Mass of a Patch of an FRW Universe

Abstract: 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.