Horizon entropy does not by itself determine a force between black holes

Black holes are often described as the ultimate cosmic traps, regions where gravity is so intense that nothing, not even light, can escape. For decades, physicists have understood these objects through a lens that combines gravity with thermodynamics, the branch of physics dealing with heat and energy. A famous rule established in the 1970s states that the surface of a black hole, known as its horizon, possesses an amount of disorder called entropy that is directly tied to its size. Just as a hot cup of coffee has more thermal energy than a cold one, a larger black hole has more entropy than a smaller one. This connection suggests that if you could somehow change the shape or size of a black hole's surface, you might be able to calculate a force, much like how stretching a rubber band creates tension. Some researchers have proposed that this entropic tension could act as a new kind of force between two black holes, potentially pushing them apart or pulling them together in ways that standard gravity does not predict.

A team of researchers has now investigated this idea with extreme precision, asking a simple but difficult question: if you have two black holes, does the way their surface entropy is distributed create a measurable force between them? They set out to test a specific theory that suggested such a force exists, one that would appear as a subtle repulsion between the two objects. By examining the exact mathematical description of two black holes sitting still in space, the team discovered that the proposed force is not a real physical interaction at all. Instead, they found that the calculation leading to this force relies on an arbitrary choice of how to label points on the black hole's surface. When the researchers accounted for the fact that these labels can be shifted without changing the physical reality of the black hole, the supposed force vanished. The study concludes that the horizon's entropy, by itself, does not determine a force between black holes, and that previous reports of such a force were the result of a mathematical artifact rather than a new law of nature.

To understand how they reached this conclusion, one must first look at how physicists describe the surface of a black hole. Imagine the horizon not as a solid shell, but as a flexible, two-dimensional skin floating in space. In the scenario the team studied, two black holes are placed near each other in a snapshot of time where they are momentarily at rest. The researchers wanted to see what happens to the entropy on the surface of one black hole as the distance to the other changes. If the entropy on the near side of the black hole were to change differently than the entropy on the far side, it might create a pressure difference, similar to how wind blowing on one side of a sail pushes a boat. The theory they tested assumed that you could track specific patches of this surface as the black holes moved, calculating how the entropy of each patch changed to find a resulting force.

However, the researchers found a fundamental problem with this tracking method. The surface of a black hole is a mathematical object that does not come with a built-in map or a set of street signs telling you which point on the surface corresponds to which point at a different moment. To perform the calculation, one must invent a way to match points on the first black hole to points on the second. The team showed that if you choose one way to match these points, you get a non-zero force. But if you choose a different, equally valid way to match them, the force changes completely. In fact, by simply shifting the labels on the surface, the calculated force could be made to disappear entirely or even flip its direction. This means the force is not a property of the black holes themselves, but a property of the arbitrary map used to describe them.

The team went further to identify exactly where this "ghost" force came from. They discovered that the apparent repulsion was entirely due to a specific distortion in how the surface area was distributed, a distortion that looked like a dipole, or a separation of positive and negative charges. In the language of the study, this dipole was not a real physical bulge or dent in the black hole's shape. Instead, it was an illusion created by the way the researchers were viewing the black hole from a distance. The companion black hole's gravity creates a uniform field that stretches the space around the first black hole, but this stretching is purely a matter of perspective. When the researchers adjusted their view to a "balanced frame"—a way of looking at the black hole where the surface area is distributed as evenly as possible given its shape—the dipole vanished. In this balanced view, the force disappeared, confirming that the repulsion was never there to begin with.

The study also checked whether this force might appear in a more realistic, dynamic situation where the black holes are actually moving and interacting over time. They simulated the evolution of the black holes using two different, well-established methods for tracking how the horizon changes. In both cases, the actual movement of the surface area did not match the pattern required to create the proposed force. The real physical changes in the black hole's surface were far too small and occurred at a much slower rate than the theory required. The researchers found that the specific pattern of area redistribution needed to generate the force simply does not happen in the way the theory predicted. The horizon does not transport the necessary "dipole" of entropy across its surface to create a push or a pull.

This work resolves a conflict between the idea of entropic forces and the established laws of gravity. If the proposed force were real, it would alter the way two black holes orbit each other, changing the timing of the gravitational waves they emit in a way that has not been observed. The researchers calculated that if this force existed, it would shift a specific coefficient in the orbital energy by a precise amount, which would have been detectable in recent observations of merging black holes. Since no such shift has been found, and since the mathematical derivation of the force relies on an arbitrary labeling choice, the conclusion is robust: the force does not exist. The entropy of a black hole horizon is a powerful concept that helps us understand the nature of gravity and information, but it does not, on its own, generate a mechanical force between two black holes. The apparent push was a mirage created by the way the calculation was set up, not a new feature of the universe.