A no-go on strictly stationary spacetimes in four/higher dimensions
  We show that strictly stationary spacetimes cannot have non-trivial configurations of form fields/complex scalar fields and then the spacetime should be exactly Minkowski or anti-deSitter spacetimes depending on the presence of negative cosmological constant. That is, self-gravitating complex scalar fields and form fields cannot exist. 
Tetsuya Shiromizu, Seiju Ohashi and Ryotaku Suzuki
gr-qc hep-th