Black holes with supertranslation field, "large transformations" and Israel theorem

TL;DR

This paper investigates a vacuum solution with a supertranslation field related to the Schwarzschild metric, demonstrating that the Israel theorem's proof extends to this case despite boundary condition differences.

Contribution

It shows that the Israel theorem applies to metrics with supertranslation fields, clarifying the role of large transformations in the theorem's applicability.

Findings

The solution contains a supertranslation field diffeomorphic to Schwarzschild.

The proof of Israel theorem extends to metrics with supertranslation fields.

Large transformations affect the boundary conditions but not the core proof.

Abstract

An axial-symmetric vacuum solution of the Einstein equations containing a supertranslation field diffeomorphic to the Schwarzschild solution is discussed in the context of 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 supertranslation field and the (transformed) metric used in the proof is spherically symmetric. We explain the source of the seeming discrepancy connected with the use of "large" transformations changing supertranslation field in the metric.