Reformulation and Interpretation of SMS Braneworld

TL;DR

This paper reformulates the SMS braneworld equations within five-dimensional Einstein theory, highlighting the non-uniqueness of Weyl term splitting and emphasizing the importance of studying the full brane-bulk system rather than truncated versions.

Contribution

It introduces new formulations of the SMS braneworld equations that are second-order and clarifies the non-uniqueness of Weyl term splitting, advocating for full brane-bulk analysis.

Findings

Different formulations are equivalent but can lead to different results when truncated.

Truncation of Weyl terms can cause inaccuracies in anisotropic or inhomogeneous models.

New second-order formulations may facilitate the study of brane-bulk systems.

Abstract

We reformulate the Shiromizu, Maeda and Sasaki (SMS) braneworlds within the framework of the five-dimensional Einstein equations. In many applications of the braneworld Einstein field equations, the Weyl term is attributed to the bulk, thus splitting the non-Einsteinian terms into `bulk' and `brane' terms. Here by employing standard geometrical identities, we show that such a split is non-unique, since these terms get mixed up in different formulations. An important consequence of this non-uniqueness is that even though the full brane-bulk systems in all such formulations are completely equivalent, important differences can arise were one to truncate different formulations by throwing away the associated `bulk' terms. This is particularly likely to be the case in more general anisotropic/inhomogeneous settings with non-AdS bulks, in which the usual truncation of the SMS (which throws away the Weyl term) would not coincide with the full system. We emphasize that rather than providing support for any truncation, these differences show clearly the dangers of using any truncated equations and provide a strong argument in favour of studying the full brane-bulk system. The different formulations we provide also permit different ways of approaching the full brane-bulk system which may greatly facilitate its study. An example of this is the second-order nature of the formulations given here as opposed to the SMS's formulation which is third-order.