Black holes with a supertranslation field and the Israel theorem

An axial-symmetric vacuum solution of the Einstein equations containing a supertranslation field diffeomorphic to the Schwarzschild solution is discussed in the context of the Israel theorem. The metric satisfies all conditions of the Israel theorem except for the condition on the form of the metric at spatial infinity. Nevertheless, following the steps of the proof of the theorem, we show that the proof applies to the metric with a supertranslation field and that the (transformed) metric used in the proof is spherically symmetric. We explain the source of the seeming discrepancy.