A black hole is not an eternal vacuum cleaner. Surround it with a non-ideal fluid (matter with internal friction), and the geometry starts to lead a life of its own. The authors of the paper showed that a special geometric flow, the almost η-Ricci–Yamabe soliton, acts like an invisible sculptor. It reshapes the curved spacetime around a static, spherically symmetric black hole.

This flow is no accident. Its scale is governed by thermodynamic friction along the radial direction, and at the event horizon it is directly linked to the Hawking temperature. A simple formula, α($r_H$) $S_{tt}$ = 2π $T_H$, fixes this connection. The kinematics of the flow depend on the fluid's equation of state: ρ = γ σ, where ρ is the energy density, σ is the pressure, and γ is a parameter that specifies the type of matter.

When the parameter γ drops to -1, the fluid enters the era of dark energy and violates the null energy condition (the sum of energy density and pressure becomes negative). At that moment, the geometric flow reshapes the black hole into something else — a traversable wormhole. The time coordinate flattens out, and the spatial cross-section expands outward, forming a throat. The key condition is that the soliton must dominate the local curvature: ω'($r_H$) > f''($r_H$).

At infinity, the soliton smoothly fades away, preserving the usual cosmological spacetime. An analysis of small perturbations showed that the flow introduces dissipation: the evolution of perturbations reduces to a damped wave equation. The geometry acquires a kind of friction that stabilizes the wormhole.