A classical black hole has a temperature and other thermodynamic properties. When physicists began accounting for changes in the cosmological constant, the energy of empty space, a strange coupling emerged in the equations. Changing the size of the universe simultaneously changes two seemingly different things: the theory's color charge (the number of 'colors,' or particle species) and its spatial volume. In ordinary thermodynamics, these are independent quantities, but here they go hand in hand. It's as if heating a kettle always changed the volume and color of the water together, strictly in lockstep.

In the new work, the scientists considered not an ordinary but a quantum black hole in so-called double holography. The idea is that our three-dimensional universe with gravity may be equivalent to a flat two-dimensional theory without gravity on its boundary: this is the holographic principle. In double holography, an extra layer is added: the boundary itself becomes a physical membrane with its own gravity. Then a classical black hole in the bulk turns into a quantum black hole of lower dimensionality, where the backreaction of matter on geometry is already taken into account.

It is precisely this backreaction that cuts the Gordian knot. Matter on the membrane provides an additional independent variable, the color charge, which is no longer rigidly tied to volume. The authors tested the mechanism on a concrete example: a quantum BTZ black hole in three-dimensional spacetime. It turned out that quantum corrections naturally introduce the missing degree of freedom. Thus, nature itself, without artificial contrivances, resolves the contradiction that recently attracted theorists' attention.