Classical general relativity predicts that black holes harbor spacetime singularities, a sign that the classical description breaks down in the strong-field regime. Regular black holes have long been studied as effective models of singularity resolution, arising from nonlinear electrodynamics, effective matter sources, or quantum-gravity-inspired corrections. In these geometries the Schwarzschild behavior is recovered at large radii while the deep interior is modified to avoid a central singularity. Yet removing the central singularity is only part of the problem. Most regular black holes also contain an inner, or Cauchy, horizon that is famously sensitive to perturbations: ingoing and outgoing radiation are exponentially blueshifted as they approach it, driving rapid growth of the effective internal mass. This phenomenon, mass inflation, can destabilize the interior and undermine the semiclassical description, so a viable regular black hole must control the inner horizon as well as the core.

The new geometry is static and spherically symmetric, with an asymptotically Schwarzschild exterior, a non-extremal outer horizon, and a degenerate inner horizon whose surface gravity vanishes exactly. The metric function is built so that the outer horizon is a simple root while the inner horizon is an effective triple root, achieved through an auxiliary function that behaves quadratically near the inner horizon and grows rapidly toward the center. The authors note that a double zero would already give vanishing surface gravity but would not restore the correct sign of the metric function inside the core, so the effective cubic order is essential. The same cubic structure turns out to govern everything that follows, from the curvature bound to the response of the geometry to perturbations.

The headline result concerns the Kretschmann scalar, a curvature invariant whose value marks the strength of spacetime curvature. At the center the spacetime approaches flat Minkowski space, with the Ricci scalar, the quadratic Ricci invariant, and the Kretschmann scalar all vanishing as the radius goes to zero, independently of the black hole’s mass. More importantly, in the large-mass regime where the outer horizon radius equals twice the ADM mass, the maximal Kretschmann scalar becomes nearly independent of the mass and is controlled mainly by the inner-horizon radius. Rewriting the metric in dimensionless variables shows that the leading curvature contribution scales as the inverse fourth power of the inner-horizon radius, with mass dependence entering only as subleading corrections. Numerically, for a fixed inner-horizon radius of 20 in Planck units, the maximal curvature converges to about 0.00866 as the mass grows from 10,000 to 1,000,000 Planck masses, with the curves for the two largest masses nearly indistinguishable.

This mass-independence yields a simple sufficient condition for global regularity in the strongest sense. Because the curvature vanishes both at the center and at infinity, its global maximum must occur at a finite radius, which the calculations place inside the inner horizon. In Planck units, choosing the inner-horizon radius to be at least approximately 6.10174 guarantees that the Kretschmann scalar stays below 1, the Planck value, everywhere in the spacetime. Since Planck-scale curvature marks the regime where higher-curvature and quantum corrections can no longer be assumed negligible, this limiting-curvature property means the geometry never enters territory where the effective description would fail. The location of the maximum also tracks the inner-horizon radius, remaining inside it throughout the parameter range studied.

The authors also examined the effective stress-energy tensor reconstructed from the geometry through the null energy condition, a diagnostic of whether the source matter behaves reasonably. Along radial null directions the condition is identically saturated, with the radial pressure exactly canceling the energy density. The nontrivial angular component is strictly satisfied at the inner horizon and at the regular center, and it approaches zero at large distances. For a representative configuration with mass 1,000,000 in Planck units and inner-horizon radius 20, the most significant violation is confined to a finite shell between roughly 2.54 and 9.46, well below the inner horizon, and is strongly suppressed elsewhere. The model therefore does not evade energy-condition scrutiny entirely, but it confines the violation to a controlled interior region.

The second major result addresses mass inflation directly. A massless scalar test field propagated on the fixed background reveals how the cubic zero of the metric function reshapes the near-horizon physics. For a generic nondegenerate inner horizon, outgoing null rays approach the horizon exponentially and the geometric blueshift factor grows exponentially with advanced time, the mechanism behind runaway mass inflation. In the new geometry, the cubic degeneracy forces outgoing rays to approach the inner horizon only algebraically, and the blueshift factor grows as a power law, proportional to the three-halves power of advanced time. The effective potential governing the scalar modes is likewise cubically suppressed near the horizon, so wave propagation directly reflects the same near-horizon structure that controls the curvature bound.

