Gravitational collapse of a degenerate wormhole

TL;DR

The paper investigates the gravitational collapse of a degenerate, vacuum wormhole using the Klinkhamer metric and extends the equivalence principle to include matter-free gravitational sources. By mapping the wormhole's radial dynamics to the free-fall of a test particle in a Schwarzschild field, it derives a unique evolution for the throat radius b(t) and analyzes the phase-space behavior via a Hamilton-Jacobi framework. It shows that bound wormhole states inevitably collapse to a nontraversable Einstein-Rosen throat (b = 2M), while higher-energy states can undergo long-lived expansion, thereby providing a consistent dynamical picture within regularized Einstein equations. The work also addresses criticisms about non-uniqueness (Feng) by linking the local metric to global topological constraints through a least-action construction, highlighting a viable, physically meaningful evolution for degenerate wormholes.

Abstract

The dynamics of a degenerate spherically symmetric wormhole in a vacuum is considered. An extension of the equivalence principle to matter free objects that are the source of a gravitational field is proposed. Using the Klinkhamer metric as an example, it is shown that a degenerate wormhole is precisely such an object. Application of the extended equivalence principle reduces the radial dynamics of the Klinkhamer wormhole to the dynamics of the radial fall of a test particle in a Schwarzschild gravitational field. It is proven that any bound state of the traversable Klinkhamer wormhole eventually collapses into a nontraversable Einstein-Rosen wormhole. An estimate is presented showing that the traversable Klinkhamer wormhole, although nonstationary, is a longlived state.

Figures (3)

• Figure 1: By virtue of the extended equivalence principle, the collapse of a degenerate wormhole of mass $M$ is synchronous with the free fall of a test particle of mass $m$ located near the surface of the wormhole.
• Figure 2: Phase portrait of the Klinkhamer wormhole dynamics. The lower part of the phase portrait describes the gravitational contraction of the wormhole, and the upper part describes its inertial expansion. The separatrix, shown by the thick line, bounds the region $E<M$, where the wormhole radius can vary only within finite limits. In this region, as in the region of gravitational contraction, the evolution of the Klinkhamer wormhole always ends with gravitational collapse into an Einstein-Rosen wormhole.
• Figure 3: During gravitational collapse, the traversable Klinkhamer wormhole $(b>2M)$ transforms into the non-traversable Einstein-Rosen wormhole $(b=2M)$. The Flamm paraboloid has a kink at the throat, which disappears as a result of collapse (i.e., as $b \rightarrow 2M$). The three Flamm paraboloids shown are assumed to have identical values of the mass parameter $M=1$ and successively decreasing values of the throat radius: $b=6$, $b=4$, and $b=2$.