Light passing by a black hole swirls like water down a bathtub drain. The most intriguing part is the thin ring where rays can orbit forever, neither falling in nor escaping. This is the photon sphere — an invisible boundary that paints the hole's shadow on the sky.

The authors of the study proposed a new way to locate this boundary — not by solving cumbersome equations, but by measuring the curvature of space itself, much as a surveyor measures the bend of a hill. They used Gaussian curvature (a quantity familiar from saddle-shaped chips or the surface of a ball) and derived radii for light and massive particles. For particles with mass, this calculation is done for the first time.

The method turned out to be not only elegant but also powerful: it allows tracking how the photon sphere changes as a black hole gains or loses mass. It's like watching how much wider a funnel gets when more water pours in. Such calculations will be useful for new telescopes that capture black hole shadows and will help verify whether we truly understand gravity in its most extreme manifestations.