Black holes in the expanding Universe

Abstract: The McVittie metric does not describe a physical black hole in an expanding universe because the curvature scalar and pressure at its event horizon are infinite. We show that extending this metric to an inhomogeneous scale factor, which depends on both the time and radial coordinate, removes those infinities by imposing the constancy of the Hubble parameter at the horizon. We also show that the Hubble parameters at the event horizons of all centrally symmetric black holes without accreting matter are equal to the same constant $H_\textrm{hor}=(\Lambda/3)<sup>{1/2}$. Because of this equality and the equivalence to the Kottler metric near the horizon, black holes do not grow with the universe expansion. Black holes with accretion are described by this extension with different values of $H_\textrm{hor}$.