3 v 1 4 J un 1 99 6 Tension as a perturbative parameter in non – linear string equations in curved space – time

A perturbation theory with respect to the tension parameter γ/α ′ for the non–linear equations of string, moving in curved space–time, is considered. Obtained are linearized motion equations for the functions of the n−th degree of approximation (n = 0, 1, 2). The study of string dynamics in curved space–time has evoked considerable interest [1–3]. The investigation of this problem is hampered by a non–linear character of the string equations exactly solvable for a restricted class of special metrics [1– 8]. Therefore in [2,3,7] it was proposed to study the approximate solutions of these equations using a perturbative theory. A rigorous realization of this program implies the presence of a small parameter in the string equations, and the search for their solutions in the form of a series expansion in terms of this small parameter. Here it will be shown that the string tension may play the role of a small parameter in the string motion equations. We will prove that the solution of the string equations and constraints can be sought in the form of a series expansion in terms of the tension parameter. The functions of the zero and subsequent approximations depend on the singular world–sheet variable ξ = σ/ε, where ε is a small dimensionless parameter proportional to the string tension. To investigate the described problem it is convenient to use the reparametrization invariant representation [9] for the bosonic string action 1