A black hole is like a perfect vacuum cleaner: anything that crosses its horizon vanishes without a trace. But this vacuum cleaner has a strange feature: its "dust bag" grows, and by the size of the bag you can tell how much has been sucked in. In black hole physics, this size is linked to entropy entropy — a measure of hidden information about what fell inside.
The authors of the new work took not just an ideal black hole but a dynamic one — one that is actively accreting matter. Their method was reminiscent of studying ripples on water from a thrown stone: physicists looked at small deviations (perturbations) from a calm state, but not of first order, rather of second order. It's like noticing not only the first splash but also the next, fainter one. To account for the contribution of infalling matter, they introduced a special quantity, modified canonical energy, and derived a balance equation: the change in entropy is related to the energy flux through the horizon.
The main result is this: when the infalling matter satisfies the null energy condition (that is, its energy is positive for any observer), the entropy to second order exactly equals the area of the so-called apparent horizon. And this entropy obeys the second law: it does not decrease. Even if the null condition is violated, the area law may still hold, opening the door to studying exotic matter. Thus, a black hole, like a meticulous accountant, keeps track of everything that enters it — only it records not in a ledger but in the geometry of space.
The entropy of a black hole with a mass like the Sun's is about $10^{77}$ times greater than the entropy of the Sun itself. That's a number with 77 zeros, a black hole is a true champion of disorder.