Defect wormholes are defective

The various "defect wormholes" developed by Klinkhamer have recently attracted considerable attention -- especially in view of the fact that the simplest example, the so-called "vacuum defect wormhole", was claimed to be an everywhere-vacuum everywhere-Ricci-flat exact solution to the Einstein equations. This claim has been conclusively refuted by Feng, and in the current article we take a deeper look at exactly what goes wrong. The central issue is this: Although Klinkhamer's specific representation of the metric $g_{ab}$ is smooth ($C^\infty$) his inverse metric $g^{ab}$ is not even everywhere continuous ($C^0$), being undefined at the wormhole throat. This situation implies that one should very carefully investigate curvature tensors at the throat using the Israel--Lanczos--Sen thin-shell formalism. Doing so reveals the presence of a delta-function energy-condition-violating thin shell of matter at the wormhole throat. The "defect wormholes" are thus revealed to be quite ordinary "cut-and-paste" thin-shell wormholes, but represented in a coordinate system which is unfortunately pathological at exactly the same place that all the interesting physics occurs. To help clarify the situation, we shall focus on the behaviour of suitable coordinate invariants -- the Ricci scalar, the eigenvalues of the mixed $R^a{}_b$ Ricci tensor, and the eigenvalues of the mixed $R^{ab}{}_{cd}$ Riemann tensor.