Generation of Source Terms in General Relativity by differential structures

TL;DR

This paper explores how changing differential structures in general relativity can generate source terms resembling energy-momentum tensors, highlighting the impact of exotic smooth structures on gravitational equations.

Contribution

It demonstrates that modifications in differential structures induce source terms in Einstein's equations, providing a novel geometric origin for matter-like contributions.

Findings

Exotic $S^7$ structures lead to curvature corrections not removable by gauge transformations.

In four dimensions, embedded surface intersections produce connection corrections.

These corrections act as source terms analogous to energy-momentum tensors.

Abstract

In this paper the relation between the choice of a differential structure and a smooth connection in the tangential bundle is discussed. For the case of an exotic $S^7$ one obtains corrections to the curvature after the change of the differential structure, which can not be neglected by a gauge transformation. In the more interesting case of four dimensions we obtain a correction of the connection constructed by intersections of embedded surfaces. This correction produce a source term in the equation of the general relativity theory which can be interpreted as the energy-momentum tensor of a embedded surface.