Dynamics of Cosmic Strings and Other Brane Models

The supporting worldsheet of a string, membrane, or other higher dimensional brane, is analysed in terms of its first, second, and third fundamental tensors, and its inner and outer curvature tensors. The dynamical equations governing the models appropriate for phenomena such as (superconducting) cosmic strings and cosmic domain walls are developed in a general framework (allowing for both electromagnetic and Kalb Ramond background coupling). It is shown how the surface stress momentum energy density tensor determines the propagation characteristics of small " wiggle " perturbations of the worldsheet. Attention is then focussed on special features of strings (using the transonic model with tension T inversely proportional to the energy density U as a particularly important example). A quadratic Hamilton-Jacobi formulation is shown to govern equilibium states and other conservative string configurations sharing a symmetry of the (gravitational, electromagnetic, and Kalb-Ramond) background, including stable ring states that may be cosmologically important. 1 Worldsheet Curvature Analysis In preparation for the more specific study of strings in the last three sections of this course, the first three sections are intended as an introduction to the systematic study, in a classical rela-tivistic framework, of " branes " , meaning physical models in which the relevant fields are confined to supporting worldsheets of lower dimension than the background spacetime. While not entirely new[1][2], this subject is still at a rather early stage of development (compared with the corresponding quantum theory[3] which has been stimulated by the rise of " superstring theory "), the main motivation for recent work[4] on classical relativistic brane theory being its application to vacuum defects produced by the Kibble mechanism[5], particularly when of composite type as in the case of cosmic strings attached to external domain walls[6] and of cosmic strings carrying internal currents[7]. Before discussing the dynamic laws governing the evolution of a brane worldsheet it is worthwhile to devote this first section to a recapitulation of the essential differential geometric machinery[8][9] needed for the analysis of a timelike worldsheet of dimension d say in a background space time manifold of dimension n. At this stage no restriction will be imposed on the curvature of the metric – which will as usual be represented with respect to local background coordinates x µ (µ= 0, ..., n–1) by its components g µν – though it will be postulated to be flat, or at least stationary 1 string and membrane theory has …