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A black hole is like a bottomless well. You can measure how much water has flowed into it by watching the level from outside. But how do you find the mass of the water itself if you're standing inside the well? That's roughly the problem the authors of a new paper tackle.

They have built a 'scale' for the mass enclosed in a bounded region of space — say, around a black hole's horizon. Such a mass is called quasilocal (almost local). It turns out that this scale always shows a positive number if there's a black hole inside. And the reading never goes down: the mass only grows as matter falls into the hole.

The proof relies on two scenarios. The first is a compact region whose boundary is the horizon itself. The second is an infinite region that contains the horizon inside. In both cases, the mass is strictly greater than zero and does not decrease over time. This resembles a conservation law, but for the geometry of spacetime itself.

The result matters for general relativity: it provides a mathematically rigorous way to talk about mass in strong gravitational fields, without resorting to global definitions that only work at infinity.

Original abstract

as written by the authors arXiv:2606.07781

We define a quasilocal mass of Bartnik type, and establish its positivity and temporal monotonicity properties for two classes of domains associated with black holes. More precisely, we first show that the quasilocal mass is strictly positive for spacelike hypersurfaces that are: compact with apparent horizon boundary or noncompact with asymptotically flat ends and containing an apparent horizon in any admissible extension. Secondly, we show that the quasilocal mass is monotonically nondecreasing in time within evolutionary scenarios related to the two aforementioned settings.