Einstein equations for an asymmetric brane-world

We consider a brane-world of codimension one without the reflection symmetry which is commonly imposed between the two sides of the brane. Using the coordinate-free formalism of the Gauss-Codazzi 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.