To test the consequences for the internal mass, the authors employed two complementary effective descriptions. In the double-null shell model of Dray, ‘t Hooft, and Redmount, an ingoing and an outgoing null shell intersect inside the hole, and the nonlinear shell-crossing term in the matching condition is what drives standard mass inflation. Because the inner-horizon radius is held fixed while the ADM mass varies, the first nonvanishing response of the Misner-Sharp mass to a change in the mass parameter is cubic in the distance from the inner horizon, rather than quadratic as in families where the inner horizon moves with mass. Combined with the algebraic approach of the crossing point, this cubic response makes the nonlinear contribution decay as the inverse three-halves power of advanced time at late times, provided the shell amplitudes remain finite or decay sufficiently fast. The future Misner-Sharp mass then approaches the finite value r-minus over two, half the inner-horizon radius.

The Ori model, in which the outgoing perturbation is a null shell and the ingoing perturbation is a continuous stream of infalling radiation following a Price-law tail with decay exponent 12, tells the same story. Numerically solving the coupled evolution, the authors found that the internal Misner-Sharp mass remains bounded and settles onto a plateau at late times. For an inner-horizon radius of 20, the mass approaches 10, exactly half the inner-horizon radius, in agreement with the geometric expectation that the mass function tends to r-minus over two as the shell asymptotes to the degenerate horizon. Within both effective descriptions, the exponential amplification characteristic of a nondegenerate Cauchy horizon is absent, and the same inner-horizon scale that bounds the global curvature fixes the finite mass scale reached by the dynamics.

The study comes with clear caveats. The analysis treats the inner-horizon radius as an independent fixed scale; if that radius depends on the mass, additional terms appear in the mass response and the near-horizon power laws can change, requiring a separate analysis. The curvature discussion covers invariants built algebraically from the Riemann tensor, not those involving covariant derivatives of the curvature, and the shell and scalar-field probes are test-field and effective models rather than fully nonlinear evolutions. Previous fully nonlinear simulations of scalar collapse in related geometries have shown that sufficiently strong perturbations can still drive an inner horizon to zero, so the stability question is not closed. What the new work demonstrates is a proof of principle: a single explicit geometry can combine a Minkowski core, a softened inner horizon, and a globally sub-Planckian curvature bound, with one internal scale governing both the geometry and its response to perturbations.

Subject of Research: A theoretical model of a regular black hole with a Minkowski core and a degenerate inner horizon that bounds curvature and suppresses mass inflation

Article Title: Global curvature control in a regular black hole with Minkowski core and inner-extremal horizon

Article References: Feng, Z.-W., Liu, H.-L., Ling, Y., & Jiang, Q.-Q. (2026). Global curvature control in a regular black hole with Minkowski core and inner-extremal horizon. The European Physical Journal C, 86(10), Article 1205. https://doi.org/10.1140/epjc/s10052-026-16504-3

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16504-3

Keywords: regular black holes, singularity resolution, mass inflation, inner horizon, Kretschmann scalar, Planck scale curvature, surface gravity, null energy condition, Misner-Sharp mass, general relativity, theoretical physics, Cauchy horizon

Unimodular gravity is not a new idea; Einstein himself toyed with a version of it. The theory imposes a strict constraint that the determinant of the metric, the mathematical object encoding distances and times in spacetime, must remain fixed. This restriction shrinks the full freedom of general relativity down to transformations that preserve volumes, and it changes the field equations into the traceless part of Einstein’s equations. One celebrated consequence is that the cosmological constant, the mysterious energy density driving the accelerated expansion of the universe, emerges as a constant of integration rather than a parameter inserted by hand into the theory. Another, more radical consequence is that energy-momentum conservation is no longer automatically enforced, allowing matter and geometry to exchange energy in ways that general relativity forbids.

The research team focused on a classic family of wormhole solutions of the Morris-Thorne type, static and spherically symmetric geometries in which the shape of the tunnel is controlled by a power-law shape function. The exponent of that power law, which the authors call xi, is determined by two parameters alpha and beta describing a barotropic equation of state, in which the radial pressure is proportional to the energy density and the tangential pressure is proportional to the radial one. Requiring the throat of the wormhole to flare outward, the geometric signature of a traversable tunnel, restricts these parameters to specific regions of the parameter space. Within that allowed domain, the researchers proved that the combination of energy density and radial pressure entering the null energy condition is strictly negative at every radius. Both the null and weak energy conditions are violated everywhere in the spacetime, confirming that unimodular gravity does not eliminate the need for exotic matter.

