Constructing "hair" for the three charge hole

TL;DR

This paper constructs a regular, normalizable perturbation representing a 3-charge extremal system, extending the understanding of horizonless geometries and black hole 'hair' from 2-charge to 3-charge configurations.

Contribution

It introduces a new regular perturbation for the 3-charge system, supporting the idea that such states can resemble 2-charge geometries without horizons.

Findings

Constructed a regular, normalizable perturbation carrying momentum charge.

Matched solutions in inner and outer regions of the geometry.

Conjectured the general form of 'hair' for 3-charge black holes.

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.