Regular black holes do not violate the first law of thermodynamics

Deep in the heart of our universe, gravity can become so intense that it crushes matter into a point of infinite density, a place where the known laws of physics seem to break down. These are black holes, and for decades, scientists have been trying to understand what happens at their very center. A popular idea in theoretical physics suggests that if we combine Einstein's theory of gravity with a more complex version of electromagnetism—the force behind electricity and magnetism—we might find "regular" black holes. These are objects that look like black holes from the outside but have a smooth, finite center instead of a crushing singularity. To make sense of these objects, physicists often treat them like hot, glowing bodies that follow the same rules of heat and energy that govern steam engines or stars. This is the first law of thermodynamics, a fundamental rule stating that energy cannot be created or destroyed, only changed in form. For regular black holes, however, a confusing mismatch has appeared in the math, leading some researchers to believe this fundamental law of nature might need to be rewritten or that these objects possess a strange, hidden form of energy.

A team of physicists has now stepped in to resolve this puzzle, showing that the first law of thermodynamics remains perfectly intact for these regular black holes and that no new, exotic energy terms are required. The confusion arose because scientists were looking at the problem from the wrong angle. When they calculated how the mass of a regular black hole changes as it grows or shrinks, they were inadvertently changing the very rules of the universe that the black hole lives in. It is as if one were trying to measure the weight of a balloon while simultaneously changing the air pressure in the room; the result would be a number that doesn't fit the standard rules of physics, not because the rules are wrong, but because the conditions of the experiment were shifting. The researchers demonstrated that if you keep the underlying rules of the universe fixed—specifically the constants that define how gravity and electromagnetism interact—the standard thermodynamic laws hold true without any modification. The temperature of the black hole matches the heat flow, and the energy balance works exactly as expected.

The key to this discovery lies in understanding the difference between a single, specific black hole and the entire family of possible black holes. In the past, researchers often took a specific, smooth black hole solution and asked how its mass would change if they tweaked a single number in its description, like its size or charge. However, doing this in a regular black hole model actually changes the fundamental constants of the theory itself. It is like trying to study a specific type of tree by changing the soil composition every time you measure its height; you are no longer studying the tree, but a different ecosystem entirely. The authors showed that when you calculate the energy changes while keeping the "soil"—the fundamental constants of the theory—exactly the same, the math works out perfectly. The apparent mismatch, which had led some to propose a new, corrected form of internal energy, was simply an artifact of moving through different theories rather than staying within one.

Furthermore, the team proved that the strange correction factor that some scientists had tried to use to fix the energy equation is not a true measure of energy. In thermodynamics, for a quantity to represent a stored energy, it must behave in a very specific way: if you go in a circle and return to your starting point, the total change in energy must be zero. The researchers showed that this correction factor fails this test. It cannot be the differential of a local energy function, meaning it does not represent a real, physical energy stored inside the black hole. Instead, this factor is simply a mathematical ratio that describes how much of the black hole's total mass is actually located inside its event horizon compared to the total mass of the system. It is a geometric relationship, not a new form of energy.

The study also uncovered a subtle issue that arises in higher dimensions, specifically in universes with five or more spatial dimensions. In these cases, a mathematical singularity in the description of the electromagnetic field can appear just outside the black hole's event horizon for certain types of near-extreme black holes. While this does not invalidate the thermodynamic laws derived in the paper, it suggests that the physical description of the matter inside these higher-dimensional black holes might be incomplete or unstable in those specific regions. This finding highlights that while the energy balance is sound, the full picture of how matter behaves in these extreme environments still holds mysteries, particularly when we move beyond the four dimensions of our everyday experience.

Ultimately, this work restores confidence in the standard laws of physics when applied to these exotic, singularity-free black holes. The researchers have clarified that the first law of thermodynamics does not need to be modified, nor does the concept of entropy need to be changed. The apparent violations were illusions created by comparing apples to oranges—mixing different physical theories in a single calculation. By carefully separating the properties of the black hole from the properties of the theory that describes it, the authors have shown that the universe remains consistent. Regular black holes, if they exist, obey the same fundamental rules of energy and heat as the rest of the cosmos, governed by the familiar Hawking temperature and the area of their horizons. This clarity allows physicists to continue exploring the nature of these objects without the distraction of unnecessary corrections, focusing instead on the deeper questions of how gravity and matter interact at the very edge of reality.