When a balloon with a painted pattern deflates, the pattern ripples and distorts. Something similar happens to a black hole that slowly evaporates, losing energy through the radiation discovered by Stephen Hawking. But instead of rubber, it's curved spacetime, and instead of a pattern, it's the boundary beyond which even light cannot escape.

Usually, we talk about the event horizon — an imaginary sphere from which nothing can break free. But an evaporating black hole has a more tangible boundary: the apparent horizon. It depends on the observer and responds sensitively to the slightest changes in mass. In the new work, physicists derived equations that describe how this horizon oscillates and changes its area as the hole "slims down." They even accounted for the backreaction — the influence of the radiation on the hole itself.

With the shape of the horizon in hand, the scientists were able to reconstruct the full four-dimensional metric — a complete map of spacetime around the black hole. They then moved to a quantum description and averaged this metric, obtaining an effective classical spacetime. Its causal structure can be depicted on a Penrose diagram — a special plot where light paths always travel at 45 degrees. Such a "quantum-corrected" diagram shows how the notions of "earlier" and "later" change near an evaporating hole.

Ultimately, the work paves the way for a full quantum theory of black holes. The next step is to apply a quantization procedure to the variables describing the horizon and see what happens to its area on a microscopic level.