  We investigate some of the issues relating to the dynamical instability of
general static spacetimes with horizons. Our paper will be partially
pedagogical and partially exploratory in nature. After discussing the current
understanding, we generalize the proposal of Gubser and Mitra, which identifies
dynamical instability of black branes with local thermodynamic instability, to
include all product spacetimes with the horizon uniformly smeared over an
internal space. We support our conjecture by explicitly exhibiting a threshold
unstable mode for Schwarzschild-AdS_5 x S^5 black hole. Using the AdS/CFT
correspondence, this simultaneously yields a prediction for a phase transition
in the dual gauge theory. We also discuss implications for spacetimes with
cosmological horizons.
