Black holes in the expanding Universe

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 at the horizon the constancy of the Hubble parameter and a particular constraint on the gradient of the scale factor. We consider a special case of this metric, and show that the Hubble parameters at the event horizons of all centrally symmetric black holes are equal to the same constant . Because of this equality and the equivalence to the Kottler metric near the horizon, black holes do not grow with the Universe expansion.