Extended objects with edges

Abstract

We examine, from a geometrical point of view, the dynamics of a relativistic extended object with loaded edges. In the case of a Dirac-Nambu-Goto [DNG] object with DNG edges, the worldsheet $m$ generated by the parent object is, as in the case without boundary, an extremal timelike surface in spacetime. Using simple variational arguments, we demonstrate that the worldsheet of each edge is a constant mean curvature embedded timelike hypersurface on $m$, which coincides with its boundary, $\partial m$. The constant is equal in magnitude to the ratio of the bulk to the edge tension. The edge, in turn, exerts a dynamical influence on the motion of the parent through the boundary conditions induced on $m$, specifically that the traces of the projections of the extrinsic curvatures of $m$ onto $\partial m$ vanish.