Stationary Black Holes with Time-Dependent Scalar Fields

TL;DR

This paper proves that stationary black holes cannot support scalar fields with time dependence, extending no-hair theorems to more general scalar fields and spacetimes.

Contribution

It generalizes no-hair theorems by showing stationary black holes lack scalar hair even with time-dependent scalar fields.

Findings

Stationary black holes do not support time-dependent scalar fields.

The proof applies to non-canonical scalar fields and certain non-asymptotically flat spacetimes.

The results do not depend on the spacetime being a black hole.

Abstract

It has been well known since the 1970s that stationary black holes do not generically support scalar hair. Most of the no-hair theorems which support this depend crucially upon the assumption that the scalar field has no time dependence. Here we fill in this omission by ruling out the existence of stationary black hole solutions even when the scalar field may have time dependence. Our proof is fairly general, and in particular applies to non-canonical scalar fields and certain non-asymptotically flat spacetimes. It also does not rely upon the spacetime being a black hole.