Stationary black holes with stringy hair

We discuss properties of black holes which are pierced by special configurations of cosmic strings. For static black holes, we consider radial strings in the limit when the number of strings grows to infinity while the tension of each single string tends to zero. In a properly taken limit, the stress-energy tensor of the string distribution is finite. We call such matter stringy matter. We present a solution of the Einstein equations for an electrically charged static black hole with the stringy matter, with and without a cosmological constant. This solution is a warped product of two metrics. One of them is a deformed 2-sphere, whose Gaussian curvature is determined by the energy density of the stringy matter. We discuss the embedding of a corresponding distorted sphere into a three-dimensional Euclidean space and formulate consistency conditions. We also found a relation between the square of the Weyl tensor invariant of the four-dimensional spacetime of the stringy black holes and the energy density of the stringy matter. In the second part of the paper, we discuss test stationary strings in the Kerr geometry and in its Kerr-NUT-(anti-)de Sitter generalizations. Explicit solutions for strings that are regular at the event horizon are obtained. Using these solutions, the stress-energy tensor of the stringy matter in these geometries is calculated. Extraction of the angular momentum from rotating black holes by such strings is also discussed.

Figure 1

Illustration of the Gauss map. This map $\mathrm{\Psi }:M^2\to S^2$ transforms points on the deformed 2-sphere $M^2$ to points on the unit sphere $S^2$.

Figure 2

Representing the string, a line (54) is shown in polar coordinates $(\rho ,\varphi )$ in a 2-plane.

Figure 3

Illustration of the calculation of the torque. The picture shows a small piece of the sphere of large radius $r$. At a point $A_0$ with spherical coordinates $({\theta }_0,{\varphi }_0)$, one introduces a triad of unit orthonormal vectors $\{{\overrightarrow{e}}_r,{\overrightarrow{e}}_{\theta },{\overrightarrow{e}}_{\varphi }\}$. Two of these vectors, in the $\theta$ and $\varphi$ directions, span a two-dimensional plane tangent to the sphere. Because the sphere’s curvature is small, this plane practically coincides with the sphere at $A_0$. The string enters the tangent plane orthogonally. Its direction coincides with ${\overrightarrow{e}}_r$, and its position is shifted from $A_0$ by the distance $\overrightarrow{\ell }$ in the direction of ${\overrightarrow{e}}_{\varphi }$. The unit vector ${\overrightarrow{e}}_z$ is in the direction of the angular momentum of the black hole.