Induced gravity and multi-normal anisotropy on (D-E)-dimensional submanifolds: Singular split versus compactification and revision of gravitational energy problem

Not long ago it was proposed the (4+1)-dimensional brane-world cosmological scenarios where an extra dimension was assumed to be orthogonal to the Universe rather than compactified. However, all the proposed so far models in this theme are concerned with (D − 1)-dimensional embeddings into the D-dimensional spacetime, besides, it is supposed D = 5 as a rule. However, the regarding of the five-dimensional spacetime as a physical one is a step in past because the modern concepts of superstring theory require to consider our four-Universe as a region inside of a much more higher-dimensional manifold. So, it would be much more realistic to consider our four-Universe as 4-shell or 3-brane inside, e.g., 10-dimensional (or even infinite-dimensional) spacetime. In turn it immediately means that it is needed the theory of the (D−DE)-dimensional singular embeddings where the number of extra dimensions DE > 1. Hence, the aim of this work is to provide such a theory: we construct the rigorous general theory of the induced gravity on singular submanifolds. At first, we perform the decomposition of the tangent bundle into the two subbundles which will be associated later with external and visible (with respect to some low-dimensional observer) parts of the high-D manifold. Then we go to physics and perform the split of the manifold (in addition to the split of the tangent bundle) to describe both the induced internal geometry and external as-a-whole dynamics of singular embeddings, assuming matter (including the Standard Model with the corresponding fiber bundle) to be confined on the singular submanifold but gravity to be propagated through the high-D manifold. Discussing the revealed generic features of the theory (such as the multi-normal anisotropy as the new mechanism of generation of internal degrees of freedom which supplements the existing ones, reformulation of the conserved gravitational stress-energy tensor problem, etc.) we present the opposite, optimistic and pessimistic, viewpoints on the contemporary brane-world models. PACS number(s): 04.50.+h, 98.80.Hw, 11.27.+d Typeset using REVTEX ∗On leave from Department of Theoretical Physics, Dnepropetrovsk State University, Ukraine. Electronic address: [email protected]