Static horizons in cosmology
Although previous results have ruled out the possibility of a static event horizon in cosmology, we present black hole and white hole metrics that retain static event horizons while reproducing cosmological behavior at large distances. Using an appropriate coordinate choice, we demonstrate that a static event horizon can exist in a cosmological setting without introducing curvature invariant singularities at the horizon. The resulting metric reduces to the Schwarzschild de Sitter solution when the Hubble parameter is constant. We find that white hole metrics in an expanding universe, or black holes in a contracting universe, are significantly easier to construct, as a black hole in an expanding cosmology requires the velocity function to change sign. Consequently, this work initially examines white holes in expanding cosmologies as a foundation for subsequent analysis of black holes in expanding universes. In later sections, we investigate scenarios involving a white hole coupled with cosmological matter, as well as a white hole with both matter and a cosmological constant. Assuming the pressure component takes its cosmological value, we show that the physical radius of the apparent horizon can asymptotically approach a constant value at late times. This metric avoids pathologies such as a singular horizon in the limit of a vanishing Hubble parameter. Finally, we analyze the realistic case of a black hole embedded in pressureless cosmological matter with and without a cosmological constant and explore its properties. We specifically show that the velocity function can become zero and change sign in the vicinity of a black hole. This means we can smoothly transition from an expanding cosmological phase with a positive velocity function to a contracting black hole phase with a negative velocity function.