Generic junction conditions in brane-world scenarios

TL;DR

This paper derives the general junction conditions for a co-dimension one brane in arbitrary backgrounds, extending the usual conditions by including an additional law governing brane evolution, with applications to force calculations.

Contribution

It introduces a new junction condition analogous to Newton's second law, applicable beyond symmetric scenarios, and demonstrates its impact on brane dynamics.

Findings

Derived the general junction conditions for branes in arbitrary backgrounds.

Identified an additional evolution condition beyond the Darmois-Israel junction conditions.

Calculated the force on the brane due to a form field.

Abstract

We present the generic junction conditions obeyed by a co-dimension one brane in an arbitrary background spacetime. As well as the usual Darmois-Israel junction conditions which relate the discontinuity in the extrinsic curvature to the to the energy-momentum tensor of matter which is localized to the brane, we point out that another condition must also be obeyed. This condition, which is the analogous to Newton's second law for a point particle, is trivially satisfied when $Z_2$ symmetry is enforced by hand, but in more general circumstances governs the evolution of the brane world-volume. As an illustration of its effect we compute the force on the brane due to a form field.