Black Hole Universe: Time Evolution

Time evolution of a black hole lattice universe is simulated. The vacuum Einstein equations in a cubic box with a black hole at the origin are numerically solved with periodic boundary conditions on all pairs of opposite faces. Defining effective scale factors by using the area of a surface and the length of an edge of the cubic box, we compare them with that in the Einstein-deSitter universe. It is found that the behaviour of the effective scale factors is well approximated by that in the Einstein-deSitter universe. Our result suggests that local inhomogeneities do not significantly affect the global expansion law of the universe even if the inhomogeneity is extremely nonlinear.