SPHERICALLY SYMMETRICAL SOLUTIONS IN DIMENSIONALLY REDUCED SPACETIMES WITH A HIGHER-DIMENSIONAL COSMOLOGICAL CONSTANT

Certain static solutions of D-dimensional gravity with a higher-dimensional cosmological constant 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 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.