Dynamical Formation of Self-Similar Wormholes

1. Problem Statement

The paper addresses two fundamental challenges in general relativity regarding traversable wormholes:

1. Singularity Issues in Static Models: Previous static wormhole solutions (e.g., Hayward's model) supported by colliding negative-energy null dust inevitably possess spacetime singularities in both spatial and null directions, rendering them globally non-viable.
2. Dynamical Formation: While black hole formation is well-understood, the dynamical process of forming a traversable wormhole from realistic initial conditions (specifically, converting a black hole into a wormhole via negative energy injection) lacks a comprehensive, non-static model that avoids future singularities.

The authors aim to construct a non-static, self-similar wormhole solution and demonstrate a dynamical formation scenario where an initial Schwarzschild black hole evolves into a regular traversable wormhole through the injection of negative-energy flux.

2. Methodology

A. Self-Similar Wormhole Solution (Section 2)

• Matter Source: The authors utilize a model of two counter-propagating streams of negative-energy null dust. This is the simplest matter source capable of violating the Null Energy Condition (NEC) required for wormholes.
• Symmetry Assumption: To simplify the highly nonlinear Einstein equations, they impose self-similarity (homothetic symmetry). This assumes the existence of a homothetic Killing vector $\xi = u\partial_u + v\partial_v$, reducing the partial differential equations (PDEs) to a system of ordinary differential equations (ODEs).
• Coordinate Transformation: The metric is expressed in double-null coordinates $(u, v)$ and transformed to $(t, l)$ coordinates where the metric takes a conformally static form:
  $$g = e^{2t}L^2 \left[ \frac{e^{B(l)}}{\sqrt{A(l)}}(-dt^2 + dl^2) + A(l)d\Omega^2 \right]$$
  Here, $r = e^t L \sqrt{A(l)}$ is the areal radius.
• Numerical Integration: The reduced ODEs are solved numerically with boundary conditions enforcing a throat at $l=0$ (minimal areal radius, $A'(0)=0$, $A''(0)>0$) and symmetry under $l \to -l$.

B. Dynamical Formation Model (Section 3)

• Geometric Patching: The authors construct a composite spacetime by stitching together three distinct regions across null hypersurfaces (shells) using the Barrabès–Israel formalism:
  1. Schwarzschild Region: The initial state (Black Hole).
  2. Vaidya Regions: Intermediate states containing ingoing and outgoing negative-energy null dust.
  3. Self-Similar Wormhole Region: The final state.
• Junction Conditions: The matching conditions ensure continuity of the induced metric and relate the discontinuity in extrinsic curvature to the surface stress-energy tensor of the thin shells.
• Constraints: The shells are assumed to be pressureless ($P=0$), containing only energy density. This condition fixes the mass function of the Vaidya regions.

3. Key Contributions and Results

A. Properties of the Self-Similar Solution

• Regularity vs. Singularities: The global structure of the solution depends critically on a single asymptotic parameter, $\gamma$, which governs the behavior of the metric function $A(l)$ at large distances ($A(l) \sim e^{2l}/4\gamma$).
  • Case $\gamma < 2$: The spacetime is regular at both spatial infinity ($l \to \infty$) and future null infinity. Null geodesics are complete.
  • Case $\gamma > 2$: A parallelly propagated (pp) curvature singularity develops in the future null direction due to the divergence of the negative energy flux.
  • Case $\gamma > 4$: Scalar curvature invariants also diverge.
  • Past Direction: All solutions possess an unavoidable curvature singularity in the past ($t \to -\infty$), consistent with the self-similar nature of the model.
• Throat Size Control: Numerical results show that the parameter $\gamma$ is monotonically decreasing with respect to the throat radius $A_{th}$.
  • Crucial Finding: For sufficiently large throat radii ($A_{th} > 0.14$), the condition $\gamma < 2$ is satisfied. This implies that wormholes with large throats can avoid future singularities, a result not possible in the static Hayward model.

B. The Formation Scenario

• Black Hole to Wormhole Transition: The authors demonstrate that an initial Schwarzschild black hole can evolve into a wormhole by injecting negative-energy flux.
• Mass-Throat Relation: By applying the junction conditions at the intersection of the shells, they derive an explicit relation between the initial black hole mass ($M$), the injected mass change ($\Delta M < 0$), and the final throat radius ($r_0$):
  $$A_{th} = \frac{1}{2} \frac{\left(1 - \frac{2(M+\Delta M)}{r_0}\right)^2}{\left(1 - \frac{2(M+\Delta M)}{r_0}\right)^2 - \left(1 - \frac{2M}{r_0}\right)}$$
• Physical Interpretation:
  • The formation requires the throat to emerge from inside the original event horizon ($r_0 < 2M$).
  • The negative energy injection reduces the black hole mass, causing the horizon to shrink and eventually exposing the throat.
  • Unlike static models, the throat radius depends on the injection time (parameter $u_0$), reflecting the dynamic expansion of the wormhole.

4. Significance and Implications

1. Resolution of Static Limitations: The paper generalizes the static Hayward-Koyama formation model to a non-static context. It proves that the singularities plaguing static models are not inevitable; they can be avoided if the wormhole throat is sufficiently large ($\gamma < 2$).
2. Mechanism for Singularity Avoidance: The study identifies a specific mechanism where the interplay between the throat size and the negative energy flux profile allows the geometry to remain regular at future infinity.
3. Dynamical Evolution: It provides a concrete, mathematically consistent scenario for the dynamical formation of a traversable wormhole from a black hole, bridging the gap between black hole physics and wormhole topology.
4. Theoretical Framework: The work establishes a robust framework using self-similarity and the Barrabès–Israel formalism to analyze time-dependent wormhole geometries, offering a new perspective on how exotic matter might shape spacetime evolution.

5. Limitations and Future Work

• Past Singularity: The model retains a singularity in the past direction, suggesting the wormhole emerges from an initial singularity rather than a smooth vacuum state.
• Stability: The perturbative stability of these self-similar solutions has not yet been fully addressed.
• Physical Realization: The mechanism for generating the specific negative-energy flux profile required to satisfy the $\gamma < 2$ condition remains a theoretical construct requiring further physical justification.

In summary, this paper provides a rigorous mathematical demonstration that dynamical, self-similar wormholes can form from black holes and remain regular at future infinity, provided the throat is large enough to suppress the divergence of negative energy flux.