Extended objects with edges

TL;DR

This paper studies the geometry and dynamics of relativistic extended objects with loaded edges, revealing how edges with constant mean curvature influence the motion of the entire object.

Contribution

It demonstrates that the worldsheet of edges in Dirac-Nambu-Goto objects are constant mean curvature hypersurfaces, linking edge tension to bulk tension and boundary conditions.

Findings

Edges are constant mean curvature hypersurfaces.

Edge tension ratio determines mean curvature.

Edge dynamics influence the parent object's motion.

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.