Central charge and black hole entropy for regular extremal black-bounce spacetimes

Black holes have long been the ultimate cosmic laboratories, places where the laws of physics are stretched to their breaking point. For decades, scientists have struggled to reconcile two giant pillars of modern physics: the theory of gravity, which describes how massive objects bend space and time, and quantum mechanics, which governs the behavior of the tiniest particles. A major stumbling block in this unification has been the nature of the black hole's center. According to the standard equations of gravity, the center of a black hole is a singularity—a point where matter is crushed into infinite density and the known laws of physics simply cease to exist. This "singularity" suggests that our current understanding of the universe is incomplete. To fix this, physicists often look to the surface of the black hole, specifically its event horizon, the point of no return. Decades ago, a profound discovery revealed that the entropy, or the measure of hidden information, of a black hole is directly proportional to the area of this horizon. This idea sparked a revolution, suggesting that the three-dimensional volume of a black hole might be encoded on a two-dimensional surface, much like a hologram.

Building on this holographic idea, researchers have developed a powerful method to count the microscopic building blocks of a black hole's entropy. By treating the space just outside the event horizon as a distinct, simpler system, they can apply a mathematical tool known as the Cardy formula. This tool allows them to calculate the number of possible quantum states a system can have, effectively counting the "atoms" of the black hole. If this microscopic count matches the entropy calculated from the horizon's area, it proves that the holographic view is correct and that the black hole is made of discrete quantum pieces. However, this method has mostly been tested on idealized black holes that still contain the problematic singularity at their core. The question remained: does this elegant counting method still work if the black hole is "regular," meaning it has no singularity and the laws of physics remain intact all the way to the center?

A team of researchers from Lanzhou University and the Shaoxing Institute of Technology has now answered this question by applying the holographic counting method to a new class of theoretical objects called "black-bounce" spacetimes. These are not the black holes we observe in the sky, but rather mathematical models designed to be perfectly smooth and free of the infinite density singularities that plague traditional theories. In these models, the center of the black hole is not a point of destruction but a bridge to another region of space, or simply a smooth, finite point where curvature remains manageable. The researchers focused on three specific types of these regular black holes: rotating ones similar to the Kerr black hole, rotating ones with an electric charge similar to the Kerr-Newman black hole, and non-rotating, charged ones similar to the Reissner-Nordström black hole.

The team began by peering into the immediate vicinity of the event horizon for each of these three regular black holes. They found that the geometry of space and time in this region possesses a special, enhanced symmetry. For the rotating black holes, the symmetry structure is a combination of a specific type of time-space symmetry and a rotational symmetry. For the non-rotating, charged black hole, the symmetry is even richer, involving a three-dimensional spherical symmetry. This enhanced symmetry is the key that unlocks the door to the microscopic counting. By carefully defining the rules for how the space can wiggle at the very edge of the horizon, the researchers identified the "asymptotic symmetry group," a collection of transformations that leave the physics unchanged. From this group, they extracted a crucial number called the central charge, which acts as a measure of the complexity of the underlying quantum system.

The calculation revealed something remarkable. For the rotating black holes, the central charge came entirely from the geometry of space itself. However, for the rotating black hole with an electric charge, the contribution from the electromagnetic field to the central charge vanished, leaving the geometry as the sole provider of the count. In the case of the non-rotating, charged black hole, the researchers had to take a creative step: they imagined the four-dimensional space-time of the black hole as part of a five-dimensional structure, where the electric field acts as a hidden extra dimension. This allowed them to treat the electric field's influence as a geometric property, enabling them to calculate the central charge for this static case as well.

With the central charge in hand, the researchers then determined the temperature of the quantum vacuum surrounding the horizon. This temperature, known as the Frolov-Thorne temperature, describes the thermal energy of the quantum fields in the near-horizon region. By plugging the central charge and this temperature into the Cardy formula, they calculated the microscopic entropy—the total number of ways the quantum states of the black hole could be arranged. The result was a perfect match. The entropy derived from counting the microscopic quantum states was identical to the entropy calculated from the simple area of the event horizon.

This agreement is a significant finding. It demonstrates that the holographic method of counting black hole entropy is not limited to idealized, singular black holes. The approach holds true even for these "regular" black holes, where the crushing singularity at the center has been replaced by a smooth, finite geometry. The study suggests that the statistical mechanics of black holes is robust and universal, working regardless of whether the center of the object is a mathematical breakdown or a smooth, regular point. By showing that the microscopic counting works for these singularity-free models, the research provides a stronger foundation for the idea that black holes are made of discrete quantum information, offering a clearer path toward understanding how gravity and quantum mechanics might finally fit together.