On the embedding of branes in five-dimensional spaces

TL;DR

This paper explores how four-dimensional branes can be embedded in five-dimensional spaces, analyzing various models including vacuum bulk and Randall-Sundrum scenarios, and demonstrating that brane matter content doesn't solely determine its curvature.

Contribution

It applies the Campbell-Magaard theorem to construct embeddings in different contexts, revealing that brane energy-matter content isn't the only factor influencing curvature.

Findings

Embedding in vacuum bulk with delta-function energy-momentum tensor.

Inclusion of Z2 symmetry and cosmological constant in Randall-Sundrum models.

Constructed embedding of Minkowski spacetime with dust.

Abstract

We investigate the embedding of four-dimensional branes in five-dimensional spaces. We firstly consider the case when the embedding space is a vacuum bulk whose energy-momentum tensor consists of a Dirac delta function with support in the brane. We then consider the embedding in the context of Randall-Sundrum-type models, taking into account $Z_{2}$ symmetry and a cosmological constant. We employ the Campbell-Magaard theorem to construct the embeddings and are led to the conclusion that the content of energy-matter of the brane does not necessarily determine its curvature. Finally, as an application to illustrate our results, we construct the embedding of Minkowski spacetime filled with dust.