Gravity from generalized entropy

Gravity is the force that keeps our feet on the ground and the planets in their orbits, but for physicists, it remains the most stubborn puzzle in the universe. While we have a precise mathematical description of how gravity works on large scales, and another for how particles behave on the smallest scales, the two theories refuse to speak the same language. For decades, scientists have searched for a bridge between them, often looking at the strange connection between gravity, heat, and information. It turns out that black holes, the most extreme gravitational objects, carry a kind of heat and information, suggesting that the very fabric of space and time might be made of something more fundamental, like data. This idea has led researchers to ask if the equations of gravity can be derived not from forces, but from the way information is organized and lost.

A researcher has taken this concept a step further, proposing a new way to understand gravity that treats the geometry of space itself as a quantum object. In their work, they suggest that the laws governing the universe can be found by measuring the difference between two ways of describing space: one that exists on its own, and another that is shaped by the matter and energy within it. They call this difference "entropy," a measure of disorder or information content. By building a theory where gravity emerges from this information gap, they have created a framework that naturally produces the familiar laws of Einstein's general relativity, but also reveals what happens when we look closer at the quantum scale.

The researcher began by revisiting a previous idea called "Gravity from Entropy," which identified the fundamental equation of gravity with a specific type of information difference known as geometric quantum relative entropy. In this view, the metric—the mathematical tool that defines distances and angles in space—is not just a static stage for events, but a dynamic quantum operator, much like a particle in quantum mechanics. This approach successfully reproduced Einstein's equations in the limit where gravity is weak, but it relied on a specific mathematical function, the natural logarithm, to calculate the entropy. The question remained: is the natural logarithm the only possible choice, or are there other ways to measure this information that would lead to different, perhaps more complete, theories of gravity?

To answer this, the researcher expanded the theory to include three broader families of entropy functions, known as Tsallis, Rényi, and Sharma–Mittal. These are generalizations of the standard entropy formula that have been used in other areas of physics to describe complex systems. By replacing the standard logarithm with these new functions, the researcher constructed a new family of theories they call "Gravity from Generalized Entropy." They then analyzed how these theories behave, checking if they still produce the correct laws of gravity in everyday situations and what new particles or forces might appear at the quantum level.

One of the most significant findings is that all these different entropy choices lead to the same result when gravity is weak: they all reproduce the leading terms of Einstein's theory of general relativity. This means that the core structure of the theory is robust; the specific choice of entropy function does not change the fundamental way gravity interacts with matter in the world we observe. However, the choice of entropy does control what happens at the very highest energies and smallest distances. The researcher found that for most of these generalized theories, the quantum description of gravity includes a new, massive particle with a spin of two. This particle is problematic because it behaves like a "ghost," a type of entity that would make the universe unstable if it were real.

The study identified a very specific condition under which this troublesome ghost particle disappears. The researcher found that if the entropy function is chosen in a particular way—specifically, if it behaves like a flat line near a certain point, or if the curvature of space enters the equations only through a single summary number—the ghost vanishes. In the case of the Tsallis family of entropies, this happens at a specific value where the theory becomes exactly equivalent to Einstein's general relativity. For the other families, this condition leads to theories that are free of ghosts but still contain a healthy, stable particle that could play a role in the early universe.

The researcher also examined the "horizon entropy," which is the amount of information stored on the boundary of a black hole. In standard theories, this is proportional to the area of the horizon. The new analysis shows that for these generalized theories, the entropy still follows an area law, but with small corrections that depend on the curvature of space. These corrections appear as constant terms and terms that decrease as the area gets larger, rather than the strange power-law behaviors sometimes predicted by other non-standard theories. This suggests that the area law is a more fundamental feature than previously thought, surviving even when the rules of entropy are changed.

Furthermore, the researcher explored how these theories handle the cosmological constant, the energy density of empty space that drives the expansion of the universe. They found that the way this constant emerges is tied to the mass of the scalar particle mentioned earlier. In the vacuum of space, the theory predicts a dynamic energy density that acts like a cosmological constant, but its value is determined by the specific entropy function chosen. This provides a mechanism for the cosmological constant to arise naturally from the geometry of space itself, rather than being an arbitrary number inserted by hand.

The work confirms that the idea of deriving gravity from information is a powerful and flexible framework. It shows that while the specific mathematical details of the entropy function can change the high-energy behavior of the universe, the low-energy world we live in remains consistent with Einstein's predictions. The presence or absence of unstable particles depends entirely on the shape of the entropy function, offering a clear path for testing these ideas. If the universe is indeed built from information, the researcher has shown that there are many ways to write the code, but only a few that result in a stable, predictable reality. The study leaves open the question of which specific entropy function nature has chosen, but it provides the tools to narrow down the possibilities and understand the deep connection between the geometry of space and the information it contains.