N-Black Hole Stationary and Axially Symmetric Solutions of the Einstein-Maxwell Equations

TL;DR

This paper proves the existence and uniqueness of harmonic maps with prescribed singularities, leading to solutions modeling equilibrium configurations of multiple rotating charged black holes in Einstein-Maxwell theory.

Contribution

It establishes a general mathematical framework for harmonic maps with singularities, applied specifically to black hole solutions in Einstein-Maxwell equations.

Findings

Existence and uniqueness of harmonic maps with prescribed singularities.

Construction of solutions representing multiple co-axially rotating charged black holes.

Interpretation of solutions as equilibrium configurations with singular struts.

Abstract

The Einstein/Maxwell equations reduce in the stationary and axially symmetric case to a harmonic map with prescribed singularities phi: R^3\Sigma -> H^2_C, where Sigma is a subset of the axis of symmetry, and H^2_C is the complex hyperbolic plane. Motivated by this problem, we prove the existence and uniqueness of harmonic maps with prescribed singularities phi: R^n\Sigma -> H, where Sigma is a submanifold of R^n of co-dimension at least 2, and H is a classical Riemannian globally symmetric space of noncompact type and rank one. This result, when applied to the black hole problem, yields solutions which can be interpreted as equilibrium configurations of multiple co-axially rotating charged black holes held apart by singular struts.