Then comes the study’s most striking conceptual move. Because the motion of any test particle, whether a spacecraft, a photon, or a falling astronaut, is dictated entirely by the spacetime metric, the researchers showed that the geodesic structure of the wormhole is absolutely identical in unimodular gravity and general relativity. Solving the geodesic equations on the equatorial plane, using the conserved energy and angular momentum that follow from the time-translation and axial symmetries of the metric, they found two qualitatively distinct classes of trajectories. Particles with sufficiently high energy cross the throat and emerge in the opposite asymptotic region of the wormhole, while lower-energy trajectories are reflected at a finite-radius turning point before ever reaching the throat. The transition between these regimes depends only on the geometry and the conserved quantities, so no experiment based on the motion of particles through the wormhole could ever distinguish the two theories.

If the geometry and the kinematics are the same, where does unimodular gravity actually differ from Einstein’s theory? The answer, the authors demonstrate, lies in the source sector, the matter and vacuum energy required to hold the wormhole open. They explored three comparison strategies. In the first, they imposed local energy-momentum conservation as an extra condition on the unimodular solution. This turns out to be a severe restriction: the two-parameter family of barotropic wormholes collapses to a one-parameter subset in which the exponent xi is fully fixed by alpha alone. Remarkably, this restricted solution coincides exactly with the unique wormhole obtained by solving the full Einstein equations directly, a consistency check guaranteed by the Bianchi identity, the mathematical statement that underlies energy conservation in general relativity.

The second strategy is more permissive. Instead of forcing energy-momentum conservation, the researchers allowed the spacetime-dependent effective vacuum contribution, which they call the internal cosmological constant, to act as part of the source. In this picture, the same wormhole geometry can be embedded within general relativity by adding an inhomogeneous vacuum energy component whose density varies with radius and whose pressure is equal and opposite, exactly like a vacuum fluid. The price of this interpretation is that the original barotropic equation of state is replaced by a position-dependent effective fluid with radius-varying equations of state. The wormhole looks the same from the outside, but the stuff holding it open has been fundamentally reinterpreted.

To quantify how much exotic matter each scenario demands, the team employed the Volume Integral Quantifier, a standard diagnostic that integrates the null-energy-condition-violating combination over the entire wormhole spacetime. The integral converges only when the exponent xi is less than minus one, and the authors mapped the full parameter plane into three regions: one where no traversable wormhole exists at all, one where the geometry is admissible but the total exotic matter content diverges to infinity, and a physically complete sector where the throat exists and the exotic matter is finite. Intriguingly, the quantifier is blind to the vacuum contribution, because the vacuum’s density and pressure cancel exactly in the relevant combination. Both theories can share the same integrated exotic matter content while differing in how that matter is conserved, which means the amount of exotic matter alone cannot settle the debate between the two theories.

The most conceptually rich result concerns the fate of the cosmological constant itself. In unimodular gravity, the effective vacuum term acquires a radial profile whenever the non-conservation current is active. The researchers solved for this profile exactly, imposing the natural boundary condition that the vacuum energy approaches a constant value at spatial infinity, and found that the value at the wormhole throat is generally shifted away from that asymptotic constant by an offset they call delta Lambda. The sign and magnitude of this shift are controlled by the same combination of parameters that governs the energy-momentum exchange between matter and geometry. Only on the special curve where the offset vanishes, precisely the curve corresponding to the exact general-relativistic limit, does the vacuum energy remain uniform throughout the spacetime. The approach to the asymptotic value follows a rapid power-law decay, and for strongly flared geometries the throat-to-infinity offset can grow without bound.

This mechanism has no analogue in Einstein’s theory, where the cosmological constant is a fixed parameter of the action and cannot vary across spacetime. The authors are careful to note that this does not solve the cosmological constant problem, the decades-old puzzle of why vacuum energy is so absurdly small compared to quantum estimates. But it identifies a genuinely new geometric mechanism: compact objects in unimodular gravity can generate local shifts in the effective vacuum energy, raising the question of whether an ensemble of such structures could contribute to an emergent cosmological background. It is a speculative but testable-in-principle avenue for future work.

The broader lesson is profound. Two gravitational theories can produce exactly the same spacetime geometry, the same bending of light, the same particle trajectories, and the same traversable tunnel, while requiring fundamentally different matter and vacuum sectors to sustain it. Spacetime geometry alone does not uniquely determine the physical content of a gravitational system. For wormhole hunters, this means that observations based on geodesic motion are powerless to discriminate between unimodular gravity and general relativity when the geometry is shared; the discriminating evidence, if it exists, is hidden in the source sector. Wormholes, it turns out, are not just cosmic shortcuts. They are precision instruments for dissecting what we actually mean when we say a theory of gravity is different from another.

