Trapped and anti-trapped surfaces in Friedmann universes
Description
Explicit expressions for the location of trapped, and anti-trapped surfaces of cosmological origin in Friedmann type universes are obtained. Friedmann universes dominated by energy-momentum tensors of matter with an equation of state of the form $p = (\gamma - 1)\rho$ (where $\gamma > {2\over3},\ p$ is the pressure, and $\rho$ is the energy density), a decoupled mixture of matter and radiation, and a positive cosmological constant are considered. The first form includes the physically significant special cases of pressureless matter or dust ($\gamma$ = 1), electromagnetic radiation ($\gamma = {4\over3}$), and a non-interacting massless scalar field ($\gamma$ = 2). A model closed Friedmann universe with trapped surfaces in the contracting phase but with no event horizons is also considered in this study, in order to emphasize the distinction between horizons and trapped surfaces. Penrose diagrams for special cases are drawn, showing the location and nature of trapped/anti-trapped, and marginally trapped/anti-trapped surfaces. The computational results for the marginally trapped and marginally anti-trapped surfaces and the trapping horizons foliated by marginally trapped/anti-trapped surfaces are analyzed in the context of available classification schemes for such surfaces. It is demonstrated by means of counterexamples that the current classification schemes fail to properly distinguish between trapping horizons of cosmological vs. non-cosmological origin, and that event horizons and trapped surfaces need not occur together. This demonstrates that an adequate definition for black holes in closed universes does not yet exist. A joint between a dust dominated closed Friedmann universe and a horizonless closed universe is also studied. Since the latter universe has no future event horizons, such a matching will enable a closed Friedmann universe, which has future event horizons, to evolve into a universe without such horizons. Although a physical process which could accomplish this has not been postulated here, the junction conditions and the energy conditions do not preclude such a transition