(D-E)-dimensional brane worlds and de Rham distribution formalism: Singular split versus compactification, restrictions on scenario and revision of gravitational energy problem

TL;DR

This paper develops a rigorous theory for embedding our universe as a singular submanifold in higher-dimensional spaces, extending brane-world models beyond the common 5D case to 10D, and discusses implications for gravity and cosmology.

Contribution

It introduces a general framework for singular embeddings in higher dimensions, including 10D, using de Rham distributions, and analyzes restrictions and features of such models.

Findings

Singular embedding of 4-universe only possible in 5D and 6D spaces.

In 10D, extra dimensions must form a product space for embedding.

Reformulation of gravitational energy and stress-energy tensor in brane-world scenarios.

Abstract

The proposed so far brane-world cosmological scenarios are concerned with (D-1)-dimensional embeddings into the D-dimensional spacetime, besides, it is supposed D=5 as a rule. However, it would be much more realistic to consider our four-Universe as 4-shell or 3-brane inside, e.g., 10-dimensional spacetime. In turn it immediately means that the theory of the $(D-D_E)$-dimensional singular embeddings, where the number of extra dimensions $D_E > 1$, is needed. 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 being confined on the singular submanifold but gravity being propagated through the high-$D$ manifold. With the use of the de Rham axiomatic approach to delta-distributions we demonstrate that the four-Universe can be singularly embedded only in five- and six-dimensional space so if we want to consider its embedding in 10D then extra dimensions must be included as a product space only. We discuss the revealed generic features of the theory such as the multi-normal anisotropy, restrictions on an ambient space, reformulation of the conserved gravitational stress-energy tensor problem, etc.