The Embedding of Schwarzschild in Braneworld

TL;DR

This paper demonstrates a mathematical inconsistency in the Randall-Sundrum braneworld model by proving that Schwarzschild space-time cannot be embedded in a five-dimensional constant curvature bulk, challenging its geometric viability.

Contribution

It provides a theorem showing the impossibility of embedding Schwarzschild space-time in the Randall-Sundrum model's bulk, revealing a fundamental geometric limitation.

Findings

Schwarzschild space cannot be embedded in AdS-5 bulk

Highlights a geometric restriction in Randall-Sundrum models

Challenges the assumptions of braneworld embeddings

Abstract

The braneworlds models were inspired partly by Kaluza-Klein's theory, where both the gravitational and the gauge fields are obtained from the geometry of a higher dimensional space. The positive aspects of these models consist in perspectives of modifications it could bring in to particle physics, such as: unification in a TeV scale, quantum gravity in this scale and deviation of Newton's law for small distances. One of the principles of these models is to suppose that all space-times can be embedded in a bulk of higher dimension. The main result in these notes is a theorem showing a mathematical inconsistency of the Randall-Sundrum braneworld model, namely that the Schwarzschild space-time cannot be embedded locally and isometrically in a five dimensional bulk with constant curvature,(for example AdS-5). From the point of view of semi-Riemannian geometry this last result represents a serious restriction to the Randall-Sundrum's braneworld model.