EQUATIONS OF MOTION OF A STIFF GEODYNAMIC STRING OR HIGHER BRANE

Equations of motion are derived for a generic string or higher-dimensional brane of minimally non-trivial geodynamic type, meaning one that is governed by a surface action density consisting of a constant Dirac-Goto-Nambu type of contribution augmented by a 'stiffness' contribution constructed from the second fundamental tenser in the form B (KKrho)-K-rho-(CKKmu nu rho)-K-mu nu rho, where B and C are arbitrarily chosen constants. It is shown that there is a canonical (nonsymmetric) surface stress momentum-energy density tenser T-nu(mu), including terms of third differential order, and an associated three index tenser Sigma(rho)(mu nu) Of second differential order, that give a corresponding conserved surface current P mu=T-nu(mu)+2 Sigma(rho)(mu nu)del(nu)k(rho) representing the generalized momentum flux associated with any background spacetime Killing vector k(mu). The dynamic equations are generically of fourth differential order, but are shown to reduce to second order if and only if B = C, so that the 'stiffness' contribution becomes proportional to the intrinsic Ricci scalar (which for a string is just a divergence, having no effect at all).