The Dynamics of Relativistic Membranes II: Nonlinear Waves and Covariantly Reduced Membrane Equations

By explicitly eliminating all gauge degrees of freedom in the $3+1$-gauge description of a classical relativistic (open) membrane moving in $\Real^3$ we derive a $2+1$-dimensional nonlinear wave equation of Born-Infeld type for the graph $z(t,x,y)$ which is invariant under the Poincar\'e group in four dimensions. Alternatively, we determine the world-volume of a membrane in a covariant way by the zeroes of a scalar field $u(t,x,y,z)$ obeying a homogeneous Poincar\'e-invariant nonlinear wave-equation. This approach also gives a simple derivation of the nonlinear gas dynamic equation obtained in the light-cone gauge.