Brane-world creation and black holes

An inflating brane-world can be created from ``nothing'' together with its Anti-de Sitter (AdS) bulk. The resulting space-time has compact spatial sections bounded by the brane. During inflation, the continuum of KK modes is separated from the massless zero mode by the gap $m=(3/2) H$, where $H$ is the Hubble rate. We consider the analog of the Nariai solution and argue that it describes pair production of ``Black cigars'' attached to the inflating brane. In the case when the size of the instantons is much larger than the AdS radius, the 5-dimensional action agrees with the 4-dimensional one. Hence, the 5D and 4D gravitational entropies are the same in this limit. We also consider thermal instantons with an AdS black hole in the bulk. These may be interpreted as describing the creation of a hot universe from nothing, or the production of AdS black holes in the vicinity of a pre-existing inflating brane-world. The Lorentzian evolution of the brane-world after creation is briefly discussed. An additional "integration constant" in the Friedmann equation -accompanying a term which dilutes like radiation- describes the tidal force in the fifth direction and arises from the mass of a spherical object inside the bulk. This could be a 5-dimensional black hole or a "parallel" brane-world of negative tension concentrical with our brane-world. In the case of thermal solutions, and in the spirit of the $AdS/CFT$ correspondence, one may attribute the additional term to thermal radiation in the boundary theory. Then, for temperatures well below the AdS scale, the entropy of this radiation agrees with the entropy of the black hole in the AdS bulk.