Has the black hole equilibrium problem been solved?

TL;DR

The paper surveys the history of classifying pure vacuum black hole equilibrium states in four-dimensional general relativity, with electrovac generalisations, arguing that while the Kerr family is strongly supported as the generic stationary equilibrium under non-degeneracy and analyticity, a fully rigorous, assumption-free proof remains elusive. It traces progress from the preclassical Schwarzschild era through the classical phase's reduction to a 2D elliptic boundary-value problem with horizon data and the no-hair uniqueness, to the post-classical phase's staticity theorems and robust foundational work by Wald, Sudarsky, and Chrusciel. Key contributions include the reduction to boundary data $(M,c)$ with $c=\kappa\mathcal{A}/(4\pi)$, the Kerr-based no-hair paradigm, and modern staticity theorems that tighten the assumptions, but open problems persist in degenerate horizons ($\kappa=0$), multi-hole configurations, and causality-free formulations. The discussion clarifies the current status and maps out mathematical challenges necessary for a complete four-dimensional classification of pure vacuum black hole equilibria, with implications for both the mathematics of GR and astrophysical black hole modelling.

Abstract

When the term ``black hole'' was originally coined in 1968, it was immediately conjectured that the only pure vacuum equilibrium states were those of the Kerr family. Efforts to confirm this made rapid progress during the ``classical phase'' from 1968 to 1975, and some gaps in the argument have been closed during more recent years. However the presently available demonstration is still subject to undesirably restrictive assumptions such as non-degeneracy of the horizon, as well as analyticity and causality in the exterior.