Geometric Constraint in Brane-World

TL;DR

This paper investigates the geometric constraints of brane-world models, showing that Schwarzschild spacetime cannot be embedded in a five-dimensional constant curvature bulk, which impacts the theoretical foundations of such models.

Contribution

It demonstrates a geometric limitation in brane-world models by proving Schwarzschild spacetime cannot be embedded in a 5D constant curvature bulk.

Findings

Schwarzschild spacetime cannot be embedded in a 5D constant curvature bulk.

Theorem of Collinson-Szekeres is used to establish embedding constraints.

Constraints impact the theoretical viability of brane-world models.

Abstract

The brane-worlds model was inspired partly by Kaluza-Klein's theory, where the gravitation and the gauge fields are obtained of a geometry of higher dimension (bulk). Such a model has been showing positive in the sense of we find perspectives and probably deep modifications in the physics, such as: Unification in a scale TeV, quantum gravity in this scale and deviation of Newton's law for small distances. One of the principles of this model is to suppose a space-time embedded in a bulk of high dimension. In this note it is shown, basing on the theorem of Collinson-Szekeres, that the space-time of Schwarzschild cannot be embedded locally and isometrically in a bulk of five dimensions with constant curvature,(for example ADS-5). From the point of view of the semi-Riemannian geometry this last result consists constraints to the model brane-world.