Black holes cannot support conformal scalar hair

TL;DR

This paper proves that static, asymptotically flat black holes cannot support conformal scalar hair of finite length, implying the Schwarzschild solution is unique under these conditions.

Contribution

The work demonstrates the non-existence of conformal scalar hair for static, asymptotically flat black holes with bounded scalar fields on the horizon.

Findings

Schwarzschild is the only solution under the specified conditions

Black holes cannot have finite-length conformal scalar hair

Scalar field boundedness on the horizon is crucial for the proof

Abstract

It is shown that the only static asymptotically flat non-extrema black hole solution of the Einstein-conformally invariant scalar field equations having the scalar field bounded on the horizon, is the Schwarzschild one. Thus black holes cannot be endowed with conformal scalar hair of finite length.