On the Stability of Axisymmetric Systems to Axisymmetric Perturbations in General Relativity. III. Vacuum Metrics and Carter's Theorem

The analysis of Paper II is specialized to vacuum metrics appropriately for a discussion of their stability. And Carter's theorem, that asymptotically fiat axisymmetric vacuum metrics, external to black holes, cannot allow nontrivial axisymmetric neutral deformations, is deduced.