Gravitational Field of a Rotating Mass on an Expanding Universe

We present a new exact solution to Einstein's field equations that unifies the Kerr black hole with Friedmann-Lema\^itre-Robertson-Walker cosmology. This metric reduces to Kerr-de Sitter in the appropriate limit and accounts for cosmological expansion dynamics both inside and outside the black hole. The model predicts a stationary mass, as well as a contracting ergosphere and event horizon with respect to the expanding cosmic rest frame. This result correctly generalises the McVittie metric to rotating masses or Kerr-de Sitter to arbitrary scale factors $a(t)$. Additionally, we find that the ergosphere tends to fade away with respect to the universe's expansion and no further interactions of dark energy with respect to the black hole's rotation.