Microscopic entropy of regular BTZ black holes in 3d Lovelock gravity at all orders

Deep in the theoretical landscape of physics, where scientists try to reconcile the smooth curves of gravity with the jittery rules of the quantum world, there exists a persistent puzzle: where does the entropy of a black hole come from? Entropy is a measure of disorder, or more precisely, the number of ways a system can be arranged internally while looking the same from the outside. For decades, the standard view held that a black hole's entropy is a direct reflection of its surface area, but understanding the microscopic ingredients that create this entropy has remained a major challenge. In three-dimensional space, a simplified version of our universe, physicists have long used a powerful mathematical tool called the Cardy formula to count these microscopic states. This tool works beautifully for the simplest black holes, but it has historically stumbled when the black holes become more complex or when the laws of gravity are tweaked with extra terms. The question has been whether this counting method can survive in a more complicated, "regularized" version of gravity that removes the infinite, crushing singularities usually found at the heart of black holes.

A researcher has now answered this question by extending the analysis to a theory of gravity that includes an infinite tower of corrections, a framework known as three-dimensional Lovelock gravity. These corrections are designed to smooth out the violent infinities that typically plague black hole solutions, replacing the central singularity with a calm, inner horizon. The researcher set out to see if the microscopic counting method still holds true in this new, smoother environment. They did not rely on string theory or supersymmetry, which are often required for such calculations, but instead worked within a specific, manageable slice of the theory known as the mini-superspace formalism. This approach allowed them to reduce the complex equations of motion into a simpler form that could be solved exactly, even with the infinite number of corrections included.

The core of their work involved calculating the properties of a special, quiet state of space called a soliton. In this context, a soliton is a stable, non-rotating configuration of space that serves as the "ground state" or the lowest energy version of the universe in this theory. By performing a specific mathematical transformation on a standard black hole, the researcher derived the mass of this soliton. They found that, regardless of the strength of the various corrections in the theory, this ground state has zero entropy. This result is crucial because it confirms that the ground state is unique and non-degenerate, a necessary condition for the counting method to work. With the mass of this quiet state in hand, they applied the modified Cardy formula, which relies on the energy of the ground state rather than the central charge of the theory.

The results were strikingly consistent. When the researcher plugged the mass of the soliton and the mass of the black hole into the formula, the result matched perfectly with the semi-classical entropy calculated from the black hole's surface area. This agreement held true for all possible values of the coupling constants that define the strength of the gravitational corrections, including the specific cases that produce regular, singularity-free black holes. The study also extended these findings to rotating black holes. By applying a mathematical boost to the static solutions, they calculated the mass and spin of rotating black holes and verified that the counting formula still worked, accurately predicting the entropy even when the black hole was spinning.

A significant part of the discovery lies in what the formula does not use. In many previous attempts to explain black hole entropy, physicists relied on a quantity called the central charge, which describes how the symmetries of the theory behave at the edge of space. However, in this specific theory, the central charge does not match the entropy. The researcher showed that the correct counting relies instead on an "effective central charge" derived directly from the mass of the soliton. This distinction is vital because it proves that the theory, while possessing scale invariance, does not possess the full conformal symmetry that usually makes the central charge the key to the puzzle. The fact that the entropy can be reproduced without the central charge, using only the energy of the ground state, suggests a more robust mechanism for counting quantum states that works even when the underlying symmetries are less perfect.

The researcher also explored the nature of the "regular" black holes, where the infinite tower of corrections prevents the formation of a singularity. They found that their method works seamlessly for these objects as well. The thermodynamic properties, such as temperature and entropy, depend only on the geometry outside the event horizon, meaning the internal resolution of the singularity does not disrupt the counting process. The study confirms that the microscopic derivation of entropy is stable across the entire range of this theory, from the simplest black holes to the most complex, regularized versions.

In the end, the work provides a consistent picture of how gravity and quantum mechanics might interact in three dimensions. It demonstrates that the microscopic origin of black hole entropy can be understood through the properties of a quiet, soliton-like state, even in a theory with an infinite number of gravitational corrections. The findings rule out the idea that the standard central charge is the only way to count these states, showing instead that the energy of the ground state is the true driver. This clarity offers a new perspective on how to approach the quantum nature of gravity, suggesting that the rules for counting states are more flexible and resilient than previously thought, capable of surviving even when the geometry of space is smoothed out and the symmetries are altered.