5 Dynamically generated embeddings of spacetime
نویسنده
چکیده
We discuss how embeddings in connection with the CampbellMagaard (CM) theorem can have a physical interpretation. We show that any embedding whose local existence is guaranteed by the CM theorem can be viewed as a result of the dynamical evolution of initial data given in a four-dimensional spacelike hypersurface. By using the CM theorem, we establish that for any analytic spacetime, there exist appropriate initial data whose Cauchy development is a five-dimensional vacuum space into which the spacetime is locally embedded. We shall see also that the spacetime embedded is Cauchy stable with respect these the initial data. The development of the braneworld scenario [1, 2, 3, 4], in which our ordinary spacetime is viewed as a hypersurface of a higher-dimensional space, has greatly contributed to increase recent interest on embeddings theorems. The so-called non-compact Kaluza-Klein (NKK) models [5, 6, 7] have also motivated the study of embedding problems of the spacetime. In this context, the Campbell-Magaard (CM) theorem [8, 9, 10] and its variants [11, 12, 13, 14, 15] are of special interest when the embedding spaces possess only one extra dimension as is the case of the Randall-Sundrum braneworld scenario [3, 4] and the NKK models [5, 6, 7]. The CM theorem ensures the existence of local and analytic isometric embedding of any n-dimensional analytic manifoldM into a Ricci-flat (n+ 1)Henceforth embedding for us will mean analytic and isometric embedding.
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