  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.
