Turning Point Instabilities for Relativistic Stars and Black Holes

In the light of recent results relating dynamic and thermodynamic stability of relativistic stars and black holes, we re-examine the relationship between ``turning points''---i.e., extrema of thermodynamic variables along a one-parameter family of solutions---and instabilities. We give a proof of Sorkin's general result---showing the existence of a thermodynamic instability on one side of a turning point---that does not rely on heuristic arguments involving infinite dimensional manifold structure. We use the turning point results to prove the existence of a dynamic instability of black rings in $5$ spacetime dimensions in the region where $c_J > 0$, in agreement with a result of Figueras, Murata, and Reall.