# The extreme orbital period in scalar hairy kerr black holes

## TL;DR

This paper extends Hod's theorem on the extreme orbital period being linked to null circular geodesics from Kerr black holes to those with scalar hair, suggesting a broader applicability in axially symmetric spacetimes.

## Contribution

It demonstrates that the extreme orbital period in scalar hairy Kerr black holes coincides with the null circular geodesic, generalizing Hod's theorem.

## Key findings

- The extreme orbital period circle matches the null circular geodesic.
- Hod's theorem may hold in any axially symmetric spacetime with reflection symmetry.
- Scalar hair does not alter the fundamental relation between orbital period and geodesics.

## Abstract

In a very interesting paper, Hod has proven that the equatorial null circular geodesic provides the extreme orbital period to circle a kerr black hole, which is closely related to the Fermat's principle. In the present paper, we extend the discussion to kerr black holes with scalar field hair. We show that the circle with the extreme orbital period is still identical to the null circular geodesic. Our analysis also implies that the Hod's theorem may be a general property in any axially symmetric spacetime with reflection symmetry on the equatorial plane.