Subject of Research: Traversable wormholes in unimodular gravity versus general relativity

Article Title: On the geometrical and dynamical distinction between unimodular and general relativistic wormholes

Article References: Bosquez, M., Pastén, E., Cataldo, M., & Cruz, N. (2026). On the geometrical and dynamical distinction between unimodular and general relativistic wormholes. The European Physical Journal C, 86(10), Article 1178. https://doi.org/10.1140/epjc/s10052-026-16472-8

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16472-8

Keywords: unimodular gravity, general relativity, traversable wormholes, exotic matter, null energy condition, cosmological constant, geodesics, Morris-Thorne wormholes, vacuum energy, energy-momentum conservation, theoretical physics, spacetime geometry

To appreciate why this matters, one has to understand what ModMax actually is. Maxwell’s equations, formulated in the 1860s, are remarkably special: they are conformally invariant, meaning they look the same at every scale, and they possess a continuous electric–magnetic duality that swaps electric and magnetic fields in an elegant rotation. For decades, theorists wondered whether any nonlinear extension of electromagnetism could preserve both of these prized symmetries simultaneously. The answer, established by Bandos, Lechner, Sorokin and Townsend, is that exactly one such theory exists in four dimensions: ModMax. It depends on a single deformation parameter, conventionally written as γ, and when γ is set to zero the theory collapses smoothly back to Maxwell. For nonzero γ, however, the electromagnetic response of the vacuum becomes genuinely nonlinear, and the familiar solution-generating machinery of Maxwell theory largely breaks down.

That breakdown is precisely what makes exact solutions so valuable and so rare. A substantial catalog of Einstein–ModMax solutions has accumulated in recent years — Reissner–Nordström-type black holes, gravitational waves, Taub–NUT geometries, dyonic extensions with NUT wormholes, accelerating black holes, Melvin–Bonnor electromagnetic universes, and even exact multi-black-hole and black-dihole spacetimes. But there is a catch, and it is a deep one. As Ortaggio has shown, broad classes of non-null Einstein–Maxwell configurations can be converted into solutions of conformally invariant nonlinear electrodynamics by a simple constant rescaling of the electromagnetic field. This means many of the known ModMax solutions are, in a sense, Maxwell solutions in disguise. The truly interesting question is whether the nonlinear theory supports exact solutions that have no Maxwell ancestor at all — and the rotating sector, in particular, has proven notoriously rigid. Even in the slow-rotation limit, the standard Maxwell-inspired ansatz generally fails to remain consistent for nontrivial nonlinear electrodynamics, and Newman–Janis-type transformation tricks, often used to manufacture rotating solutions, generally fail to preserve the full field equations.

The difficulty multiplies when a dilaton — a scalar field coupled to the electromagnetic sector — enters the picture. In Einstein–Maxwell–dilaton theory itself, asymptotically flat, charged, rotating black holes are known in closed form only for very special choices of the dilatonic coupling; generic couplings have historically required perturbative treatments and, only recently, fully numerical constructions. Against this backdrop, Bixano and Matos started from a framework they had developed in earlier work: a generalized Ernst-potential formulation for Einstein–ModMax–scalar-field systems. The Ernst formalism, a classic tool of stationary axisymmetric gravity, reduces the coupled field equations to a more tractable structure, and the authors’ extension of it to the ModMax setting provided the technical key. The Lagrangian they work with is written in the Einstein frame, combining the Ricci scalar, the kinetic term of the scalar field with a sign parameter that distinguishes an ordinary dilaton from a phantom field, and an exponentially coupled ModMax Lagrangian built from the two electromagnetic invariants F and G, the latter being the dual invariant involving the dual Faraday tensor.

The new solution belongs to a specific nonlinear sector in which the ratio of the two electromagnetic invariants remains constant across the spacetime. It is stationary and axisymmetric, carrying a nontrivial gravitomagnetic potential ω that encodes frame dragging, together with both a time component A_t and an azimuthal component A_φ of the electromagnetic four-potential, and a nonconstant scalar field φ. Written in spheroidal coordinates, the family comes in an oblate branch and a prolate branch, and it admits configurations with NUT charge — the gravitational analogue of a magnetic monopole’s twist — as well as a NUT-free, asymptotically flat limit. Crucially, the solution exists only in the dilatonic sector, with the sign parameter fixed to the ordinary, non-phantom value.

