Last Updated: 2017/8/10
We consider arbitrary stationary and axisymmetric black holes in general relativity in $(d +1)$ dimensions (with $d \geq 3$) that satisfy the vacuum Einstein equation and have a non-degenerate horizon. We prove that the canonical energy of axisymmetric electromagnetic perturbations is positive definite. This establishes that all black holes are stable to axisymmetric electromagnetic perturbations. Our results also hold for asymptotically AdS black holes that satisfy the vacuum Einstein equation with a negative cosmological constant. Our results also apply to extremal black holes provided that the initial perturbation vanishes in a neighborhood of the horizon.