PDE System Approach to Large Extra Dimensions

TL;DR

This paper investigates the mathematical foundations of large extra dimensions in braneworld models, comparing timelike and spacelike hypersurface splits, and proposing methods for constructing initial data in these higher-dimensional theories.

Contribution

It provides a comprehensive analysis of hypersurface splitting in D-dimensional Einstein field equations, emphasizing the advantages of spacelike splits and extending initial data construction methods to braneworld scenarios.

Findings

Spacelike splits are well-behaved and preferable for initial data formulation.

Timelike splits face mathematical and physical challenges, especially regarding singularities.

Algorithms for constructing initial data with branes are developed, including boundary conditions and thick brane considerations.

Abstract

We explore some fundational issues regarding the splitting of D-dimensional EFE's w.r.t timelike and spacelike (D-1)-dimensional hypersurfaces, first without and then with thin matter sheets such as branes. We begin to implement methodology, that is well-established for the GR CP and IVP, in the new field of GR-based braneworlds, identifying and comparing many different choices of procedure. We abridge fragmentary parts of the literature of embeddings, putting the Campbell--Magaard theorem into context. We recollect and refine arguments why York and not elimination methods are used for the GR IVP. We compile a list of numerous mathematical and physical impasses to using timelike splits, whereas spacelike splits are known to be well-behaved. We however pursue both options to make contact with the current braneworld literature which is almost entirely based on timelike splits. We look at the Shiromizu-Maeda -Sasaki braneworld by means of reformulations which emphasize different aspects from the original formulation. We show that what remains of the York method in the timelike case generalizes heuristic bulk construction schemes. We formulate timelike (brane) versions of the thin sandwich conjecture. We discuss whether it is plausible to remove singularities by timelike embeddings. We point out how the braneworld geodesic postulates lead to further difficulties with the notion of singularities than in GR where these postulates are simpler. Having argued for the use of the spacelike split, we study how to progress to the construction of more general data sets for spaces partially bounded by branes. Boundary conditions are found and algorithms provided. Working with (finitely) thick branes would appear to facilitate such a study.