Stationary Black Holes with Time-Dependent Scalar Fields

TL;DR

This paper extends the no-hair theorems by proving that stationary, asymptotically flat black holes cannot support time-dependent hair from a real, non-canonical scalar field, under broad assumptions about the action and spacetime. The proof leverages the Einstein equations, the form of the metric in an axisymmetric, asymptotically flat setting, and asymptotic behavior to show that any time dependence in a real scalar field is generically incompatible with stationarity, forcing the time-dependence parameter to vanish. The result does not extend to complex scalar fields, where the dynamics permit time dependence without violating stationarity, aligning with known boson-star phenomena and numerically reported hairy black holes. Overall, the work fills a gap in the no-hair program, with implications for scalar-tensor theories and potential extensions to alternative gravity models.

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.