Static Black Hole Solutions without Rotational Symmetry

TL;DR

This paper constructs static black hole solutions lacking rotational symmetry in theories with charged vector mesons, revealing new non-spherical black hole configurations with vector hair and analyzing their properties near instability thresholds.

Contribution

It develops a perturbative method to find non-spherical static black hole solutions with vector hair in theories including charged vector mesons, especially near the instability point of Reissner-Nordstrom black holes.

Findings

Black holes with vector meson hair can be non-spherical and lack rotational symmetry.

Solutions exhibit higher multipole moments in magnetic fields and metrics.

Vector meson fields decay exponentially, while magnetic and metric fields have power-law tails.

Abstract

We construct static black hole solutions that have no rotational symmetry. These arise in theories, including the standard electroweak model, that include charged vector mesons with mass $m\ne 0$. In such theories, a magnetically charged Reissner-Nordstrom black hole with horizon radius less than a critical value of the order of $m^{-1}$ is classically unstable against the development of a nonzero vector meson field just outside the horizon, indicating the existence of static black hole solutions with vector meson hair. For the case of unit magnetic charge, spherically symmetric solutions of this type have previously been studied. For other values of the magnetic charge, general arguments show that any new solution with hair cannot be spherically symmetric. In this paper we develop and apply a perturbative scheme (which may have applicability in other contexts) for constructing such solutions in the case where the Reissner-Nordstrom solution is just barely unstable. For a few low values of the magnetic charge the black holes retain a rotational symmetry about a single axis, but this axial symmetry disappears for higher charges. While the vector meson fields vanish exponentially fast at distances greater than $O(m^{-1})$, the magnetic field and the metric have higher multipole components that decrease only as powers of the distance from the black hole.