Static spacetimes haunted by a phantom scalar field. I. Classification and global structure in the massless case

We discuss various novel features of $n(\ensuremath{\ge}4)$-dimensional spacetimes sourced by a massless (non)phantom scalar field in general relativity. Assuming that the metric is warped product static two-dimensional Lorentzian spacetime and an ($n\ensuremath{-}2$)-dimensional Einstein space ${K}^{n\ensuremath{-}2}$ with curvature $k=0,\ifmmode\pm\else\textpm\fi{}1$, depends only on radial variable, we present complete classification solutions for both signs kinetic term. Contrary to case nonphantom field, Fisher solution not unique, there exist two additional metrics corresponding generalizations Ellis-Gibbons Ellis-Bronnikov solution. explore maximal extension these detail analysis null/spacelike geodesics singularity. For phantom solutions, find inevitably appear parallelly propagated (p.p) singularities parameter region where are no singularities. Interestingly, areal radius blows up at p.p singularities, which nevertheless accessible within finite affine time along null geodesics. It follows describes regular wormhole two-sided asymptotically flat spacetime. Using transformation relating Jordan frames, also same symmetry coupled conformal field. Additionally, solving equations frame, prove this genuinely complete.