A new type of multi-branch periodic orbits in dyonic black holes

Gravity is the invisible architect of the cosmos, shaping the paths of everything from falling apples to the light of distant stars. Around a black hole, this force becomes so intense that it warps space and time into a complex landscape. For decades, physicists have studied how objects move through this landscape, particularly focusing on "orbits"—the paths a particle takes as it circles a massive object. In the standard view of black holes, these paths are predictable and orderly. If you know the energy and spin of a particle, you can calculate exactly where it will go, and for any specific type of orbit, there is usually only one possible path. This understanding has been the bedrock of our knowledge about how black holes eat matter and how they ripple the fabric of space with gravitational waves.

However, a new study suggests that this orderly picture might be incomplete. Researchers have discovered that under certain conditions, a black hole can host a much stranger reality where a single type of orbit can split into multiple, distinct paths. By examining a specific theoretical model of a black hole that carries both electric and magnetic charges, the team found that the geometry of space around it can create "valleys" in the gravitational landscape. These valleys allow a particle to settle into different orbits that look completely different from the outside but share the same fundamental rhythm. This discovery challenges the long-held assumption that the number of paths a particle can take is determined solely by the number of boundaries, or horizons, surrounding the black hole. Instead, the study reveals that the shape of the space itself—whether it rises and falls smoothly or has bumps and dips—is the true controller of orbital behavior.

The researchers, working with a theoretical model known as quasi-topological electromagnetism, investigated a black hole that possesses both electric and magnetic charges. In this model, the strength of the interaction between these charges and the fabric of space is controlled by a specific parameter. By adjusting this parameter, the team could change the shape of the space surrounding the black hole without altering its mass or total charge. They focused on how a massive particle, like a small star or a speck of dust, would move in this environment. The key to their investigation was the "effective potential," a concept that describes the gravitational energy a particle feels as it moves closer to or farther from the black hole. In most known black holes, this energy landscape looks like a smooth, single bowl. A particle rolling in this bowl can only settle into one stable position for a given speed and spin.

The team found that when they tweaked the interaction parameter, the smooth bowl could transform into a landscape with two or even three separate valleys separated by hills. This happens because the metric function, which describes how space is stretched and curved, stops rising and falling in a simple, straight line and instead develops bumps and dips outside the black hole's event horizon. When this non-monotonic shape occurs, the gravitational potential develops multiple wells. A particle with the right amount of energy and spin can now choose to orbit in the inner valley, the middle valley, or the outer valley. Crucially, the researchers showed that these different orbits can share the exact same "rational number," a mathematical label that describes the ratio of how many times the particle circles the black hole versus how many times it moves in and out.

This means that for a single, specific type of orbital rhythm, there can be up to three different physical paths existing at the same time. One path might be a tight, nearly circular loop close to the black hole, while another might be a highly stretched, elongated ellipse that reaches far out into space. Even more surprisingly, the team found that as you increase the energy of the particle, the innermost orbit can become more circular, while the outer orbits become more stretched out. This is the opposite of what happens in standard black holes, where adding energy usually makes all orbits more eccentric. The study also revealed that these multiple paths can exist even when the particle has more energy than is typically required to escape the black hole's grasp, a phenomenon that would be impossible in a standard single-valley landscape.

A critical finding of the paper is that this complex behavior is not caused by the number of event horizons the black hole possesses. The researchers carefully distinguished between the number of horizons—the boundaries from which nothing can escape—and the shape of the space outside them. They demonstrated that a black hole could have four distinct horizons and still behave like a standard black hole with only one type of orbit, provided the space outside the outermost horizon remains smooth and monotonic. Conversely, a black hole with only one or two horizons could exhibit this complex, multi-path behavior if the space outside it is bumpy. This proves that the topology of the orbit is dictated by the geometry of the space, not by the count of the black hole's internal boundaries.

The implications of this discovery extend to how we might observe the universe. Just as light can form rings around a black hole, creating a "photon ring," the existence of multiple valleys in the gravitational landscape suggests that light could also form multiple rings. While the study focused on massive particles, the same geometric principles apply to light, suggesting that future telescopes might see more than one ring of light around certain black holes. Furthermore, for gravitational wave astronomy, which listens to the chirps of black holes merging, this finding is significant. If a small object is spiraling into a larger black hole, it could be following one of these multiple paths. Even if two objects are in orbits that look topologically identical, their different shapes and sizes would produce slightly different gravitational wave signals. This could provide a new way to test whether the black holes we observe in the universe follow the standard rules of Einstein's theory or if they possess these more exotic, multi-valley geometries.

The researchers did not find this behavior in all black holes, but specifically in those where the interaction between electric and magnetic charges creates a specific distortion in space. They simulated these conditions using a fixed set of parameters, varying only the strength of the electromagnetic coupling. Their results show that when this coupling is within a certain range, the space outside the black hole becomes non-monotonic, creating the conditions for these multiple orbits. When the coupling is outside this range, the space smooths out, and the black hole returns to having a single, standard orbit for any given rhythm. This precise control over the geometry suggests that the universe might contain black holes that are far more dynamic and structurally complex than previously imagined, offering a new window into the fundamental nature of gravity and spacetime.