It is shown that previous asymptotically Einstein–de Sitter cosmological black hole spacetimes violate the energy conditions in some region of spacetime. Thus, cosmological black hole spacetimes (for the Schwarzschild and Reissner–Nordström cases) are obtained that satisfy the energy conditions throughout spacetime. The solutions are obtained by performing conformal transformations on the isotropic forms of the isolated black hole metrics.

The ultimate application of Einstein's field equations is to empirically determine the geometry of the Universe from its matter content, rather than simply assuming the Universe can be represented by a homogeneous model on all scales. Choosing a Lemaître-Tolman-Bondi model as the most convenient inhomogeneous model for the early stages of development, a data reduction procedure was recently validated using perfect test data. Here, we simulate observational uncertainties and improve the previous numerical scheme to ensure that it will be usable with real data as soon as observational surveys are sufficiently deep and complete. Two regions require special treatment-the origin and the maximum in the areal radius. To minimize numerical errors near the origin, we use a Lemaître-Tolman-Bondi series expansion to provide the initial values for integrating the differential equations. We also use an improved method to match the numerical integration to the series expansion that bridges the region near the maximum in the areal radius. Because the mass enclosed within the maximum obeys a specific relationship, we show that it is possible to correct for a fixed systematic error in either the distance scale or the redshift-space mass density, such that the integrated values are consistent with the data at the maximum.

Based on general relativity, it can be argued that deviations from a uniform Hubble flow should be thought of as variations in the Universe's expansion velocity field, rather than being thought of as peculiar velocities with respect to a uniformly expanding space. The aim of this paper is to use the observed motions of galaxies to map out variations in the Universe's expansion, and more importantly, to investigate whether real variations in the Hubble expansion are detectable given the observational uncertainties. All-sky maps of the observed variation in the expansion are produced using measurements obtained along specific lines-of-sight and smearing them across the sky using a Gaussian profile. A map is produced for the final results of the HST Extragalactic Distance Scale Key Project for the Hubble constant, a comparison map is produced from a set of essentially independent data, and Monte Carlo techniques are used to analyse the statistical significance of the variation in the maps. A statistically significant difference in expansion rate of 9 km s −1 Mpc −1 is found to occur across the sky. Comparing maps of the sky at different distances appears to indicate two distinct sets of extrema with even stronger statistically significant variations. Within our supercluster, variations tend to occur near the supergalactic plane, and beyond our supercluster, variations tend to occur away from the supergalactic plane. Comparison with bulk flow studies shows some concordance, yet also suggests the bulk flow studies may suffer confusion, failing to discern the influence of multiple perturbations.

By matching across a surface of constant time, it is demonstrated that the spacetime for a radiation-dominated Einstein-de Sitter universe can be directly matched to the spacetime for a matter-dominated Einstein-de Sitter universe. Thus, this can serve as a model of a universe filled with radiation that suddenly is converted to matter and antimatter, or a universe filled with matter and antimatter that suddenly annihilates to leave radiation. This matching is shown to hold for asymptotically Einstein-de Sitter cosmological black hole spacetimes, yielding simplistic models of primordial black holes that evolve between being in radiation-dominated universes and matter-dominated universes.