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We consider arbitrary stationary and axisymmetric black holes in generalrelativity in $(d +1)$ dimensions (with $d \geq 3$) that satisfy the vacuumEinstein equation and have a non-degenerate horizon. We prove that thecanonical energy of axisymmetric electromagnetic perturbations is positivedefinite. This establishes that all vacuum black holes are stable toaxisymmetric electromagnetic perturbations. Our results also hold forasymptotically deSitter black holes that satisfy the vacuum Einstein equationwith a positive cosmological constant. Our results also apply to extremal blackholes provided that the initial perturbation vanishes in a neighborhood of thehorizon.

Stability of stationary-axisymmetric black holes in vacuum general relativity to axisymmetric electromagnetic perturbations