No hair theorem for massless scalar fields outside asymptotically flat horizonless reflecting compact stars

Abstract

In a recent paper, Hod started a study on no scalar hair theorem for asymptotically flat spherically symmetric neutral horizonless reflecting compact stars. In fact, Hod's approach only rules out massive scalar fields. In the present paper, for massless scalar fields outside neutral horizonless reflecting compact stars, we provide a rigorous mathematical proof on no hair theorem. We show that asymptotically flat spherically symmetric neutral horizonless reflecting compact stars cannot support exterior massless scalar field hairs.

Analytical investigations on formations of hairy neutral reflecting shells in the scalar-Gauss–Bonnet gravity

“…Hod firstly proved that massive static scalar field hairs cannot form in the background of neutral horizonless reflecting objects [24]. This no hair theorem for neutral horizonless reflecting objects was also extended to the case of massless scalar field hairs [25,26]. Considering a positive cosmological constant, it was found that the no hair theorem still holds in the background of neutral horizonless reflecting objects [27].…”

No scalar hair theorem for neutral Neumann stars: Static massive scalar fields nonminimally coupled to gravity

“…In fact, it was shown that static scalar fields cannot condense outside charged reflecting shells of large radii [34][35][36]. Large charged reflecting stars also cannot support static scalar field hairs [37][38][39][40][41][42][43][44]. Moreover, it was found that scalar fields cannot exist outside compact stars with Neumann surface boundary conditions [45,46].…”

Scalarization of compact stars in the scalar-Gauss-Bonnet gravity

“…Hod firstly showed that massive static scalar fields cannot exist outside asymptotically flat neutral horizonless compact stars with a scalar reflecting surface [53]. For massless static scalar fields outside such asymptotically flat neutral horizonless reflecting compact stars, the no hair theorem was firstly analysed in [54] and another proof was provided based on Cauchy-Kowalevski theorem [55]. In the asymptotically dS gravity, static scalar hairs still cannot form outside the neutral horizonless reflecting compact stars [56].…”