Static spacetimes haunted by a phantom scalar field: classification and global structure in the massless case

Understand

We discuss various novel features of $n(\ge 4)$-dimensional spacetimes sourced by a massless (non-)phantom scalar field in general relativity.

• Assuming that the metric is a warped product of static two-dimensional Lorentzian spacetime and an $(n-2)$-dimensional Einstein space $K^{n-2}$ with curvature $k=0, \pm 1$, and that the scalar field depends only on the radial variable, we present a complete classification of static solutions for both signs of kinetic term.
• Contrary to the case with a non-phantom scalar field, the Fisher solution is not unique, and there exist two additional metrics corresponding to the generalizations of the Ellis-Gibbons solution and the Ellis-Bronnikov solution.
• We explore the maximal extension of these solutions in detail by the analysis of null/spacelike geodesics and singularity.