Essentials of Classical Brane Dynamics

TL;DR

This work provides a comprehensive, gauge-friendly treatment of relativistic brane geometry and dynamics, unifying worldsheet kinematics with extrinsic motion through the fundamental tensors ${\eta}^{\mu}_{\ \nu}$, ${\mathrm{K}}_{\mu\nu}{}^{\rho}$, and ${\mathrm{C}}_{\mu\nu}{}^{\rho}$. It derives the general extrinsic motion equation ${\overline{T}}^{\mu\nu}{\mathrm{K}}_{\mu\nu}^{\rho}=f^{\rho}$, encompassing external background forces such as electromagnetism and Kalb–Ramond fields via couplings $e_{\{p\}}$, and addresses regular vs distributional action formulations for brane complexes. The text also develops perturbation theory and the extrinsic characteristic equation, showing that perturbations propagate with speeds determined by the surface stress-energy ${\overline{T}}^{\mu\nu}$ through the condition ${\overline{T}}^{\mu\nu}{\chi}_{\mu}{\chi}_{\nu}=0$, with implications for cosmic strings and brane-world cosmologies. Overall, the article provides a rigorous, conformally aware framework for analyzing brane dynamics in diverse backgrounds, linking intrinsic geometry with extrinsic evolution and perturbations.

Abstract

This article provides a self contained overview of the geometry and dynamics of relativistic brane models, of the category that includes point particle, string, and membrane representations for phenomena that can be considered as being confined to a worldsheet of the corresponding dimension (respectively one, two, and three) in a thin limit approximation in an ordinary 4 dimensional spacetime background. This category also includes ``brane world'' models that treat the observed universe as a 3-brane in 5 or higher dimensional background. The first sections are concerned with purely kinematic aspects: it is shown how, to second differential order, the geometry (and in particular the inner and outer curvature) of a brane worldsheet of arbitrary dimension is describable in terms of the first, second, and third fundamental tensor. The later sections show how -- to lowest order in the thin limit -- the evolution of such a brane worldsheet will always be governed by a simple tensorial equation of motion whose left hand side is the contraction of the relevant surface stress tensor $ bar T^{μν}$ with the (geometrically defined) second fundamental tensor $K_{μν}{^ρ}$, while the right hand side will simply vanish in the case of free motion and will otherwise be just the orthogonal projection of any external force density that may happen to act on the brane.