Lorentzian and signature changing branes

TL;DR

This paper develops a formalism for describing brane-worlds and shell cosmologies with hypersurfaces that can change signature, ensuring regularity and unifying timelike and signature-changing branes, with applications to static bulks and anti de Sitter spaces.

Contribution

It introduces a consistent, regular framework for signature-changing branes that unifies different types of hypersurfaces and explores their physical properties and relations to fundamental constants.

Findings

Signature change can occur smoothly without singularities.

Energy density vanishes near signature change.

Upper bound on energy density in asymmetric shells.

Abstract

General hypersurface layers are considered in order to describe brane-worlds and shell cosmologies. No restriction is placed on the causal character of the hypersurface which may thus have internal changes of signature. Strengthening the results in our previous letter [1], we confirm that a good, regular and consistent description of signature change is achieved in these brane/shells scenarios, while keeping the hypersurface and the bulk completely regular. Our formalism allows for a unified description of the traditional timelike branes/shells together with the signature-changing, or pure null, ones. This allows for a detailed comparison of the results in both situations. An application to the case of hypersurface layers in static bulks is presented, leading to the general Robertson-Walker geometry on the layer --with a possible signature change. Explicit examples on anti de Sitter bulks are then studied. The permitted behaviours in different settings ($Z_{2}$-mirror branes, asymmetric shells, signature-changing branes) are analysed in detail. We show in particular that (i) in asymmetric shells there is an upper bound for the energy density, and (ii) that the energy density within the brane vanishes when approaching a change of signature. The description of a signature change as a `singularity' seen from within the brane is considered. We also find new relations between the fundamental constants in the brane/shell, its tension, and the cosmological and gravitational constants of the bulk, independently of the existence or not of a change of signature.