Degenerate Geometries as Matter-Free Physical Configurations in General Relativity: Three Examples

We examine three degenerate spacetime configurations with wormhole topology, obtained via branching coordinate transformations of the Rindler, Minkowski, and Schwarzschild vacuum metrics. These configurations are, respectively, a Rindler wormhole with a planar throat, and the Klinkhamer and Schwarzschild–Klinkhamer wormholes with spherical throats. Within the framework of the Einstein–Palatini–Cartan formulation, we demonstrate that these degenerate configurations are matter-free. In this regard, they differ fundamentally from their non-degenerate wormhole counterparts in the thin-shell model, which require exotic matter. Nevertheless, the degenerate configurations considered here exhibit distinct physical manifestations, ranging from purely topological structures to genuine gravitational effects. Furthermore, in all three examples, we demonstrate the absence of a limiting transition from the non-degenerate spacetime to the matter-free degenerate configuration. This suggests that degenerate geometries constitute an independent sector of the configuration space of general relativity.