No long hair behaviors of ultra-compact objects

Abstract

We investigate distributions of matter fields outside spherically symmetric ultra-compact objects with null circular orbits in the asymptotically flat background. Based on the dominant energy and the non-negative trace conditions, we prove the existence of effective radius of matter field hairs outside horizonless ultra-compact objects. We also analytically find a no long hair behavior, which states that the effective radius of matter field hairs cannot extend beyond the outermost null circular orbit of horizonless ultra-compact objects.