# A covariant simultaneous action for branes

## TL;DR

This paper introduces a covariant, simultaneous action principle for branes in curved spacetimes, unifying equations of motion and Jacobi equations, with applications to Dirac-Nambu-Goto branes and potential extensions to extrinsic geometry.

## Contribution

It presents a novel covariant action depending on independent variables that simplifies deriving brane equations and extends previous particle models to branes in curved backgrounds.

## Key findings

- Unified derivation of brane equations of motion and Jacobi equations.
- Application to Dirac-Nambu-Goto branes in curved spacetime.
- Framework potentially extendable to extrinsic geometry models.

## Abstract

A covariant simultaneous action for branes in an arbitrary curved background spacetime is considered. The action depends on a pair of independent field variables, the brane embedding functions, through the canonical momentum of a reparametrization invariant geometric model for the brane, and an auxiliary vector field. The form of the action is analogous to a symplectic potential. Extremization of the simultaneous action produces at once the equations of motion and the Jacobi equations for the brane geometric model, and it also provides a convenient shortcut towards its second variation. In this note, we consider geometric models depending only on the intrinsic geometry of the brane worldvolume, and discuss briefly the generalization to extrinsic geometry dependent models. The approach is illustrated for Dirac-NambuGoto [DNG] branes. For a relativistic particle, a simultaneous action was introduced by Bazanski, that served as an inspiration for the present work.