Generalization of the model of a~relativistic string in a~geometrical approach

B. M. Barbashov ; V. V. Nesterenko ; A. M. Chervyakov

Teoretičeskaâ i matematičeskaâ fizika, Tome 45 (1980) no. 3, pp. 365-376

Résumé

A model of a one-dimensionally extended relativistic object is proposed. Its dynamics is determined by the requirement that its covering surface in Minkowski space have constant mean curvature $h$ with respect to each normal direction. A special case of such surfaces is the world surface of a relativistic string (minimal surface with $h=0$). The methods of differential geometry are used to investigate the most interesting cases when the enveloping pseudo-Euclidean space-time has dimensions $D=3,4$. In the ease $D=3$, the proposed model is described by the single nonlinear equation $\square\varphi=h\sh\varphi$. In fourdimensional space-time, the dynamics of the model is determined by the system of two equations $$ \square\varphi=\frac{1}{2}h(e^\varphi-e^{-\varphi}\cos\theta), \quad \square\theta=\frac{1}{2}he^{-\varphi}\sin\theta. $$ A Lax representation for this system is obtained in a geometrical approach, and the use of the inverse scattering technique is briefly discussed.