On the symmetry orbits of black holes in non-linear sigma models

TL;DR

This paper examines the claim that all four-dimensional asymptotically flat black holes are related by symmetries to Schwarzschild or Kerr black holes, showing that a key transformation used in previous proofs does not always exist.

Contribution

The paper identifies limitations in the previous symmetry-based classification of black holes and proposes specific boundary conditions under which the earlier results hold.

Findings

The H-transformation used to remove charges does not always exist.

Certain boundary conditions on scalar fields allow the previous symmetry results to be valid.

The general proof of black hole equivalence under symmetries has exceptions.

Abstract

Breitenlohner, Maison and Gibbons claimed some time ago that all bona-fide four dimensional asymptotically flat non-degenerate black holes are in a symmetry orbit of the Schwarzschild/Kerr black hole in a large set of theories of gravity and matter. Their argument involved reducing the theory on a time-like Killing vector field and analysing the resulting three dimensional sigma model of maps to a symmetric space $G/H$. In the construction of their proof, they conjectured the existence of a suitable $H$-transformation that always remove the electromagnetic charges of the four dimensional black hole solution. We show in this short note that such a transformation does not exist in general, and discuss a set of boundary conditions on the horizon for the scalar fields in the sigma model that yield black holes for which the result by Breitenlohner, Maison and Gibbons can be applied.