Brane Variation Dirac Style

TL;DR

This paper demonstrates a Dirac-style variation method for branes, deriving key equations like the Israel junction condition and energy conservation, with applications to gravity in higher-dimensional embeddings.

Contribution

It introduces a novel Dirac-style variation approach for branes, providing new derivations of fundamental brane equations in gravitational contexts.

Findings

Re-derivation of Snell's law using the method

Derivation of Israel junction condition in a new framework

Formulation of energy-momentum conservation and geodetic equations for branes

Abstract

Dirac's method for variations of a brane embedded in co-dimension one is demonstrated. The variation in the location of the brane invokes a rest frame formulation of the 'sandwiched' brane action. We first demonstrate the necessity of this method by re-deriving Snell's law. Second, we apply the method to a general $N$-dimensional brane embedded in co-dimension one bulk in the presence of gravity. We re-derive the brane equations: (i) Israel junction condition, (ii) Energy/momentum conservation on the brane, and (iii) Geodetic-type equation for the brane.