Dynamical Formation of Self-Similar Wormholes

Abstract: We study spherically symmetric, self-similar wormhole solutions supported by colliding streams of negative-energy null dust, and their dynamical formation. Under the assumption of self-similarity, the Einstein equations reduce to a system of ordinary differential equations, which we solve numerically under boundary conditions enforcing the existence of a minimal areal radius (the throat) on constant-time hypersurfaces. For a sufficiently large throat radius, the resulting geometries remain regular at both spatial and future null infinity, while a singularity is retained in the past direction. We then construct a dynamical formation scenario by patching together three regions: a Schwarzschild black hole, negative-energy Vaidya spacetimes, and the self-similar wormhole geometry. These regions are joined across null shells using the Barrabes--Israel formalism, which provides explicit relations among the throat radius, the black hole's mass and the energy injection by the shell, demonstrating that an initial black hole can evolve into a wormhole. Our analysis generalizes the formation model for static wormhole solutions proposed by Hayward and Koyama in 2004 to non-static wormhole solutions, offering a novel perspective on the formation of regular traversable wormholes.

• The paper constructs time-dependent, spherically symmetric traversable wormholes supported by colliding negative-energy null dust and reduces Einstein’s equations to a single nonlinear ordinary differential equation through self-similarity.
• Numerical solutions show that throat size controls asymptotic regularity: for A_th > 0.14, wormholes avoid spatial and future-null pp-curvature singularities, although a past singularity remains unavoidable.
• The paper matches Schwarzschild, Vaidya, and wormhole regions with thin-shell junction conditions, demonstrating that negative mass injection shrinks the horizon and exposes a throat inside the original black hole.

The paper "Dynamical Formation of Self-Similar Wormholes" (2602.13609) constructs time-dependent, spherically symmetric wormhole solutions supported by colliding streams of negative-energy null dust, and embeds them into an explicit formation scenario in which a Schwarzschild black hole is converted into a traversable wormhole by thin null shells of negative energy. The work extends the static Hayward wormhole [gr-qc/0202059] and the Hayward–Koyama formation model [gr-qc/0406080, gr-qc/0406113] to non-static geometries.

Self-similar reduction of the Einstein equations

The authors begin with the general spherically symmetric ansatz in null coordinates $(u,v)$ sourced by two counter-propagating null dust streams, $T_{uu}$ and $T_{vv}$. Conservation of the stress tensor fixes the dust profiles in terms of arbitrary functions $\tau_u(u)$ and $\tau_v(v)$, and the Einstein equations reduce to two coupled wave equations for the metric functions $A(u,v)$ and $B(u,v)$. Eliminating $B$ yields a single closed, highly nonlinear PDE for $A$, whose solutions generate exact null-dust spacetimes.

Imposing a homothetic Killing vector $\xi = u\,\partial_u + v\,\partial_v$ (future-directed timelike, so $T_{uu}$0) and passing to coordinates $T_{uu}$1 with $T_{uu}$2 and $T_{uu}$3 renders the metric conformally static, with areal radius $T_{uu}$4. The field equations collapse to ODEs in $T_{uu}$5. The dust functions $T_{uu}$6 become constant; fixing them to $T_{uu}$7 encodes the null energy condition violation required for a traversable throat, and the coordinate rescaling freedom is thereby fixed, making the normalizations of $T_{uu}$8 and $T_{uu}$9 physically meaningful. A single second-order ODE for $T_{vv}$0 remains, with $T_{vv}$1 determined algebraically.

Throat conditions and asymptotic structure

The throat is defined as the minimal areal radius on constant-$T_{vv}$2 hypersurfaces, a choice shown in the appendix to coincide with the Maeda–Harada–Carr criterion for cosmological wormholes (0901.1153). Restricting to solutions symmetric under $T_{vv}$3, the throat sits at $T_{vv}$4 with boundary conditions $T_{vv}$5, $T_{vv}$6; the flare-out condition $T_{vv}$7 is then automatic. Requiring $T_{vv}$8 to remain timelike restricts the throat parameter to $T_{vv}$9.

