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Constructing "hair" for the three charge hole

Samir D. Mathur, Ashish Saxena, Yogesh K. Srivastava

TL;DR

The work argues that black hole entropy arises from a large ensemble of horizonless microstate geometries, with the 2-charge D1-D5 system providing a concrete realization parameterized by a profile $\vec{F}(v)$. By constructing a regular, normalizable perturbation that carries one unit of momentum on the 2-charge background and matching inner AdS$_3\times$S$^3$ and outer asymptotically flat regions, the authors demonstrate a viable 3-charge hair for at least one microstate. They develop a perturbative scheme in $\epsilon$, perform spherical-harmonic expansions, and show consistent inner/outer matching up to $O(\epsilon^3)$, arguing for the existence of an exact solution. The results imply a nontrivial interior structure for 3-charge extremal holes and support broader implications for black hole interiors, coarse-graining, and the role of fractionation in setting the horizon scale, while calling for a complete classification of generic 3-charge bound-state geometries.

Abstract

It has been found that the states of the 2-charge extremal D1-D5 system are given by smooth geometries that have no singularity and no horizon individually, but a `horizon' does arise after `coarse-graining'. To see how this concept extends to the 3-charge extremal system, we construct a perturbation on the D1-D5 geometry that carries one unit of momentum charge $P$. The perturbation is found to be regular everywhere and normalizable, so we conclude that at least this state of the 3-charge system behaves like the 2-charge states. The solution is constructed by matching (to several orders) solutions in the inner and outer regions of the geometry. We conjecture the general form of `hair' expected for the 3-charge system, and the nature of the interior of black holes in general.

Constructing "hair" for the three charge hole

TL;DR

The work argues that black hole entropy arises from a large ensemble of horizonless microstate geometries, with the 2-charge D1-D5 system providing a concrete realization parameterized by a profile . By constructing a regular, normalizable perturbation that carries one unit of momentum on the 2-charge background and matching inner AdSS and outer asymptotically flat regions, the authors demonstrate a viable 3-charge hair for at least one microstate. They develop a perturbative scheme in , perform spherical-harmonic expansions, and show consistent inner/outer matching up to , arguing for the existence of an exact solution. The results imply a nontrivial interior structure for 3-charge extremal holes and support broader implications for black hole interiors, coarse-graining, and the role of fractionation in setting the horizon scale, while calling for a complete classification of generic 3-charge bound-state geometries.

Abstract

It has been found that the states of the 2-charge extremal D1-D5 system are given by smooth geometries that have no singularity and no horizon individually, but a `horizon' does arise after `coarse-graining'. To see how this concept extends to the 3-charge extremal system, we construct a perturbation on the D1-D5 geometry that carries one unit of momentum charge . The perturbation is found to be regular everywhere and normalizable, so we conclude that at least this state of the 3-charge system behaves like the 2-charge states. The solution is constructed by matching (to several orders) solutions in the inner and outer regions of the geometry. We conjecture the general form of `hair' expected for the 3-charge system, and the nature of the interior of black holes in general.

Paper Structure

This paper contains 35 sections, 122 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Black hole 'hair'
  3. The three charge case
  4. The 2-charge system: review
  5. Generating the 'correct' D1-D5 geometries
  6. The geometry for $|0\rangle_R$
  7. The 'inner' region
  8. The 'outer' region
  9. The spectral flow map
  10. The perturbation carrying momentum
  11. The equations
  12. The $(B,w)$ perturbation at leading order ($O(\epsilon^0)$)
  13. Inner region: The chiral primary $|\psi\rangle_{NS}$
  14. Inner region: The state $J^-_0|\psi\rangle_{NS}~\leftrightarrow~J^-_{-1}|\psi\rangle_R$
  15. Outer region: Continuing the perturbation $J^-_{-1}|\psi\rangle_R$
  16. ...and 20 more sections

Figures (7)

  • Figure 1: (a) The usual picture of a black hole. (b) If the microstates represented small deformations of (a) then each would itself have a horizon.
  • Figure 2: The D1-D5 CFT is represented by a smooth geometry in the dual representation.
  • Figure 3: The naive geometry of the extremal D1-D5 system.
  • Figure 4: Actual geometries for different microstates of the extremal D1-D5 system.
  • Figure 5: Obtaining the 'horizon' by 'coarse-graining'.
  • ...and 2 more figures