Black holes can have curly hair

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 noninteracting 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.