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Small Hairy Black Holes in Global AdS Spacetime

Pallab Basu, Jyotirmoy Bhattacharya, Sayantani Bhattacharyya, R. Loganayagam, Shiraz Minwalla, V. Umesh

TL;DR

The paper shows that small charged RNAdS black holes in global AdS$_5$ develop hairy endpoints through a superradiant tachyonic instability. It analytically constructs hairy black holes and horizonless solitons via a double perturbative expansion in the hole radius $R$ and scalar amplitude $\epsilon$, using far-field and near-field matching. The microcanonical phase diagram reveals a second-order-like transition between RNAdS and hairy BH phases, with solitons forming the lower bound of the existence region and excited hairy BHs occupying a subdominant role. The results offer intuitive, analytic insight into hairy black hole dynamics in AdS, and lay groundwork for applying similar methods to more realistic theories such as IIB supergravity on AdS$_5\times S^5$.

Abstract

We study small charged black holes in global AdS spacetime in the presence of a charged massless minimally coupled scalar field. In a certain parameter range these black holes suffer from well known superradiant instabilities. We demonstrate that the end point of the resultant tachyon condensation process is a hairy black hole which we construct analytically in a perturbative expansion in the black hole radius. At leading order our solution is a small undeformed RNAdS black hole immersed into a charged scalar condensate that fills the AdS `box'. These hairy black hole solutions appear in a two parameter family labelled by their mass and charge. Their mass is bounded from below by a function of their charge; at the lower bound a hairy black hole reduces to a regular horizon free soliton which can also be thought of as a nonlinear Bose condensate. We compute the microcanonical phase diagram of our system at small mass, and demonstrate that it exhibits a second order `phase transition' between the RNAdS black hole and the hairy black hole phases.

Small Hairy Black Holes in Global AdS Spacetime

TL;DR

The paper shows that small charged RNAdS black holes in global AdS develop hairy endpoints through a superradiant tachyonic instability. It analytically constructs hairy black holes and horizonless solitons via a double perturbative expansion in the hole radius and scalar amplitude , using far-field and near-field matching. The microcanonical phase diagram reveals a second-order-like transition between RNAdS and hairy BH phases, with solitons forming the lower bound of the existence region and excited hairy BHs occupying a subdominant role. The results offer intuitive, analytic insight into hairy black hole dynamics in AdS, and lay groundwork for applying similar methods to more realistic theories such as IIB supergravity on AdS.

Abstract

We study small charged black holes in global AdS spacetime in the presence of a charged massless minimally coupled scalar field. In a certain parameter range these black holes suffer from well known superradiant instabilities. We demonstrate that the end point of the resultant tachyon condensation process is a hairy black hole which we construct analytically in a perturbative expansion in the black hole radius. At leading order our solution is a small undeformed RNAdS black hole immersed into a charged scalar condensate that fills the AdS `box'. These hairy black hole solutions appear in a two parameter family labelled by their mass and charge. Their mass is bounded from below by a function of their charge; at the lower bound a hairy black hole reduces to a regular horizon free soliton which can also be thought of as a nonlinear Bose condensate. We compute the microcanonical phase diagram of our system at small mass, and demonstrate that it exhibits a second order `phase transition' between the RNAdS black hole and the hairy black hole phases.

Paper Structure

This paper contains 55 sections, 169 equations, 18 figures.

Table of Contents

  1. Introduction
  2. The basic setup for the Hairy BHs
  3. Basic equations of motion
  4. RNAdS Black Holes and their superradiance
  5. Setting up the perturbation theory
  6. Perturbation theory at ${\cal O}(\epsilon)$
  7. Far Field Region
  8. Near Field Region
  9. Matching
  10. Perturbation theory at ${\cal O}(\epsilon^2)$
  11. Far Field Region
  12. Near Field Region
  13. Iteration
  14. The Soliton
  15. Perturbation theory for soliton
  16. ...and 40 more sections

Figures (18)

  • Figure 1: Micro Canonical Phase Diagram at small mass and charge. The overlapping region is dominated by the hairy blackhole.
  • Figure 2: Microcanonical ensemble for $e= 5.4$ : Existence region of ground state and excited state hairy black holes.
  • Figure 3: Proposed Microcanonical phase diagram for large $M$ and $Q$.
  • Figure 4: $f(r)$ for $\epsilon=0.4$ and $e=4$
  • Figure 5: $f(r)$ for $\epsilon=0.7$ and $e=4$
  • ...and 13 more figures