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.

Figures (8)

• Figure 1: The Penrose diagrams of the geometry for $\gamma<2$ (left panel) and $\gamma>2$ (right panel). For $\gamma<2$, the geometry is free from singularity in the spatial and the future null infinity, while there is singularity for $\gamma>2$. In the past region, there is curvature singularity regardless of the value of $\gamma$. The figure also includes $t$-constant and $l$-constant lines, drawn as gray curves.
• Figure 2: Plot of the solution of Eq. \ref{['Aeq']} with $A_{\rm{th}}=1/4$. (blue solid curve). The red dashed curve is the fitted line $\mathrm{e}^{2|l|}/4 \gamma$ with $\gamma \simeq 1.474$. $A(l)$ takes minimum at $l=0$, and increases monotonically with as $l$ increases.
• Figure 3: The embedded diagram of the geometry around the throat at some moment ($A_{\rm{th}}=1/4$) into the three-dimensional spaces. The orange region is embedded geometry in the Euclid space, while the blue colored region is the Minkowski spacetime.
• Figure 4: Plot of $- g_{tt}/\mathrm{e}^{2t}L^2 = \mathrm{e}^{B(l)}/\sqrt{A(l)}$ for the wormhole solutions with $A_{\rm{th}} = 1/4$. The solution satisfy $- g_{tt} > 0$ throughout the entire domain.
• Figure 5: The value of fitted parameter $\gamma$ in Eq. \ref{['Ainf']}. $\gamma$ decreases as the throat radius $A_{\rm{th}}$ increases, and asymptotes to $\gamma=1$ as $A_{\rm{th}}$ approaches to $1/2$.