Quasinormal modes response to thermodynamic phase transitions in the charged AdS black hole surrounded by perfect fluid dark matter

The Cosmic Dance of Black Holes and the Invisible Ocean

Imagine the universe not as a static stage, but as a bubbling pot of energy where gravity, heat, and light play a complex game of chess. For decades, scientists have realized that black holes aren't just cosmic vacuum cleaners; they are also thermodynamic systems, much like a cup of coffee cooling down or a balloon expanding. They have a temperature, they have pressure, and they can even undergo "phase transitions." You might know phase transitions from everyday life: water turning into ice, or steam condensing into rain. In the exotic world of black holes, a similar thing happens, but instead of water, the substance is the black hole itself. It can suddenly switch from being a tiny, dense object to a massive, sprawling one, or vice versa, depending on its temperature and the pressure of the space around it.

To understand this, we need two main characters. First, there's the Anti-de Sitter (AdS) black hole. Think of this as a black hole living in a special kind of universe that acts like a giant, curved mirror box. This "box" keeps the black hole in thermal equilibrium, allowing scientists to study its behavior without it just flying off into the void. Second, there are Quasinormal Modes (QNMs). If you tap a bell, it doesn't just ring once; it vibrates at a specific pitch and slowly fades away. A black hole does the same thing when it gets "tapped" by a disturbance, like a passing wave of energy. These vibrations have a specific frequency (the pitch) and a damping rate (how fast the sound dies out). By listening to these cosmic "ringdowns," scientists can deduce the black hole's size, spin, and even what kind of invisible stuff might be surrounding it.

The big question scientists have been asking is: Can we "hear" a black hole changing its phase? If a black hole suddenly jumps from a small state to a large state, does its ringing sound change abruptly, like a car engine sputtering and shifting gears? This is where the mystery deepens, because real black holes in our universe aren't alone; they are likely swimming in a sea of Dark Matter. Dark matter is the invisible stuff that holds galaxies together, but we don't know exactly what it is. To solve this, physicists use a model called "Perfect Fluid Dark Matter" (PFDM), which treats this invisible ocean as a smooth, flowing fluid that gently pushes and pulls on the black hole. The paper you are about to read explores what happens when we combine these ideas: How does this invisible dark matter ocean affect the black hole's phase transitions, and more importantly, how does it change the "song" the black hole sings when it shifts gears?

The Paper's Discovery: Listening to the Shift

In this study, the authors, Long-Xiang Li and Zhong-Wen Feng, set out to simulate a charged black hole sitting inside this dark matter fluid and see how it behaves when it undergoes a phase transition. They treated the pressure of the universe (the cosmological constant) as a real, adjustable knob, just like the pressure in a steam engine. They also treated the strength of the dark matter fluid as another knob they could turn.

The Main Finding: The "Gear Shift" is Audible
The researchers found that when the black hole is below a certain critical temperature, it can exist in two distinct states: a "Small Black Hole" and a "Large Black Hole." These are like two different gears in a transmission. As the black hole moves from one state to the other (a first-order phase transition), its "song" changes dramatically.

• The Jump: Just as a car engine makes a distinct noise when shifting gears, the black hole's vibration frequency (the pitch) and its damping rate (how fast the sound fades) jump abruptly. The small black hole and the large black hole have completely different "voices."
• The Dark Matter Effect: The presence of the dark matter fluid (controlled by a parameter called $\alpha$) acts like a tuner for this song. The authors found that increasing the amount of dark matter doesn't just change the volume; it shifts the entire pitch. Specifically, a stronger dark matter presence makes the black hole vibrate at a higher frequency and dampen (fade out) more quickly. It also changes the "critical point"—the specific temperature and pressure where the phase transition happens—shifting it to higher temperatures and pressures while making the critical black hole size smaller.

The Critical Point: When the Song Smooths Out
The paper also looked at what happens exactly at the "critical point," the precise moment where the distinction between the small and large black holes disappears (like the point where water and steam become indistinguishable). Here, the results were very clear: The abrupt jump in the song disappears.

• As the black hole approaches this critical point, the two distinct "voices" (the small and large branches) merge into one smooth, continuous melody. There is no sudden gear shift, no sharp jump in the frequency. The song just glides smoothly from one note to another.
• The authors explicitly argue that while the dark matter parameter changes how loud or how fast the black hole vibrates, it cannot create a sharp signal at this critical point. Even with the dark matter, the transition remains smooth and continuous, offering no "dynamical signature" of a phase change at this specific second-order point.

The Coexistence Curve: The Tug-of-War
The researchers also traced the path where the small and large black holes can coexist (like ice and water existing together). They found that as you move along this path toward the critical point, the difference between the two "voices" gradually shrinks. The "swallowtail" shape of the energy diagram (a fancy way of describing the energy landscape) gets smaller, and the gap between the two QNM frequencies closes up until they meet at the critical point. The dark matter parameter changes the size of this gap, but the overall behavior—two distinct paths merging into one—remains the same.

How They Did It
To get these results, the authors didn't just guess; they performed detailed numerical simulations. They used a sophisticated mathematical technique called the Chebyshev pseudospectral method to solve the complex equations governing how a massless scalar field (a type of wave) vibrates around the black hole. They mapped out the black hole's horizon radius and the surrounding pressure, calculating the exact frequencies for different scenarios. They also broke down the isothermal process (where temperature is constant) to show that the change in the black hole's song isn't just due to the black hole getting bigger or smaller; it's a complex competition between the size of the black hole and the pressure of the surrounding space.

What This Means
The paper concludes that the fundamental scalar Quasinormal Modes are excellent "dynamical probes." They are sensitive enough to detect the sudden, first-order jump between small and large black holes, even when the black hole is surrounded by a dark matter fluid. However, they are not sensitive enough to detect the smooth, second-order transition at the critical point. The dark matter environment changes the scale of the vibrations and the location of the phase transition, but it doesn't change the fundamental rule: you can hear the gear shift, but you can't hear the smooth glide at the critical point.

In short, if we could listen to the ringing of a black hole in our universe, and if that black hole were surrounded by dark matter, we might be able to tell when it suddenly switches from a small state to a large one. But if it were slowly evolving toward a critical point where the two states become one, the song would just fade smoothly, offering no sudden clue that a major change was happening. This gives us a new, dynamic way to look at the thermodynamics of black holes, complementing the traditional "heat and pressure" maps with a "sound and vibration" map.