Circulation and operational sparsity of analogue Hawking radiation in rotating acoustic horizons

In the vast silence of space, black holes are often imagined as cosmic vacuum cleaners, swallowing everything that crosses their edge. But in the 1970s, a revolutionary idea suggested that these objects are not entirely silent. They should actually emit a faint, ghostly radiation, a whisper of particles leaking out from the very edge of their event horizons. This phenomenon, known as Hawking radiation, arises from the strange rules of quantum mechanics playing out in the extreme gravity of a black hole. The temperature of this radiation is determined by the strength of the gravity at the horizon; the stronger the pull, the hotter the emission. However, observing this directly from a real black hole is nearly impossible because the signal is too weak and the distances too great. To understand how this radiation works, scientists have turned to "analogue gravity," creating miniature models in laboratories using fluids or sound waves. In these systems, a flowing medium can create a "sonic horizon" where sound waves cannot escape, mimicking the behavior of light near a black hole. These tabletop experiments allow researchers to test the fundamental physics of emission without needing to travel to the edge of a galaxy.

A specific question has long puzzled researchers: how crowded is this stream of emitted particles? Is it a steady, continuous flow, or is it so sparse that the particles arrive one by one with long gaps in between? This concept, called temporal sparsity, asks whether the time between two emitted particles is longer or shorter than the time it takes for a single particle to oscillate or vibrate. If the gaps are huge, the radiation is "sparse" and looks like individual particles. If the gaps are tiny, the waves overlap, and the radiation begins to look like a continuous classical wave. While previous studies looked at non-rotating black holes, a new study by Fernando M. Belchior, João A. A. S. Reis, and Edilberto O. Silva investigates what happens when the black hole spins. They focused on a rotating model known as a "draining bathtub," which is a common way to simulate a spinning black hole using water flowing down a drain. In this setup, the water drains radially inward while also swirling around the center. The researchers wanted to see how this swirling motion, or circulation, changes the timing and spacing of the emitted sound particles.

The team began by establishing a baseline prediction based purely on the geometry of the system. They found that the temperature of the radiation depends only on how fast the water drains, not on how fast it swirls. However, the swirling motion does change the width of the "capture zone," the area from which particles can be emitted. As the swirl gets stronger, this capture zone widens. A simple geometric calculation suggested that as the swirl increases, the particles should arrive closer together in time, eventually overlapping to form a continuous flow. The researchers then moved beyond this simple picture to solve the complex scattering problem, calculating exactly how the sound waves interact with the rotating flow. They accounted for the fact that not all frequencies pass through equally and included the effects of "superradiance," a phenomenon where the rotation of the black hole actually amplifies certain waves, adding extra energy to the emission.

The results revealed a more nuanced reality than the simple geometric model predicted. The researchers calculated the emission using two different ways of measuring time. The first method used a fixed clock based on the average temperature of the system. Using this clock, they found that as the swirl increased, the particles did indeed arrive closer together. At a specific ratio where the swirl speed matched the drain speed, the time between particles dropped to less than half the duration of a single particle's vibration. This indicated that, for this specific clock, the radiation had become dense enough to overlap. However, the second method used a clock based on the actual average frequency of the particles being emitted. This method told a different story. Because the rotation also "hardened" the spectrum, pushing the emitted particles to higher, faster frequencies, the time between them remained longer than the vibration time of a single particle. Even at the highest swirl speeds they tested, the radiation remained sparse according to this spectrum-based measure.

The study also addressed a technical complication regarding how to count the particles. In two-dimensional systems like their model, the emission of very low-frequency particles can theoretically become infinite if not limited by the size of the detector or the observation time. The researchers showed that any measurement of sparsity in such a system must inherently depend on the limits of the observation, such as how long the experiment runs or how sensitive the detector is. They further tested the robustness of their findings by removing the superradiant particles from their calculations. Even without these amplified waves, the conclusion held: the rotation made the particle count increase, but it also shifted the energy to higher frequencies. This meant that whether the radiation appeared as a dense stream or a sparse trickle depended entirely on which "clock" the observer chose to use.

Ultimately, the paper demonstrates that in a rotating acoustic black hole, the sparsity of Hawking radiation is not a single, fixed property of the geometry. Instead, it is an operational quantity that changes based on how one chooses to measure it. The rotation enhances the number of particles emitted, making them arrive more frequently, but it simultaneously shifts their energy to higher frequencies, keeping the gaps between them wide when viewed through the lens of their own vibration. The researchers conclude that to truly understand the nature of this emission, one must look at both the number of particles and their energy distribution together. The findings confirm that while the temperature of the radiation remains steady regardless of the spin, the way the radiation is delivered to the outside world is deeply influenced by the rotation, creating a complex interplay between the flow of particles and the rhythm of their arrival.