Kleinian black holes

We prove there is a unique vacuum solution in split-signature spacetimes with Kleinian SO(2,1) spherical symmetry. We extend our analysis to accommodate a positive or negative cosmological constant and we prove the Kleinian spherically symmetric solutions to Einstein's equation are locally isomorphic to the split-signature analogues of Schwarzschild-(Anti)-de Sitter or Nariai spacetimes. Our analysis provides a Kleinian extension of Birkhoff's theorem to metrics with split-signature. Axisymmetric vacuum solutions are also considered, including (2,2) signature formulations of the Kerr and Taub-NUT metrics.