Essentials of Classical Brane Dynamics

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 $bar{T}^{\mu \nu }$ with the (geometrically defined) second fundamental tensor $K_{\mu \nu }{}^{\rho }$, 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.

37 pages, Latex, revised to update references. Based on talk on 5 June, 2000 at Meudon Colloquium ``Motivations for and Implications of Extra Cosmological Dimensions'' (transparencies visible on line at http://darc.obspm.fr/col/M00.html) Contribution to proc. Peyresq 5 meeting, June 2000, ``Quantum spacetime, Brane Cosmology, and Stochastic Effective Theories''

References in corpus (9)

• The Einstein Equations on the 3-Brane World
• Non-conventional cosmology from a brane-universe
• Brane-World Black Holes
• Cosmological dynamics on the brane
• Supergravity on the Brane
• Curvature Corrections to Dynamics of Domain Walls
• Gravitational Perturbations of Relativistic Membranes and Strings
• Radiative Backreaction on Global Strings
• Non-divergence of gravitational self-interactions for Goto-Nambu strings

Cited by in corpus (15)

• Essentials of Blackfold Dynamics
• Cosmic strings and their induced non-Gaussianities in the cosmic microwave background
• Derivation of the blackfold effective theory
• Gravitational Chern-Simons Lagrangians and black hole entropy
• Quasilocal equilibrium condition for black ring
• Simulated gravity without true gravity in asymmetric brane-world scenarios
• Electroelasticity of Charged Black Branes
• The essence of the Blandford-Znajek process
• Open/closed string duality and relativistic fluids
• Relativistic Elasticity of Stationary Fluid Branes
• Black holes and black strings in plane waves
• Rotating elastic string loops in flat and black hole spacetimes: stability, cosmic censorship and the Penrose process
• Non-Abelian current oscillations in harmonic string loops: existence of throbbing vortons
• Fields in nonaffine bundles. IV. Harmonious non-Abelian currents in string defects
• Entanglement, anomalies and Mathisson's helices