Einstein equations for an asymmetric brane-world

TL;DR

This paper derives effective Einstein equations for an asymmetric brane-world without reflection symmetry, showing Einstein gravity emerges naturally in the weak field limit and connecting to cosmological solutions.

Contribution

It introduces a coordinate-free formalism to derive Einstein equations for asymmetric brane-worlds, extending previous symmetric models and linking to cosmological solutions.

Findings

Einstein gravity emerges naturally in the weak field limit of asymmetric brane-worlds.

The derived equations agree with known cosmological solutions.

Models with a bulk form field can realize such asymmetric brane-worlds.

Abstract

We consider a brane-world of co-dimension one without the reflection symmetry that is commonly imposed between the two sides of the brane. Using the coordinate-free formalism of the Gauss-Codacci equations, we derive the effective Einstein equations by relating the local curvature to the matter on the brane in the case when its bare tension is much larger than the localized matter, and hence show that Einstein gravity is a natural consequence of such models in the weak field limit. We find agreement with the recently derived cosmological case, which can be solved exactly, and point out that such models can be realized naturally in the case where there is a minimally coupled form field in the bulk.