The most striking feature emerges when one asks whether the solution can be deformed back to Maxwell. Within the authors’ parametrization, the ModMax deformation parameter η₀ is not free: the field equations force it to satisfy the constraint η₀ = (2α₀² − 5)/(2√2 α₀), where α₀ is the dilatonic coupling. Maxwell theory corresponds to η₀ = 0, which is reached only at the isolated coupling α₀² = 5/2. For every established value of the coupling — the low-energy effective superstring value α₀² = 1, the Kaluza–Klein value α₀² = 3, the Einstein–ModMax value α₀ = 0, and the entanglement-relativity value — the parameter η₀ is unavoidably nonzero. In other words, for all the physically motivated dilatonic theories, this rotating configuration is intrinsically ModMax. A Maxwell description, if it exists at all, is confined to a single isolated point in coupling space corresponding to a theory that is not among the standard models. This is the sense in which the solution has no Maxwell analogue: it is not a rescaled Einstein–Maxwell–dilaton spacetime but a genuinely nonlinear creature.

The authors then confronted the singularities. Curvature invariants reveal a ring singularity at the origin of the spheroidal coordinates, two surface singularities along x = ±y in the prolate branch, and an additional surface singularity whose location depends on the parameters τ₀ and λ₀. A careful analysis shows that when the parameter λ₀ is smaller than τ₀, this third singularity is confined to the region x < 1, outside the physical exterior domain of the prolate branch. This matters enormously, because in the prolate sub-extreme regime the surface x = 1 turns out to be an event horizon. The dragging function evaluated on that surface is constant, which means the Killing vector combining time translation and rotation with the horizon angular velocity Ω_H = f₁/(Lλ₀) becomes null there — the defining signature of a genuine event horizon. In Boyer–Lindquist-type coordinates, the horizon sits at r_H = l₁ + L, where l₁ is the NUT-related offset.

Even more remarkably, the exterior region of this prolate black hole satisfies the null energy condition — the requirement that T_{μν} l^μ l^ν ≥ 0 for every null vector l^μ — throughout the entire domain 0 < λ₀ ≤ τ₀ < 1, x ≥ 1, and −1 ≤ y ≤ 1. The proof is a piece of mathematical craftsmanship: the authors show that each bracketed polynomial factor in the numerator of the energy expression is non-negative over the physical domain, and that the remaining term, viewed as a quadratic polynomial in the variable u = y², is concave, so its minimum occurs at the endpoints where it evaluates to manifestly non-negative quantities. With the denominator strictly positive in the same parameter range, the null energy condition holds. Consequently, the black hole respects Penrose’s weak cosmic censorship conjecture: all curvature singularities remain hidden behind the event horizon. The authors note that in the limiting case λ₀ = τ₀ the energy condition is violated at the poles and some singularities touch the horizon there, so the inequality must be strict for the physically clean sector.

The conserved charges reveal one final, structurally elegant surprise. The NUT parameter, the Komar mass, the Einstein–Maxwell–dilaton-type electric charge, and the ModMax electric and magnetic charges are all proportional to a single parameter τ₀, while the Komar angular momentum J is independently controlled by λ₀. This produces an unusually clean separation between the monopolar and rotational sectors of the solution. Setting τ₀ = 0 simultaneously switches off the mass, the NUT charge, and all the electric and magnetic monopole charges, yet leaves the angular momentum nonzero — yielding an asymptotically flat spacetime that is still nontrivial, describing a pure mass dipole accompanied by ModMax electric and magnetic dipole moments. Moreover, the EMD-type charge and the ModMax electric charge can never be made equal, which the authors take as a strong indication that no analogous solution exists in the Einstein–Maxwell–dilaton framework. A complete linear stability analysis, including coupled gravitational, electromagnetic, and dilatonic perturbations and their dependence on the ModMax parameter, is currently in progress and will be presented separately. For now, theorists have a new benchmark: an exact rotating dilatonic black hole that exists only because Maxwell’s theory was left behind.

Subject of Research: Exact rotating dilatonic black hole solutions in Einstein–ModMax nonlinear electrodynamics without a Maxwell analogue

Article Title: Exact rotating dilatonic branch in ModMax electrodynamics without Maxwell analogue

Article References: Bixano, L., & Matos, T. (2026). Exact rotating dilatonic branch in ModMax electrodynamics without Maxwell analogue. The European Physical Journal C, 86(8), Article 1020. https://doi.org/10.1140/epjc/s10052-026-16268-w

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16268-w

Keywords: ModMax electrodynamics, black holes, nonlinear electrodynamics, dilaton, Ernst potentials, event horizon, null energy condition, cosmic censorship, Kaluza–Klein theory, superstring theory, NUT charge, general relativity

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