Stationary generalizations for the Bronnikov-Ellis wormhole and for the vacuum ring wormhole

We analyze possibilities to obtain a globally regular stationary generalization for the ultrastatic wormhole with a repulsive scalar field found by Bronnikov and by Ellis in 1973. The extreme simplicity of this static solution suggests that its stationary version could be obtainable analytically and should be globally regular, but no such generalization has been found. We analyze the problem and find that the difficulty originates in the vacuum theory, since the scalar field can be eliminated within the Eris-Gurses procedure. The problem then reduces to constructing the spinning generalization for the vacuum wormhole sourced by a thin ring of negative tension. Solving the vacuum Ernst equations determines the $g_{00}$, $g_{0\varphi}$ metric components and hence the AMD mass ${\rm M}$ and angular momentum ${\rm J}$, all of these being specified by the ring source. The scalar field can be included into consideration afterwords, but this only affects $g_{rr}$ and $g_{\vartheta\vartheta}$ without changing the rest. Within this approach, we analyze a number of exact stationary generalizations for the wormhole, but none of them are satisfactory. However, the perturbative expansion around the static vacuum background contains only bounded functions and presumably converges to an exact solution. Including the scalar field screens the singularity at the ring source and renders the geometry regular. This solution describes a globally regular spinning wormhole with two asymptotically flat regions. Even though the source itself is screened and not visible, the memory of it remains in $g_{00}$ and $g_{0\varphi}$ and accounts for the ${\rm M}\propto {\rm J}^2$ relation typical for a rotating extended source. Describing stationary spacetimes with an extended source is a complicated problem, which presumably explains the difficulty in finding the solution.