerr solution is minimal across all tested profiles.

Significance and Claims
The paper concludes that for the Kerr-like black holes studied, the black hole shadow is not a suitable tool for distinguishing the nature of the dark matter distribution in galactic centers. Despite the theoretical increase in shadow size due to DM, the effect is too small to be observationally significant with current or near-future capabilities. Furthermore, the shadow cannot effectively differentiate between "cored" profiles (like King and Burkert) and "cuspy" profiles (like Moore and NFW). The authors assert that the distinction between these DM distributions based solely on BH shadow imaging is nearly impossible, as the variations are negligible compared to the standard Kerr metric.