Gilbert Weinstein
N-Black Hole Stationary and Axially Symmetric Solutions of the Einstein-Maxwell Equations
The paper is published: Commun. Partial Differ. Equations 21, 9-10 (1996) 1389-1430

MSC:

83C57 Black holes

35Q75 PDE in relativity

58E20 Harmonic maps

Abstract: It is well-known that the
Einstein-Maxwell equations reduce in the stationary and axially symmetric
case to an axially symmetric harmonic map with prescribed singularities
$\vp\colon\R^3\bs\Sigma\to\HC$, where $\Sigma$ is a subset of the axis of
symmetry, and $\HC$ is the complex hyperbolic plane.
Motivated by this problem, we prove the existence and uniqueness of harmonic
maps with prescribed singularities $\vp\colon\R^n\bs\Sigma\to\Hy$, where
$\Sigma$ is a submanifold of $\R^n$ of co-dimension
$\geq 2$, and $\Hy$ is a
classical Riemannian
globally symmetric space of noncompact type and rank one. This
result, when applied to the black hole problem yields
solutions of the reduced equations which
can be interpreted as equilibrium configurations of
multiple co-axially rotating charged
black holes held apart by singular struts.

Keywords: Einstein-Maxwell equations, rotating charged black holes, harmonic maps