Hairy Wormholes and Bartnik-McKinnon Solutions

We consider Lorentzian wormholes supported by a phantom field and threaded by non-trivial Yang-Mills fields, which may be regarded as hair on the Ellis wormhole. Like the Bartnik-McKinnon solutions and their associated hairy black holes, these hairy wormholes form infinite sequences, labeled by the node number $k$ of their gauge field function. We discuss the throat geometry of these wormholes, showing that odd-$k$ solutions may exhibit a double-throat, and evaluate their global charges. We analyze the limiting behavior exhibited by wormhole solutions as the gravitational coupling becomes large. The even-$k$ solutions approach smoothly the Bartnik-McKinnon solutions with $k/2$ nodes, while the odd-$k$ solutions develop a singular behavior at the throat in the limit of large coupling. In the limit of large $k$, on the other hand, an embedded Abelian wormhole solution is approached, when the throat is large. For smaller throats the extremal Reissner-Nordstr\"om solution plays a fundamental role in the limit.