McVittie metric information

In the general theory of relativity, the McVittie metric is the exact solution of Einstein's field equations describing a black hole or massive object immersed in an expanding cosmological spacetime. The solution was first fully obtained by George McVittie in the 1930s, while investigating the effect of the, then recently discovered, expansion of the Universe on a mass particle.

The simplest case of a spherically symmetric solution to the field equations of General Relativity with a cosmological constant term, the Schwarzschild-De Sitter spacetime, arises as a specific case of the McVittie metric, with positive 3-space scalar curvature ${\displaystyle \kappa =+1}$ and constant Hubble parameter ${\displaystyle H(t)=H_{0}}$.