  In 1963, Zel'dovich devised a method to define the mass of a closed
Friedmann-Robertson-Walker (FRW) universe, showing that by this definition it
is exactly zero. Rounding out this result, we show that the masses of flat and
open universes are (unsurprisingly) divergent. We also present closed-form
solutions for the trajectory of the boundary of a finite spherical patch of
homogeneous pressureless dust for each class of curvature, exploring the
dynamics of the boundary in detail. In all cases, the FRW patch emerges from a
white hole. In the closed case, the patch expands to a maximum radius before
contracting and entering a black hole, while flat and open FRW patches expand
without bound. We compare our results to the classical expectations of
Newtonian cosmology, showing that for small radii the Newtonian energy gives
the leading correction to the rest mass energy.

General Relativity and Quantum Cosmology