Spherically symmetric solutions in dimensionally reduced spacetimes with a higher-dimensional cosmological constant.

Abstract

Certain static solutions of $D$-dimensional gravity with a higher-dimension\`al cosmological constant $\ensuremath{\Lambda}$ are studied. The solutions are taken to be spherically symmetric in the physical ($m+2$)-dimensional spacetime, where $D=m+n+2$ (or more generally the $m$-sphere is replaced by an arbitrary Einstein space), while the internal space is an arbitrary $n$-dimensional Einstein space. The global properties of all such solutions are derived by considering the equivalent dimensionally reduced system in $m+2$ dimensions, and by using techniques from the theory of dynamical systems after a judicious choice of variables. All solutions with a nonzero $\ensuremath{\Lambda}$ are either found to contain naked singularities or not be asymptotically flat, as would be expected from the "no-hair" theorems. A recent "black-hole" solution derived by Kim and Cho in the context of these models is shown to be incorrect.