Wormholes have fascinated physicists since Morris and Thorne’s landmark 1988 paper, which described tunnels connecting distant regions of the universe that matter and light could safely traverse. The catch has always been exotic matter: to hold a wormhole throat open against its own tendency to collapse, the spacetime must violate the Null Energy Condition, meaning the sum of energy density and pressure must go negative. Ordinary matter never does this. Most previous approaches therefore began by assuming a mathematically convenient shape function for the throat and then calculated what strange substance would be needed to support it. The new study flips that logic entirely.

Singh and Jaswant start from a four-dimensional, static, spherically symmetric spacetime — the classic setting for a charged black hole embedded in a universe with a cosmological constant, described by the Reissner–Nordström–de Sitter metric. Rather than treating the surrounding matter as an idealized perfect fluid, they model it as an imperfect fluid, complete with radial heat flux and anisotropic shear stress, a more faithful picture of the swirling, friction-laden accretion disks that feed real black holes. Onto this geometry they impose an almost eta-Ricci–Yamabe soliton, a hybrid of the Ricci and Yamabe geometric flows that smooth and deform curved spaces, with all coupling parameters allowed to vary smoothly with the radial coordinate.

The soliton equation balances a deformation term, the Ricci curvature, a Yamabe-type scalar curvature contribution, and a scaling factor omega multiplied by a one-form aligned with the radial direction. Because the spacetime is static and spherically symmetric, these parameters can depend only on radius, allowing the geometric flow to dynamically adapt to the thermodynamic state of the surrounding fluid. Solving the temporal and radial components of the soliton equation yields explicit expressions for the expansion parameter lambda and the scaling factor omega in terms of the metric function, the Ricci tensor, and the coupling functions — the mathematical machinery that drives everything that follows.

The first major result concerns the black hole’s apparent horizon, the surface where the metric function vanishes. For the geometric flow to remain finite there, the authors prove using L’Hôpital’s rule that a precise cancellation must occur: the product of the Ricci coupling and the temporal curvature component at the horizon must equal half the metric’s radial slope. Remarkably, that quantity is exactly the surface gravity, which Hawking’s famous relation ties to the horizon temperature. The result is a striking thermogeometric identity — alpha times the temporal curvature equals two pi times the Hawking temperature — linking the soliton’s regularization of the horizon directly to black hole thermodynamics.

The pivotal transition arrives when the accretion fluid enters a dark-energy-like regime. Under a barotropic equation of state relating pressure to density, the fluid’s state parameter gamma classifies cosmic eras: dust, radiation, and dark energy. When gamma reaches minus one, the fluid develops extreme negative radial pressure and violates the Null Energy Condition. The authors show that in this regime the soliton’s scaling factor becomes strictly positive at the horizon, and through an exact mapping onto the Morris–Thorne wormhole metric, this positivity forces the flare-out condition — the requirement that the throat geometry opens outward rather than pinching shut. Simultaneously, the temporal coordinate is regularized so the redshift function stays finite, preventing an event horizon from forming. Both criteria together constitute a rigorous topological transition from black hole to traversable wormhole, generated endogenously by the flow rather than assumed beforehand.

The analysis goes further, establishing that the transition is stable and localized. At the throat, the radial derivative of the soliton’s scaling factor strictly dominates the local curvature gradient, providing the structural repulsion needed to keep the passage open. Far away, the exotic stress decays exponentially, and the soliton’s influence fades smoothly: in an asymptotically flat universe its effect vanishes entirely, while in a de Sitter background it converges to the ordinary cosmological expansion. The wormhole’s exotic geometry is thus confined to the throat, leaving the deep-space universe untouched — a property the authors call cosmological safety.

Stability under gravitational waves is addressed through a tensorial perturbation analysis. Introducing a first-order metric perturbation and working in the transverse-traceless gauge, the authors show that the soliton flow converts the wave equation into a damped wave equation, with a friction term proportional to the flow’s coupling parameter. Provided that parameter remains positive, incoming perturbations decay rather than grow. In a concrete numerical example with mass one, charge 0.60, and cosmological constant 0.05, the apparent horizon sits at roughly 1.93 units, the scaling factor at the throat evaluates to about 0.37, and the perturbation frequency squared is positive — confirming that the throat dampens shocks and remains linearly stable.

The authors are careful to state the framework’s central limitation. Because the model operates within standard general relativity, the topological transition still depends on real matter entering the dark energy era and violating the Null Energy Condition — the long-standing exotic matter problem of classical wormhole physics. Their proposed remedy is to integrate the endogenous geometric flow into modified f(R) gravity, where higher-curvature terms could play the role of effective exotic matter and allow the flare-out condition to be satisfied with ordinary, positive-pressure matter. Extensions to rotating Kerr-like geometries and global stability checks via quasinormal mode analysis are also on the agenda.

For now, the result stands as a compelling piece of mathematical physics: a demonstration that geometric flows of the Ricci–Yamabe family, coupled to realistic imperfect fluids, can de-singularize a black hole horizon and organically sculpt a traversable wormhole throat, with thermodynamic fingerprints of Hawking radiation woven into the mechanism. No one expects a passable spacetime tunnel to appear in the sky tomorrow, but the study reframes wormhole formation as an emergent consequence of geometric evolution — and offers a concrete mathematical pathway that future work in modified gravity may be able to travel without exotic matter at all.

Subject of Research: Geometric soliton flows transforming black hole spacetimes into traversable wormholes in general relativity

Article Title: Traversable wormhole de-singularization: almost (\eta )-Ricci–Yamabe solitons in static spherically symmetric imperfect fluid spacetimes

Article References: Singh, J. P., & Jaswant (2026). Traversable wormhole de-singularization: almost $$\eta $$-Ricci–Yamabe solitons in static spherically symmetric imperfect fluid spacetimes. The European Physical Journal C, 86(10), Article 1141. https://doi.org/10.1140/epjc/s10052-026-16321-8

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16321-8

Keywords: traversable wormholes, black holes, Ricci–Yamabe solitons, geometric flow, imperfect fluid, Null Energy Condition, Hawking temperature, Morris–Thorne metric, general relativity, dark energy, apparent horizon, stability analysis