Black holes can have curly hair

TL;DR

This paper investigates the equilibrium conditions between static, spherically symmetric black holes and classical matter, revealing that black holes can have 'curly hair' characterized by specific pressure-to-density ratios near the horizon.

Contribution

It introduces new conditions under which black holes can coexist with matter having particular equations of state, expanding understanding of black hole 'hair' beyond traditional no-hair theorems.

Findings

Equilibrium possible with vacuum-like matter (w→ -1) at the horizon.

Equilibrium possible with matter where w→ -1/(1+2k), with k>0.

A mixture of these matter types can also be in equilibrium with a black hole.

Abstract

We study equilibrium conditions between a static, spherically symmetric black hole and classical matter in terms of the radial pressure to density ratio p_r/\rho = w(u), where u is the radial coordinate. It is shown that such an equilibrium is possible in two cases: (i) the well-known case w\to -1 as $u\to u_h (the horizon), i.e., "vacuum" matter, for which \rho(u_h) can be nonzero; (ii) w \to -1/(1+2k) and \rho \sim (u-u_h)^k as u\to u_h, where k>0 is a positive integer (w=-1/3 in the generic case k=1). A non-interacting mixture of these two kinds of matter can also exist. The whole reasoning is local, hence the results do not depend on any global or asymptotic conditions. They mean, in particular, that a static black hole cannot live inside a star with nonnegative pressure and density. As an example, an exact solution for an isotropic fluid with w = -1/3 (that is, a fluid of disordered cosmic strings), with or without vacuum matter, is presented.