The existence of null circular geodesics outside extremal spherically symmetric asymptotically flat hairy black holes

The existence of null circular geodesics has been proved in the background of non-extremal spherically symmetric asymptotically flat black holes in previous works. Then it is an interesting question that whether extremal black holes possess null circular geodesics outside horizons. In the present paper, we pay attentions to the extremal spherically symmetric asymptotically flat hairy black holes. We show the existence of the fastest trajectory to circle a extremal black hole. As the fastest trajectory corresponds to the position of null circular geodesics, we prove that null circular geodesics exist outside extremal spherically symmetric asymptotically flat hairy black holes. We also point out that our proof also works for non-extremal black holes.