Horizon closeness bounds for static black hole mimickers

We consider the question whether a wormhole can be converted into a non-extremal quasi-black black hole by continuous change of parameters. In other words, we ask whether "black" wormholes can exist as end points of families of static wormhole geometries. The answer is negative since the corresponding limit is shown to be singular. Similar conclusions are valid also for other types of black hole mimickers such as gravastars and quasi-black holes without wormhole behavior. Our treatment is model-independent and applies to any static geometries without requirement of special symmetries. We also find an asymptotic expression for the Kretschmann scalar for wormholes on the threshold of horizon formation that can be used as an the bound on proximity of the configuration to the would-be horizon. The derived bound is very weak for astrophysical black holes but becomes relevant for microscopic ones. We point out complementarity between ability of wormholes to mimic black holes and their ability to be traversable "in practice".