Divergence equations and uniqueness theorem of static spacetimes with conformal scalar hair

TL;DR

This paper presents a new divergence identity and a simplified proof for the uniqueness of static spacetimes with conformal scalar hair, extending the classical Israel proof to include scalar fields.

Contribution

It introduces a systematic derivation of a divergence identity with three parameters, enabling a novel proof of the uniqueness theorem in Einstein-conformal scalar systems.

Findings

New divergence identity derived systematically

Proof of uniqueness theorem extended to conformal scalar hair

Simplified and generalized proof method provided

Abstract

We reexamine the Israel-type proof of the uniqueness theorem of the static spacetime outside the photon surface in the Einstein-conformal scalar system. We derive in a systematic fashion a new divergence identity which plays a key role in the proof. Our divergence identity includes three parameters, allowing us to give a new proof of the uniqueness.