Somewhere in the center of a galaxy, a black hole is spinning. It doesn't hang in a void — around it swirls an invisible swarm of dark matter particles. Until now, this swarm was described for non-rotating holes, but in reality, holes always rotate. The authors of the new work have, for the first time, built a unified picture for a rotating black hole inside a cloud of arbitrary density.

They took a simple spherical solution and "spun it up" using the Newman–Janis algorithm. The result was a black hole resembling a Kerr hole but with a correction for the environment. And then something strange emerged. In an ordinary rotating hole, a singularity lurks at the center — a point where the laws of physics break down. But if the dark matter profile is sufficiently flat (index γ less than or equal to two), this curvature catastrophe disappears. Spacetime remains smooth even at the core.

Next, the authors studied how dark matter affects the spectrum of oscillations around the hole. To do this, they "placed" a test scalar field near the hole — something like a ubiquitous background of light particles. It turned out that the dark matter cloud acts like a damping spring: the denser it is, or the sharper its central peak, the more strongly particles are bound to the hole and the faster they lose energy.

The most striking effect concerns superradiance. A rotating hole can give energy to waves, amplifying them, like a swing that gets pushed higher by your movements. But dark matter narrows the "window" for this amplification and lowers its peak. A dense halo muffles superradiance, like a noise suppressor. And if the hole bathes in a thermal field of particles, the efficiency of energy extraction drops even more at high temperatures and sharp density profiles. A cool, fluffy cloud around the hole is the best ally for extracting energy.