The large-$\tau_u(u)$0 asymptotics are controlled by a single exponent $\tau_u(u)$1. The exact solution $\tau_u(u)$2 is degenerate ($\tau_u(u)$3), but perturbing about it yields the physical asymptotic behavior, with corrections decaying as $\tau_u(u)$4. Null geodesic analysis and curvature scalars then classify the global structure by the value of $\tau_u(u)$5:

Regime	Null geodesic	Spatial infinity	Future null infinity	Past null direction
$\tau_u(u)$6	complete	finite	finite	diverges
$\tau_u(u)$7	incomplete	diverges	finite	diverges
$\tau_u(u)$8	incomplete	diverges	diverges	diverges

For $\tau_u(u)$9, the Ricci component $\tau_v(v)$0 diverges along outgoing null geodesics at finite affine parameter, producing a pp-curvature singularity. Crucially, the authors trace this singularity to the divergent flux of negative energy in the outgoing null direction, $\tau_v(v)$1 — the singularity is not a geometric artifact but a consequence of unbounded negative-energy flux. In the past direction ($\tau_v(v)$2 or past null infinity), all solutions develop a scalar curvature singularity regardless of $\tau_v(v)$3: self-similarity forces the wormhole to emerge from a singular past.

Numerical solutions and the role of throat size

Numerical integration of the ODE with the symmetric boundary conditions confirms the analytic picture. For $\tau_v(v)$4, the fitted asymptotic parameter is $\tau_v(v)$5, and $\tau_v(v)$6 grows monotonically away from the throat, with embedding diagrams confirming the minimal-surface structure. The key quantitative finding is that $\tau_v(v)$7 decreases monotonically with $\tau_v(v)$8, asymptoting to $\tau_v(v)$9 as $A(u,v)$0. Since regularity at spatial and future null infinity requires $A(u,v)$1, the authors conclude that wormholes with sufficiently large throats ($A(u,v)$2) are free of singularities in both the spatial and future null directions. This is a substantive departure from the static Hayward solution and its charged extension (Koga et al., 26 May 2025), where singularities persist generically; here, throat size acts as a control parameter for singularity avoidance. The past-direction singularity, however, is unavoidable within the self-symmetric class.

Dynamical formation via the Barrabès–Israel formalism

The formation scenario patches three exact solutions — a Schwarzschild black hole of mass $A(u,v)$3, negative-energy Vaidya regions on both sides, and the self-similar wormhole — across spherically symmetric thin null shells using the Barrabès–Israel junction formalism [Barrabes:1991ng]. The shells are required to be pressureless dust, and left–right symmetry is assumed.

For the shell joining the black hole to the right Vaidya region, the pressureless condition is automatic, and the shell energy density is $A(u,v)$4, defining the (constant) injected mass $A(u,v)$5. The shell joining the Vaidya region to the wormhole is less trivial: the pressureless condition $A(u,v)$6 becomes a first-order ODE for the Vaidya mass function $A(u,v)$7, fully determining the flux profile in terms of wormhole data. Near the throat, $A(u,v)$8 at leading order, consistent with the negative-energy injection.

Continuity of the surface stress tensor at the intersection of the two shells yields the central relation of the paper:

$A(u,v)$9

where $B(u,v)$0 is the throat areal radius and $B(u,v)$1 fixes the shell insertion time. Two consequences follow. First, $B(u,v)$2: wormhole formation requires net negative energy injection, as expected. Second, $B(u,v)$3 implies $B(u,v)$4 — the future throat lies inside the original event horizon, so the negative-energy flux shrinks the horizon and exposes a throat that was previously hidden in the black hole interior. Moreover, $B(u,v)$5, so larger mass loss produces larger throats.

A structural difference from the static Hayward–Koyama construction deserves emphasis: the throat radius here depends not only on the black hole mass but also on the insertion time $B(u,v)$6, directly reflecting the time-dependent growth of the throat in the self-similar geometry.

Limitations and open questions

Several caveats qualify the results. The past-direction curvature singularity is unavoidable in the self-similar class, so these geometries do not constitute globally regular wormholes; the formation scenario inherits an initial singularity. The analysis is restricted to spherical symmetry and to the specific null-dust source with fixed $B(u,v)$7; whether other matter content or symmetries admit singularity-free time-dependent wormholes is not established. The physical mechanism by which large throats evade the pp-curvature singularity — beyond the empirical correlation between $B(u,v)$8 and $B(u,v)$9 — remains unexplained. Finally, the linear stability of the self-similar wormholes is untested, leaving their dynamical viability as solutions open.

Conclusion

The paper provides a tractable, self-similar class of time-dependent wormholes supported by negative-energy null dust, classifies their singular structure through the asymptotic parameter $B$0, and demonstrates numerically that sufficiently large throats ($B$1) yield geometries regular at spatial and future null infinity. The Barrabès–Israel matching of Schwarzschild, Vaidya, and wormhole regions yields an explicit relation between the initial black hole mass, the injected negative energy, and the resulting throat radius, generalizing the 2004 Hayward–Koyama formation model to non-static wormholes. The construction establishes that an initial black hole can evolve into a traversable wormhole via thin shells of negative energy, while leaving open the mechanism of singularity avoidance, perturbative stability, and extensions beyond spherical symmetry.