Hairy black holes in a box

Abstract

We do a systematic study of the phases of gravity coupled to an electromagnetic field and charged scalar in flat space, with box boundary conditions. The scalar-less box has previously been investigated by Braden, Brown, Whiting and York (and others) before AdS/CFT and we elaborate and extend their results in a language more familiar from holography. The phase diagram of the system is analogous to that of AdS black holes, but we emphasize the differences and explain their origin. Once the scalar is added, we show that the system admits both boson stars as well as hairy black holes as solutions, providing yet another way to evade flat space no-hair theorems. Furthermore both these solutions can exist as stable phases in regions of the phase diagram. The final picture of the phases that emerges is strikingly similar to that found recently for holographic superconductors in global AdS, arXiv:1602.07211. Our construction lays bare certain previously unnoticed subtleties associated to the definition quasi-local charges for gravitating scalar fields in finite regions.

On the thermodynamics of the black hole and hairy black hole transitions in the asymptotically flat spacetime with a box

“…When choosing m 2 = 0, our results are related to the right panel of Fig. 11 in [42]. It can easily be seen from the left panel of Fig.…”

“…Since we are interested in including the matter fields' backreaction on the background, we assume the ansatz of the geometry of the four-dimensional hairy black hole solution in the form [42] …”

“…The authors in [42] have proposed a way to calculate the free energy of the system by doing a subtraction of the flat space background. In this way, the free energy of the Minkowski box is set to zero.…”

“…On the other side, it is also meaningful to study the similarity between transitions of this Einstein-Maxwell-scalar system in a box and those of the s-wave holographic superconductors constructed with scalar fields coupled to Maxwell fields in an AdS background [39][40][41]. Recently, Basu et al initiated a thermodynamical study of such Einstein-Maxwell-scalar systems on asymptotically flat background with reflecting mirror-like boundary conditions for the scalar field [42]. For a certain range of parameters, this model admits stable hairy black hole solutions, which provides a way to evade the flat space no-hair theorems.…”

“…So it is interesting to extend the discussion in the horizonless spacetime in [45] to the background of black holes, which have been studied a lot in holographic theories. On the other hand, it is also meaningful to generalize the quasi-local black hole/hairy black hole transition model in [42] by considering a nonzero mass of the scalar field since the mass usually plays a crucial role in determining the properties of transitions.…”